Set stabilization using transverse feedback linearization. Christopher Nielsen

Transcription

1 Set stabilization using transverse feedback linearization by Christopher Nielsen A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2009 by Christopher Nielsen

2 Abstract Set stabilization using transverse feedback linearization Christopher Nielsen Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2009 In this thesis we study the problem of stabilizing smooth embedded submanifolds in the state space of smooth, nonlinear, autonomous, deterministic control-affine systems. Our motivation stems from a realization that important applications, such as path following and synchronization, are best understood in the set stabilization framework. Instead of directly attacking the above set stabilization problem, we seek feedback equivalence of the given control system to a normal form that facilitates control design. The process of putting a control system into the normal form of this thesis is called transverse feedback linearization. When feasible, transverse feedback linearization allows for a decomposition of the nonlinear system into a transverse and a tangential subsystem relative to the goal submanifold. The dynamics of the transverse subsystem determine whether or not the system s state approaches the submanifold. To ease controller design, we ask that the transverse subsystem be linear time-invariant and controllable. The dynamics of the tangential subsystem determine the motion on the submanifold. The main problem considered in this work, the local transverse feedback linearization problem (LTFLP), asks: when is such a decomposition possible near a point of the goal submanifold? This problem can equivalently be viewed as that of finding a system output with a well-defined relative degree, whose zero dynamics manifold coincides with the goal submanifold. As such, LTFLP can be thought of as the inverse problem to input-output feedback linearization. ii

3 We present checkable, necessary and sufficient conditions for the existence of a local coordinate and feedback transformation that puts the given system into the desired normal form. A key ingredient used in the analysis is the new notion of transverse controllability indices of a control system with respect to a set. When the goal submanifold is diffeomorphic to Euclidean space, we present sufficient conditions for feedback equivalence in a tubular neighbourhood of it. These results are used to develop a technique for solving the path following problem. When applied to this problem, transverse feedback linearization decomposes controller design into two separate stages: transversal control design and tangential control design. The transversal control inputs are used to stabilize the path, and effectively generate virtual constraints forcing the system s output to move along the path. The tangential inputs are used to control the motion along the path. A useful feature of this twostage approach is that the motion on the set can be controlled independently of the set stabilizing control law. The effectiveness of the proposed approach is demonstrated experimentally on a magnetically levitated positioning system. Furthermore, the first satisfactory solution to a problem of longstanding interest, path following for the planar/vertical take-off and landing aircraft model to the unit circle, is presented. This solution, developed in collaboration with Luca Consolini and Mario Tosques at the University of Parma, is made possible by taking a set stabilization point of view. iii

4 Dedication To Jessica and Amelia. iv

5 Acknowledgements Thank you Professor Manfredi Maggiore. You have undoubtedly been the greatest intellectual influence in my life and a great friend. I will always cherish the time we spent working together. The following individuals have interacted with me and helped me through my Ph.D degree and I wish to thank them for their support. Mireille Broucke, Luca Consolini, Bruce Francis, Cameron Fulford, Michael Goldstein, Louis R. Hunt, Velimir Jurdjevic, Vadim Yu Kaloshin, David Langen, Joshua Marshall, Ruth Milman, Charles Pugh, Mario Tosques, Murray Wonham and Zhigang Xu. I would also like to thank John Hauser for being the external examiner of my dissertation and for providing valuable suggestions that improved this work. v

6 Contents List of figures viii List of notation x List of abbreviations xiv 1 Introduction Motivation Transverse feedback linearization and the panorama of feedback equivalence Systems without outputs Systems with outputs Feedback linearization and the existence of virtual outputs Other literature on set stability and stabilization Organization & contribution of thesis Mathematical Preliminaries Notation Differentiable manifolds Submanifolds Vector bundles, distributions and Lie algebras Tangent space at a point of a manifold Tangent bundles vi

7 2.4.3 Derivatives and tangents Cotangent bundle Vector fields Vector bundles Distributions and codistributions Lie algebras Tubular neighbourhoods of manifolds Control systems Controlled invariance of submanifolds and distributions Classical results on feedback equivalence Set stability Local Transverse Feedback Linearization Introduction Local transverse feedback linearization problem Linear time invariant systems Standard approach to stabilizing (A,B)-invariant subspaces Alternative approach Local transverse feedback linearization solution Transverse controllability indices and preliminary results Proof of the main result (Theorem 3.4.4) Example: Synchronization Transverse feedback linearization with partial information Global Transverse Feedback Linearization Global transverse feedback linearization problem Necessary conditions On the interaction between system and set vii

8 4.4 On the geometry of Γ Feedback transformations in a tubular neighbourhood of a manifold Sufficient conditions The Path Following Problem Introduction Path following methodology Comparison of path following approaches in the literature Path following for a maglev positioning system Experimental apparatus and model Path following control design Experimental results Path following for the PVTOL aircraft model Path following problem Step 1: Path following manifold Step 2: Transverse feedback linearization Step 3: Transversal control design Step 4: Tangential control design Stability analysis Simulations Conclusions and future research 204 Index 208 Bibliography 210 viii

9 List of Figures 1.1 PVTOL aircraft model Illustration of sets from the main result in [21] Flow chart of concepts and contributions A coordinate chart Compatible coordinate charts Mappings between manifolds Embedded submanifold Tangency of curves at a point on a manifold Depiction of a vector bundle Section of a vector bundle Integral submanifolds Illustration of the tubular neighborhood theorem Block diagram of TFL normal form Transversal dynamics of TFL normal form Tangential dynamics of TFL normal form An illustration of the integer ρ 0 (p) Partial synchronization of Lorenz oscillators Phase portrait of Lorenz oscillators Behaviour of control vector fields on the set Γ ix

10 5.1 The magnetic levitation system Top view of the magnetic levitation system Platen and five degrees-of-freedom Desired path for maglev experiment Block diagram of maglev system control structure Position response of the maglev system Measured path error for maglev system PVTOL aircraft model Physical interpretation of the PVTOL s tangential states Phase portrait of the PVTOL Simulation of the PVTOL following a circle PVTOL approaching and then traversing the unit circle Typical behavior of the roll angle of the PVTOL Typical dynamics of the PVTOL following a circle C.1 Coordinated path following x

11 List of notation N The set of natural numbers, page 22. Z The set of integer numbers, page 22. R The set of real numbers, page 22. C The set of complex numbers, page 22. k The set of integers {0, 1,...,k 1}, page 22. card (S) The number of elements in the finite set S, page 22. The empty set, page S The identity map on a set S, page 22. col(x 1,...,x k ) A column vector with elements x 1,...,x k, page 22. x The transpose of the vector x, page 22., The standard inner product on R n, page 23. Euclidean norm, page 23. x A Distance from the point x to the set A, page 23. S T Function with domain S and codomain T, page 23. δ ij Kronecker delta function, page 23. xi

12 V W Internal direct sum of independent subspaces, page 23. X Dual space to the vector space X, page 23. V Orthogonal complement of the subspace V, page 23. I m The m m identity matrix, page 23. A B The direct sum of two matrices A and B, page 23. GL(n, R) The set of nonsingular n n matrices with real coefficients, page 24. C k The class of k times continuously differentiable functions, page 24. C The class of smooth functions, page 24. Diff(U) The family of diffeomorphisms with domain U, page 24. U V The sets U and V are diffeomorphic, page 24. df x The differential of the map f : R n R m evaluated at x R n, page 24. (W,ψ) A chart on a manifold with domain W and mapping ψ, page 25. f g Composition of maps, f following g, page 25. C (M) Ring of smooth real-valued functions on a manifold M, page 28. dim ( ) Dimension, page 30. T p M The tangent space to a manifold M at the point p, page 31. TM Tangent bundle to a manifold M, page 32. df p The differential of map F : M N between manifolds evaluated at p M, page 34. {e 1,...,e m } Canonical basis of R m, page 35. xii

13 ( ψ i )p Basis vector of T p W induced by a coordinate chart (W,ψ) on a manifold M, page 35. T p M Cotangent space to a manifold M at the point p, page 36. T M Cotangent bundle to a manifold M, page 36. V(M) The set of all smooth vector fields on a manifold M, page 36. φ v t(p) Integral curve through p generated by the vector field v, page 37. A B A implies B, page 37. F Push-forward map, page 38. L v λ Lie derivative of λ along v, page 40. [, ] Lie bracket, page 41. ad k fg Iterated Lie brackets of the vectors f and g, page 42. L k vλ Iterated Lie derivatives, page 42. ξ = (π,e,b) Vector bundle ξ with total space E and base space B, page 44. ξ A The restriction of the vector bundle ξ to the submanifold A, page 47. X /V Quotient space of the vector space X and V, page 47. ξ 1 ξ 2 The sum of vector bundles ξ 1 and ξ 2, page 48. The direct sum of vector bundles ξ 1 and ξ 2, page 48. ξ 1 ξ 2 The intersection of vector bundles ξ 1 and ξ 2, page 48. ξ/η algebraic normal bundle of the vector bundle η in ξ, page 48. ξ Dual bundle to the vector bundle ξ, page 49. xiii

14 ann (ξ) Annihilator of the vector bundle ξ, page 49. D Involutive closure of the distribution D, page 54. M ǫ An ǫ tubular neighbourhood of the manifold M, page 60. I (f,g, R n ) The class of closed, connected, embedded submanifolds of R n which are controlled invariant for ẋ = f + gu, page 62. F(f,g,N) The collection of maps that render N controlled invariant, page 62. A B A implies B, and B implies A, page 81. S T S is not a subset of T, page 107. T k The k-dimensional torus, page 155. S 1 The unit circle, page 172. xiv

15 List of abbreviations PVTOL := planar vertical/short take-off and landing, 160 SELP := state-space exact linearization problem, 66 SSP := set stabilization problem, 3 TFL := transverse feedback linearization, 76 VC := virtual constraint, 192 VRD := vector relative degree, 68 DOF := degrees-of-freedom, 167 GTFLP := global transverse feedback linearization problem, 146 IOFLP := input-output feedback linearization problem, 68 LQR := linear quadratic regulation, 179 LTFLP := local transverse feedback linearization problem, 82 LTFLPI := LTFLP with partial information, 138 LTI := linear time-invariant, 62 ODE := ordinary differential equation, 37 PFLP := partial feedback linearization problem, 66 PFP := path following problem, 161 PID := proportional derivative integral, 178 PMLSM := permanent magnet linear synchronous motors, 169 xv

16 Chapter 1 Introduction Imagine the following hypothetical scenario. You find yourself, as the Chair of the Engineering department of a university, required to increase student enrollment. Due to rising costs, the university has asked you to increase student enrollment from 1000 students to 1500 students in order to receive more funding. The enrollment goal must be accomplished by the next academic year. There are numerous ways to reach the 1500 student target. As the Chair you could (a) increase enrollment by 100 students in each of the 5 engineering disciplines offered by your department, or, (b) you could increase the enrollment by 500 in the most popular engineering discipline, or, something in between the extremes of options (a) and (b). The above, idealized, situation is an instance of a set stabilization problem. The set to be stabilized is the one with 1500 students. The various distributions of those 1500 students among the 5 engineering disciplines represent different points on the set. In this scenario, the goal is to quickly (before the start of the academic year) stabilize the set. Once the set has been stabilized (and pressure from the university fades), then you, as Chair, are free to move from point to point within the target set depending on fluctuating demand and changing resources. This toy example, while tacitly ignoring some practical considerations, is represen- 1

17 Chapter 1. Introduction 2 tative of the way in which common problems can be formulated in the set stabilization framework. Intuitively, the set stabilization problem is the following: given a control system and a set in the state space of the system, find, if possible, a feedback control law such that, for all initial conditions near the set, the state remains near the set and asymptotically approaches it. In control theory, most research is concerned with designing control laws that stabilize equilibria. Lyapunov s insight, that the stability of a solution of a system is reduced to the problem of stability of the zero equilibrium position for a new system, makes the equilibrium stabilization viewpoint a useful one. Indeed, many control problems, e.g., setpoint stabilization and tracking to name a couple, can be cast in the equilibrium stabilization framework. Often times, however, one is interested in stabilizing sets, rather than just equilibria. The set stabilization viewpoint is a natural way to interpret problems where certain quantities should converge to zero while other quantities are allowed to evolve freely, and even become unbounded. Set stabilization also finds application in hierarchical control design problems. By this we mean design problems where a high level, priority constraint must be satisfied, while sub-constraints are allowed to change over time within the structure of the high-level one. The approach we take to solving the set stabilization problem is based on feedback equivalence of control systems [35]. Roughly speaking, two control systems are feedback equivalent if there is a coordinate transformation and a feedback transformation that maps the solutions of one system into the second. Given a control system and a submanifold that we wish to stabilize, instead of designing controllers to stabilize the submanifold directly, we first seek a coordinate and feedback transformation that changes the form of the system into a simpler form, the transverse feedback linearization normal form, and then design the set stabilizing control law. This normal form is helpful for designing set stabilizing control laws because it converts the problem into the stabilization of an LTI system. This discussion is made precise in this body of work.

18 Chapter 1. Introduction Motivation This thesis deals with smooth, nonlinear dynamical systems modeled by finite dimensional, deterministic ordinary differential equations m ẋ = f(x) + g i (x)u i, x(0) = x 0 (1.1) i=1 in which 1 x R n is the state, R n is the state space, x 0 is the initial condition and u = (u 1,...,u m ) R m is the control input. At times we will consider systems with outputs, that is, systems that in addition to the above differential equation model have an output given by y = h(x), y R p. (1.2) The objective of this thesis is to develop a method for designing feedback controls that solve the set stabilization problem 2. Set stabilization problem (SSP): Given a control system (1.1) and a closed, controlled invariant set Γ in the state space of the system, find, if possible, a mapping ū : R n R m, such that the set Γ is asymptotically stable with respect to the closed-loop system ẋ = f(x) + m i=1 g i(x)ū i (x). Notice the requirement, in the problem statement, that the set be controlled invariant, stemming from the fact that invariance of a set is a necessary condition for its stability [15, Theorem 1.6.6]. In this thesis, we will restrict our attention to the case when Γ is a closed embedded submanifold of the state space; that is, roughly, we require Γ to be a surface in the state space of the system. The set stabilization viewpoint can be applied to many different control problems, some of which are illustrated below. 1 The results of this thesis do not rely on the assumption that the state space be Euclidean. One could replace R n by a smooth Riemannian manifold. Nonetheless, we assume x R n to avoid unnecessarily cumbersome notation. 2 Refer to Definition 2.5.3, page 62, for the definition of a controlled invariant set and refer to Definition 2.6.1, page 73, for the definition of an asymptotically stable set.

19 Chapter 1. Introduction 4 Example In Section 5.5 we study a simplified model of a planar/vertical take-off and landing aircraft (PVTOL) in hover mode, as introduced in [44]. The model is ẋ 1 = x 2 ẋ 2 = u 1 sin x 5 + ǫu 2 cos x 5 ẋ 3 = x 4 ẋ 4 = g + u 1 cos x 5 + ǫu 2 sin x 5 (1.3) ẋ 5 = x 6 ẋ 6 = µu 2 y = h(x) = col (x 1,x 3 ), where (x 1,x 3 ) is the position of the center of mass in an inertial frame, x 5 is the roll angle of the aircraft, u 1, u 2 are control inputs, while g is the acceleration constant due to gravity, µ and ǫ are positive constants (see Figure 1.1). u 1 u 2 x 5 (x 1,x 3 ) g u 2 Figure 1.1: PVTOL aircraft model. Now consider the path following problem : design a static feedback controller u such

20 Chapter 1. Introduction 5 that the closed loop system s output approaches and goes around the unit circle γ = {y R 2 : y y = 0} without stopping in a desired direction and furthermore, when properly initialized, the output never leaves the set γ. We formally introduce the path following problem in Chapter 5. The set Γ = {x : x x = 0} consists of points in the state space of (1.3) for which the corresponding output lies on the path γ. We do not seek to stabilize Γ because it is not controlled invariant. This is because there are points in Γ at which the velocity (x 2,x 4 ) of the centre of mass is not tangent to the path in the output space. At these points, the solution to (1.3), under any control law, will result in the output instantaneously leaving the path. Instead, the goal submanifold for this problem is Γ, the largest controlled invariant submanifold contained in Γ. The set Γ consists of all those trajectories of (1.3) whose associated output signals can be made to lie on the desired path for all time. For the PVTOL, Γ is the subset of Γ such that the velocity of the centre of mass, (x 2,x 4 ), is tangent to the path. We call Γ the path following manifold. Its precise definition is found in Section 5.2. Solving the path following problem for the PVTOL and the path γ, entails stabilizing the path following manifold. In the case of the PVTOL, there is an additional difficulty in that the aircraft has an internal instability that makes it roll about its longitudinal axis while moving along the circle. In this thesis we design a controller that guarantees that this phenomenon does not occur. This elementary discussion demonstrates that the path following problem for the PVTOL and the unit circle can be viewed as the problem of stabilizing the path following manifold and ensuring that the dynamics on this set are bounded.

21 Chapter 1. Introduction 6 We now turn our attention to synchronization [19], another important application in which set stabilization plays an important role. Consider two dynamical control systems ẋ 1 = f 1 (x 1,x 2 ) + g 1 (x 1,x 2 )u 1 ẋ 2 = f 2 (x 1,x 2 ) + g 2 (x 1,x 2 )u 2 (1.4) with x 1,x 2 R n, u 1,u 2 R m. We seek to design control laws u 1, u 2 such that if x 1 (0) and x 2 (0) are close, then the solutions x 1 (t), x 2 (t) to (1.4) remain close to each other and satisfy lim t + x 1 (t) x 2 (t) = 0. In other words, the diagonal subspace Γ = {(x 1,x 2 ) R 2n : x 1 = x 2 } should be made asymptotically stable. Example In Section 3.7 we study the synchronization problem for two Lorenz oscillators [68], [112]. The equations of motion are ẋ 11 = σ(x 12 x 11 ) + u 1 ẋ 21 = σ(x 22 x 21 ) + u 2 ẋ 12 = rx 11 x 12 x 11 x 13 ẋ 22 = rx 21 x 22 x 21 x 23 (1.5) ẋ 13 = bx 13 + x 11 x 12 + u 1 ẋ 23 = bx 23 + x 21 x 22 + u 2. To achieve synchronization of the Lorenz systems, we must stabilize the diagonal Γ = {(x 1,x 2 ) R 3 R 3 : x 11 = x 21,x 12 = x 22,x 13 = x 23 } which is an invariant for the open-loop system. In some applications it is neither necessary nor desirable to completely synchronize two systems. Instead, the appropriate specification is partial synchronization [90], which can also be framed as an instance of SSP. Consider, once again, the dynamical control systems (1.4) with respective outputs y 1 = h 1 (x 1 ), y 2 = h 2 (x 2 ). (1.6)

22 Chapter 1. Introduction 7 To achieve partial synchronization we seek control laws u 1 and u 2 such that, for all initial conditions x 1 (0) and x 2 (0) such that h 1 (x(0)) and h 2 (x(0)) are near each other, the resulting output signals h 1 (x(t)) and h 2 (x(t)) remain near each other and lim t + h 1 (x 1 (t)) h 2 (x 2 (t)) = 0. In other words, the set Γ = {(x 1,x 2 ) R 2n : h 1 (x 1 ) = h(x 2 )} should be asymptotically stabilized. Example Take the Lorenz oscillator systems from Example with outputs y 1 = h 1 (x 1 ) = x 11, y 2 = h 2 (x 2 ) = x 21. (1.7) In Section 3.7, we solve a partial synchronization problem where we impose that the outputs (1.7) satisfy y 2 1 +y 2 2 = 1, i.e., the outputs lie on a unit circle on the (y 1,y 2 )-plane. In other words, we stabilize the set Γ = {(x 1,x 2 ) R 3 R 3 : x x 2 2 = 1}, which, as we will show, is controlled invariant for (1.5). More generally, one can pose the partial synchronization problem for N coupled dynamical systems as follows. Given N dynamical systems of the form ẋ i = f i (x i ) + g i (x i )u i, x i R n,u i R m y i = h i (x i ), y i R p, the partial synchronization problem is that of designing control laws u i such that the set Γ = {(x 1,...,x N ) R nn : h 1 (x 1 ) = = h N (x N )} is asymptotically stable.

23 Chapter 1. Introduction Transverse feedback linearization and the panorama of feedback equivalence As mentioned earlier, the approach to set stabilization proposed in this thesis is to seek a coordinate and feedback transformation bringing the control system to a normal form, presented in Section 3.2, in which the dynamics are decomposed into two cascade-connected subsystems, which for the moment we call the driving system and the driven system. In our normal form, the driving system is linear, time-invariant, and controllable. It models the dynamics transversal to the target set in the sense that, in transformed coordinates, the target set corresponds to the origin of this linear system. In this thesis, we refer to the driving subsystem as the transversal subsystem. On the other hand, the restriction of the driven system to the target set represents the tangential motion of the control system on the set, and for this reason such restriction will be referred to as the tangential subsystem. In our normal form, the control input undergoes a partition analogous to that just described for the dynamics. Namely, the feedback transformation we seek partitions the control vector fields into tangential and transversal components to which one associates tangential and transversal control inputs. The key feature here is that the tangential and transversal subsystems are driven, respectively, by the tangential and transversal inputs. The process of bringing the original control system to the normal form just described is called transverse feedback linearization (TFL). This terminology originated with the seminal work of Andrzej Banaszuk and John Hauser in [9], which inspired this research. If it is possible to perform TFL for a given target set then, leaving aside technicalities, stabilizing the set amounts to stabilizing the origin of an LTI system, the transversal subsystem. The core contribution of this thesis, in Theorem 3.4.4, is to give geometric necessary and sufficient conditions for transverse feedback linearization to be possible for a target set given by a surface in the state space (a closed embedded submanifold).

24 Chapter 1. Introduction 9 In order to put this contribution into perspective, it is useful to place it in the context of the wide panorama of research on feedback equivalence, the feedback equivalence being established by means of a coordinate and feedback transformation, with a focus on feedback linearization. The following discussion is introductory, and is mirrored in Section by a more detailed treatment of the same concepts. The feedback equivalence problems we discuss can be partitioned into two classes. In the first class of problems, one is given a system without output and investigates whether or not the system can be brought, by means of a coordinate and feedback transformation, to a normal form where the dynamics are linear or partially linear. TFL belongs to this class of problems. The second class of problems involves systems with output. Here, one investigates whether it is possible to bring the system to a form where the input-output dynamics are linear time-invariant. As it turns out, the problems in the two classes are all related Systems without outputs The basic feedback equivalence problem is the state space exact linearization problem (SELP), posed by Brockett in [18] for single-input systems. The problem is to determine whether a control system is feedback equivalent to an LTI system. A related problem was investigated earlier on by Krener in [63]. The solution to SELP for multi-input systems was independently developed by Hunt, Su, and Meyer in [49], and Jakubczyk and Respondek in [57]. It is presented in Theorem on page 66. When SELP is not solvable, it is natural to ask whether it is possible to establish feedback equivalence to a system containing an LTI controllable subsystem. This problem, called the partial feedback linearization problem (PFLP), seems to have appeared for the first time in the work of Isidori and Krener in [55] (see Theorem on page 67). Two different transformations may yield different solutions to PFLP with linear subsystems of different dimension. Therefore, the question arises of what is the maximal dimension

25 Chapter 1. Introduction 10 of the linear subsystem. The work of Marino in [74] (see Theorem on page 68) answers this question through the definition of a set of controllability indices invariant under coordinate and feedback transformations. It is interesting to notice that TFL amounts to partial feedback linearization with the additional requirement that the linear subsystem be representative of the dynamics transversal to a given target set, and in fact it is possible to modify the result of Isidori and Krener in [55] to state necessary and sufficient conditions to solve the main problem of this thesis. This is done in Theorem on page 102, but the conditions in this theorem are not checkable. Much more work is needed to get the checkable conditions of our main result (Theorem 3.4.4). As mentioned earlier, Banaszuk and Hauser in [9] (see also [10]) were the first researchers to investigate TFL. In their setup, the system has one input and the target set is a periodic orbit of the open-loop system. Their main results are necessary and sufficient conditions for TFL near a point of the orbit, as well as sufficient conditions for TFL in a neighborhood of the entire orbit. The novelty of our contributions stems from the fact that rather than stabilizing a closed curve, we seek to stabilize a general embedded submanifold. Further, our theory applies to multi-input systems and, for single-input systems, we solve the transverse feedback linearization problem with partial information. Finally, we establish a link between TFL and the zero dynamics assignment problem, discussed below Systems with outputs When the system has an output function, the basic feedback equivalence problem is to determine whether it is possible to find a coordinate and feedback transformation bringing the system into a form where the input-output dynamics are LTI and controllable. This input-output feedback linearization problem (IOFLP), was first explicitly posed in the work of Isidori et al. in [54], although related concepts were developed by researchers as early

26 Chapter 1. Introduction 11 as It turns out that IOFLP is solvable if and only if the system possesses a welldefined relative degree, see Theorem on page 69. The notion of relative degree, presented on page 68, is the nonlinear generalization of the analogous familiar notion for LTI systems. Roughly speaking, the relative degree of a system is the number of time derivatives of the output function that one should take in order for the control input to appear nonsingularly. If the system has a well-defined relative degree, then the maximal controlled invariant subset contained in the zero level set of the output is the zero level set of the output and a few of its time derivatives (as many as the relative degree minus one). This set, called the zero dynamics manifold, is essential in generalizing the notion of transmission zeros of an LTI system. It is not necessary for a system to have a well-defined relative degree in order for the zero dynamics manifold to exist. As a matter of fact, this manifold is guaranteed to exist under mild regularity assumptions, and an algorithm to locally characterize it -the zero dynamics algorithm- was developed by Byrnes and Isidori (see [52]). This algorithm is a nonlinear version of an analogous procedure by Basile-Marro and Wonham-Morse. As by-product of the algorithm, one obtains that a very large class of nonlinear systems is locally feedback equivalent to a normal form wherein the input-output dynamics are as close as possible to being linear (see Proposition in [52]). In the special case of systems with well-defined relative degree, this normal form reduces to the normal form of IOFLP. If the zero level set of the output is controlled invariant, the zero dynamics algorithm does not iterate, and the associated normal form becomes degenerate, it being a nonlinear system with no special structure, and no closer to being linear than the original system.

27 Chapter 1. Introduction Feedback linearization and the existence of virtual outputs It turns out that the two classes of feedback equivalence problems described in the previous two sections are closely related, because solving any of the problems in the first class amounts to looking for a virtual output yielding a well-defined relative degree. Specifically, the state-space exact linearization problem is solvable if and only if there exists an output function yielding full relative degree, while the partial feedback linearization is solvable if and only if there exists an output function yielding some well-defined vector relative degree. The result by Marino on the maximal dimension of the linear subsystem can be equivalently stated in terms of the maximal relative degree achievable by choice of a suitable output. These relationships are explored in more detail in Lemmas on page 72. One of the contributions of this thesis is in establishing a precise relationship between TFL, the input-output feedback linearization problem of Section 1.2.2, and the notion of zero dynamics. For, we show in Theorem that performing TFL amounts to solving the following zero dynamics assignment problem: given a controlled invariant manifold (the target set), find an output function yielding a well-defined relative degree whose associated zero dynamics manifold coincides with the target set. If one can find the output function in question, which we call a transverse output, then TFL amounts to input-output linearization. The challenge in TFL, then, is to find the output function, and most importantly in determining whether it exists or not. Our main result in Theorem completely answers the latter question. 1.3 Other literature on set stability and stabilization This section reviews the literature on set stability/stabilization. Later, in Section 3.1, we review the literature on feedback equivalence, and in Section 5.5 we review the path

28 Chapter 1. Introduction 13 following literature. Set stability and partial stability In his study of the solar system near the end of the 18 th century, Joseph Lagrange introduced a notion of stability which became known as Lagrange stability. A point x R n is said to be positively Lagrange stable (L + -stable) for a dynamical system if the motion through x is bounded. A set B R n is L + -stable if each of its points is L + -stable. Lagrange stability of a set is strictly a property of points in the set and does not tell us about the qualitative behavior of points in a neighbourhood of it. Partial stability of a dynamical system is a concept that can be traced to Lejeune Dirichlet s work on the stability of equilibria at isolated minima of the potential function in mechanical systems. A system ẋ = f(x) is said to be partially stable if, for all ǫ > 0, there exists δ > 0 such that x(0) < δ implies that the norm of some of the states of the solution is bounded by ǫ for all t 0. Partial stability differs from set stability because its definition requires that the initial condition be within a δ-neighbourhood of the origin instead of being within a δ-neighbourhood of an invariant set. In [88], the authors point out that Dirichlet s work pertains to a type of partial stability restricted to the Lagrangian coordinates irrespective of the velocities. The late 19 th century was a fertile period for the study of dynamical systems. Poincaré studied the stability of the solar system and orbital stability of limit cycles, a special case of set stability [91]. Lyapunov introduced his definitions of stability and asymptotic stability for differential equations [69] and proposed, but did not study, the partial stability problem. The fundamental results on partial stability for finite dimensional systems of ordinary differential equations with continuous right hand sides were obtained by Rumyantsev [97]. The authors in [26], [39] and [88] further generalized stability results by adapting Lyapunov s indirect method, i.e., determining the stability of a nonlinear system by studying its linearization at a point, to partial stability problems. Partial

29 Chapter 1. Introduction 14 stability for large-scale systems was studied in [16]. In the mid 20 th century, Barbashin [11] and Zubov [125] initiated the thorough study of set stability for dynamical systems. The book by Zubov established early results and definitions, including the definition of set stability put forward in Section 2.6. Zubov considered closed, not necessarily compact, invariant sets and gave necessary and sufficient conditions for their stability. His conditions are presented in terms of qualitative properties of the trajectories of the dynamical system, and in terms of a scalar functional, a Lyapunov-like function. Concurrently, and to the best of our knowledge independently, Bhatia and Szegö [15] arrived at similar results. The book [15] summarizes the work of Zubov and also clearly points out the difficulties in extending results for compact sets to the non-compact case. The work by Roxin [96] also characterizes the relationship between set stability and Lyapunov functions. The converse problem for set stability (the existence of a Lyapunov function) has been studied by numerous authors, but we mention in particular the text [95]. More recently, Lin, Sontag and Wang [67] showed that, for a large class of systems, uniform global asymptotic stability with respect to a closed, not necessarily compact set A is equivalent to the existence of a smooth Lyapunov function with respect to A. Lyapunov theoretic set stabilization Despite the voluminous work on set stability in the dynamics community, comparatively little has been done on set stabilization within the control community. Albertini and Sontag [5] generalized an equilibrium result by Sontag [111] by showing that global asymptotic controllability to a closed, possibly non-compact, set is equivalent to the existence of a continuous control-lyapunov function (i.e., a Lyapunov function that can be made to decrease along solutions by a suitable choice of control signal) with respect to the set. The authors did not investigate whether the existence of such a control-lyapunov function implies the existence of a feedback stabilizer. More recently, Kellet and Teel

30 Chapter 1. Introduction 15 in [59], [60], and [61], proved that for a locally Lipschitz control system, uniform global asymptotic controllability to a closed, possibly non-compact set is equivalent to the existence of a locally Lipschitz control Lyapunov function. Moreover, they used this result to construct semi-global practical asymptotic stabilizing feedbacks, thus establishing the equivalence between a notion of controllability and one of set stabilization. Set stabilization for passive systems In a series of papers published in 2000 and 2001, Shiriaev [104, 105] and Shiriaev and Fradkov [106, 107] researched set stabilization for passive nonlinear systems, extending the landmark equilibrium stabilization results of Byrnes, Isidori and Willems [22] to the context of set stabilization. In Shiriaev s framework, the goal set is compact and it coincides with a level set of the storage function (the energy-like function characterizing the passivity property). In [32], El-Hawwary and Maggiore generalized Shiriaev s results by giving necessary and sufficient conditions for the passivity-based stabilization of closed, not necessarily compact, sets that are contained in a level set of the storage function. Sliding mode control When discussing set stabilization, sliding mode control often comes to mind. Sliding mode control [116] involves finding a codimension one hypersurface, the so-called sliding surface, on which the system trajectory exhibits desirable behaviour. Once the sliding surface is identified, the designer seeks feedback controls so that the system trajectory converges to it in finite time. Sliding mode control, then, involves a special type of set stabilization, whereby the set has codimension one. Recent work by Hirschorn [46, 47] on sliding mode control has some similarities to the ideas of this thesis. Hirschorn considers the situation where some of the vector fields g 1,...,g m in (1.1) span a subspace, at each point of the sliding surface, that has a non-zero intersection with the tangent space to

31 Chapter 1. Introduction 16 the surface. The author presents a framework for the design of sliding mode control variations, which generate additional controlled vector fields. This extra control action can be used to achieve a stable sliding motion for a given sliding surface or even allow one to steer the state to a sliding surface of lower dimension. Linear systems, positive systems and virtual constraints The authors in [21] aim to solve SSP using not necessarily smooth controls for systems of the form ẋ = Ax + u, where u belongs to a closed convex cone K in the state space and the set to be stabilized is an invariant, convex, closed cone M in the state space. Their main result is that the problem is solvable if and only if the matrix A does not have eigenvectors corresponding to nonnegative eigenvalues in the cone K M, or nontrivial invariant subspaces corresponding to eigenvalues with nonnegative real parts. In the above, K is the dual cone K = {x R n : x, y 0, y K} and M = {x R n : x, y 0, y M} (see Figure 1.2). K M K M (a) K and K in the plane (b) M and M in the plane Figure 1.2: Illustration of sets from the main result in [21] Imsland and Foss [51] study SSP for a class of nonlinear systems formally identical

32 Chapter 1. Introduction 17 to (1.1), except that the control inputs can only take on positive values. This class of systems includes positive systems as a special case. The authors use a Lyapunov condition for asymptotic stability of compact sets taken from [95], introduce a novel control law, and then provide sufficient conditions for it to asymptotically stabilize a compact set. In [102], the authors present a constructive tool for the stabilization of closed curves in the state space of a class of mechanical systems underactuated by one control. Their method relies on the notion of a virtual constraint of a mechanical system, and on the transversal linearization of the system around the closed curve. 1.4 Organization & contribution of thesis This thesis is organized as follows. In Chapter 1 we present examples that serve to motivate this study and review pertinent literature on set stability/stabilization. Chapter 2 presents a reasonably complete review of the mathematical background needed to understand the main results of this thesis. The most important results of the thesis are in Chapter 3, which presents the local transverse feedback linearization problem (LTFLP) (in Section 3.2) and its solution. The main result, presented in Theorem 3.4.4, gives necessary and sufficient conditions for LTFLP to be solvable. Chapter 3 also discusses linear time invariant systems (in Section 3.3) and the new notion of transverse controllability indices (in Section 3.5). In Section 3.8 we investigate LTFLP in the context of output feedback for single input systems. We investigate the global transverse feedback linearization problem (GTFLP) in Chapter 4 and give restrictive sufficient conditions for its solution in Theorem Chapter 5 looks at transverse feedback linearization in relation to the path following problem. We frame the path following problem as a set stabilization problem and develop a methodology (in Section 5.2) based on transverse feedback linearization, for designing path following control laws. Our approach is successfully applied experimentally, in collaboration with Cameron Fulford, on a mag-

33 Chapter 1. Introduction 18 netically levitated positioning system in Section 5.4. Finally, we present the first solution to the path following problem for the planar/vertical take-off and landing (PVTOL) aircraft model and the unit circle. This solution was developed in collaboration with Luca Consolini and Mario Tosques at the University of Parma. Statement of contributions The following list constitutes the contributions that this thesis makes to the body of knowledge in systems control. 1. Theorem Checkable necessary and sufficient conditions that determine whether or not the local transverse feedback linearization problem is solvable [main result]. 2. Section 3.5. The introduction of transverse controllability indices of a system with respect to a set. 3. Corollary The transverse controllability indices of a system with respect to a set are invariant under coordinate and feedback transformations. 4. Section 3.4. We outline a method for finding the coordinate and feedback transformation needed in order to bring a control system into the normal form of Chapter Section 3.8. Introduce the local transverse feedback linearization problem with partial information. This is a version of the main problem that uses output feedback. A complete solution is presented in the single-input case (Theorem 3.8.1). 6. Chapter 4. Characterization of the fundamental obstacles to solving the global transverse feedback linearization problem. 7. Theorem Restrictive sufficient conditions for global transverse feedback linearization when the set to be stabilized is diffeomorphic to Euclidean space.

34 Chapter 1. Introduction Section 5.2. A general method for designing path following controllers based on transverse feedback linearization. 9. Section 5.5. The first reported solution to a problem of longstanding interest: the path following problem for the PVTOL and the unit circle. The following diagram is a flowchart that shows the concepts contained in this thesis and their interdependence.

35 Chapter 1. Introduction 20 LTFLP Chapter 3 LTI systems Section 3.3 Nonlinear systems Section 3.4 Transverse controllability indices Section 3.5 Feedback & coordinate invariance Corollary GTFLP Chapter 4 Necessary & sufficient conditions for LTFLP Section 3.4 LTFLP with partial information Section 3.8 PFP Chapter 5 Design methodology Section 5.2 Necessary conditions Lemma Obstacles to a global solution Sections 4.3, 4.4 PVTOL Section 5.5 Experimental work Section 5.4 Sufficient conditions Theorem Figure 1.3: Flow chart of concepts and contributions

36 Chapter 2 Mathematical Preliminaries This chapter reviews the mathematical foundations used in this thesis. In Section 2.1 we present the basic notation used throughout, and review some elementary notions from algebra and calculus taken from [40] and [113]. In Section 2.2 we review the concept of a differentiable manifold, while in Section 2.3 submanifolds are discussed. The material of Sections 2.2 and Section 2.3 is based on the exposition from [1], [6], [17], and [23]. We begin Section 2.4 by introducing the concept of the tangent space to a manifold at a point. The discussion here is based on [1] and [114]. This naturally leads to the definition of a tangent bundle to a manifold in Section 2.4.2, taken from [6] and [38]. Next, Section discusses the notion of the derivative of a smooth map between manifolds, based on [1]. Once the derivative of a map has been introduced, we give a generalized version of the inverse function theorem as presented in [38]. Next, in Section 2.4.4, we review the meaning of the cotangent bundle associated to a manifold. The main references for Section are [6] and [114]. In Section we introduce smooth vector fields on a manifold. The notation used to represent operations on smooth vector fields is also presented. Our main reference for this section is [17] as well as [52] and [84]. In Section the notion of a tangent bundle is generalized and leads to the idea of a vector bundle. Vector bundles play an important role in many of the results in this 21

37 Chapter 2. Mathematical Preliminaries 22 thesis. The discussion, terminology and notation of Section is standard and is based on [45], [101], and [114]. The tubular neighbourhood of a manifold is defined in Section and its existence is assured by the tubular neighbourhood theorem, as presented in [38]. Section reviews the notions of a smooth distribution on a manifold, of involutivity, and the Frobenius theorem. The main references for that section are [38] and [52]. In Section 2.4.8, a Lie algebra is introduced, based on the definition from [24]. In Section 2.5 we introduce the class of control system that is studied in this thesis. The main references for this section are [52], and [84]. Finally, we review some basic notions used in characterizing the stability of sets. This material is taken from [53], [62] and [125]. 2.1 Notation In this thesis, N denotes the set of natural numbers, Z denotes the set of integers, R denotes the set of real numbers, and C denotes the set of complex numbers. If k is a positive integer, k denotes the set of integers {0, 1,..., k 1}. If S is a finite set, then card (S) denotes the number of elements in the set S. The symbol means the empty set. On any set S the identity map is denoted 1 S. Let R k, k N, denote the k-fold Cartesian product k {}}{ R R. Elements of R k are vectors, ordered k-tuples of real numbers (x 1,...,x k ). If x R k, we denote by x i the i th component of x. We let col(x 1,...,x k ):= [x 1... x k ] where denotes transpose. Let a and b be two column vectors, define col(a,b) := [a b ]. We will treat R n as a Euclidean space, i.e., an inner product space with the standard inner product x, y := n x i y i, i=1

38 Chapter 2. Mathematical Preliminaries 23 which induces the Euclidean norm x := ( n i=1 x 2 i )1 2, and hence a metric on R n. If A R n is any subset, the point-to-set distance of a point p to A is defined as x A := inf x a. a A The function sgn : R { 1, 0, 1} is defined as sgn(t) := t/ t if t 0, else sgn(t) = 0. The symbol δ ij represents the Kronecker delta function: δ ij : Z Z {0, 1}, 0, if i j δ ij = 1, if i = j. Unless otherwise stated, all vector spaces discussed in this thesis are finite-dimensional. Let V and W be two subspaces of the same vector space X, the notation V W (internal direct sum) is used to represent the subspace V + W when V and W are linearly independent. Let A : V W be a linear map, we will not notationally distinguish between the map and its matrix representation. The context of the discussion will make it clear which one we mean. For a vector space X, we denote by X, its dual space, i.e., the set of all linear functionals X R on X. For a subspace V of the real vector space X, we denote by V the orthogonal complement of V, i.e., the set of all linear functionals on X annihilating V. Thus, V is a subspace of the dual space X. However, by endowing X with the an inner product, we may identify X with its dual, and hence consider V as a subspace of X. More explicitly, V = {w X : ( v V ) w, v = 0}. We denote by I m the m m identity matrix. Following [70], the direct sum of two matrices A and B is the block-diagonal matrix A B := A 0 0 B,

39 Chapter 2. Mathematical Preliminaries 24 where the zeros denote matrices of suitable size. Let GL(n, R) := {M R n n : detm 0} denote the set of all nonsingular n n matrices with real coefficients. The symbol GL(n, R) is used because the set of nonsingular n n matrices with real coefficients can be endowed with the algebraic structure of a group [70] and this group is called the general linear group. Let f be a scalar-valued function from an open set U R n into R. The function is said to be k times continuously differentiable at u U if it possesses continuous partial derivatives of all orders less than or equal to k. If f is k times continuously differentiable at every u U, then we say it is of differentiability class C k on U. If f is C k for all k, then f is C or smooth. A map f : U R n V R m is C k if each of its component scalar functions is C k. Let U R n and V R n be open sets. A map f : U V is a diffeomorphism if it is bijective and both f and f 1 are of class C. If U is an open set of R n, let Diff(U) denote the family of diffeomorphisms with domain U. Two sets U and V are diffeomorphic if there exists a diffeomorphism between them. If U and V are diffeomorphic sets, we write U V. Let f : U R n V R m be a C 1 map, for each x U, the derivative of f at x, denoted df x, is a linear map df x : R n R m. Its matrix representation is the Jacobian matrix of f evaluated at x. 2.2 Differentiable manifolds The definition of a smooth manifold generalizes the familiar notion of a surface in R 3 to more general spaces. The key idea of this generalization is to define spaces that, locally around each point, can be viewed as being Euclidean by using suitable coordinate charts.

40 Chapter 2. Mathematical Preliminaries 25 Definition A coordinate chart of a set M is a pair (W,ψ), where W is a subset of M and ψ : W U is a bijection from W onto the domain U R n. ψ W p U ψ(p) M R m Figure 2.1: A coordinate chart. Let ψ 1,...,ψ m denote the components of ψ so that ψ = col(ψ 1,...,ψ m ). The functions ψ i : W R are called local coordinate functions and, for each point p W, the values ψ 1 (p),...,ψ m (p) are the local coordinates of p (see Figure 2.1). Having defined a coordinate chart, the next step is to define the notion of compatibility of two coordinate charts which, loosely speaking, guarantees that overlapping charts are related by a differentiable map. Definition Two coordinate charts (W 1,ψ 1 ), (W 2,ψ 2 ) are C k -compatible if (i) ψ 1 (W 1 W 2 ), ψ 2 (W 1 W 2 ) are open sets in R m and (ii) when W 1 W 2, the coordinate changes are both of differentiability class C k. ψ 1 ψ 1 2 : ψ 2 (W 1 W 2 ) ψ 1 (W 1 W 2 ), ψ 2 ψ 1 1 : ψ 1 (W 1 W 2 ) ψ 2 (W 1 W 2 ) Definition is illustrated in Figure 2.2. With the notion of compatible coordinate charts in hand, the next step is to define an atlas for M, i.e., a collection of compatible charts that cover M.

41 Chapter 2. Mathematical Preliminaries 26 M W 1 W 2 ψ 1 ψ 2 ψ 2 ψ 1 1 U 1 U 2 ψ 1 (W 1 W 2 ) ψ 2 (W 1 W 2 ) R m R m Figure 2.2: Compatible coordinate charts. Definition An atlas A on M is a countable family of charts {(W i,ψ i ) : i N} such that (i) M = i N W i. (ii) Any two charts (W i,ψ i ), (W j,ψ j ) in A are C compatible. Generally, a set M may admit more than one atlas. For instance, consider the unit circle M = {(x 1,x 2 ) R 2 : x x 2 2 = 1}. An atlas on M is given by the two coordinate charts (W 1,ψ 1 ), (W 2,ψ 2 ) defined as follows W 1 = M {(1, 0)}, ψ 1 (x 1,x 2 ) = arg1 (x 1 + ix 2 ) W 2 = M {( 1, 0)}, ψ 2 (x 1,x 2 ) = arg2 (x 1 + ix 2 ),

42 Chapter 2. Mathematical Preliminaries 27 where arg1 and arg2 denote the branches of the complex argument function such that ψ 1,2 (i) = π and ψ 2 1(W 1 ) = (0, 2π), ψ 2 (W 2 ) = ( π,π). One obtains infinitely many other atlases on M by removing different points from M and choosing different branches of the argument function. All such atlases, however, are equivalent in the sense that all the resulting coordinate charts are C -compatible. Based on this example, we now define the notion of a differentiable structure on M, given by the collection of all equivalent atlases on M. Definition Two atlases A 1 and A 2 on M are equivalent if their union is an atlas on M. Definition A differentiable structure D on M is an equivalence class of equivalent atlases on M. The coordinate charts in all atlases in D are called admissible. Thus, if A is an atlas on M and (V,ψ) is a coordinate chart that is compatible with every (W i,ψ i ) A, then (V,ψ) is part of the differentiable structure on M which includes A. In the example of the unit circle, the chart (V,ψ) with V = M {(0, 1)} and ψ the branch of arg (x 1 + ix 2 ) with image ( 3π, π ) and such that ψ(0) = 0, is a member of the 2 2 same differentiable structure D which includes (W 1,ψ 1 ), (W 2,ψ 2 ) defined earlier. We are now ready to define a smooth manifold. Definition A smooth manifold or, more concisely, a manifold, is a set M equipped with a differentiable structure D with the property that, given any two distinct points p 1,p 2 M, there exist two coordinate charts (W 1,ψ 1 ), (W 2,ψ 2 ) in D such that p 1 W 1, p 2 W 2, and W 1 W 2 =. The integer m in the definition of coordinate chart is the dimension of the manifold. The extra property in the definition above is referred to as the Hausdorff separability condition, and it is used to avoid topological pathologies.

43 Chapter 2. Mathematical Preliminaries 28 Definition A subset V M is said to be open in M if for any p V there is an admissible coordinate chart (W,ψ), with p W, such that W V. Definition Suppose that M and N are smooth manifolds. We say that a map f : M N is of class C r, 0 r, if for each p M and each chart (V,ϕ) of N with f(p) V, there exists a chart (W,ψ) of M with p W and f(w) V, such that the local representation of f ϕ f ψ 1 : ψ(w) ϕ(v ) is of class C r. The maps in Definition are illustrated in the following commutative diagram W ψ ψ(w) f V ϕ ϕ f ψ 1 ϕ(v ) and in Figure 2.3. When f : M N is smooth (i.e., C ) we call it a differentiable mapping between manifolds. Definition A map f : M N between manifolds M and N is a diffeomorphism if it is a smooth bijection and f 1 : N M is smooth. If a diffeomorphism exists between two manifolds, they are called diffeomorphic, written M N. In the special case when f : M R is a real-valued function, the local representation of f in a coordinate chart (W,ψ) of M is the function f ψ 1 : ψ(w) R. The symbol C (M) denotes the ring of smooth real-valued functions on M. Let f : M N be a map between manifolds, we define the rank of f at a point p M using the rank of the Jacobian of its local representation. Definition The rank of a map f : M N between manifolds M and N at a point p M, where (W,ψ) is a chart on M, p W and (V,ϕ) a chart on N with f(w) V, is defined to be the rank of the Jacobian of ϕ f ψ 1 evaluated at ψ(p).

44 Chapter 2. Mathematical Preliminaries 29 M f N V W ψ ϕ ϕ f ψ 1 ϕ(v ) ψ(w) R m R n Figure 2.3: Mappings between manifolds. It can be shown that the rank of a map is coordinate-invariant, i.e., it does not depend on the charts (W,ψ), (V,ϕ) chosen in the definition. 2.3 Submanifolds Definition Let M be an m-dimensional differentiable manifold. A subset N M is called an embedded submanifold of dimension n m if for each p N there exists a coordinate chart (W,ψ) of M, with p W, such that N W = {q W : ψ n+1 (q) = = ψ m (q) = 0}. Henceforth, we will refer to N simply as a submanifold of M. The chart (W,ψ) is

45 Chapter 2. Mathematical Preliminaries 30 said to be adapted to N, see Figure 2.4. A submanifold N of M inherits a manifold M W ψ x n+1,...,x m ψ(w) N ψ(n W) R m x 1,...,x n Figure 2.4: Embedded submanifold. structure from M via the charts adapted to N. Let π : R m R n be the projection π : (x 1,...,x m ) (x 1,...,x n ), and let (W,ψ) be a coordinate chart adapted to N. Take (N W, π ψ N ) as a coordinate chart of N. These charts endow N with a differentiable structure. A Submanifold often arises as a level set of a map as follows. Suppose that f : M N is a map between manifolds and that dim(m) > dim(n). A point q N is called a regular value of f if the rank of f at every point in f 1 (q) is equal to the dimension of N. If every point in the image of f is a regular value, then f is called a submersion. Theorem If q is a regular value of f : M N, then the preimage f 1 (q) = {p M : f(p) = q} is a submanifold of M and dim (f 1 (q)) = dim (M) dim (N).

46 Chapter 2. Mathematical Preliminaries Vector bundles, distributions and Lie algebras Tangent space at a point of a manifold The notion of tangent space at a point of a manifold generalizes the concept of the tangent plane at a point of a surface in R 3. There are various equivalent ways to characterize T p M, the tangent space at a point p M. Here we define T p M as an equivalence class of curves on M. This definition has the advantages of not requiring M to be a subset of some Euclidean space R N and of retaining the strong geometric intuition inherited from surfaces in R 3. Definition Let M be a manifold and p M. A curve at p is a C 1 map c : I M, t c(t) from an open interval I R into M, with 0 I and c(0) = p. Let c 1 and c 2 be curves at p and (W,ψ) a chart, with p W. Then, c 1 and c 2 are tangent at p with respect to ψ if and only if d (ψ c 1 ) dt = d (ψ c 2) t=0 dt. t=0 In other words, two curves are tangent at a point p of a manifold if the tangent vectors of the curves in local coordinates coincide, see Figure 2.5. It can be shown that the tangency of curves at p M is a coordinate-independent notion (i.e., it doesn t depend on the chart chosen in the definition). Tangency at p M is an equivalence relation among curves at p. Let [c] p denote one such equivalence class. Definition For a manifold M and a point p M, the tangent space to M at p, T p M, is the set of equivalence classes at p: T p M := {[c] p : c is a curve at p}. (2.1) Each equivalence class [c] p is called a tangent vector to M at p.

47 Chapter 2. Mathematical Preliminaries 32 ψ(w) c 1 ψ c 1 R 0 M W p ψ R m ψ c 2 c 2 Figure 2.5: Tangency of curves at a point on a manifold. The tangent space T p M can be turned into a real vector space as follows. Let λ R and let [c 1 ] p, [c 2 ] p be two tangent vectors. Define the tangent vector [c 1 ] p + λ [c 2 ] p as [c 1 ] p + λ [c 2 ] p := [ψ 1 (ψ c 1 + λψ c 2 )] p, where c 1, c 2 are elements in [c 1 ] p and [c 2 ] p, respectively, and (W,ψ) is a coordinate chart with p W Tangent bundles Definition The tangent bundle TM of a manifold M given by TM := p M T p M (disjoint union). An element of TM can be taken to be a pair (p, [c] p ), with p M and [c] p T p M. The map π : TM M, (p, [c] p ) p, is called the natural projection of TM onto M. The tangent bundle is a manifold in its own right. It has a natural manifold structure induced by the differentiable structure of M. Let (W,ψ) be a chart on M and let dim (M) = m. Suppose p W and [c] p T p M. By definition, all curves in the equivalence class [c] p share the same tangent vector in local coordinates, t ([c] p ) R m, where ( c [c] p ) t ([c] p ) := d (ψ c) dt. t=0

48 Chapter 2. Mathematical Preliminaries 33 Using this fact, it is easy to show that the pair (TW,dψ), where TW = T p M (disjoint union) p W and dψ : TW ψ(w) R m R 2m (2.2) (p, [c] p ) (ψ(p),t([c] p )) is a coordinate chart of TM. Therefore, the coordinate charts of M induce coordinate charts of TM. It can be shown that the latter charts are C compatible and give rise to a differentiable structure on TM, turning it into a 2m-dimensional manifold. Moreover, with this differentiable structure, the map dψ defined above is a diffeomorphism of TW onto ψ(w) R m and therefore we say that TM is locally diffeomorphic to M R m. The relationship between ψ and dψ is exposed by the following commutative diagram, where π 1 (w,v) = w and π is the natural projection defined earlier: TW dψ ψ(w) R m (2.3) π W ψ π 1 ψ(w) By the way in which, in Section 2.4.1, we endowed T p M with a vector space structure, it follows that the map [c] p t ([c] p ) is an isomorphism T p M R m Derivatives and tangents The notion of the derivative of a vector function can be generalized to the setting of manifolds. Definition Let f : M N be a smooth map between manifolds M and N, with dim (M) = m and dim (N) = n. Let p M, the derivative (or differential) of f at p is the linear 1 map df p : T p M T f(p) N 1 The linearity of df p follows from its definition and the vector space structure for T p M defined on page 31

49 Chapter 2. Mathematical Preliminaries 34 defined as df p ( [c] p ) = [f c] f(p). )) The map df : TM TN defined as (p, [c] p ) (f(p),df p ([c] p is called the derivative of f or the tangent map induced by f. In other words, the derivative of f at p maps the velocity vector of a curve at p into the velocity vector of the image of the curve under f at f(p). More precisely, let (W,ψ) be a coordinate chart of M with p W. Let c be a curve in [c] p and t ([c] p ) its velocity vector at p in local coordinates, t ([c] p ) = d(ψ c) dt. Let (V,ϕ) be a coordinate chart of N t=0 with f(w) V. Then, we have the commutative diagram below, where ˆf = ϕ f ψ 1 : W f V (2.4) ψ ψ(w) ˆf ϕ ϕ(v ) The image of c under f in local coordinates is t ϕ f c(t) and its velocity vector is d(ϕ f c) d( ˆf ψ c) dt. Using (2.4), we rewrite the velocity vector as dt t=0. Notice that ˆf is t=0 a map between open subsets of R m and R n, while ψ c maps an open interval of the real line into an open subset of R m, so we can use the chain rule of vector calculus to express the velocity vector as d ˆf ψ(p) d (ψ c) dt = d ˆf ψ(p) t ([c] p ). t=0 In conclusion, the representation of df p in local coordinates coincides with the derivative at p of the local representation of f, ˆf, as one would expect. Therefore, it doesn t come as a surprise that the derivative map obeys the familiar chain rule Theorem (Chain rule). If f : M N and g : N R are smooth maps between manifolds, then for each p M d(g f) p = dg f(p) df p.

50 Chapter 2. Mathematical Preliminaries 35 Theorem (Inverse function theorem). Suppose that f : M N is a smooth map between manifolds. Then f is a local diffeomorphism in some neighbourhood of p if and only if df p is an isomorphism. In this thesis we will need a generalization of the inverse function theorem as discussed in [38]. Theorem Suppose that f : M N is a smooth map between manifolds. Let P M be a submanifold of M and assume that (i) df p is an isomorphism for every p P and (ii) f P maps P diffeomorphically onto f(p). Then, f maps a neighbourhood of P diffeomorphically onto a neighbourhood of f(p). We conclude this section with the following observation. A coordinate chart (W, ψ) of M induces a natural basis of the tangent space T p M at each p W as follows. Let U = ψ(w), and notice that ψ 1 is a diffeomorphism of U R m onto W. Let {e 1,...,e m } denote the canonical basis of R m and define ( ) := d ( ψ 1) ψ (e ψ(p) i), i {1,...,m}. i p ( In other words, ψ i is the image of e i under the differential of ψ )p 1 at ψ(p). It ( follows that ψ i is a tangent vector in T p M. Moreover, since ψ )p 1 is a diffeomorphism, { ( Theorem guarantees that d (ψ 1 ) ψ(p) is an isomorphism, and therefore ψ i, i } )p {1,...,m} is a basis of T p M. The next theorem characterizes the tangent space to submanifolds that arise as the level set of a map. Theorem Let φ : M R m n be a smooth map, and 0 be a regular value of φ. Letting N = φ 1 (0), at each p N we have T p N = ker (dφ p ).

51 Chapter 2. Mathematical Preliminaries Cotangent bundle Definition Let p be a point of a manifold M. The cotangent space T p M is the dual space of T p M, i.e., the set of all linear functionals σ(p) :T p M R. Elements of the cotangent space are called tangent covectors. The cotangent bundle of M is defined as T M := p M T p M (disjoint union). The cotangent bundle can be given a manifold structure using the differentiable structure on M, in much the same way as the tangent bundle Vector fields Definition Let M be a manifold. A vector field on M is a mapping v : M TM such that π v = 1 M, where π : TM M is the natural projection. That is, to each p M a vector field v assigns a vector v(p) T p M. As discussed in Section 2.4.3, any chart (W,ψ) of M induces a natural basis for T p M { ( } at each p W, denoted, i {1,...,m}. A vector field v may thus be locally represented in W as ψ i )p v(p) = m i=1 ( ) v i (p), ψ i p where v i : W R are scalar functions. The vector field is said to be of differentiability class C k if each of the functions v i : W R is of class C k. In this thesis, we only consider smooth vector fields, i.e., vector fields of class C. Denote the set of all C -vector fields on M by V(M).

52 Chapter 2. Mathematical Preliminaries 37 Definition Let v V(M), an integral curve of v through a point p M is a curve c(t) at p such that the tangent vector of the curve at every point q = c(t) coincides with v(q). Within each coordinate chart (W,ψ) of M, integral curves are determined as solutions to an ordinary differential equation (ODE). We have seen that, at each q W, the vector field can be represented as v(q) = ( m i=1 v i(q) ψ i )q. Therefore, at each q W, the tangent vector v(q) is given in local coordinates by the vector ˆv(x) := col (v 1 ψ 1 (x),...,v m ψ 1 (x)). Consider the ODE in local coordinates ẋ = ˆv(x) x(0) = ψ(p), (2.5) and note that ˆv(x) is smooth because v i (q), i = 1,...,m, are smooth. Let x(t) denote the unique solution 2 of (2.5). Then, the curve ψ 1 (x(t)) is an integral curve of v through p. By using compatibility of coordinate charts and ODE representations of v in each coordinate chart, it is possible to patch together segments of integral curves to define the maximal integral curve through p. Such an integral curve is unique and we henceforth denote it φ v t(p). The maximal integral curve φ v t(p) will also be called the flow generated by the vector field v. Definition A vector field v V(M) is complete (respectively, forward complete) if, for every p M, the maximal integral curve through p of v is defined for all t R (respectively, t 0). Definition A set N M is said to be invariant under v V(M) if (p N) ( t 0)(φ v t(p) N). 2 The solution of (2.5) through any x(0) ψ(w) is unique because ˆv(x) is smooth.

53 Chapter 2. Mathematical Preliminaries 38 Remark The property of invariance in Definition is sometimes called positive invariance because N is invariant for t 0. When N is a closed submanifold 3, invariance for t 0 is equivalent to invariance for t R. If N is an n-dimensional submanifold of M expressed as N = {p M : φ(p) = 0}, where φ(p) = col(φ 1 (p),...,φ m n (p)) is a smooth map M R m n, and 0 is a regular value of φ, then there is a particularly simple criterion for invariance. Theorem Let φ : M R m n be a smooth map, and 0 be a regular value of φ. Let v V(M), then, N = φ 1 (0) is invariant under v if, and only if, (dφ i ) p (v(p)) = 0 for all i {1,...,m n} and all p φ 1 (0). Geometrically, the theorem asserts that N is invariant under v if and only if v is tangent to N, everywhere on N. The same is true for general closed submanifolds of M. Theorem Let N be a closed submanifold of M, and v V(M), Then, N is invariant under v if, and only if, ( p N) v(p) T p N or, equivalently, v N : N TN. Definition If F : M N is a diffeomorphism between two manifolds, and if v V(M), then the differential of F defines a vector field F v V(N) at each q N by means of the push-forward map F : V(M) V(N), defined as F v(q) := (df p v(p)) p=f 1 (q). 3 A submanifold N M is closed if the set M N is open.

54 Chapter 2. Mathematical Preliminaries 39 This corresponds to the usual change of coordinates in a differential equation. Lemma If F : M N is a diffeomorphism between manifolds, and v V(M), then F(φ v t(p)) = φ F v t (F(p)). In other words, if F is a diffeomorphism, the flow generated by F v is the image under F of the flow generated by v. We can define an object that dualizes the notion of a vector field. Definition A covector field or one-form on M is a map σ : M T M such that π σ = 1 M, where π : T M M is the natural projection. While, in local coordinates, a vector field on M assigns to a point in M a column vector col (ˆv 1 (x),..., ˆv m (x)), a covector field can be thought of as the assignment to a point in M of a row vector. More precisely, let (W,ψ) be a chart on M and let { ( } ψ i, i {1,...,m} be the induced basis for T p M at each p W. The dual { )p } { ( } basis (dψ i ) p. i {1,...,m} of, i {1,...,m} is defined by the local relationship ψ i )p ( ) (dψ i ) p = δ ij, i,j {1,...,m}. ψ j p As a consequence, every covector field σ can be represented in W as σ(p) = m σ i (p) (dψ i ) p. Therefore, the action of σ on a vector field v(p) = m j=1 v j(p) σ(p) (v(p)) = = i=1 ( m m σ i (p) (dψ i ) p i=1 m σ i (p)v i (p) i=1 j=1 ( ψ j )p ( ) ) v j (p) ψ j p is given by

55 Chapter 2. Mathematical Preliminaries 40 which is the multiplication of the row vector (σ 1 (p),...,σ m (p)) by the column vector col (v 1 (p),...,v m (p)). The differentiability class of σ is determined by that of the functions σ i : W R, i {1,...,m}. Notice that if λ : M R is a smooth real-valued function on M, then we have dλ p : T p M T λ(p) R for each p M. Since, for all t R, T t R R, we may identify T λ(p) R with R and write dλ p : T p M R. Thus, the map p dλ p is a covector field on M. For obvious reasons, it is called the differential of λ and, with some abuse of notation, it is denoted dλ. Let (W,ψ) be a chart on M, and define ˆλ(x) := λ ψ 1 (x). Then, for all p W, dλ p = m i=1 ˆλ x i (ψ(p)) (dψ i ) p. Definition A one-form σ is called exact if it is the differential of some λ C (M), i.e., if σ(p) = dλ p at each p M. Definition If v V(M) and λ C (M) then the derivative of λ along v is a function L v λ : M R defined by 1 L v λ(p) = lim h 0 h [λ(φv h(p)) λ(p)] and called the Lie derivative of λ along v at p. It is an element of C (M). The Lie derivative of λ along v can be shown to coincide with the action of the covector field dλ on the vector field v: ( p M) L v λ(p) = dλ p (v(p)). Therefore, if (W,ψ) is a chart on M, we compute L v λ(p) as follows L v λ(p) = m i=1 ˆλ x i (ψ(p)) v i (p), where ˆλ(x) = λ ψ 1 (x) and the R-valued functions v i (p) are the coordinates of v in the { ( } basis, i {1,...,m}. ψ i )p

56 Chapter 2. Mathematical Preliminaries 41 Definition If f,g V(M), then the Lie bracket of f and g is a vector field [f, g] V(M) defined by the relation L [f,g] λ = L f (L g λ) L g (L f λ), λ C (M). Therefore, the Lie bracket is a binary operator [, ] : V(M) V(M) V(M). The Lie bracket operation is skew-commutative and bilinear over R, i.e., [f, g] = [g, f], [αf 1 + βf 2, g] = α [f 1, g] + β [f 2, g], [f, αg 1 + βg 2 ] = α [f, g 1 ] + β [f, g 2 ], where α,β R, and f,f 1,f 2,g,g 1,g 2 V(M). However, the bracket is not bilinear over the ring of smooth functions C (M). This is because, if α C (M), then [f, αg] = α [f, g] + (L f α) g. (2.6) Finally, the Lie bracket satisfies the Jacobi identity [f, [g, h]] + [g, [h, f]] + [h, [f, g]] = 0. Definition is implicit in that it does not directly indicate how to compute the Lie bracket. If (W,ψ) is a chart on M, and f,g V(M) are represented as f(p) = ( i f i(p) ψ i )p, g(p) = ( i g i(p) ψ i )p, then, [f, g] (p) = m i=1 ( ) (L f g i (p) L g f i (p)). ψ i p For instance, consider the vector fields on R 2, f(x 1,x 2 ) = sin (x 1) x 2, g(x 1,x 2 ) = x2 2 x 1 cos (x 2 ).

57 Chapter 2. Mathematical Preliminaries 42 Then, [f, g] (x 1,x 2 ) = (dg) (x1,x 2 ) f(x 1,x 2 ) (df) (x1,x 2 ) g(x 1,x 2 ) = = 1 2x 2 0 sin (x 2 ) sin (x 1) x 2 sin (x 1) + 2x 2 2 cos (x 1 ) (x 2 2 x 1 ) x 2 sin (x 2 ) cos (x 2 ) cos (x 1) x2 2 x 1 cos (x 2 ) In general, if f,g V(R n ), the Lie bracket of f and g is computed as follows [f, g] (x) = dg x (f(x)) df x (g(x)), where df x, dg x are the derivative maps of the vector functions f,g : R n R n. We will use the following standard notation for iterated Lie derivatives and Lie brackets : L g L f λ := L g (L f λ), L 0 gλ := λ, ad 0 fg := g, L k gλ := L g (L k 1 g λ) ad k fg := [ f,ad k 1 f g ], k 1. Lemma ([114]). Let M be a manifold and let f,g V(M). Then [f, g] = 0 if and only if φ f t φ g s(p) = φ g s φ f t (p) for all p M and all t,s R such that the flows in the above equation are defined Vector bundles The tangent bundle TM of a manifold M discussed in Section is an important example of a more general object called a vector bundle. Definition A n-dimensional (real) vector bundle is a map π : E B

58 Chapter 2. Mathematical Preliminaries 43 of manifolds E and B such that, for any b B, the inverse image π 1 (b) has the structure of the n-dimensional vector space R n having the following property of local triviality: For each b B, there exists a neighbourhood U of b in B and a diffeomorphism h : π 1 (U) U R n such that for every b U the assignment of x π 1 (b ) to h(x) = (b,ĥ(x)) is an isomorphism of π 1 (b ) to {b } R n. The manifold E is called the total space, B is called the base space and the vector space E b := π 1 (b) is called the fibre over b. The property of local triviality in Definition is exhibited by the commutative diagram below, where π 1 (u,v) = u: h π 1 (U) U R n π π 1 U Not surprisingly, the tangent bundle TM of a manifold M is a vector bundle π : TM M. To see this, recall from Section that any coordinate chart (W,ψ) of M gives rise to the diffeomorphism dψ defined in (2.2) with the property that the diagram in (2.3) is commutative. We use dψ to define a map h : π 1 (W) W R m as follows: h : ( p, [c] p ) dψ (ψ(p),t([c] p )) (ψ 1, 1 R m) (p,t([c] p )). The map h is a diffeomorphism because it is the composition of two diffeomorphisms. Moreover, since as discussed in Section 2.4.2, the map [c] p t ([c] p ) is an isomorphism T p M R m, it follows that for each fixed p W, the map [c] p (p,t([c] p )) is an isomorphism T p M {p} R m, or, what is the same, π 1 (p) {p} R m. Therefore, having shown that TM satisfies the property of local triviality, we conclude that TM M is a vector bundle. Analogously to tangent bundles, vector bundles π : E B can be given a manifold structure, whereby coordinate charts are defined using the pairs (π 1 (U),h) from Defini-

59 Chapter 2. Mathematical Preliminaries 44 tion , and the differentiable structure of B. Each pair (π 1 (U),h) is called a vector bundle chart. We will often write a vector bundle as π : E B or (π,e,b) or even denote the bundle by E alone. Intuitively, a smooth n-dimensional vector bundle ξ = (π,e,b) can be thought of as a family {E b } b B of disjoint n-dimensional vector spaces with origin b, that are parameterized by a space B. The vector spaces E b vary smoothly as b is varied over B. The union of these vector spaces is the total space E. The map π : E B, E b b is a smooth surjective submersion and is called the vector bundle projection. See Figure 2.6 for a depiction of a vector bundle. E b E π b B Figure 2.6: A vector bundle (π,e,b). In this figure, the fibre E b = π 1 (b) is twodimensional. Diffeomorphic vector bundles Definition Let ξ = (π,e,b) and ξ = (π,e,b) be two vector bundles over the same base space B. A fibre map F : E E is a map such that the following diagram

60 Chapter 2. Mathematical Preliminaries 45 commutes E F E π B 1 B Additionally, if F is smooth and, at each b B, the map F π 1 (b) B π is an isomorphism, then F is a vector bundle diffeomorphism and the bundles ξ, ξ are diffeomorphic, written ξ ξ. The trivial n-dimensional bundle over a base space B is ε n B := (π,b Rn,B). Definition An n-dimensional vector bundle ξ = (π,e,b) is called trivial or trivializable if it is diffeomorphic to ε n B. In the case of tangent bundles, if TM is trivial, then the manifold M is called parallelizable. For example, the tangent bundle of the unit circle S 1 = {(x 1,x 2 ) R 2 : x x 2 2 = 1} is trivializable. To see that, let F : TS 1 ε 1 S 1 be defined as F ((x 1,x 2 ),λcol( x 2,x 1 )) = ((x 1,x 2 ),λ). Then, F is clearly smooth and the restriction F π 1 (x 1,x 2 ) is the map λ col( x 2,x 1 ) λ which is clearly an isomorphism. On the other hand, it can be shown that the tangent bundle to the 2-sphere S 2 is not trivializable. Trivial vector bundles are important because they admit a global basis, as discussed in Theorem An important class of trivial vector bundles is given by bundles over contractible spaces. Definition A set S is contractible if there exists a point s S and a continuous map H : S [0, 1] S such that, for all s S, H(s, 0) = s, H(s, 1) = s. The following result, adapted from [1, Theorem ], gives sufficient conditions for a vector bundle to be trivializable. Theorem Let ξ = (π,e,b) be a C -bundle over a contractible space B. Then E is trivial.

61 Chapter 2. Mathematical Preliminaries 46 Smooth sections of a vector bundle A smooth section of a vector bundle generalizes the notion of a vector field on a manifold. Recall that a vector field v on a manifold M is a map v : M TM such that π v = 1 M. If we replace M with a general base space B, and TM with the total space E we are lead to the definition of a section. Definition Let ξ = (π,e,b) be a vector bundle. A C local section of ξ is a C map v : U E, where U is open in B, such that π v = 1 U. If U = B, v is called a C global section, or simply a smooth section of ξ. Figure 2.7 illustrates the definition of a section. E b b v(b) v π b Figure 2.7: A section v of a vector bundle (π,e,b) generalizes the idea of a vector field. This figure demonstrates the properties v π(e b ) = v(b) and π v(b) = b. In Section we have shown that a coordinate chart (W,ψ) of a manifold M induces m local sections of TM defined on W. When an n-dimensional vector bundle is trivial, it admits n global sections. Theorem An n-dimensional vector bundle ξ = (π,e,b) is trivial if and only if

62 Chapter 2. Mathematical Preliminaries 47 there exist n sections v 1,...,v n : B E such that, at each b B, E b = span {v 1 (b),...,v n (b)}. Restrictions and subbundles of vector bundles Definition If ξ = (π,e,b) is a vector bundle and A B is a submanifold, then, the restriction of ξ to A is ξ A := ( π E A, E A,A ) = ( ) π π 1 (A),π 1 (A),A. Definition A k-dimensional subbundle of the bundle ξ = (π,e,b) is another bundle η = (π F,F,B), over the same base space B, such that F E, and π F = π F. Additionally, for each b B, there exists a vector bundle chart (π 1 (U),h), with b U B, such that h(π 1 (U) F) = U R k {0}. If η is a subbundle of ξ, we write η ξ. In other words, a subbundle of ξ assigns at each point b B a k-dimensional subspace of E b which varies smoothly as b varies. If N is a submanifold of M, then TN := {(p, [c] p ) TM N : [c] p T p N}, is a subbundle of the restriction TM N. Definition The algebraic normal bundle of N in M is the subbundle over N whose fibres are the quotient spaces T p M/T p N. It is denoted TM N /TN. Theorem Let N M be a submanifold with m = dim (M), n = dim (N). The bundle TM N /TN is trivial if, and only if, there exist m n sections v 1,...,v m n : N TM such that ( p N) T p M = T p N span {v 1 (p),...,v n m (p)}.

63 Chapter 2. Mathematical Preliminaries 48 Operations on subbundles Definition Let ξ 1 = (π E1,E 1,B) and ξ 2 = (π E2,E 2,B) be smooth bundles. If the dimension of the sum E 1,b + E 2,b is constant for every b B, then the vector bundle sum ξ 1 + ξ 2 is the bundle over B whose fibres are (E 1 + E 2 ) b. Definition Let ξ 1 = (π E1,E 1,B) and ξ 2 = (π E2,E 2,B) be smooth bundles. If, for each b B, the vector spaces E 1,b and E 2,b are independent, we write ξ 1 ξ 2 for their sum. It is the direct sum of ξ 1 and ξ 2. The direct sum of vector bundles is sometimes called the Whitney sum. Definition Let ξ 1 = (π E1,E 1,B) and ξ 2 = (π E2,E 2,B) be smooth bundles. If the dimension of the fibre E 1,b E 2,b is constant for every b B, then the vector bundle intersection ξ 1 ξ 2 is the bundle over B whose fibres are (E 1 E 2 ) b. Definition Let η = (π F,F,B) be a subbundle of the smooth vector bundle ξ = (π,e,b). The algebraic normal bundle ξ/η of η in ξ is the subbundle of ξ whose fibres are the quotient spaces E b /F b. Note that TM N /TN in Definition is an instance of the above definition. Definition Let ξ = (π,e,b) be a smooth vector bundle. A C inner product or orthogonal structure on ξ is a family of functions {α b } b B where each α b is an inner product on E b and the map (b,e 1,e 2 ) α b (e 1,e 2 ) defined on {(b,e 1,e 2 ) B E E : b = π(e 1 ) = π(e 2 )} is C. The pair (ξ,α) is called an orthogonal vector bundle. If M is a manifold, a C orthogonal structure on TM is called a Riemannian metric. In this thesis, orthogonal structures always arise in subbundles of TR n N, where N is a submanifold of R n, wherein the standard inner product on R n is used. Suppose (ξ,α) is an orthogonal vector bundle. If e 1,e 2 are in the same fibre E b, we write e 1, e 2 for α b (e 1,e 2 ) because, in this thesis, e 1, e 2 does not depend on the base point b.

64 Chapter 2. Mathematical Preliminaries 49 Definition Let η = (π F,F,B) be a subbundle of an orthogonal vector bundle ξ = (π,e,b). The orthogonal complement η ξ of η in ξ is the subbundle of ξ with fibres Fb, the orthogonal complement of F b in E b. Note that η is diffeomorphic to ξ/η. Of particular interest to us will be the case when N M is a submanifold and M has a Riemannian metric. In this case, TN TM N is called the geometric normal bundle of N in M. Theorem Let N R m be an n-dimensional submanifold of R m. Then, TN is trivial, i.e. TN N R m n, if and only if there exists a submersion s : U R m n, where U is an open subset of R m containing N, such that N = s 1 (0). Let ξ 1, ξ 2 be two smooth subbundles of the orthogonal bundle (ξ,α). If their intersection ξ 1 ξ 2 is a smooth vector bundle, then (ξ 1 ξ 2 ) = ξ1 + ξ2. Finally, given a bundle ξ = (π,e,b), for each b B we can replace the fibre π 1 (b) with different vector spaces. The simplest, but most important, case for this thesis occurs when we replace each vector space π 1 (b) with its dual space. Definition Let ξ = (π,e,b) be a vector bundle. The dual bundle to ξ, is ξ = (π,e,b) where E := ( π 1 (b) ), b B and π : E B is the natural projection π : (π 1 (p)) p. When this construction is applied to the tangent bundle TM of a manifold M, the resulting bundle is the cotangent bundle of M, already introduced in Section Definition Let η = (π F,F,B) be a subbundle of the smooth vector bundle ξ = (π,e,b). The annihilator ann (η) of η, is the subbundle of ξ whose fibres are defined at each b B by ann (F b ) := {e Eb : e (f) = 0, f F b }.

65 Chapter 2. Mathematical Preliminaries 50 Recall that, if X is a finite dimensional vector space, then (X ) =: X is canonically isomorphic to X. Using this fact, and applying Definition twice to the vector bundle ξ, we obtain the following. Proposition Let ξ = (π,e,b) be a smooth vector bundle over B. Then ann (ann (ξ)) = ξ. Proposition implies that, if η is a subbundle of the cotangent bundle T M, then ann (η ) is a subbundle of TM, the tangent bundle to M. Proposition Let ξ = (π,e,b), ξ 1 = (π E1,E 1,B) and ξ 2 = (π E2,E 2,B) be vector bundles such that ξ 2 ξ 1 ξ, then, ann (ξ) ann (ξ 1 ) ann (ξ 2 ) ξ. Proposition Let ξ 1 = (π E1,E 1,B), ξ 2 = (π E2,E 2,B) be subbundles of the smooth vector bundle ξ = (π,e,b). If ξ 1 + ξ 2 = (π F,F,B) is also a subbundle of ξ, then ann (ξ 1 + ξ 2 ) = ann (ξ 1 ) ann (ξ 2 ) Distributions and codistributions Definition A smooth distribution, or more concisely, a distribution, D on a manifold M is an assignment to each p M of a subspace D(p) T p M which varies

66 Chapter 2. Mathematical Preliminaries 51 smoothly as a function of p. More precisely, for each p M, there exists a neighbourhood W of p in M and smooth vector fields f 1,...,f d on W such that, ( q W) D(q) = span{f 1 (q),...,f d (q)}. The vector fields f 1,...,f d are called local generators around p. A point p M is a regular point of the smooth distribution D if there exists a neighbourhood U containing p for which dim (D(q)) is constant for all q U. In this case, D is said to be nonsingular on U. Similarly, a codistribution Ω on M assigns at each p M a subspace Ω(p) Tp M. A nonsingular distribution D on a manifold M is a subbundle of TM. For the remainder of this section we discuss distributions, but all concepts subsequently introduced for distributions can be defined for, or applied to, codistributions, by changing vector fields to covector fields. Definition A collection of vector fields {f 1,...,f d } defined on an open set W in M is linearly independent if, at each q W, the vectors {f 1 (q),...,f d (q)} are linearly independent. If p is a regular point of a distribution D with dim (D(p)) = d, then there exist d linearly independent local generators around p, and we will write D = span{f 1,...,f d } on the domain of definition of the generators. Theorem on trivial vector bundles implies that nonsingular distributions on contractible manifolds have global generators. That is, if D is smooth with dimension d for all p on the contractible manifold M, then there exist d vector fields f 1,...,f d in V(M) such that ( p M) D(p) = span{f 1 (p),...,f d (p)}. Definition Let D be a distribution on a manifold M, we say that the vector field τ V(M) belongs to D, written τ D, if τ(p) D(p) for all p M.

67 Chapter 2. Mathematical Preliminaries 52 If D has local generators f 1,...,f d on an open set W of the manifold M, then the property τ D on W is equivalent to the existence of d functions α i C (W) such that τ = d α i f i. i=1 Hence, when writing D = span{f 1,...,f d }, we mean the span of f 1,...,f d over the ring C (W). Definition A distribution D 1 contains another distribution D 2, written D 2 D 1, if D 2 (p) D 1 (p) for all p. The next result is basic but useful in this thesis. Proposition Let F : R n R m m be a matrix-valued map. The function rank F : R n Z, x rank (F(x)) is upper semi continuous, that is, if rank (F(x)) = d, then rank (F(y)) d for all y in a neighbourhood of x. Proof. Pick x R n, and suppose that rank (F(x)) = d m. Since the matrix F(x) R m m has rank d, we can find a d d sub matrix of F(x) which is nonsingular. Without loss of generality, suppose that the d d sub matrix of F(x) consisting of the first d rows and d columns of F(x) is nonsingular. Denote this sub matrix as F d (x), and note det (F d (x)) 0. Since the determinant is a continuous function of the matrix entries, there exists some neighbourhood U of x where, for each y U, det (F d (y)) 0. Therefore in U rank (F(x)) d, as required. Definition A distribution D on M is called involutive if the Lie bracket of any pair of vector fields τ 1, τ 2 V(M) belonging to D is a vector field in D, i.e., τ 1 D, τ 2 D [τ 1,τ 2 ] D.

68 Chapter 2. Mathematical Preliminaries 53 Lemma A distribution D with local generators f 1,...,f d is involutive if and only if [f i,f j ] D, i,j {1,...,d}. Note that if D is a constant distribution, i.e., its generators are constant vectors, then it is involutive because its generators satisfy [f i, f j ] = 0. Operations on distributions Since a distribution D on M assigns to each p M the subspace D(p) T p M, we can introduce a variety of operations on distributions inherited from operations on subbundles. If D 1 and D 2 are two smooth distributions then D 1 + D 2 is another distribution defined point-wise by (D 1 + D 2 )(p) := D 1 (p) + D 2 (p). The sum D 1 + D 2 may fail to have constant dimension, and hence it may not be a subbundle of TM. When TM has an orthogonal structure and D is a nonsingular distribution on M, we will use the notation D to indicate the orthogonal complement of D in TM. The distribution D is a subbundle of TM and satisfies, for each p M, T p M = D(p) D (p). This stands in contrast to the notation ann(d) which we use to denote the annihilator of D, a codistribution defined as ann(d)(p) := {σ(p) T p M : σ(p) (v(p)) = 0, v(p) D(p)}. The reader may find it useful to compare this discussion with that at the end of Section

69 Chapter 2. Mathematical Preliminaries 54 The intersection of D 1 and D 2, D 1 D 2, defined by (D 1 D 2 )(p) := D 1 (p) D 2 (p), may fail to be a smooth distribution. The next lemma guarantees conditions for smoothness. Lemma Let p be a regular point of the smooth distributions D 1 and D 2. If p is also a regular point of D 1 D 2 then there exists a neighbourhood W of p such that the restriction of D 1 D 2 to W is smooth. Definition If D is a distribution, the involutive closure of D, written D, is a distribution containing D with the property that if ˆD is an involutive distribution containing D, then D ˆD. It can be shown, using Zorn s Lemma 4, that D exists and is unique. If D is a distribution defined on a manifold M and N M is a submanifold we will at times consider the subbundles TN + D and TN D of TM N defined fibre-wise, for each p N, by T p N + D(p) and T p N D(p), respectively. Lemma Let N M be an n-dimensional submanifold of the m-dimensional manifold M. Let p N be a regular point of a d-dimensional distribution D on M. Suppose there exists an open neighbourhood V of p in N such that k = dim(t q N D(q)) is constant for all q V. Then, there exists a neighbourhood U of p in V such that TN D is smooth on U. Proof. Let (W,ψ) be a coordinate chart of M adapted to N, that is, such that ψ(n W) = {x ψ(w) : x n+1 = = x m = 0}, and let {f 1,...,f d } be a set of local generators of D around p. Let π : (x 1,...,x m ) (x 1,...,x n ) be the projection onto the first n factors. By making W smaller, we can assume that f 1,...,f d are linearly independent on W. 4 Zorn s Lemma: Let P be a non-empty partially ordered set with the property that every linearly ordered subset of P has an upper bound in P. Then P contains at least one maximal element [123].

70 Chapter 2. Mathematical Preliminaries 55 Recall that ˆψ := π ψ : N W R n is a diffeomorphism onto its image, and let ˆf i := ˆψ (f i ), i {1,...,d}. The vector fields ˆf i are defined on an open set of R n. Letting {e 1,...,e n } denote the natural basis of R n, for each q N W we have d ˆψ q (T q N D(q)) = d ˆψ q (T q N) d ˆψ q (D(q)) = span{e 1,...,e n } span{ ˆf 1 (ψ(q)),..., ˆf d (ψ(q))}. Hence, d ˆψ(TN D) is a distribution on an open set of R n. By assumption, and since d ˆψ q is an isomorphism at each q N W, it is the intersection of two smooth nonsingular distributions, and it has constant dimension near ψ(p). Therefore, by Lemma , it is smooth. This implies that TN D is also smooth on a neighbourhood V of p. If the submanifold N in Lemma is contractible and the regularity conditions hold over all of N, then the bundle TN D is trivializable. This follows from Theorem , see also [33] and [37]. If D 1 and D 2 are distributions and f is a vector field, then we use the following notation: [D 1,D 2 ] := span{[τ 1,τ 2 ] : τ 1 D 1,τ 2 D 2 }, [f,d 1 ] := span{[f,τ] : τ D 1 }. Integral manifolds and the Frobenius theorem Definition Let D be a smooth nonsingular distribution on a manifold M. A submanifold N of M is called an integral submanifold of the distribution D if, for every p N, D(p) = T p N. A maximal integral submanifold, N, of D, is an integral submanifold of D with the property that any other integral submanifold of D that contains N is equal to N. The distribution D is integrable if through every point p M there passes an integral submanifold of D.

71 Chapter 2. Mathematical Preliminaries 56 The terms of Definition are illustrated in Figure 2.8. The properties of a distribution being integrable and involutive are closely related. This is the subject of the Frobenius theorem. M N 2 W T p N = D(p) p N N 1 Figure 2.8: The manifold M has a submanifold N. The set N W has two connected components: N 1 and N 2. If the distribution D is such that D(p) = T p N for every p N, then N 1 and N 2 are integral submanifolds of D and N is the maximal integral submanifold of D through p. If D is nonsingular and involutive on M, then through every point p M there passes an integral submanifold of D. Theorem (Frobenius). A smooth nonsingular distribution is integrable if and only if it is involutive. It turns out that a smooth, nonsingular, d-dimensional distribution D on an m- dimensional manifold M is integrable if and only if there exist m d functions λ i C (M) such that ann (D) = span{dλ 1,...,dλ m d }. In other words, we have the following: Proposition A smooth nonsingular distribution is involutive if and only if its annihilator is spanned by smooth, exact one-forms.

72 Chapter 2. Mathematical Preliminaries 57 Theorem ([114]). Let D be a smooth, nonsingular, integrable distribution on a smooth manifold M. Then, M is partitioned by integral manifolds of D. Each component of this partition is a maximal integral manifold of D. The partition of M made by the maximal integral manifolds of D of equal dimension is called a foliation of M. Each maximal integral manifold of the partition is called a leaf of the foliation Lie algebras Definition Let F be a field, and let g be a vector space over F. Suppose moreover that there is given a binary operation [, ] : g g g, with the following properties: ( α,β F), ( X,Y,X 1,X 2,Y 1,Y 2 g) (i) it is bilinear, i.e., [αx 1 + βx 2, Y ] = α [X 1, Y ] + β [X 2, Y ] [X, αy 1 + βy 2 ] = α [X, Y 1 ] + β [X, Y 2 ] (ii) it is skew commutative, i.e, [X, Y ] = [Y, X] (iii) it satisfies the Jacobi identity, i.e., [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0. Then g, equipped with the binary operation [, ], is a Lie algebra over F. Definition Let (g, [, ]) be a Lie algebra. A subspace h g is called a Lie subalgebra if h 1,h 2 h implies [h 1,h 2 ] h. The set V(M) of smooth vector fields on a manifold M can be equipped with different algebraic structures. It can be given the structure of a R-vector space since, for every

73 Chapter 2. Mathematical Preliminaries 58 f,g V(M), α,β R, the sum αf + βg is also an element of V(M). As a vector space, V(M) is not finite-dimensional over R. If, in addition to its R-vector space structure, we equip V(M) with the Lie bracket, [, ] from Definition , then (V(M), [, ]) is a Lie algebra. This is the only Lie algebra used in this thesis. Definition Let (V(M), [, ]) be a Lie algebra and let f 1,...,f d V(M) be a family of vector fields. The Lie algebra generated by {f 1,...,f d } is the smallest subalgebra of (V(M), [, ]) containing the vector fields f 1,...,f d. It is denoted by Lie({f 1,...,f d }). In general, Lie({f 1,...,f d }) is an infinite-dimensional subspace of V(M). An element of Lie({f 1,...,f d }) is a finite R-linear combination of vector fields of the form [ fjk, [ f jk 1,, [f j 2,f j1 ] ]], (2.7) 1 j k d, 1 k <. It turns out that Lie({f 1,...,f d }) is the smallest subspace S of V(M) that satisfies [f i, s] S for any s S and any i {1,...,d}. The set V(M) can also be given the structure of a module over the ring C (M) if, for every f,g V(M), α,β C (M), we define (αf + βg)(p) := α(p)f(p) + β(p)g(p). We denote this module by (V(M),C (M)). Recall from Section that, given a family of vector fields f 1,...,f d V(M), the distribution D = span{f 1,...,f d } is the set of all vector fields that are C (M)- linear combinations of f 1,...,f d. In other words, a distribution is a submodule of (V(M),C (M)). Definition Let (V(M), [, ]) be a Lie algebra and let f 1,...,f d V(M) be a family of vector fields. The distribution spanned by the vector fields in Lie({f 1,...,f d })

74 Chapter 2. Mathematical Preliminaries 59 is a finite C -linear combination of vector fields of the form (2.7). It is denoted by Lie C (M)({f 1,...,f d }). Definitions and imply that, if f Lie({f 1,...,f d }), then f Lie C (M)({f 1,...,f d }). The reverse implication does not hold. Finally, given f 1,...,f d V(M), let D = span{f 1,...,f d } be the associated distribution and let Lie({f 1,...,f d }) be the associated Lie algebra. Let τ 1, τ 2 be any two vector fields that are C -linear combinations of vector fields of the form (2.7). By Definition , these vector fields are elements of Lie C (M)({f 1,...,f d }). By Definitions and , τ 1, τ 2 are also elements of D, the involutive closure of D. Therefore, by Definition , τ 1 Lie C (M)({f 1,...,f d }), τ 2 Lie C (M)({f 1,...,f d }) [τ 1,τ 2 ] D. However it is not true, in general, that [τ 1,τ 2 ] Lie C (M)({f 1,...,f d }). The reason is that neither τ 1 nor τ 2 are necessarily elements of Lie({f 1,...,f d }), because the Lie bracket operator is not bilinear over C (M) (see (2.6)). Therefore, [τ 1, τ 2 ] may fail to be in Lie({f 1,...,f d }) which in turn means that there is no guarantee that it is an element of Lie C (M)({f 1,...,f d }) Tubular neighbourhoods of manifolds Let M be an m-dimensional submanifold of R n with an orthogonal structure, (TM,, ), on TM induced by the standard inner product on TR n. As reviewed in Section 2.4.6, the geometric normal bundle of M in R n is the subbundle TM TR n M defined fibrewise as (T p M) = {v T p R n : v, w = 0, w T p M}. If ε > 0 and p M, let D p (ε) = {v T p M : v < ε}.

75 Chapter 2. Mathematical Preliminaries 60 Definition If ǫ : M R >0 is a smooth function, let D(ǫ) := {p} D p (ǫ(p)). p M Then D(ǫ) TM and M {0} = {(p,v) D(ǫ) : v = 0} D(ǫ). D(ǫ) is referred to as the disk sub-bundle. Theorem (Tubular neighborhood theorem) If M is a closed submanifold of R n, then there exists a smooth function ǫ : M R >0 and a diffeomorphism t : D(ǫ) R n onto an open neighborhood of M in R n such that t M {0} : (p, 0) p. The map t is called a tubular map and its image M ǫ := t(d(ǫ)) is called a tubular neighborhood of M in R n, see Figure 2.9. It is an open set in R n. When M is compact, T p M M 0 x n p v t t(p, v) M D p (ǫ(p)) D(ǫ) M ǫ x 1 Figure 2.9: Illustration of the tubular neighborhood theorem. there exists a constant ε > 0 such that M ε is a tubular neighborhood of M. A tubular neighborhood of a contractible submanifold is a contractible manifold. Remark The statement of Theorem holds if M is a closed submanifold of an n-dimensional manifold N, rather than R n. Definition If A M is any subset of the manifold M, then a smooth map r : M A such that r A = 1 A is called a (smooth) retraction of M onto A.

76 Chapter 2. Mathematical Preliminaries 61 The tubular neighborhood theorem implies that any closed submanifold M of R n admits a retraction of a tubular neighborhood of M, M ǫ, onto M. Such a retraction is defined by the commutative diagram t D(ǫ) M ǫ R n π π t 1 M where π is the natural projection of the vector bundle TM. The following corollary is a local version of the tubular neighborhood theorem. Corollary Let N R n be a submanifold of R n. Then, for every p N there exist a neighbourhood U of p in R n and a smooth retraction r : U N U. 2.5 Control systems The central theme of this thesis is the study of a class of control systems that are modeled by equations of the form m Σ : ẋ = f(x) + g i (x)u i =: f(x) + g(x)u. (2.8) i=1 Here x R n is the state, and u = (u 1,...,u m ) R m is the control input. The vector fields f,g 1,...,g m : R n TR n are smooth (C ). We assume throughout this thesis that g 1,...,g m are linearly independent. When the independent variable t R does not appear explicitly in the right side (2.8), the system is called time-invariant or autonomous. An excellent introduction to nonlinear control systems is [62]. For more information on nonlinear control systems from a geometric point of view please refer to the texts [52] and [84]. Additional resources include [58] for geometric nonlinear control, [118] for linear systems. For texts dealing with nonlinear control design, refer to [53] and [65]. Definition Let U R n be an open set, a regular static feedback on U, denoted (α,β), for the control system (2.8) is a relation u = α(x) + β(x)v,

77 Chapter 2. Mathematical Preliminaries 62 where α : U R m and β : U GL(m, R) are smooth mappings. We denote by f := f + gα and g := gβ the vector fields obtained after the application of (α,β). In this thesis, by feedback transformation we always mean a regular static feedback transformation. Definition Two control systems, Σ : ẋ = f(x) + g(x)u and ˆΣ : ˆx = ˆf(ˆx) + ĝ(ˆx)û, are locally feedback equivalent near x 0 R n if there exist a neighbourhood U of x 0, a regular static feedback (α,β), defined on U, and a diffeomorphism Ξ Diff(U), such that ˆf = Ξ (f + gα), ĝ = Ξ (gβ) on U. If this is the case, Σ and ˆΣ are said to be feedback equivalent on U Controlled invariance of submanifolds and distributions Definition A closed connected submanifold N R n is called controlled invariant for (2.8) if there exists a smooth feedback u : N R m making N an invariant set for the closed-loop system. Following [118], we denote the class of closed, connected, embedded submanifolds of R n which are controlled invariant for (2.8) by I (f,g, R n ). If N I (f,g, R n ), we write F(f,g,N) for the collection of maps that render N controlled invariant. Often, one is given a set M, perhaps defined by virtual constraints or design goals, and then one must pare away pieces of M until all that remains is the largest controlled invariant submanifold M contained in Γ. Consider for a moment a deterministic, finitedimensional, linear time-invariant (LTI) control system modeled as ẋ(t) = Ax(t) + Bu(t). (2.9) The vector x X is the state of (2.9) and the state space X is a real n-dimensional vector space. The vector u U is the control input and the control space U is a real

78 Chapter 2. Mathematical Preliminaries 63 m-dimensional vector space. This system is a special case of (2.8) with f(x) = Ax and g(x) = B. For this class of systems the analogous concept to an invariant manifold is an A- invariant subspace. This is defined as a subspace V X such that AV V. As a matter of fact, the latter inclusion is precisely the condition that is required in order for the set V to be invariant under the vector field Ax. The concept of a controlled invariant manifold also has a linear counterpart, an (A,B)-invariant subspace. Definition Let A : X X and B : U X. A subspace V X is (A,B)-invariant if there exists a linear map F : X U such that (A + BF)V V. Denote the class of (A,B)-invariant subspaces for (2.9) by I (A,B,X ). Lemma ([118]). Let V X and write B := Im B. Then V I (A,B,X ) if and only if AV V + B. It is shown in [118, Lemma 4.4] that given an arbitrary subspace K X, the subclass of (A,B)-invariant subspaces contained in K, denoted by I (A,B,K ), contains a supremal element V := sup{v : V I (A,B,K )}. In other words, V contains any other subspace in I (A,B,K ). The maximal (A,B)- invariant subspace V can be computed via an algorithm, independently developed in [12] and [119]. For nonlinear systems, this last idea has been generalized. Given a system (2.8) and a submanifold M R n, let I (f,g,m) := {N : N I (f,g, R n ), N is a submanifold of M.}.

79 Chapter 2. Mathematical Preliminaries 64 In general, viability theory [8] guarantees that M has a maximal controlled invariant subset M, but this set need not be a submanifold. Therefore, I (f,g,m) may not contain a supremal element in terms of manifold inclusion. However, under mild regularity assumptions, a locally maximal element of I (f,g,m), (that is, an element that is maximal in a neighbourhood of p M), is guaranteed to exist, and it can be computed by means of the zero dynamics algorithm of Isidori-Moog [56]. This algorithm is a nonlinear enhancement of the algorithm of Basile-Marro and Wonham-Morse. Definition Given a control system (2.8), an output map h : R n R p, and a point x 0 h 1 (0), the maximal, in a neighbourhood of x 0, controlled invariant submanifold of R n contained in h 1 (0), is called the zero dynamics manifold of the map h with respect to (2.8). The notion of (A, B)-invariant subspace has a second nonlinear generalization, that of a controlled invariant distribution. Definition A distribution D defined on an open set U is called locally controlled invariant for the dynamics (2.8) if for each x 0 U there exist a neighbourhood U 0 of x 0 and a regular static feedback (α,β) on U 0 such that (i) [ f,d] D, (ii) [ g i,d] D, i {1,...,m}, where f = f + gα and g = gβ. Controlled invariant distributions allow one to construct quotient control systems, in much the same way as (A,B)-invariant subspaces do for LTI systems. For, if D is a regular involutive distribution which is locally controlled invariant for (2.8), then there exists a local coordinate transformation x (z 1,z 2 ) such that in new coordinates, and after application of the feedback transformation u = α(x) + β(x)v in Definition 2.5.7,

80 Chapter 2. Mathematical Preliminaries 65 system (2.8) has the following representation (see [52]) ż 1 = f m 1 (z 1,z 2 ) + g i 1 (z 1,z 2 )v i ż 2 = f 2 (z 2 ) + i=1 m g i 2 (z 2 )v i. i=1 Thus, in new coordinates, both f and g take on a triangular structure. Condition (ii) in Definition 2.5.7, responsible for the triangular structure of g in new coordinates, is automatically satisfied for LTI systems when D is an (A,B)-invariant subspace, since in this case the vector fields g i and D are constant, and hence [ g i,d] = 0. The z 2 subsystem is a well-defined control system in its own right, in that it is decoupled from the z 1 subsystem. If we define an equivalence relation by stating that two states are equivalent if they belong to the same integral manifold of D, then the z 2 subsystem represents the quotient dynamics with respect to this equivalence relation. Theorem ([48], [52], [54], [82], [84]). Let D be an involutive distribution. Suppose D and D+span{g 1,...,g m } are nonsingular on U. Then D is locally controlled invariant for the dynamics (2.8) if and only if [f,d] D + span{g 1,...,g m }, [g i,d] D + span{g 1,...,g m }, i {1,...,m} Classical results on feedback equivalence Next, we present some classical results on feedback equivalence of nonlinear control systems. The various feedback equivalence problems and their solutions can be partitioned into two classes. In the first class of problems, associated to systems without outputs, one asks whether or not the system is feedback equivalent to a linear or partially linear system. The second class, associated to systems with outputs, one asks whether the system is feedback equivalent to a system whose input-output dynamics are LTI.

81 Chapter 2. Mathematical Preliminaries 66 As it turns out, the two classes of problems are closely related, in that all problems in the former class can be equivalently restated in terms of the existence of suitable output functions yielding LTI input-output dynamics, after a suitable feedback transformation. Systems without outputs Consider the following distributions G 0 := span{g 1,...,g m } G 1 := span{g 1,...,g m,ad f g 1,...,ad f g m } G i := span{ad j f g k : 0 j i, 1 k m}. The state-space exact linearization problem (SELP) is to determine whether or not (2.8) is feedback equivalent to a controllable LTI system, in which case the system is said to be feedback linearizable. This problem, originally formulated by Brockett in [18], was solved in full generality in [49], [57]. Theorem (state-space exact linearization). Consider system (2.8) and suppose that the matrix g(x) has rank m at x 0 R n. Then, (2.8) is locally feedback equivalent to a controllable LTI system (2.9) near x 0 if and only if (i) the distribution G n 1 has dimension n at x 0 ; (ii) for each i n 1, the distribution G i is nonsingular and involutive near x 0. When (2.8) is not feedback linearizable, it is natural to ask whether it is feedback equivalent to a system containing an LTI subsystem. This problem, referred to as partial feedback linearization (PFLP), was solved in [55, Theorem 2.1]. The next theorem is adapted from [55]. Theorem (partial feedback linearization). Let be a nonsingular, involutive, d-dimensional, distribution on a neighbourhood U of x 0 in R n that is locally controlled invariant for system (2.8). Assume that the following conditions are satisfied on U

82 Chapter 2. Mathematical Preliminaries 67 (i) For all i n d, + G i = + Ḡi. (ii) For all i n d, dim( + G i ) = constant. (iii) dim( + G n d 1 ) = n. Then then system (2.8) is locally feedback equivalent near x 0 to ẋ 1 = ˆf(x 1,x 2 ) + ĝ(x 1,x 2 )v ẋ 2 = A 2 x 2 + B 2 v, (2.10) with dim(x 1 ) = d, dim(x 2 ) = n d and (A 2,B 2 ) a controllable pair. Furthermore, in (x 1,x 2 )-coordinates the integral submanifolds S of are expressed as S = {(x 1,x 2 ) : x 2 = constant.}. Theorem characterizes the solution of PFLP in terms of the existence of a suitable controlled invariant distribution. If (2.8) is not feedback linearizable, but it is partially feedback linearizable, it is natural to look for a coordinate and feedback transformation yielding a linear subsystem of maximal dimension. The work of Marino [74] characterizes the existence and dimension of such a maximal subsystem. In order to present Marino s result, we need the distributions G f := f + G 0 = {f + g : g G 0 }, G i := G i 1 + [G f,g i 1 ], G 0 := G 0, i = 1, 2,..., (2.11) S i := G i 1 + ad i fg 0, S 0 := G 0, i = 1, 2,..., and the integers Define the controllability integers r 0 := dimg 0, r i := dim S i dim G i 1. (2.12) k i := card {ρ j i : j 0}, i {1,...,ρ 0 }. (2.13)

83 Chapter 2. Mathematical Preliminaries 68 Theorem (maximal feedback linearizable subsystem). Suppose that the distributions (2.11) are nonsingular at a point x 0. Then, system (2.8) is feedback equivalent to a system of the form (2.10), where the pair (A 2,B 2 ) has controllability indices k1..., kµ > 0, µ m, and the dimension of the linear subsystem is i k i. Moreover, this dimension is maximal in the sense that any coordinate and feedback transformation defined in a neighborhood of x 0 yielding a normal form of the type (2.10) has a linear subsystem of dimension i k i. Systems with outputs We now turn our attention to control-affine systems with outputs, modeled as ẋ = f(x) + y = h(x), m g i (x)u i =: f(x) + g(x)u i=1 (2.14) where h : R n R p, p m. The input-output feedback linearization problem (IOFLP) is to find a feedback transformation yielding a system whose input-output dynamics are LTI. The key notion in characterizing IOFLP is that of vector relative degree, defined next. Definition System (2.14) has a vector relative degree of {r 1,...,r p } at a point x 0 if the control input appears nonsingularly in the time derivatives: y (r i) i, and it does not appear in lower-order time derivatives of y. More precisely, (VRD1) L gj L k f h i(x) = 0 for all 1 j m, for all 0 k r i 2, for all 1 i p and for all x in a neighbourhood of x 0. (VRD2) The p m matrix D(x) = L g1 L r 1 1 f h 1 (x) L gm L r 1 1 h 1 (x) L g1 L r 2 1 f h 2 (x) L gm L r 2 1 f h 2 (x) L g1 L rp 1 f h p (x) L gm L rp 1 h p (x) f f (2.15)

84 Chapter 2. Mathematical Preliminaries 69 is full-rank at x = x 0. Using [52, Lemma 4.1.2], conditions (VRD1) and (VRD2) can be stated equivalently in terms of the Lie brackets generated by system (2.14). The equivalent conditions are (VRD1) L ad k f g j h i (x) = 0 for all 1 j m, for all 0 k r i 2, for all 1 i p and for all x in a neighbourhood of x 0. (VRD2) The p m matrix D(x) = L ad r 1 1 f g 1 h 1 (x) L r ad 1 1 f g m h 1 (x) L ad r 2 1 f g 1 h 2 (x) L r ad 2 1 g m h 2 (x) L ad rp 1 f g 1 h p (x) L rp 1 ad f g m h p (x) f (2.16) is full-rank at x = x 0. The matrix in condition (VRD2) is called the decoupling matrix of (2.14). System (2.14) is called square if the number of inputs equals the number of outputs, i.e., m = p. For square systems, the following solution to IOFLP was given in [54], [108]. Theorem (input-output feedback linearization for square systems). Suppose that system (2.14) has vector relative degree {r 1,...,r m } at x 0. Then, (2.14) is locally feedback equivalent near x 0 to a system of the form η = q(η,ξ) + p(η,ξ)v ξ = Aξ + Bv, (2.17) where p is a smooth matrix-valued map, (A,B) is a controllable pair in Brunovský normal form, v R m, and dim (ξ) = r r m. Moreover, in new coordinates the output function is given by h(η,ξ) = Cξ where C = c 1 c m and c i = [1 0 0] 1 ri, i {1,...,m}. For non-square systems, an analogous result to Theorem can be proven.

85 Chapter 2. Mathematical Preliminaries 70 Theorem (input-output feedback linearization for non-square systems). Suppose that system (2.14) has vector relative degree {r 1,...,r p } at x 0. Then, (2.14) is feedback equivalent near x 0 to a system of the form η = f 0 (η,ξ) + g 1 (η,ξ)v 1 + g 2 (η,ξ)v 2 ξ = Aξ + Bv 1, (2.18) where v = col (v 1,v 2 ) R m, f 0, g 1 and g 2 are smooth matrix-valued maps, B is full rank, (A,B) is a controllable pair, and dim (ξ) = r r p. Proof. Let r := r r p. By [52, Proposition 5.1.1], the r functions h 1 (x),...,l r 1 1 f h 1 (x);...;h p (x),...,l rp 1 h p (x) f have linearly independent differentials at x 0 and so, by the inverse function theorem (Theorem 2.4.6), it is always possible to find n r additional functions φ i : U R, i {1,...,n r}, with U a neighbourhood of x 0, such that the mapping Ξ :U Ξ(U) ( x Ξ(x) = col φ 1,...,φ n r,h 1 (x),...,l r 1 1 f h 1,...,h p,...,l rp 1 f h p ) is a local diffeomorphism at x 0. Set η = col(φ 1,...,φ n r ), ξ i = col ( ξ i 1,...,ξ i r i ) = col ( ) h i (x),...,l r i 1 f h i for i {1,...,p}, and ξ = col(ξ 1,...,ξ p ). Then, in (η,ξ) coordinates, the system reads as η = q(η,ξ) + ξ i 1 = ξ i 2 m p k (η,ξ)u k k=1 ξ r i i 1 = ξr i i m ξ r i i = b i (η,ξ) + a ij (η,ξ)u j j=1

86 Chapter 2. Mathematical Preliminaries 71 with i {1,...,p}, where q and p k, k {1,...,m}, are suitable smooth functions, and a ij (η,ξ) = L gj L r i 1 f h i (x) x=ξ 1 (η,ξ), i {1,...,p} b i (η,ξ) = L r i f h i(x) x=ξ 1 (η,ξ), j {1,...,m}. Therefore, the coefficient multiplying u j in the equation for ξ i r i is the (i,j) th entry of the decoupling matrix D(x) in (2.15). Let (η 0,ξ 0 ) := Ξ(x 0 ) and recall that, by assumption, the decoupling matrix is full rank at (η 0,ξ 0 ), and hence, by Proposition , it is full rank in a neighbourhood of (η 0,ξ 0 ). Let β(x) = [M(x) N(x)], where M(x) := D (x)(d(x)d (x)) 1 is a m p rightinverse of D(x), and N(x) is a m (m p) smooth matrix-valued map whose columns span the kernel of D(x) for all x near x 0. Notice that β(x) just defined is nonsingular near x 0. Let α(x) = β(x) col(l r 1 f h 1,...,L rp f h p, 0 (m p) 1 ). Finally, consider the feedback transformation u = α(x) + β(x) x=ξ 1 (η,ξ) v, where v = col(v 1,...,v m ). After this feedback transformation, the system in (η,ξ) coordinates has the desired form. In Theorems and , without loss of generality, let h(x 0 ) = 0. Then, in both theorems, the zero dynamics manifolds at x 0 are expressed in (η,ξ) coordinates as Ξ (Z ) = {(η,ξ) : ξ = 0}. In light of Theorem and Theorem , when an output map h has a well-defined relative degree and the trajectories of the closed-loop system are bounded, stabilizing Z amounts to stabilizing the ξ subsystem. Feedback linearization and the existence of virtual outputs The two classes of feedback equivalence problems presented in the previous two sections can be related to each other by noting that all problems in the former class can be stated in terms of the existence of virtual output functions yielding a well-defined relative degree. This point of view was championed by Isidori [52].

87 Chapter 2. Mathematical Preliminaries 72 Lemma (state-space linearization and virtual outputs). The conditions of Theorem hold if and only if there exists an output function h : R n R m yielding a vector relative degree {r 1,...,r m } with r i = n. Lemma (partial feedback linearization and virtual outputs). There exists a nonsingular and involutive distribution which is controlled invariant for system (2.8) and satisfies the assumptions of Theorem , if and only if there exists an output function h : R n R p, p m, yielding a vector relative degree {k 1,...,k p } such that i k i = n dim ( ). Lemma (maximal linearizable subsystem and virtual outputs). Under the assumptions of Theorem , there exists an output function h : R n R µ, µ m, yielding a vector relative degree {k 1,...,k µ}, where the k i s are defined in (2.13). Limitations of feedback linearization-based techniques Model uncertainties in (2.8) potentially limit the practical usage of feedback linearization based techniques. For input-output feedback linearization, one must account for the effect of model uncertainties in the successive differentiations of the output of interest. The theory of feedback linearization depends essentially on differentiating functions, perhaps to high order depending on the application. If the uncertainty does not affect the relative degree of the system, then it is not difficult to attenuate its effect or even reject it. For simplicity, consider a single-input control system of the form (2.14), and suppose an unmodeled uncertainty p(x,t) affects the system, ẋ = f(x) + g(x)u + p(x,t) y = h(x). Furthermore, suppose that the nominal system (2.14) has a well-defined relative degree r at a point x 0. If the uncertainty does not affect the relative degree of the system, then

88 Chapter 2. Mathematical Preliminaries 73 the r th derivative of h along the uncertain system has the form h (r) (x) = L r fh(x) + L g L r 1 f h(x)u + ψ(x,t), where ψ(x,t) is an unmodeled term. The control law yields u(x) = Lr f h(x) + v L g L r 1 f h(x) h (r) (x) = v + ψ(x,t). A high-gain controller or, if the uncertain term ψ(x, t) has a parametric structure, an adaptive controller, can attenuate the effect of ψ to any desired degree of accuracy. A more problematic situation occurs when the uncertainty affects the relative degree by making the control input u appear in a derivative h (k) (x), with k < r. While, in some cases, this problem can be tolerated (see for example [43]), in general it may be difficult to overcome it using feedback linearization. Analogous comments hold for other feedback linearization results presented in this section, as well as for the technique presented in this thesis. 2.6 Set stability Here we review some basic notions of set stability. These definitions are standard and can be found in [15], [125]. See also [30]. Definition A closed set M R n, invariant for ẋ = f(x), is called stable if for any ǫ > 0, there exists a neighborhood U of M such that, for all initial conditions x 0 U, φ f t (x 0 ) M < ǫ for all t 0. If, furthermore, ( x 0 U) φ f t (x 0 ) M 0 as t +, then M is called asymptotically stable. As pointed out in Section 1.3, set stability can also be characterized by the existence of Lyapunov functions, see for instance [15]. The notion of a class K function comes up in the sequel.

89 Chapter 2. Mathematical Preliminaries 74 Definition A continuous function α : [0,a) [0, ) is said to be of class K if it is strictly increasing and α(0) = 0. If a = and lim r α(r) =, the function is said to belong to class K.

90 Chapter 3 Local Transverse Feedback Linearization Starting with an autonomous control-affine system and a controlled invariant submanifold, necessary and sufficient conditions are presented for local feedback equivalence to a system whose dynamics transversal to the submanifold are linear and controllable. A key ingredient used in the analysis is the new notion of transverse controllability indices of a control system with respect to a set. The exposition begins by examining the case of linear time-invariant systems. Then, the approach is extended to the more general nonlinear case. Portions of this chapter have appeared in modified form in [81]. 3.1 Introduction The problem studied in this chapter is to find conditions under which a class of nonlinear systems is locally feedback equivalent to a particular normal form. The normal form that we seek is useful for designing control laws that stabilize a controlled invariant set in the system s state space. In such normal form, the system s transverse dynamics relative to the set (intuitively, that portion of a systems dynamics that tend to move the system away from or closer to the set) are modeled by a controllable LTI subsystem. Another 75

91 Chapter 3. Local Transverse Feedback Linearization 76 way to think geometrically about the problem is this. We seek to find conditions for the existence of a foliation of a neighbourhood of the state space with the following properties (i) One leaf of the foliation is the set we wish to stabilize. (ii) The foliation enjoys an invariance property, with respect to the control system, guaranteeing the existence of a well-defined quotient control system. The equivalence relation inducing the quotient dynamics is given by the foliation itself: two points are equivalent if and only if they belong to the same leaf. (iii) The quotient system is LTI and controllable. We refer to the process of bringing the control system to the normal form above as transverse feedback linearization (TFL). Set stabilization, as discussed in Chapter 1, is a common objective in many control applications. If, given a system and a set to be stabilized, one can perform TFL, then designing stabilizers for the set amounts to stabilizing the origin of a controllable linear system. As a result, in this chapter we focus on finding necessary and sufficient conditions for TFL and so rather than directly looking at the set stabilization problem, we focus on the equivalence problem. Nevertheless, the reader should bear in mind the big picture and in particular the utility of this normal form for stabilizing sets. Upon first glance, the normal form we seek appears very similar to the normal form from Theorem The difference is explained in detail in Section 3.2, but essentially it lies in the fact that the data in our problem differ fundamentally from the data in Theorem We now summarize the literature on feedback equivalence relevant to this thesis, reminding the reader that the most salient results are reviewed in Section Ever since Poincaré s seminal work [92], the problem of equivalence of vector fields has been a central question in the field of dynamics. In his 1879 work, Poincaré found sufficient conditions for an analytic vector field to be locally equivalent to a linear one by means of an analytic transformation. Poincaré s key insight in formulating this problem was that, rather than

92 Chapter 3. Local Transverse Feedback Linearization 77 trying to solve a differential equation, it is convenient to seek a coordinate transformation reducing the associated vector field to its simplest form, the normal form. In control theory, the problem of equivalence of a control system to a linear controllable system by means of smooth coordinate transformations was first formulated by Krener in 1973 [63]. In 1978, Brockett [18] formulated and solved the feedback linearization problem for singleinput systems, whereby the equivalence to a linear controllable system is established by means of a smooth coordinate transformation and a regular feedback transformation; this is referred to as feedback equivalence (see Definition 2.5.2). The multi-input, multi-output extension of Brockett s work was carried out by Jakubczyk and Respondek in [57] and, independently, by Hunt, Su, and Meyer in [49]; see also [115]. These results are contained in Theorem When a control system is not feedback linearizable, it is natural to ask whether it admits a feedback linearizable subsystem. This problem, first posed by Isidori and Krener in [55], is referred to as partial feedback linearization and the main result is presented in Theorem (2.10). For single-input systems, Krener, Isidori, and Respondek [64] investigated partial feedback linearization yielding a linear subsystem of maximal dimension. This result was extended by Marino to the multi-input case, and it is presented in Theorem ; see also [75], [94]. For systems with outputs, Xu and Hunt [121], [122], consider a similar problem. The above cited work on control systems focuses on the equivalence problem near equilibrium points of the given system. The work in this chapter follows in the footsteps of these important results but instead of looking for equivalence around an equilibrium point, we consider equivalence about an invariant set. An important precursor to our work is that of Andrzej Banaszuk and John Hauser. In [9], Banaszuk and Hauser formulated and solved the transverse feedback linearization problem (TFLP) for periodic orbits of single-input control-affine systems. If Γ is a periodic orbit of the open loop system, the problem entails finding conditions for feedback equivalence to a control system whose dynamics transversal to Γ are linear and controllable. In [80], we generalized Banaszuk

93 Chapter 3. Local Transverse Feedback Linearization 78 and Hauser s results to the case when Γ is an arbitrary controlled invariant embedded submanifold of the state space. In this chapter, we present the complete solution to the local transverse feedback linearization problem (LTFLP) for multi-input systems, relying on a mild regularity assumption. A key ingredient used in the analysis of the problem is the new notion of transverse controllability indices of a control system with respect to a set. The transverse controllability indices are an adaptation of those, presented in Section (see (2.13)), introduced by Marino for the case of equilibria as a generalization of Brunovský s notion of controllability indices of an LTI system. We also consider the situation where one wants to stabilize the set by output feedback. Our results in this case concern single-input systems. When the set Γ is an equilibrium point, the problem considered in this chapter (see Section 3.2) reduces to the classical state-space exact linearization problem. In this special case, our conditions coincide with those of the classical results on feedback equivalence to linear, time-invariant, controllable systems [49], [57], and the transverse controllability indices coincide with the controllability indices introduced by Marino [74]. When the system is LTI and the set is an (A,B)-invariant subspace, our results recover those in geometric linear control theory found in [118]. We do not talk about output stabilization in this chapter. It is true that the given controlled invariant submanifold to be stabilized is given by the zero level set of a vector function. This function could be interpreted as the output of the system, in which case the set stabilization problem can be framed as an output stabilization problem. We do not push this view because in most cases the zero level set of the output of a system is not controlled invariant. This means that in general, output stabilization entails stabilizing the zero dynamics manifold associated with the output. This chapter is organized as follows: Section 3.2 presents the formal problem statement. Section 3.3 specializes the problem to the case of linear time-invariant systems. Section 3.4 presents the statement of our main result, Theorem 3.4.4, and a compar-

94 Chapter 3. Local Transverse Feedback Linearization 79 ison of our result to the solution of the classical state-space exact linearization problem [49], [57]. A relationship to the partial feedback linearization problem is established in Theorem In Section 3.5 we introduce the notion of transverse indices, compare them to the controllability indices of Marino [74] in Lemma 3.5.1, and establish their feedback invariance. The proof of the main result is presented in Section 3.6 and Section 3.7 contains an example of these results applied to a synchronization problem. We conclude the chapter by looking at LTFLP in the context of output feedback for single input systems in Section Local transverse feedback linearization problem In this section we present the main problem studied in this chapter. Consider a control system of the form (1.1) ẋ = f(x) + m g i (x)u i. (3.1) i=1 Suppose we are given a pair (Γ,u ), where Γ I (f,g, R n ), dim Γ = n, and u F(f,g, Γ ). We are ultimately interested in stabilizing the set Γ. Since Γ is a submanifold of R n, it can be expressed in an open set U R n as Γ U = {x R n : γ 1 (x) = = γ n n (x) = 0} where γ i : U R. In order to render Γ a stable set for system (3.1) a natural approach is to perform input-output linearization using some combination of the functions γ 1,...,γ n n as the linearizing output. This is illustrated in the next example. Example Consider the point-mass system ẋ 1 = u 1 ẋ 2 = u 2.

95 Chapter 3. Local Transverse Feedback Linearization 80 Here f = col(0, 0) and g = I 2. Suppose we are interested in stabilizing the unit circle Γ = {x R 2 : x x = 0} which is a controlled invariant set with u equal to, for instance, u = (x 2, x 1 ). A natural approach to stabilizing the circle is to treat the function γ(x) = x x as an output of the system and to check whether or not it yields a well-defined relative degree. If it does, then stabilizing Γ becomes, locally at least, the problem of stabilizing a controllable linear system. Simple calculations reveal γ(x) = x x [ ] L g γ(x) = 2x 1 2x 2. Therefore, the function γ yields a well defined relative degree of 1 at every x Γ because the decoupling matrix D(x) = L g γ(x) is full rank there. Following the procedure in the proof of Theorem , let φ(x) = arg (x 1 + ix 2 ) and use γ(x) = x x to obtain a coordinate transformation Ξ : R 2 \{0} [0, 2π) ( 1, + ), x (η,ξ) := col (φ(x),γ(x)). For our feedback transformation, let u = α(x)+β(x)v, where α(x) = β(x)l f γ(x) = 0, v = col (v,v ) R 2, and [ β(x) = M(x) N(x) ] = x 1 2x x2 2 x 2 2x x2 2 x 2 x 1, with M(x) R 2 1, a right inverse of D(x), N(x) R 2 1 spans the kernel of D(x). After the coordinate and feedback transformation, the point-mass system has the form η = v ξ = v and Ξ(Γ ) = {(η,ξ) : ξ = 0}.

96 Chapter 3. Local Transverse Feedback Linearization 81 Clearly, (η,ξ) Ξ(Γ ) x Γ. Therefore, stabilizing the set Γ is equivalent to stabilizing ξ = 0. If we choose the feedback v = kξ, k > 0, then the set Ξ(Γ ) is exponentially stable. Furthermore, on Ξ(Γ ) the dynamics reduce to η = v. Therefore, the control input v, which only affects the motion on the set Ξ(Γ ), can be designed to meet additional specifications for the motion on the set. However, the existence of such a linearizing output, from the above example, is not obvious since it is not clear, in general, that any of the constraints defining Γ U yield a well-defined relative degree. The following simple example illustrates this fact. Example Consider the system ẋ = 0 x 4 x 2 x 3 x 1 x 3 x 5 x 2 x x 2 0 u u 2. and the pair (Γ,u ) = ({x : x 1 = x 2 = x 4 = x 5 = 0}, 0). Here, Γ is a one-dimensional subspace. We now check whether or not any of the constraints defining Γ can be used as a linearizing output. It is easy to check that the functions x 2 and x 4 do not yield well-defined relative degrees anywhere on Γ because the decoupling matrices [ L g L f x 2 = x 2 0 [ ] L g x 4 = x 2 0 ], are not full rank there. The functions x 1, x 5, taken individually, each yield a welldefined relative degree of 1 on Γ. Taken as a vector output y = col (x 1,x 5 ) they yield

97 Chapter 3. Local Transverse Feedback Linearization 82 vector relative degree {1, 1}. Therefore, if we take y = col (x 1,x 5 ) as our output, in light of Theorem , we will obtain a normal form in which the feedback linearized subsystem has dimension 2. Clearly this linearized subsystem does not correspond to the transverse dynamics of Γ because the dimension of the linearized subsystem does not equal the codimension of Γ which is 4. Therefore, it is not clear whether or not LTFLP is solvable for this system and this set. The main results of this chapter prove that, yes, LTFLP is solvable for this system and this set. Furthermore, the linearizing outputs for this simple example are given by λ(x) = col(x 2 e x 3,x 1 ). Therefore, using output y = λ(x), we can employ an input-output feedback linearization approach to stabilizing Γ. In this chapter, we consider the controlled invariant set Γ as data. For example, in a mechanical system this might correspond to a motion planning task being solved in order to obtain the shortest path between two given points. Often, however, one is given a set Γ, perhaps defined by virtual constraints or design goals, and then one must pare away pieces of Γ until all that remains is the maximal controlled invariant subset Γ contained in Γ. The main problem investigated in this chapter, stated next, builds upon the intuition developed in Example and concerns the decomposition of the system dynamics into a subsystem describing the motion on Γ and one describing the motion transversal to Γ, with the essential requirement that the transversal subsystem be feedback linearizable. Local Transverse Feedback Linearization Problem (LTFLP): Given Γ I (f,g, R n ), u F(f,g, Γ ) and a point p 0 Γ, find, if possible, a neighbourhood U of p 0 in R n, a transformation Ξ Diff(U), Ξ : U R n R n n, x (η,ξ), and a regular feedback

98 Chapter 3. Local Transverse Feedback Linearization 83 transformation (α,β), such that (3.1) is feedback equivalent on U to η = f 0 (η,ξ) + g 1 (η,ξ)v + g 2 (η,ξ)v ξ = Aξ + Bv (3.2) where v = col(v,v ) R m, g 1 and g 2 are smooth matrix-valued functions, B is full column rank, the pair (A,B) is controllable, and Ξ(Γ U) = {(η,ξ) R n R n n : ξ = 0}. In words, we seek to characterize conditions under which (3.1) is feedback equivalent to a system whose dynamics transversal to the set Γ are LTI and controllable. LTFLP asks for a coordinate and feedback transformation on U generating a normal form with two types of decompositions. Firstly, system dynamics near Γ U are decomposed into a tangential subsystem, the η-dynamics, and a transversal subsystem, the ξ-dynamics. Secondly, the feedback transformation decomposes the input space so that the original m control inputs are split into transversal and tangential components, v and v, respectively. We refer to the normal form (3.2) as the transverse feedback linearization normal (v,v ) v Transversal subsystem ξ = Aξ + Bv ξ Tangential subsystem η = f 0 (η, ξ) + g 1 (η, ξ)v + g 2 (η, ξ)v η ξ Figure 3.1: Block diagram of the normal form (3.2). form. Observe that the level sets {ξ = constant} foliate a neighbourhood of the point Ξ(p 0 ). One of the leaves of the foliation, ξ = 0, corresponds to the set Ξ(Γ ) that we want to stabilize. The foliation in question is invariant under (3.2), in the sense that, for any

99 Chapter 3. Local Transverse Feedback Linearization 84 Γ ξ Γ ξ Figure 3.2: The ξ = constant level sets of the LTFLP normal form foliate a neighbourhood of Γ. The flow across the leaves is described by the transversal dynamics ξ = Aξ + Bv. Figure 3.3: Tangential dynamics η = f 0 (η, 0) + g 2 (η, 0)v of the LTFLP normal describe the motion on the set Γ. input signal, the flow of (3.2) maps leaves to leaves in the following sense. All initial conditions on any one leaf are mapped by the flow of (3.2) to one and only one leaf for any fixed flow time. In the TFL normal form (3.2), the ξ subsystem represents a quotient system factoring out the motion on the set Ξ(Γ ). As the value of ξ changes, the state travels across leaves of the foliation. The control system (3.2) in the problem statement is formally identical to that in (2.18) resulting from input-output feedback linearization, but LTFLP differs from IOFLP in a fundamental way. While, in IOFLP, the normal form (2.18) is induced by a given output function, and the set {(η,ξ) : ξ = 0} is the zero dynamics manifold associated with this output, in LTFLP one is given a controlled invariant manifold Γ and asks whether a normal form as in (3.2) exists with the property that Ξ(Γ ) = {(η,ξ) : ξ = 0}. Hence, in IOFLP the independent variable is the output function, while in LTFLP it is the manifold Γ. Later, with Theorem 3.4.1, we will shed additional light on the relationship between the two problems. As mentioned earlier, transverse feedback linearization finds application in the stabilization of Γ. If a transversal controller v is designed that stabilizes ξ = 0, and the trajectories of the closed-loop system are bounded and remain in the transformation neighbourhood, then the controller stabilizes Γ in original coordinates. On the other

100 Chapter 3. Local Transverse Feedback Linearization 85 hand, if the trajectories of the closed-loop system are not all bounded, then stabilization of ξ = 0 implies the stabilization of Γ in original coordinates if and only if there exist class-k functions α 1, α 2 such that α 1 ( x Γ ) ξ(x) α 2 ( x Γ ). See [30, Theorem IV.1] for more details. 3.3 Linear time invariant systems In order to motivate subsequent theoretical developments, we first look at the case of LTI systems. The aim of this section is to motivate and elucidate the approach to be taken for the case of nonlinear control-affine systems. Consider an LTI system as in (2.9), ẋ(t) = Ax(t) + Bu(t), (3.3) with x X, u U. Assume throughout that B is full rank. Let B := Im B, S j := B + AB + A j B, j N {0}. Suppose that P X is a given n -dimensional (A,B)-invariant subspace for (3.3). Our goal is to stabilize the set P. For LTI systems, the necessary and sufficient conditions for the stabilizability of P are encapsulated in the solution to the output stabilization problem from [118, Chapter 4]. First, we recall the standard solution to the problem of stabilizing P for LTI systems, and then outline an alternative approach which lends itself to a generalization for control-affine nonlinear systems Standard approach to stabilizing (A, B)-invariant subspaces The next result is an obvious consequence of [118, Theorem 4.4] Theorem Consider the linear time-invariant control system (3.3) with an (A, B)- invariant subspace P X. For any r > 0, there exists a static feedback F : X U

101 Chapter 3. Local Transverse Feedback Linearization 86 and a constant k > 0 such that, for all t 0, x(t) P ke rt x(0) P, if and only if To understand this theorem, let P + S n 1 = X. (3.4) X := X /P, P : X X be the canonical projection. By (A,B)-invariance, there exists F : X U such that (A + BF)P P. Let u = Fx + v, with v U an external input. The map Ā : X X induced by A + BF is well-defined. Setting PB = B, we have the following commutative diagram X B U B X P A+BF X P Ā X Stabilizing P is equivalent to stabilizing the quotient system with matrices (Ā, B). This is possible if and only if the pair (Ā, B) is stabilizable. If, as in Theorem 3.3.1, we want to assign the rate at which x(t) P 0, we require (Ā, B) to be controllable. This requirement is encapsulated in (3.4). It is easy to show that (3.4) holds if and only if P + S n n 1 = X. (3.5) We say that (3.3) is transversely controllable with respect to P if (3.5) holds. Theorem can also be explained using coordinates. Let T X be any subspace such that P T = X. Take as a basis for X the union of a basis {p 1,...,p n } for P and a basis {t n +1,...,t n } for T. Set, as before, u = Fx + v. In this basis and with this choice for the control law, the matrix representations of the maps A + BF and B take the form A + BF = A 1 A 3 0 A 2, B = B 1 B 2 (3.6) with A 1 R n n, A 2 R (n n ) (n n ), A 3 R n (n n ), B 1 R n m, B 2 R (n n ) m. If we partition the coordinates of X into two groups (x 1,x 2 ) in accordance with the

102 Chapter 3. Local Transverse Feedback Linearization 87 matrices (3.6), the target subspace is P = {x X : x 2 = 0}. This elementary discussion reveals that the problem of stabilizing P is equivalent to the problem of stabilizing the subsystem with matrices (A 2,B 2 ). Thus, P can be stabilized if and only if the pair (A 2,B 2 ) is stabilizable. If, in addition to stabilizing the set P, one wishes to assign the rate at which x(t) P 0, then (A 2,B 2 ) must be controllable. Theorem states that (A 2,B 2 ) is controllable if and only if (3.4) (equivalently (3.5)) holds. Remark For an LTI system ẋ = Ax, x X, the existence of an A-invariant subspace P is both necessary and sufficient for there to exist a basis for X in which the matrix A exhibits the triangular form (3.6). For a nonlinear system ẋ = f(x), x R n, the existence of an invariant set is only necessary (and far from being sufficient) for there to exist, locally, an analogous triangular decomposition. See [83] for a discussion of some of the issues that arise in the nonlinear case. Finally, for design purposes, we impose a decomposition of the external input v into two disjoint subvectors v and v such that v = (v,v ) with only the subset v affecting the (A 2,B 2 ) subsystem. To this end, we seek an isomorphism G : U U such that, by setting u = Fx + Gv, the matrix BG in the basis for X detailed above takes the appropriate form. Set ρ 0 = dim ((B + P)/P), so that rank(b 2 ) = ρ 0. Let G = [G 1 G 2 ], where G 1 is a m ρ 0 matrix such that B 2 G 1 has full rank ρ 0. Let G 2 span the m ρ 0 dimensional kernel of B 2. Then, G is nonsingular and B [ ] 1 G 1 G 2 = B 2 B 1G 1 B 1 G 2 B 2 G 1 0 =: B 11 B 12 B 21 0 In conclusion, letting η = x 1 and ξ = x 2, the LTI system reads as. (3.7) η = A 1 η + A 3 ξ + B 11 v + B 12 v ξ = A 2 ξ + B 21 v. (3.8) Comparing (3.8) to the desired normal form (3.2) we have the following:

103 Chapter 3. Local Transverse Feedback Linearization 88 Theorem For an LTI system and an (A,B)-invariant subspace P, LTFLP is solvable (globally) if and only if P + S n n 1 = X where n = dim (P) Alternative approach The preceding approach to stabilizing P is appealing due to its simplicity. Namely, the problem of stabilizing a subspace is reduced to the problem of stabilizing a reduced order quotient system. Unfortunately, the manner in which the reduced model was obtained does not generalize to the nonlinear case in a straightforward way. The underlying reason is that, as discussed in Section 2.5.1, the notion of an (A,B)-invariant subspace has two nonlinear generalizations: (a) a controlled invariant submanifold and (b) a controlled invariant distribution. Given a point p 0 on a controlled invariant submanifold, in general it is not clear whether there exists a controlled invariant, nonsingular, and involutive distribution defined in a neighbourhood of p 0 such that, at p 0, the distribution coincides with the tangent space of the submanifold. Without a controlled invariant distribution, it is not possible to define a quotient control system as in Section Here, we derive a Brunovský normal form 1 for the quotient system induced by the invariant subspace P. We do this by following the procedure in [118], but we do it in a way that can be generalized to nonlinear systems of the form (3.1). Specifically, we find a basis for X and a regular static feedback transformation u = Fx+Gv so that the system matrix A + BF has the form (3.6), and BG has the form (3.7). We do this by finding a set of output functions yielding a well-defined relative degree, and using Theorem In linear systems theory, controllability indices, see [20] and [118], describe those properties of (A,B) which are invariant under coordinate and nonsingular feedback transfor- 1 A controllable pair of matrices (A,B) are in Brunovský normal form or canonical form if they take the form of [118, Corollary 5.3], see also the matrices on page 96 or page 115.

104 Chapter 3. Local Transverse Feedback Linearization 89 mations, and serve to categorize controllable linear systems. We wish to adapt these ideas to the problem considered here and understand the relationship between the controllability indices of the quotient system induced by P and some of the properties of the original system (3.3). To this end, following [118], let B := Im B, S j := B +Ā B + Āj B, and ρ 0 := dim ( B) ρ j := dim ( S j ) dim ( S j 1 ), j N. (3.9) The controllability indices of the pair (Ā, B) are defined as k i := card { ρ j i : j 0}, i {1,..., ρ 0 }. (3.10) Proposition Let P be an (A,B)-invariant subspace for system (3.3). Let F : X U be a linear map such that (A+BF)P P and let (Ā, B) be the maps induced by P. Then the integers { ρ 0,..., ρ i,...} defined by (3.9) are the same as the integers ρ 0 := dim (P + B) dimp ρ j := dim (P + S j ) dim (P + S j 1 ), j N. (3.11) Proof. If P : X X /P then (see [118, Section 0.5]) for arbitrary T X PT = P + T P. It follows that B = PB = (P + B)/P and hence ρ0 = ρ 0. Furthermore since P(A + BF) = ĀP, S j = B + Ā B + Āj B = PB + ĀPB + + Āj PB = P ( B + (A + BF)B + + (A + BF) j B ) = P ( B + AB + + A j B ) (by [118, Lemma 2.1]) = PS j = P + S j P. Therefore, dim ( S j ) = dim (P + S j ) dim (P) = dim (S j ) dim (P S j ), and it immediately follows that ρ j = ρ j for all j 1.

105 Chapter 3. Local Transverse Feedback Linearization 90 Proposition motivates us to call the integers k i := card {ρ j i,j 0}, i {1,...,ρ 0 }. (3.12) the transverse controllability indices of (3.3) with respect to P. As we have seen, the transverse controllability indices coincide with the integers (3.10). The property (3.5) of being transversely controllable with respect to P is equivalent to ρ 0 i=1 k i = n n. From the proof of Proposition 3.3.4, P (P + S n n 1) = S n n 1. If the system is transversally controllable with respect to P, then this identity becomes P(X ) = X /P = S n n 1, implying that the quotient system (Ā, B) is controllable. Vice-versa, if the quotient system (Ā, B) is controllable, then S n n 1 = P (P + S n n 1) = X /P which gives that P +S n n 1 = X, proving that the original system (A,B) is transversally controllable. In conclusion, system (3.3) is transversely controllable with respect to P if and only if the quotient system (Ā, B) is controllable. Throughout the rest of this section, we assume (3.3) is transversely controllable with respect to P with the list {ρ 0,...,ρ n n 1} defined by (3.11) and transverse controllability indices {k 1,...,k ρ0 } defined by (3.12). Let B := (P B) B and let B := P B. Then B = B B and B has dimension ρ 0. Let {b 1,...,b ρ 0 } be a basis for B and {b ρ 0 +1,...,b m} a basis for B. By construction, we have that P + B = P B and AB P + B so that A i B P + B + AB + + A i B, i N {0}. (3.13)

106 Chapter 3. Local Transverse Feedback Linearization 91 Let {p 1,...,p n } be a basis for P. Since (3.5) and (3.13) hold, there are n linearly independent vectors in the list p 1,...,p n ; b 1,...,b ρ 0 ; Ab 1,...,Ab ρ 0 ;...; A n n 1 b 1,...,A n n 1 b ρ 0. (3.14) Working from left to right, delete each vector that is linearly dependent on its predecessors. By relabeling the b i, if necessary, the list can be arranged so that terms involving b ρ 0,...,b 1, respectively, disappear from the list in that order. The reduced list then looks like b 1,...,b ρ 0 ;...; A kρ 0 1 b 1,...,A kρ 0 1 b ρ 0 ; A kρ 0 b 1,...,A kρ 0 b ρ0 1;...;A k ρ b 1,...,A k ρ b ρ 0 1; A k 3 b 1,A k 3 b 2;...; A k 2 1 b 1,A k 2 1 b 2; (3.15) A k 2 b 1;...; A k 1 1 b 1; p 1,...,p n. By construction, any vector A i b j, A i b j not appearing in the list is a linear combination of p 1,...,p n and all vectors A k b l in the list with k {0,...,i} and l {1,...,ρ 0}. Next, we generate an isomorphism by composing the flow generated by each constant vector field in the list (3.15) in the reverse order. Let x 0 = 0 P X, and consider the map S := (s 1,...,s n ) Φ S (x 0 ), defined as Φ (x S 0 ) := φ p n φ p 1(x s 0 ). (3.16) n Since the vector fields p 1,...,p n are constant, their flows take on a particularly simple form φ p i(x) = p s i s i + x, i {1,...,n }, i s 1 and hence [ Φ (x S 0 ) = p 1 p n ] s 1.. s n

107 Chapter 3. Local Transverse Feedback Linearization 92 Similarly, with the remaining n n vector fields in (3.15), we can generate another map S := (s 1,...,s n n ) Φ S (x), defined as Φ S (x) := φ Ak 1 1 b 1 s 1 φ Ak 2b 1 φ b ρ 0 φ b 1 (x) (3.17) s k 1 k 2 1 s n n s ρ 0 +1 n n where the flow generated by each vector field above has the form φ X i(x) = X s i s i + x, i {1,...,n n } i and X i is the appropriate vector field from (3.15). Let s := (S,S ) and consider the map s Φ s (x 0 ), R n X, defined as Φ s (x 0 ) := Φ S Φ S (x 0 ). (3.18) The map Φ s (x 0 ) is obtained by first applying the map (3.16) followed by the map (3.17). Explicitly, (3.18) has the form [ ] Φ s (x 0 ) = p n p 1 A k1 1 b 1 A kρ 0 1 b ρ 0 b ρ 0 b 1 }{{} M 1 s 1. s n s 1. s n n. By construction, the map Φ s (x 0 ) is an isomorphism since the vectors in (3.15) are linearly independent. Let M 1 be the matrix above such that Φ s (x 0 ) = M 1 s. The columns of M 1 are, from left to right, the vectors of (3.15) in reverse order. Partition the matrix M as M = M M, so that S = M x and S = M x. Intuitively, the map Φ s (x 0 ) takes a set of times s R n and maps these times uniquely to a point x X. The benefit of this construction is that

108 Chapter 3. Local Transverse Feedback Linearization 93 we can now easily identify ρ 0 virtual output functions λ 1 (x),...,λ ρ0 (x) that, along with their derivatives along the vector field of the open loop system (3.3), will lead to a basis for X, and a feedback transformation u = Fx + Gv so that the system matrix A + BF has the form (3.6) and BG has the form (3.7) with (A 2,B 21 ) in Brunovský normal form. Before proceeding further, notice that P = {x X : M x = 0}. Take as virtual output functions those times associated with the flows generated by A k1 1 b 1,...,A kρ 0 1 b ρ 0, i.e., the right-most vector fields appearing in each row of (3.15), not including the final row. In other words, let λ 1 (x) := s 1(x) = m 1 x λ 2 (x) := s k 1 k 2 +1(x) = m k1 k 2 +1x (3.19) λ ρ0 (x) := s n n ρ 0 k ρ0 +1(x) = m n n ρ 0 k ρ0 +1x where m i is the i th row of M. Since P = {x X : M x = 0}, we have λ 1 (P) = 0,...,λ ρ0 (P) = 0. (3.20) For notational simplicity, re-label c 1 := m 1,c 2 := m k1 k 2 +1,...,c ρ0 := m n n ρ 0 k ρ0 +1. We now show that system (3.3) with output λ(x) = (λ 1 (x),...,λ ρ0 (x)) has relative degree {k 1,...,k ρ0 }, that is (see Section 2.5.2) λ 1 (B) = 0,..., λ 1 (A k1 2 B) = 0 λ 2 (B) = 0,..., λ 2 (A k 2 2 B) = 0 (3.21) λ ρ0 (B) = 0,..., λ ρ0 (A kρ 0 2 B) = 0 and the ρ 0 m decoupling matrix λ 1 (A k1 1 b 1 ) λ 1 (A k1 1 b ρ0 ) λ 1 (A k1 1 b m ) λ 2 (A k2 1 b 1 ) λ 2 (A k2 1 b ρ0 ) λ 2 (A k2 1 b m ) λ ρ0 (A kρ 0 1 b 1 ) λ ρ0 (A kρ 0 1 b ρ0 ) λ ρ0 (A kρ 0 1 b m ) (3.22)

109 Chapter 3. Local Transverse Feedback Linearization 94 has full rank ρ 0. We first show that (3.21) holds for the first output λ 1 (x) = c 1 x. Since MM 1 = M 1 M = I n we have that λ 1 annihilates every column of the matrix M 1 except for A k 1 1 b 1, i.e., λ 1 ( P + B + AB + + A k 1 2 B ) = 0 λ 1 (A k 1 1 b 1 ) = 1. By (3.13), for all i N {0}, A i B P+B +AB + +A i B, and thus λ 1 (A i B) = 0 for all i k 1 1. Analogously, for each j {1,...,ρ 0 }, λ j (A i B) = 0 for all i k j 1. Besides proving (3.21), these arguments also show that (3.22) has the structure and therefore, it has full rank. Having shown that the outputs λ 1,...,λ ρ0 yield vector relative degree {k 1,...,k ρ0 }, we follow the procedure in the proof of Theorem to generate a coordinate and feedback transformation. Generating the coordinate & feedback transformation Having shown that the outputs λ 1,...,λ ρ0 yield vector relative degree {k 1,...,k ρ0 }, in this section we repeat the procedure in the proof of Theorem to create a coordinate transformation Ξ : X X and a feedback transformation u = Fx + Gv so that the resulting system takes on the form 2.18 (see also (3.8)). Start with the system (3.3). It can be shown that, in light of the identities (3.21), (3.22), the linear functionals {λ i A j : X R, i {1,...,ρ 0 },j k i } are linearly independent. As a result, we can use them to form a partial basis of X. Augment these functionals with n more linearly independent linear functionals η(x) := (η 1 (x),...,η n (x)) so that, together, they generate

110 Chapter 3. Local Transverse Feedback Linearization 95 an isomorphism Ξ : X X, x (η,ξ) Ξ :x col (η,ξ) = η 1 (x) η n (x) ξ 1 (x) ξ ρ 0 (x), where ξ i (x) = ξ1(x) i ξ2(x) i = ξk i i 1 (x) λ i (x) λ i (Ax) = λ i (A ki 2 x) c i x c i Ax c i A k i 2 for i {1,...,ρ 0 }. Using (3.21), it is straightforward to show that. (3.23) ξ i 1 =ξ i 2 ξ 2 i =ξ3 i ξ k i i = λ i (A k i x) + λ x=ξ 1 (η,ξ) i(a ki 1 B)u for 1 i ρ 0. The matrix of coefficients that multiply the input u in the ξ subsystem λ 1 (A k1 1 b 1 ) λ 1 (A k1 1 b m ) λ 2 (A k2 1 b 1 ) λ 2 (A k2 1 b m ) D :=, λ ρ0 (A kρ 0 1 b 1 ) λ ρ0 (A kρ 0 1 b m ) is the decoupling matrix discussed in Section The matrix D above is not exactly the same as the matrix (3.22). In (3.22), we chose a special basis for B that resulted in (3.22) taking the form [ u ], where u is an upper-triangular matrix with ones along the diagonal. Since we haven t done that here, we do not expect the matrix to retain any special structure except for the important property that it is full-rank.

111 Chapter 3. Local Transverse Feedback Linearization 96 As for the remaining states, since the linear functions η(x) were chosen to complete the isomorphism Ξ(x), their corresponding dynamical equations in (η,ξ) coordinates do not take on any special form. Choose G : U U, G = [G 1 G 2 ] so that DG 1 = I ρ0, i.e., G 1 is a right inverse of D and DG 2 = 0, i.e., the columns of G 2 span the kernel of D. Then let F : X U be defined as F = G ( col ( c 1 A k 1,...,c ρ0 A kρ 0, 0(m ρ0 ) 1)). Finally, choose u = Fx + Gv = FΞ 1 (η,ξ) + Gv and set v = (v,v ) U where v and v correspond to the decomposition of the matrix G. The result of this feedback transformation is that in (η,ξ) coordinates system (3.3) takes the form in (3.8), η = A 1 η + A 3 ξ + B 11 v + B 12 v ξ = A 2 ξ + B 21 v (3.24) where now (A 2,B 21 ) is in Brunovský normal form with controllability indices {k 1,...,k ρ0 }, i.e., A 2 = A A ρ 0 2, B 21 = b 1... b ρ 0 where A i 2 = , b i = k i k i k i 1 Lastly, we will show that {(η,ξ) R n R n n : ξ = 0} = Ξ(P). Since the subspaces have the same dimension, it suffices to show that Ξ(P) {(η,ξ) R n R n n : ξ = 0}. By (3.20), λ 1 (P) = = λ ρ0 (P) = 0. This fact, together with (3.13), (3.21), yields λ i (A j x) = 0 for all i {1,...,ρ 0 } and j {0,...k i 1}, proving the inclusion Ξ(P) {(η,ξ) R n R n n : ξ = 0}.

112 Chapter 3. Local Transverse Feedback Linearization 97 Theorem For a linear time-invariant system (3.3) and an (A,B)-invariant subspace P, let the integers {ρ i } and {k i } be defined by (3.11) and (3.12) respectively. Then the following conditions are equivalent: (a) System (3.3) is transversely controllable with respect to P, i.e., P+S n n 1 = X. (b) There exist ρ 0 linearly independent functionals λ i X, i {1,...,ρ 0 } yielding a well-defined relative degree of {k 1,...,k ρ0 } on P and such that P {x X : λ 1 (x) = = λ ρ0 (x) = 0}. (c) LTFLP is solvable on X. Example Consider the LTI system A = , B = and the (A,B)-invariant subspace P = span {p 1 } = span {col (0, 1, 0, 1)} with dimension n = dim (P) = 1. Calculating the integers (3.11) we obtain ρ 0 = dim (P + B) dim (P) = rank rank = In the same way we obtain ρ 1 = ρ 2 = 1 so that {ρ 0,ρ 1,ρ 2 } = {1, 1, 1} and the transverse controllability indices (3.12) are {k 1 } = {3}. Since k 1 = n n, the system is transversely

113 Chapter 3. Local Transverse Feedback Linearization 98 controllable with respect to P. This is not surprising since this system is controllable. Simple matrix calculations yield B = (P B) B = span {col(1, 0, 0, 0)}. Let b 1 = col (1, 0, 0, 0), and consider the matrix [ M 1 = p 1 A 2 b 1 Ab 1 b = ] and M = M M, M R 1 4, M R = Since ρ 0 = 1, we have only one output, given by the second row of M, λ 1 (x) = [ ] x =: c 1 x. Therefore, we obtain [ ] ξ1 1 := c 1x = x [ ] ξ2 1 := c 1Ax = x [ ] ξ3 1 := c 1A 2 x = x,

114 Chapter 3. Local Transverse Feedback Linearization 99 [ ξ 1 := col(ξ1,ξ 1 2,ξ 1 3). 1 Choose η(x) := Ξ(x) = η(x) ξ 1 (x) ] x so that the map Ξ : X X = x [ ] [ is an isomorphism. The decoupling matrix D = λ 1 (A 2 b 1 ) λ 1 (A 2 b 2 ) = 2 1 full rank. Let G : U U be [ ] G = G 1 G 2 = so that DG 1 = 1 and DG 2 = 0. Let F : X U be given by F = G c 1A 3 0 = ] is Let v = (v,v ) and set u = Fx + Gv = FΞ 1 (η,ξ) + Gv to obtain the control system [ ] η = 1.4η ξ v + v ξ 1 1 = ξ 1 2 ξ 1 2 = ξ 1 3 ξ 1 3 = v. 3.4 Local transverse feedback linearization solution We now return to the nonlinear version of LTFLP. Consider the control system (3.1) on R n and the controlled invariant submanifold Γ R n. Hereafter, we assume that the

115 Chapter 3. Local Transverse Feedback Linearization 100 preliminary regular feedback (u,i m ) is applied to (3.1) so that Γ is an invariant set for the vector field f. Next, we present a technical result which is useful in proving the main theorem. The next result is a nonlinear version of parts (b) and (c) of Theorem Theorem LTFLP is solvable at p 0 if and only if there exist ρ 0 smooth R-valued functions λ 1,...,λ ρ0, defined on a neighbourhood of p 0 in R n satisfying (i) for some neighbourhood U of p 0 in R n, U Γ {x U : λ i (x) = 0, i = 1,...,ρ 0 }, and (ii) the system m ẋ = f(x) + g i (x)u i y = col(λ 1 (x),...,λ ρ0 (x)) has vector relative degree {k 1,...,k ρ0 } at p 0, for some k 1,...,k ρ0 such that k k ρ0 = n n. Moreover, if LTFLP is solvable, then there exists a neighbourhood V U of p 0 such that, on V, a connected component Z of the zero dynamics manifold of (3.25) coincides with Γ : Z V = Γ V. The proof of Theorem uses Lemma and so we postpone its presentation until Section 3.5, page 115. Definition Let λ 1,...,λ ρ0 be ρ 0 smooth R-valued functions satisfying the conditions of Theorem The map λ(x) := col (λ 1 (x),...,λ ρ0 (x)) is called a local transverse output of (3.1) with respect to Γ. i=1 (3.25) Remark In the LTI setting (Section 3.3.2), we found ρ 0 linearly independent functionals λ 1,...,λ ρ0 and we used these functions and their derivatives along the vector field of the open loop system (3.3) to find a coordinate and a feedback transformation to obtain the desired normal form. These linear functions satisfy the conditions of Theorem 3.4.1, as stated in Theorem (b).

116 Chapter 3. Local Transverse Feedback Linearization 101 In Section 3.5 we give a coordinate-free definition of transverse controllability indices for nonlinear systems. It turns out (see Lemma 3.5.3) that {k 1,...,k ρ0 } in Theorem are precisely the transverse controllability indices of (3.1) with respect to Γ. When specialized to the LTI setting, i.e., when we have a system of the form (3.3) and Γ is an (A,B)-invariant subspace, these indices are exactly the indices defined by (3.11) and (3.12). Theorem characterizes the solvability of LTFLP in terms of the existence of a local transverse output function λ : U R ρ 0 satisfying (i) and (ii). Once a transverse output function is known, a coordinate and feedback transformation yielding (3.2) is found constructively following the procedure in the proof of Theorem However, Theorem does not provide a way of finding an output function or even to determine whether one exists. Hence, it is of limited value for constructing the coordinate and feedback transformation. The theorem also shows that LTFLP is equivalent to the zero dynamics assignment problem with relative degree mentioned in Chapter 1: LTFLP is solvable if and only if Γ can be made into the zero dynamics manifold of (3.1) induced by a suitable output, col (λ 1 (x),...,λ ρ0 (x)), yielding a well-defined vector relative degree. This observation helps to better understand the relationship between IOFLP and LT- FLP. While, in IOFLP, one is given an output function and gets a zero dynamics manifold, in LTFLP one is given a controlled invariant manifold and looks for an output function for which the given manifold is the zero dynamics manifold. Hence, in some sense, the two problems are inverse of each other. Consider the distributions G i := span{ad j f g k : 0 j i, 1 k m}, (3.26) and recall from Section that Ḡi denotes the involutive closure of G i.

117 Chapter 3. Local Transverse Feedback Linearization 102 Theorem (main result). Suppose that the distributions Ḡi, i n n 1, are regular at p 0 Γ. Then, LTFLP is solvable at p 0 if and only if (i) T p0 Γ + G n n 1(p 0 ) = T p0 R n, and (ii) there exists an open neighbourhood U of p 0 in R n such that, for all i n n 1, ( p Γ U) dim(t p Γ + G i (p)) = dim(t p Γ + Ḡi(p)) = constant. Compare Theorem to Theorem In the LTI setting condition (ii) of Theorem automatically holds because all of the distributions in question are constant and hence also involutive (see page 53). On the other hand, condition (i) of Theorem is exactly the condition for transverse controllability (3.5). The assumptions of Theorem are checkable. By this we mean that checking condition (i) amounts to checking the rank of a matrix at the point p 0. Checking condition (ii) amounts to checking that the rank of a suitable matrix is constant in some neighbourhood of p 0 in Γ. While there may be cases when the latter check cannot be carried out analytically, in practice one can check condition (ii) numerically. The proof of Theorem does not provide a constructive procedure for finding the virtual outputs described in Theorem The next result sheds additional light on LTFLP by relating it to the partial feedback linearization problem. The result isn t a viable solution to LTFLP because its assumptions are not checkable. On the other hand, when used in conjunction with Theorem the theorem provides guidelines for finding the output function in Theorem 3.4.1, as discussed below. Theorem Suppose that Ḡi, i n n 1 are regular at p 0 Γ. Then, LTFLP is solvable at p 0 if and only if there exist a neighbourhood U of p 0 and a smooth, involutive, and nonsingular distribution on U such that (i) Γ = TΓ. (ii) is locally controlled invariant for (3.1).

118 Chapter 3. Local Transverse Feedback Linearization 103 (iii) ( p Γ U) T p Γ + G n n 1(p) = T p R n. (iv) ( i n n 1) + G i is nonsingular and involutive on U. Proof. Suppose that LTFLP is solvable at p 0. The necessity of conditions (i) - (iv) can be easily shown by considering the normal form (3.2) and taking { } = span,...,. (3.27) η 1 η n Conversely, suppose conditions (i)-(iv) hold. These conditions imply the conditions of Theorem In particular, since ( + G i = + G i ) ( + Ḡ i = + G i ), condition (iv) implies conditions (i) and (ii) of Theorem Conditions (i) and (iii) above imply that, on a neighbourhood of Γ U in R n, +G n n 1 has dimension n, and hence condition (iii) of Theorem holds. Therefore, by Theorem and Lemma , we obtain a system whose dynamics in transformed coordinates is given by (2.10) and where is given by (3.27). The integral submanifolds of foliate a neighbourhood U of p 0 and are locally given, in (η,ξ) coordinates by {(z,ξ) : ξ = constant}. Condition (i) above means that one of the leaves of the foliation is precisely Γ U. Without loss of generality, this leaf can be taken as the zero level set {(z,ξ) : ξ = 0}. Finally, let ρ 0 be the rank of B in (2.10) and let G be a m m matrix whose first ρ 0 columns span Im (B 2 ), while the remaining ones span ker (B 2 ). After the feedback transformation u = Gv, (2.10) takes the desired form (3.2). Note that the distribution in Theorem is not unique. Together, Theorems 3.4.1, 3.4.4, and can be used to find the coordinate and feedback transformation needed in order to bring (3.1) into the normal form (3.2). The following steps outline the typical procedure one may follow in searching for the output function from Theorem Represent Γ in a neighbourhood of p 0 as the zero level set of n n R-valued functions. Following Theorem 3.4.1, check if there exists a subset of ρ 0 of these functions, with ρ 0 defined in (3.30), yielding the correct vector relative degree.

119 Chapter 3. Local Transverse Feedback Linearization If the above step fails, check the conditions of Theorem to verify whether or not the problem is solvable. In simple cases, the procedure described in the proof of Theorem may yield the desired output functions. In the LTI case, when Γ is an (A, B)-invariant subspace, the procedure in the proof of Theorem is the one that was followed in Section It yields the same result as in Theorem If Theorem establishes that the problem is solvable, then there exists a distribution satisfying the conditions in Theorem If is found then, after computing the controllability indices defined in (3.31), the output functions are obtained by finding those exact one-forms that span the codistributions ann ( + G i ), i = 0,...,k 1 2, and arranging them in the order illustrated below, with the integers ρ i defined in (3.30). ann ( + G k1 2) dλ 1.. ann ( + G k1 2 j) dl j f λ 1.. ann ( + G k2 2) dl k 1 k 2 f λ 1 dλ 2. ann ( + G k2 2 j) dl k 1 k 2 +j f λ 1 dl j f λ 2... ann ( + G k3 2) dl k 1 k 3 f λ 1 dl k 2 k 3 f λ 2 dλ 3. ann ( ) + G kρ0 2 ann ( + G 0 )..... dl k 1 k ρ0 f λ 1 dl k 2 k ρ0 f λ 2 dλ ρ0.. dl k 1 2 f λ 1 dl k 2 2 f λ 2 dl kρ 0 2 f λ ρ0. In each row of the above table, the codistribution in the left column is locally spanned by all the differentials in that row plus all the differentials in the rows..

120 Chapter 3. Local Transverse Feedback Linearization 105 above. For example, locally we have that ann ( + G k2 2) = span{dλ 1,dL f λ 1,...,dL k 1 k 2 f λ 1,dλ 2 }. The functions λ 1,...,λ ρ0, resulting from the integration of the exact one-forms dλ 1,...,dλ ρ0 along the diagonal of the table, are the required outputs. They can be used to obtain the normal form (3.2) by following the procedure from the proof of Theorem Next, we present an example to illustrate the use of Theorems 3.4.1, 3.4.4, and Example Consider the system x 2 x 1 ẋ = f(x) + g(x)u = x 3 x x 3 1 u 1 + x 2 x u 2 (3.28) along with the pair (Γ,u ) I (f,g, R 4 ) F(f,g, Γ ) Γ = { x R 4 : x x 2 2 x 3 = x 4 = 0 } u = col(0, 0). The set Γ is an elliptic paraboloid embedded in the subspace {x R 4 : x 4 = 0}. Let γ 1 (x) = x x 2 2 x 3 and γ 2 (x) = x 4. To check that Γ is indeed controlled invariant, note that, letting v = f + gu, L v γ 1 (x) = dγx(f(x)) 1 = 2x 1 x 2 + 2x 2 x 1 x 3 x 4 = x 3 x 4 L v γ 2 (x) = dγx(f(x)) 2 = 0 both vanish on Γ, and so by Theorem , Γ is invariant under v. We want to perform transverse feedback linearization of (3.28) with respect to Γ near p 0 = col(4, 0, 2, 0). In this example, n = 4 and n = 2, so we seek to feedback

121 Chapter 3. Local Transverse Feedback Linearization 106 linearize a subsystem of dimension n n = 2. The natural approach to solving this problem is to check if one of the two constraints which define Γ satisfy the conditions of Theorem In this case, both constraints γ 1 (x) and γ 2 (x), taken individually as scalar outputs, yield a well-defined relative degree near p 0 of 1 which does not equal n n. Taken together, as a vector output γ = col(γ 1,γ 2 ), the constraints do not yield a well-defined vector relative degree. In both cases, the conditions of Theorem are not satisfied by these constraints. Next, we check whether or not LTFLP is solvable for (3.28) using Theorem Checking condition (i), one first finds that for all x Γ \ {x R 4 : x 1 = x 2 = 0}), T x Γ = span{t 1,t 2 }(x) = span x 2 x 1 0 0, x 1 x 2 2 (x x 2 2) 0 and G 1 = span{g 1,g 2,ad f g 1,ad f g 2 } = span 0 0 x 3 1, x 2 x 1 0 0, 0 0 x 3 0, so that dim (T p0 Γ + G 1 (p 0 )) = 4. Also, since [g 1,g 2 ] = 0, it follows that G 0 = Ḡ0 everywhere. It is then an easy matter to check that, for any p Γ, dim (T p Γ + G 0 (p)) = dim ( T p Γ + Ḡ0(p) ) = 3. Thus, condition (ii) of Theorem holds and LTFLP is solvable despite the fact that the constraints which locally define Γ do not satisfy Theorem The fact that Theorem holds for this system implies that there exists a distribution satisfying the conditions of Theorem Seeking this distribution we recall

122 Chapter 3. Local Transverse Feedback Linearization 107 that condition (i) of Theorem asks that Γ = TΓ. Since the above basis for T p Γ is well-defined at points off Γ, = span{t 1,t 2 } could conceivably be used as a candidate for the distribution. However, it is easy to check that, for = span{t 1,t 2 }, condition (ii) of Theorem (i.e. is locally controlled invariant) does not hold. To see this, we observe that [f, ] + G 0 and so by Theorem 2.5.8, is not locally controlled invariant and condition (ii) of Theorem is violated. Another choice for can be found by looking at all linearly independent solutions v 1,v 2 : R 4 TR 4 of the equations dγ 1 x(v 1 ) = dγ 2 x(v 2 ) = 0, x Γ, and, among those, choose the vector fields satisfying the conditions for controlled invariance in Theorem By doing that, one finds that the distribution = span{v 1,v 2 } = span x 2 x 1 0 0, x 1 x 2 2x 3 0 satisfies the conditions of Theorem Then, by the Frobenius theorem and Proposition , there exists an exact one-form dλ which spans ann( + G 0 ). For instance, the function ( ) x3 λ(x) = ln x x x yields a well-defined relative degree of 2 at p 0 and satisfies Theorem As a result, the coordinate transformation Ξ : x col(η 1,η 2,ξ 1,ξ 2 ) defined as η 1 = x 1, η 2 = x 2,

123 Chapter 3. Local Transverse Feedback Linearization 108 ξ 1 = λ, ξ 2 = L f λ = x 4, directly yields the normal form (3.2), with v = u 1 and v = u 2 η 1 = η 2 η 2 v η 2 = η 1 + η 1 v ξ 1 = ξ 2 (3.29) ξ 2 = v The coordinate transformation Ξ(x) is valid on R 4 \ ( {x R 4 : x 1 = x 2 = 0} {x R 4 : x 3 0} ). It is useful to specialize Theorem to the case when Γ is an equilibrium, because in this special case LTFLP coincides with the state-space exact linearization problem whose solution was presented in Theorem on page 66. Corollary Assume that Γ = {p 0 } is an equilibrium point of the open-loop system ẋ = f(x) and that Ḡi, i n 1, with G i defined by (3.26), are regular at p 0. Then, LTFLP is solvable at p 0 if and only if (i) dimg n 1 (p 0 ) = n, and (ii ) G i, i n 1, are involutive and regular at p 0. Proof. It suffices to show that, under the assumption that the distributions Ḡi are regular, (ii ) is equivalent to condition (ii) in Theorem Assume that (ii) in Theorem holds, i.e., G i (p 0 ) = Ḡi(p 0 ). For all p in a neighborhood of p 0, one has dim(g i (p 0 )) dim(g i (p)) dim(ḡi(p)) = dim(ḡi(p 0 )) = dim(g i (p 0 )), where the first inequality follows from Proposition Therefore, all inequalities above are equalities, and dim(g i (p)) = dim(g i (p 0 )) and dim(g i (p)) = dim(ḡi(p)), proving that (ii) = (ii ). The converse implication is obvious.

124 Chapter 3. Local Transverse Feedback Linearization 109 For convenience, we restate Theorem Theorem (state-space exact linearization [49, 57]). Assume that Γ = {p 0 } is an equilibrium point of the open-loop system ẋ = f(x). Then, the state-space exact linearization problem is solvable at p 0 if and only if conditions (i) and (ii ) in Corollary hold. Note that conditions (i) and (ii ) in Corollary imply that the distributions Ḡi, i n 1, are regular at p 0. Thus, in the special case when Γ is an equilibrium point, the conditions of our main result coincide with those of the state-space exact linearization problem. The difference between Corollary and Theorem is that the former relies on the preliminary assumption that the distributions Ḡi are regular at p 0, while the latter shows that regularity of Ḡi at p 0 is actually necessary for the solvability of LTFLP, and hence there is no need to impose it as a preliminary requirement. It is instructive to note that, in light of Theorem , a necessary condition for LTFLP to be solvable is that system (3.1) has a linear subsystem of dimension at least n n. In other words, it is necessary that the sum of the controllability indices ki in (2.13) be greater than or equal to n n. However, this condition is not sufficient for LTFLP to be solvable, because it does not guarantee that Γ is the zero level set of the linear states after the coordinate and feedback transformation. 3.5 Transverse controllability indices and preliminary results In this section we will introduce the notion of transverse controllability indices for system (3.1) with respect to Γ I (f,g, R n ). This generalizes the indices we introduced in Section 3.3. We then present results that clarify the relationship between our indices and analogous indices found in the literature. We show that the transverse controllability

125 Chapter 3. Local Transverse Feedback Linearization 110 indices of a system with respect to a set are coordinate and feedback invariant. Finally, we use these indices to find a feedback transformation β : R n GL(m, R) that allows us to generate the nonlinear counterpart to the array of vector fields (3.15). In the nonlinear setting, controllability indices have been used to characterize the largest feedback linearizable subsystem of a nonlinear system [74], and conditions under which a system is feedback linearizable, see [76], [84]. In the context of LTFLP, let V be an open subset of Γ and for each p V, let ρ 0 (p) := dim(t p Γ + G 0 (p)) dim(t p Γ ) ρ i (p) := dim(t p Γ + Ḡi 1(p) + ad i fg 0 (p)) dim(t p Γ + Ḡi 1(p)), (3.30) i = 1, 2,..., where G i are as defined in (3.26), and ad i fg 0 = span{ad i fx : X G 0 }, i = 0, 1,... Geometrically, at each p Γ, the integers ρ i (p) represent the number of linearly independent vectors in ad i f G 0(p) which are not in T p Γ + Ḡi 1(p), see Figure 3.4. In the LTI setting, when Γ is an (A,B)-invariant subspace P, it is useful to compare these integers to the integers (3.11). It is common to identify T p P with P itself, since they are isomorphic for all p P. Moreover, for the LTI system (3.3) we note that f(x) = Ax and g(x) = B so that ad f G 0 = [Ax, B] = AB ad i fg 0 = [ ] Ax, ad i fg 0 = ( 1) i A i B and G 0 = Ḡ0 = B G i = Ḡi = B + AB + + A i B = S i. Therefore, (3.30) reduces exactly to (3.11). Associated to the list {ρ 0 (p),...,ρ i (p),...} is a set of ρ 0 (p) integers, {k 1 (p),...,k ρ0 (p)}, which we refer to as the transverse controllability indices of (3.1) with respect to Γ, defined as (we omit the argument p) k i := card {ρ j i : j 0}, i {1,...,ρ 0 }. (3.31)

126 Chapter 3. Local Transverse Feedback Linearization 111 g 1 (p) G 0 (p) g 2 (p) p T p Γ Γ Figure 3.4: An illustration of the integer ρ 0 (p). In this figure ρ 0 (p) = 2. Note that k 1 k 2 k ρ0. We show in Corollary that the transverse controllability indices are invariant under coordinate and feedback transformations. Condition (ii) of Theorem implies that ρ 0,ρ 1,...,ρ n n 2 are constant, while condition (i) implies that i ρ i = n n. In the special case when Γ is an equilibrium point, it is useful to compare our definition of controllability indices with the definition by Marino in [74]. In that paper, the author considers a system of the form (3.1) around an equilibrium point x 0, and seeks to characterize the largest feedback linearizable subsystem in (3.1) around the equilibrium point x 0. The formulation from [74] relies on the distributions (2.11) defined in Section For the reader s convenience we present the definition of these distributions again G f = f + G 0 = {f + g : g G 0 }, G i = G i 1 + [G f,g i 1 ], G 0 = G 0, i = 1, 2,..., S i = G i 1 + ad i fg 0, S 0 = G 0, i = 1, 2,...

127 Chapter 3. Local Transverse Feedback Linearization 112 In [74] the integers r 0 = dimg 0, r i = dim S i dim G i 1 are used in place of the integers ρ i in the definition of controllability indices. We now show that, when Γ is an equilibrium point, the integers ρ i and r i are identical, and thus the notion of transverse controllability indices reduces to the classical notion of controllability indices. Lemma For all non-negative integers i, Ḡ i = G i. Thus, when Γ = {p 0 } is an equilibrium point, ρ i = r i. Proof. By definition, G 0 = G 0, so the lemma trivially holds for i = 0. We first show that Ḡ i G i for all i N. By definition, G i = G i 1 + [G f,g i 1 ]. Since f G f, it follows that G i G i for all nonnegative integers i, which implies Ḡi G i. Next, we show that G i Ḡi for all i N which implies Gi Ḡi. To this end, it suffices to prove that G i Lie C (R n )(G i ) since Lie C (R n )(G i ) Ḡi (see Section 2.4.8). It is obvious that G 0 Lie C (R n )(G 0 ). In fact, G 1 = G 0 + [G f,g 0 ] = span{g 1,...,g m,ad f g 1,...,ad f g m, [g,g 1 ],...,[g,g m ] : g G 0 } Lie C (R n )(G 1 ). For the induction, assume that, for some positive integer I 2, G i 1 Lie C (R n )(G i 1 ), i {2,...,I}. We must show that G i = G i 1 + [G f,g i 1 ] Lie C (R n )(G i ) for i {2,...,I}. It is enough to prove that [G f,g i 1 ] = [f,g i 1 ] + [G 0,G i 1 ] Lie C (R n )(G i ). However, since [G 0,G i 1 ] Lie C (R n )(G i 1 ) Lie C (R n )(G i ), all we are left to show is that [f,g i 1 ] Lie C (R n )(G i ).

128 Chapter 3. Local Transverse Feedback Linearization 113 Let τ 1 = g 1, τ 2 = g 2,..., τ im 1 = ad i 1 f g m 1, τ im = ad i 1 f g m. Then a general vector field in Lie C (R n )(G i 1 ) is a C (R n )-linear combination of vector fields of the form ϑ = [ τ jk, [ τ jk 1,, [τ j2,τ j1 ] ]], (3.32) 1 j k im, 1 k <. By assumption, any vector field in G i 1 can also be expressed in this way. Take any vector field h G i 1 and consider [ [f,h] = f, ] c i ϑ i = [f,c i ϑ i ], i I i I where I is some finite index set, c i C (R n ), and ϑ i are of the form (3.32). Each term in the above summation can be expressed as [f,c i ϑ i ] = c i [f,ϑ i ]+(L f c i ) ϑ i, so it is enough to show that [f,ϑ] Lie C (R n )(G i ), where ϑ is of the form (3.32). When k = 1, i.e. ϑ = τ j1, then [f,τ j1 ] G i Lie C (R n )(G i ). Next assume that ϑ = [ τ jk 1, [ τ jk 2,, [τ 2,τ 1 ] ]] is such that [f,ϑ] Lie C (R n )(G i ). We will show that [f, [τ jk,ϑ]] Lie C (R n )(G i ). Clearly, [τ jk,ϑ] Lie C (R n )(G i 1 ) for any 1 j k im. By the Jacobi identity, [f, [τ jk,ϑ]] = [[ϑ,f],τ jk ] + [[f,τ jk ],ϑ], and since [ϑ,f] Lie C (R n )(G i ) and τ jk G i 1, it follows that [[ϑ,f],τ jk ] Lie C (R n )(G i ). Also, [f,τ jk ] G i so that [[f,τ jk ],ϑ] Lie C (R n )(G i ). This induction argument shows that [f,g i 1 ] Lie C (R n )(G i ) Ḡi as required. Corollary The transverse controllability indices of system (3.1) with respect Γ are invariant under coordinate and feedback transformations. Proof. Let U R n be an open set with U Γ. The push-forward map F associated with any F Diff(U) is an isomorphism at each p Γ U. It follows from the definition of the integers ρ 0,...,ρ i,... that they do not change under coordinate transformations. By Lemma 3.5.1, ρ i (p) = dim(t p Γ + S i (p)) dim(t p Γ + G i 1 (p)).

129 Chapter 3. Local Transverse Feedback Linearization 114 In [74, Proposition 2] it is shown that S i and G i 1 are feedback-invariant, and so the integers ρ 0,...,ρ i,... are also invariant under feedback transformations. Lemma Suppose that LTFLP is solvable at p 0 Γ. Then the transverse controllability indices of (3.1) with respect to Γ coincide with the controllability indices of (A,B) in (3.2). Proof. In light of Corollary 3.5.2, we can prove this lemma by direct calculation of the integers ρ i in (η,ξ) coordinates. Let V = Ξ(Γ U). By the properties of the normal form (3.2), for any p Γ U, Ξ( p) = col(p, 0). Hence, in (η,ξ) coordinates we have that for any p V and any i n n, T p V + G i (col(p, 0)) = Im I n.... (3.33) 0 n n n B AB... Ai B In (η,ξ) coordinates, consider the collection of constant distributions i, i n n, given by ( [ i = Im I n B AB A i B ]). At each p V, i (p) = T p V + G i (col(p, 0)). Furthermore, since each i is constant, and hence involutive, and G i V i, it follows that Ḡi V i. This shows that for all i n n TV + Ḡi i = TV + G i TV + Ḡi, and so TV + G i = TV + Ḡi. Calculating the integers ρ i at any p V we have ρ i (p) = dim(t p Γ + Ḡi 1(p) + ad i fg 0 (p)) dim(t p Γ + Ḡi 1(p)) = dim(t p Γ + G i 1 (p) + ad i fg 0 (p)) dim(t p Γ + G i 1 (p)) = dim(t p Γ + G i (p)) dim(t p Γ + G i 1 (p)) = rank( i ) rank( i 1 ) = rank([b A i B]) rank([b A i 1 B]). The lemma follows from the definition of the integers {k 1,...,k ρ0 }.

130 Chapter 3. Local Transverse Feedback Linearization 115 Proof of theorem Proof. ( ) Suppose that LTFLP is solvable and let Ũ = Ξ(U). By Lemma 3.5.3, the pair (A,B) has controllability indices k 1,...,k ρ0. Thus, without loss of generality, we can assume that the pair (A,B) is in Brunovský normal form A = A 1... A ρ 0, B = b 1... b ρ 0, with A i R k i k i and B i R ki 1 given by A i = , b i = We define λ i s in (η,ξ) coordinates. Define λ := (λ 1,...,λ ρ0 ) : Ũ Rρ 0, (z,ξ) C ξ, where C = c 1... c ρ 0, c i = [ ] 1 ki. This choice of λ 1,...,λ ρ0 satisfies conditions (i) and (ii). ( ) The existence of smooth functions λ 1,...,λ ρ0 : U R yielding a vector relative degree {k 1,...,k m } (with i k i = n n ) at p 0 implies, by Theorem , that there exists an open connected neighbourhood V U of p 0 and a local coordinate transformation Ξ Diff(V ) yielding the normal form (3.2), where Ẑ V := {x V : λ i (x) = = L k i 1 f λ i (x) = 0, 1 i ρ 0 } is the zero dynamics manifold of system (3.25). On V, Ẑ V is the maximal controlled invariant subset of λ 1 (0). Thus, since Γ is controlled invariant and Γ V λ 1 (0), we have Γ V Ẑ V. Let Z V denote the connected component of Ẑ V containing Γ V. We now show that Z V = Γ V.

131 Chapter 3. Local Transverse Feedback Linearization 116 Since Γ has the same dimension as Z, Γ V is an open subset of Z V. Moreover, Γ V is closed in V because Γ is closed in R n. By [78, Theorem 17.2], it follows that Γ V is also closed in Z V. In conclusion, Γ V is both open and closed in Z V. Being nonempty and connected, we have that Z V = Γ V as required. Preparatory lemmas for main result When the transverse controllability indices are constant on an open subset of Γ and the distributions Ḡi are nonsingular, the next two lemmas establish the existence of a feedback transformation yielding a particularly useful set of local generators for each Ḡi. Recall, in the LTI setting of Section 3.3.2, that starting from the list (3.14) we deleted redundant vector fields and relabeled vectors so that terms involving b ρ 0,...,b 1, respectively, disappear from the list in that order. In Lemma we will find a feedback transformation that achieves a similar objective, this time in the nonlinear setting. Lemma is needed because, in the nonlinear case, the vector fields ad i f g j are not constant and hence a simple relabeling will not work. The second Lemma in this subsection, Lemma 3.5.5, is used to handle another complication not present in the LTI case. In the nonlinear case, in order to apply the Frobenius theorem (Theorem ), we cannot simply delete the vector fields in the distributions G i that are tangent to Γ at each p Γ. As a result, we need to find a specialized basis for the bundle TΓ that is partially formed by those vector fields from the distributions Ḡ i that belong to T p Γ at each p Γ. Together, Lemmas and will lead to the array (3.47), the nonlinear version of the array (3.15) in the LTI setting. The vector fields in this array will then be used in the proof of the main result, in Section 3.6. Lemma Let Ũ be an open subset of Rn such that Ṽ := Ũ Γ. Assume that,

132 Chapter 3. Local Transverse Feedback Linearization 117 for all i n n, ( p Ṽ ) dim(t pγ + G i (p)) = dim(t p Γ + Ḡi(p)) = constant, ( p Ũ) dim(ḡi(p)) = ν i = constant. Then, ρ 0 ρ 1 ρ n n 1 and there exist an open set U Ũ and a regular static feedback (α,β) on U such that, letting V := U Γ, for all p V and for all i n n, the following holds: T p Γ + Ḡi(p) = T p Γ ( i j=0 ) span {ad j f g } k : 1 k ρ j (p). (3.34) Proof. Choose an open set U Ũ such that V := U Γ, and that V is covered by a coordinate chart in the atlas of Γ. Apply the preliminary feedback transformation (u,i m ) defined on V, and let f = f+gu. On V, define the distribution (i.e. a subbundle of TR n V defined using the natural orthogonal structure on R n ) G 0 = [Ḡ0 TV ] Ḡ0. On V, Ḡ 0 TV is constant dimensional since dim(ḡ0 TV ) = dim(tv ) + dim(ḡ0) dim(tv + Ḡ0), and the last two terms in the sum are constant dimensional. Then, since Ḡ0 is a nonsingular distribution, it follows from Lemma that, by possibly shrinking U (and hence V ), Ḡ 0 TV is smooth, and so too is [Ḡ0 TV ]. Thus, G 0 is the intersection of smooth, nonsingular distributions. Furthermore, G 0 has constant dimension on V since, for each p V, dim(g 0 (p)) = n dim(ḡ0(p) T p V ) + dim(ḡ0(p)) dim ( [Ḡ0(p) T p V ] + Ḡ0(p) ) = dim(ḡ0(p)) dim(ḡ0(p) T p V ) = ρ 0.

133 Chapter 3. Local Transverse Feedback Linearization 118 Since G 0 Ḡ0 and G 0 TV = (Ḡ 0 + TV ) (Ḡ 0 + TV ) = 0, we have ( p V ) T p V G 0 (p) = T p V + Ḡ0(p) = T p V + G 0 (p). By construction, V is covered by a coordinate chart, so (see Section 2.4.3) there exist n independent vector fields on V such that at each p V, T p V = span{v 1,...,v n }(p). Moreover, by possibly shrinking U (and hence V ), there exist ρ 0 vector fields w 1,...,w ρ0 : V TR n V such that G 0 = span{w 1,...,w ρ0 } so that, on V, TV G 0 = span{v 1,...,v n } span{w 1,...,w ρ0 }. Using the fact that G 0 TV + G 0, we write n m w j = α j k v k + β j k g k, j = 1,...,ρ 0, (3.35) k=1 k=1 where α j k : V R, βj k : V R are C (V ) functions. Let β 0 be the m ρ 0 matrix of real-valued functions whose (k,j) th element is β j k and let [ ] [ g 1 g ρ0 = g 1 g m ]β 0. We now show that g 1,..., g ρ0 are V(V )-linearly independent, which implies that β 0 is full rank. Suppose there exist ρ 0 functions c i C (V ) such that ρ 0 i=1 c i g i = 0. Then, by (3.35), ρ 0 i=1 c iw i TV, which implies c i = 0, i = 1,...,ρ 0, since G 0 TV = 0. Note that this argument also shows that span{ g 1,..., g ρ0 } TV = 0. For, if this were false, then there would exist a linear combination of the w i s in (3.35) which belongs to TV. Next, we seek m ρ 0 vector fields g ρ0 +1,..., g m which belong to TV and such that, on V, span{ g 1,..., g m } = G 0. By possibly shrinking U (and hence V ), there exists a set of smooth local generators, g ρ0 +1,..., g m, for G 0 TV. We now have the desired decomposition on V, TV + G 0 = TV + Ḡ0 = TV span{ g 1,..., g ρ0 },

134 Chapter 3. Local Transverse Feedback Linearization 119 where, in the new basis for G 0, at each p V, g 1 (p),..., g ρ0 (p) G 0 (p), g ρ0 +1(p),..., g m (p) T p V. (3.36) On V, f(p) Tp V, so we have that ad j f g k TV + Ḡj 1, ρ k m, j = 0, 1,..., and hence ρ 0 ρ 1,...,ρ n n 1. Now we perform the induction step. Assume that, for some positive integer I, and any i {0,...,I}, there exists a basis {ĝ 1,...,ĝ m } for G 0 such that (a) TV + Ḡi 1 = TV ( i 1 j=0 { span ad j fĝ } ) k : 1 k ρ j, (b) ( k {ρ i 1 + 1,...,m}) ad i 1 ĝ f k TV + Ḡi 2. Property (b) implies that ρ i 1 ρ i,...,ρ n n 1. We now seek a basis { g 1,..., g m } for G 0 such that for any i {0,...,I}, (a) (b) TV + Ḡi = TV ( i j=0 { span ad j f g } ) k : 1 k ρ j, ( k {ρ i + 1,...,m}) ad i f g k TV + Ḡi 1. On V, define the distribution G i = [ Ḡ i (TV + Ḡi 1) ] Ḡ i. Note that Ḡi (TV + Ḡi 1) is constant dimensional on V since dim(ḡi (TV +Ḡi 1)) = dim(ḡi) + dim(tv + Ḡi 1) dim(tv + Ḡi), and each term in the sum is constant dimensional by assumption. The distribution Ḡi is nonsingular on U and TV + Ḡi 1 is constant dimensional on V. Since their intersection is constant dimensional, it follows from Lemma that the orthogonal complement of their intersection is smooth, and thus G i, being the intersection of two smooth and

135 Chapter 3. Local Transverse Feedback Linearization 120 nonsingular distributions, is smooth. Furthermore, dim G i = n dim(ḡi) dim(tv + Ḡi 1) + dim(tv + Ḡi) ( + dim(ḡi) dim [Ḡi (TV + Ḡi 1) ] ) + Ḡ i = dim(tv + Ḡi) dim(tv + Ḡi 1) = ρ i. By construction, G i Ḡi and G i (TV + Ḡi 1) = 0, so by dimensionality we have that ( TV + Ḡ i 1 ) Gi = TV + Ḡi = TV + G i. (3.37) By possibly shrinking U (and hence V ), there exist ρ i smooth independent vector fields w 1,...,w ρi such that on V, G i = span{w 1,...,w ρi }. Hence, by (3.37) we can write ρ i 1 m w j = w + β j k adi fĝ k + β j k adi fĝ k, j {1,...,ρ i }, (3.38) k=1 k=ρ i 1 +1 where w TV + Ḡi 1 and each β j k : V R is a C (V ) function. By property (b), for all k {ρ i 1 + 1,...,m}, ad i fĝ k TV + Ḡi 1. Let ρ i 1 ŵ j := β j k adi fĝ k, j = 1,...,ρ i (3.39) k=1 denote the second term in the sum (3.38). Notice that span{ŵ 1,...,ŵ ρi } ( TV + Ḡi 1) = 0. For, if this were false, then there would exist a C (V )-linear combination of the w j belonging to TV + Ḡi 1 which, by (3.37), is not possible. Furthermore, ŵ 1,...,ŵ ρi are V(V )-linearly independent because if there existed ρ i functions c i C (V ) such that, on V, c 1 ŵ c ρi ŵ ρi = 0, then, for some w TV + Ḡi 1, c 1 w c ρi w ρi w = 0 and thus, by (3.37), c 1 w c ρi w ρi = 0, implying c 1 = = c ρi = 0. Now let β i be the ρ i 1 ρ i matrix of smooth functions whose (k,j) th element is β j k obtained from (3.39) so that [ ] [ ŵ 1 ŵ ρi = ad i fĝ 1 ad i fĝ ρi 1 ]β i.

136 Chapter 3. Local Transverse Feedback Linearization 121 The vector fields ŵ 1,...,ŵ ρi are linearly independent and are generated as the image of ρ i 1 linearly independent vector fields under β i. Therefore, β i is full rank. We can now write TV + Ḡi = ( TV + Ḡi 1) span{ŵ1,...,ŵ ρi }. Let [ ] [ g 1 g ρi = ĝ 1 ĝ ρi 1 ] β i. The vector fields { g 1,..., g ρi } are linearly independent because the vector fields {ĝ 1,...,ĝ ρi 1 } are linearly independent and β i is full rank. Moreover TV + Ḡi = (TV + Ḡi 1) span{ad i f g 1,...,ad i f g ρi }. (3.40) This is because, for all j {1,...,ρ 0 }, ad f g j = [ f, ρ i 1 l=1 β j lĝl ] = ρ i 1 l=1 Iterating the Lie bracket operation, we obtain ad i f g j = ρ i 1 l=1 β j l [ ] f, ĝl + ρ i 1 l=1 β j l adi fĝ l + terms in TV + Ḡi ( ) L fβ j l ĝl. and hence [ ] [ ad i f g 1 ad i f g ρi = [ = ad i fĝ 1 ad i fĝ ρi 1 ] β i + terms in TV + Ḡi ŵ 1 ŵ ρi ] + terms in TV + Ḡi, thus proving (3.40). Next, since span{ g 1,..., g ρi } span{ĝ 1,...,ĝ ρi 1 } and both are constant dimensional, we can find (making U, and hence V, smaller if necessary) ρ i 1 ρ i vector fields g ρi +1,..., g ρi 1 such that span{ g 1,..., g ρi 1 } = span{ĝ 1,...,ĝ ρi 1 } (hence preserving property (a) from the induction assumption) and ad i f g ρ i +1,..., ad i f g ρ i 1 TV + Ḡi 1. These are found by taking the local generators for the smooth distribution span{ g 1,..., g ρi } span{ĝ 1,...,ĝ ρi 1 },

137 Chapter 3. Local Transverse Feedback Linearization 122 which has constant dimension ρ i 1 ρ i. Finally, let g ρi 1 +1 = ĝ ρi 1 +1,..., g m = ĝ m. We have therefore obtained a basis { g 1,..., g m } for G 0 in which properties (a) and (b) hold. In summary, the induction process gives a basis for G 0 in which the input vector fields are arranged in such a way that, for i n n, ( TV + Ḡ i 1 + ad i fg 0 ) / ( TV + Ḡ i 1 ) span{ad i f g 1,...,ad i f g ρi }. We are left to show that this arrangement can be achieved using a regular static feedback and that the arrangement is valid on an open set U of R n, and not just on an open subset [ ] V of Γ, as is presently the case. To this end, let g = g 1 g m and define a regular static feedback defined on V by (ˆα, ˆβ) where ˆα = u and ˆβ = ( g g ) 1 g g. To obtain a feedback transformation defined off Γ, we can, by possibly shrinking U (and hence V ) and applying Corollary , introduce a retraction r : U V of U onto V. Then, let α = ˆα r and β = ˆβ r. The regular static feedback (α,β) has the desired properties. In order to identify directions in the intersection T p V Ḡi(p) which are not contained in the intersection T p V Ḡi 1(p), it is useful to define the integers µ 0 (p) := dim(t p V Ḡ0(p)), µ i (p) := dim(t p V Ḡi(p)) dim(t p V Ḡi 1(p)), for i N, and let n i (p) := i µ j (p), j=0 so that dim(t p V Ḡi(p)) = n i (p). Under the assumption of Lemma 3.5.4, we have that dim(t p V Ḡi(p)) = n + ν i dim(t p V + Ḡi(p)), and hence the µ i are constant for all i n n, and we have the following result. Lemma Let Ũ be an open subset of Rn such that Ṽ := Ũ Γ. Assume that, for all i n n, the conditions of Lemma hold. Then, there exist an open set U Ũ and n n 1 vector fields v j l V(U), 0 j n n 1, 1 l µ j, such

138 Chapter 3. Local Transverse Feedback Linearization 123 that, after the feedback transformation of Lemma 3.5.4, letting V := U Γ, and G i := span{v0 1,...,v 0 µ 0,...,v i 1,...,v i µ i }, one has that, for all i n n, on U, ( i { Ḡ i = G i span ad j f g } ) k : 1 k ρ j j=0 and G i = TV Ḡi. V Proof. Suppose the feedback transformation (α,β) of Lemma 3.5.4, valid on U Ũ, has been applied. Since every point p of U is a regular point for the distributions Ḡi, we can, by possibly shrinking U (and hence V ), find a set of local generators X1,...,X i ν i i for Ḡi on U, i n n. On V, define the distribution Q 0 = TV Ḡ0. By assumption, Q 0 has constant dimension µ 0. Moreover, since Q 0 is the intersection of two smooth, nonsingular distributions and is constant dimensional, it is, by Lemma , smooth. By shrinking U (and hence V ) we can find a basis such that, on V, Q 0 = span{ˆv 1,..., ˆv µ0 }. By construction, Q 0 Ḡ0 so that each ˆv k Q 0 can be expressed as ν 0 ˆv k = ĉ k j0xj 0, k {1,...,µ 0 }, j=1 where each ĉ k j0 : V R is a C (V ) function. Next, we apply Corollary and, by possibly shrinking U (and hence V ), introduce a retraction r : U V of U onto V. Let c k j0 = ĉ k j0 r so that the vector fields ν 0 vk 0 := c k i0xi 0, k {1,...,µ 0 }, i=1 are now defined on U, and let G 0 := span{v0 1,...,vµ 0 0 }. It follows that G 0 Ḡ0 in U and G 0 = Q 0. By Lemma 3.5.4, TV + Ḡ0 = TV span{ g 1,..., g ρ0 } and so Q 0 V

139 Chapter 3. Local Transverse Feedback Linearization 124 span{ g 1,..., g ρ0 } = 0. The distribution Ḡ0 has dimension ν 0 = n 0 + ρ 0 throughout U so that ( p U) Ḡ0(p) G 0(p) + span{ g 1,..., g ρ0 }(p) ( p V ) Ḡ0(p) = G 0(p) span{ g 1,..., g ρ0 }(p) where G 0 TV. V The vector fields v1,...,v 0 µ 0 0, g 1,..., g ρ0 are linearly independent on V and therefore, by Proposition , they remain linearly independent in some open neighborhood of V in R n, without loss of generality, U. Therefore, we conclude that ( p U) Ḡ0(p) = G 0(p) span{ g 1,..., g ρ0 }(p). Next, we perform the induction step. Assume that, for some positive integer I, and any i {0,...,I}, there exist distributions G 0,...,G i 1 on U such that ( i 1 { Ḡ i 1 = G i 1 span ad j f g } ) k : 1 k ρ j j=0 and G i 1 = TV Ḡi 1. V (3.41) We want to show the existence of µ i vector fields v i 1,...,v i µ i such that, for any i {0,...,I}, letting G i = G i 1 span{vi 1,...,vµ i i }, one has that, on U, ( i ) Ḡ i = G i span{ad j f g k : 1 k ρ j } j=0 and G i = TV Ḡi. V (3.42) On V, define the distribution Q i by Q i = (TV Ḡi) (TV Ḡi 1). The distribution Q i is the intersection of two smooth, nonsingular distributions. Furthermore, for all p V, dim(q i (p)) = dim(t p V Ḡi(p)) dim(t p V Ḡi 1(p)) = µ i is constant by assumption, and thus, by Lemma , Q i is a smooth nonsingular distribution. Locally, by making U (and hence V ) smaller if necessary, there exist local

140 Chapter 3. Local Transverse Feedback Linearization 125 generators ˆv k, k {1,...,µ i }, for Q i. By construction, Q i Ḡi so that each ˆv k Q i can be expressed as ν i ˆv k = ĉ k jixj, i k {1,...,µ i }, j=1 where each ĉ k ji : V R is a C (V ) function. Let c k ji = ĉ k ji r so that ν i vk i := c k jixj, i k {1,...,µ i } j=1 are vector fields in Ḡi defined on U and G := i/i 1 span{vi 1,...,vµ i i } G i := G i 1 + G i/i 1. It follows that G i/i 1 Ḡi and G i/i 1 = Q i. By the definition of Q i and by (3.41), it V follows that G i 1 G i/i 1 = ( TV Ḡi 1) Qi = 0. Therefore, V V G i = G i 1 G i/i 1 = ( ) TV Ḡi 1 G TV Ḡi. V V V V i/i 1 By dimensionality G i = TV Ḡi V proving the second identity in (3.42). By Lemma 3.5.4, and dimensionality on V we have Ḡ i V = G i ( i j=0 { span ad j f g } ) k : 1 k ρ j. (3.43) Since G i Ḡi on U, (3.43) also holds in a neighborhood of V, without loss of generality, U, and so the first identity in (3.42) holds. Lemmas and elucidate the fact that when the Ḡi are nonsingular and the integers ρ i are constant, it is possible to find a local basis for each Ḡi distinguishing between the tangential component and transversal component. Specifically, we have that, letting Ḡ0/ 1 := G 0, for i n n, on U, and after feedback transformation, Ḡ i = G i G i ( ) = G 0 + G 1/0 + + G i/i 1 ( ) (3.44) G 0 + G 1/0 + + G i/i 1,

141 Chapter 3. Local Transverse Feedback Linearization 126 where and, by construction, G i/i 1 = span{vi j : 1 j µ i }, G i/i 1 = span{ad i f g j : 1 j ρ i }, G i = TV Ḡi. V (3.45) Thus, the distributions G i/i 1 and G i/i 1 span, respectively, the tangential and transversal directions to TΓ contained in Ḡi but not contained in Ḡi 1. An immediate consequence of Lemma is that, when k i = n n, i.e., TV + Ḡk 1 1 = TR n, then TV span {ad j f g } k : 0 j n n 1, 1 k ρ j = TR n. (3.46) As a result, Lemma yields the following array 2 of n independent vector fields on U: 1 G 0,G 0;...;G k ρ0 1/k ρ0 2,G k ρ0 1/k ρ0 2 ; 2 G k ρ0 /k ρ0 1,G k ρ0 /k ρ0 1 ;...;G k ρ0 1 1/k ρ 1 2,G k ρ0 1 1/k ρ0 1 2 ; ρ 0 1 G k 3 /k 3 1,G k 3 /k 3 1 ;...;G k 2 1/k 2 2,G k 2 1/k 2 2 ; (3.47) ρ 0 G k 2 /k 2 1,G k 2 /k 2 1 ;...;G k 1 1/k 1 2,G k 1 1/k 1 2 ; ρ v 1,...,v n n. k1 1 In (3.47), each block delimited by semicolons in rows 1 to ρ 0 contains independent vector fields in some Ḡk which are not contained in Ḡk 1. The vector fields in rows 1 through j, 1 j ρ 0, span Ḡk ρ0 j+1 1. The vector fields in row ρ are solely defined on V Γ and are not contained in any of the Ḡi s so that (TV + Ḡk 1 1)/Ḡk 1 1 span{v 1,...,v n n k1 1 }. 2 In the array we use the symbols G i/i 1 and G i/i 1 distribution spanned by the vector fields. to mean a family of vector fields and not the

142 Chapter 3. Local Transverse Feedback Linearization 127 They are chosen to complete the basis for T p V so that ( p V ) T p V = span{v 1,...,v n n k1 1 }(p) G k 1 1 (p). The vector fields in the array 3.47 embody all the geometric structure that is needed to solve LTFLP. In the next section, we prove Theorem 3.4.4, the main result. 3.6 Proof of the main result (Theorem 3.4.4) Suppose that LTFLP is solvable at p 0 Γ. Let V = Ξ(Γ U) and consider the expression (3.33) for TV + G i in local coordinates. It is clear from (3.33) that the subspace T p V + G i (col(p, 0)) has constant dimension n + rank([b A i B]). Since the pair (A,B) is controllable, we have that rank([b A n n 1 B]) = n n and condition (a) holds. As far as condition (b) is concerned, we have already shown in the proof of Lemma that TV + G i = TV + Ḡi. This concludes the necessity part of the proof. Conversely, suppose conditions (a) and (b) in Theorem hold. These two conditions, along with the regularity of Ḡi, i n n 1, allow one to invoke Lemmas and Specifically, there exist a neighbourhood Ũ of p 0 in R n, a regular static feedback (α,β) on Ũ, and n k 1 1 vector fields defined on Ũ such that, letting Ṽ := Ũ Γ, the distributions Ḡi have the representation given in (3.44), (3.45) on Ũ, and at each p Ṽ the n vector fields of the array (3.47) are linearly independent. Having applied the static feedback (α,β), we will denote f and g by, respectively, f and g to simplify notation. We construct ρ 0 functions λ i : U R satisfying Theorem by composing the flows generated by the vector fields in (3.47) and selecting suitable flow times as output functions. The procedure is entirely analogous to the one followed in the LTI case in Section Compose the flows generated by the vector fields in (3.47) in reverse order starting from the bottom row. Consider the mapping

143 Chapter 3. Local Transverse Feedback Linearization 128 S := ( ) s 1,...,s n n k1 Φ 1 S (p 0 ), defined as Φ S (p 0 ) := φ v n n k1 1 s n n k1 1 φ v 1(p s 0 ). 1 We continue by moving upwards in the array (3.47) to generate a sequence of mappings similar to Φ S (p 0 ). To each pair (G i/i 1,G i/i 1 ), 0 i k 1 1, we associate a set of times 3 S i/i 1 = (S i/i 1 ;S i/i 1 ) := (s (i/i 1,1),...,s (i/i 1,ρ i ) ; s (i/i 1,1),...,s (i/i 1,µ i ) ) and a mapping S i/i 1 Φ i/i 1 S i/i 1 (p), defined as Φ i/i 1 S i/i 1 (p) := φ vi µ i φ vi 1 φ adi f gρ i φ adi f g 1 s s s s (i/i 1,µ i ) (i/i 1,1) (i/i 1,ρ i ) (i/i 1,1)(p). Note that the vector fields appearing in Φ i/i 1 S i/i 1 (p) span the distributions G i/i 1, and G i/i 1. Let s := col(s ;S k1 1/k 1 2;...;S 1/0 ;S 0 ). (3.48) and let W R n be a neighbourhood of s = 0, sufficiently small, to ensure that the map s Φ s (p 0 ), W Φ W (p 0 ), defined as Φ s (p 0 ) := Φ 0 S 0 Φ 1/0 S 1/0 Φ k 1 2/k 1 3 S k1 2/k 1 3 Φk 1 1/k 1 2 S k1 1/k 1 2 Φ S (p 0 ). (3.49) is a diffeomorphism onto its image and that Φ W (p 0 ) Ũ. The existence of W is guaranteed by the inverse function theorem (Theorem 2.4.6), and the fact that the differential of Φ s (p 0 ) at s = 0, [ (dφ s (p 0 )) s=0 = v 1 v n n k1 1 ad k 1 1 f g 1 g 1 g ρ0 v 0 1 v 0 µ 0 ] (p 0 ), (3.50) is an n n square matrix whose columns span the subspace T p0 Γ + G k1 1(p 0 ) which, by condition (a), has dimension n. As candidate (virtual) output functions, let λ i, i {1,...,ρ 0 } be the time spent flowing along ad k i 1 f g i, i.e, λ i (x) = s (k i 1/k i 2, i)(x), i {1,...,ρ 0 }. (3.51) 3 We define i/(i 1) := 0 when i = 0 to be consistent with the array (3.47).

144 Chapter 3. Local Transverse Feedback Linearization 129 Thus the candidate output functions are certain components of (Φ s (p 0 )) 1. The image of Ṽ under (Φ s(p 0 )) 1 is the hyperplane (Φ s (p 0 )) 1 (Ṽ ) = {s W : S 0 = 0, S1/0 = 0,..., Sk 1 1/k 1 2 = 0}. Since the chosen functions λ 1,...,λ ρ0 are a subset of the functions whose zero level set defines (Φ s (p 0 )) 1 (Ṽ ), the λ i are identically zero on (Φ s (p 0 )) 1 (Ṽ ), and hence condition (i) of Theorem is satisfied. Next, we must show that λ = col(λ 1,...,λ ρ0 ) yields a well-defined vector relative degree of {k 1,...,k ρ0 } at p 0 = Φ 0 (p 0 ). As per Section 2.5.2, this entails showing that (VRD1) L ad k f g j λ i (x) = 0 for all 1 j ρ 0, for all 0 k k i 2, for all 1 i ρ 0 and for all x in a neighbourhood of p 0. (VRD2) The ρ 0 ρ 0 matrix L ad k 1 1 f g 1 λ 1 (p 0 ) L ad k 1 1 f g ρ0 λ 1 (p 0 ) L ad k 2 1 f g 1 λ 2 (p 0 ) L ad k 2 1 f g ρ0 λ 2 (p 0 ) L ad kρ 0 1 f g 1 λ ρ0 (p 0 ) L ad kρ 0 1 f g ρ0 λ ρ0 (p 0 ) (3.52) is non-singular at p = p 0 (if this matrix is non-singular, then the decoupling matrix has full rank). First we show that (VRD1) holds. Fix a set of times S = c, S k1 1/k 1 2 = c k1 1/k 1 2,..., S ki 1/k i 2 = c ki 1/k i 2, where each c j is a constant vector, to fix a hyperplane H i = {s W : S = c,s k1 1/k 1 2 = c k1 1/k 1 2,...,S ki 1/k i 2 = c ki 1/k i 2}. Consider the point s = col(c,...,c ki 1/k i 2, 0,...,0) H i and let x = Φ s (p 0 ) Ũ. By the Frobenius theorem (Theorem ), through x there passes an integral submanifold of each Ḡi, i k 1 1, which we denote L i (x). Consider the map (S ki 2/k i 3,...,S 0 ) Φ 0 S 0 Φ 1/0 S 1/0 Φ k i 2/k i 3 S ki 2/k i 3 (x).

145 Chapter 3. Local Transverse Feedback Linearization 130 This map is the composition of flows generated by vector fields that locally span Ḡk i 2. Since Ḡk i 2 is integrable, the set of points reachable from x under the action of this map is an integral submanifold of Ḡk i 2 passing through x. This is precisely the ν ki 2- dimensional manifold L ki 2(x) Φ W (p 0 ). This shows that H i = (Φ s (p 0 )) 1 (L ki 2(x) Φ W (p 0 )). Therefore for each s H i, T s H i = d (Φ s (p 0 )) 1 x=φ s(p 0 ) Ḡk i 2(x) = Im 0 I νki 2 In s-coordinates, the function λ i is one of those fixed times that define the hyperplane H i. Therefore, dλ i ann(ḡk i 2) ann(g ki 2) ann(g 0 ), and hence (VRD1) holds in a sufficiently small neighbourhood of p 0. Next we show that (VRD2) holds. The value of the (i,j) th entry of (3.52) is equal to. dλ i (p 0 ) ( ad k i 1 f g j (p 0 ) ). Denote the vector fields in the natural basis for T s R n, induced by s-coordinates, by s (i/i 1,a), s (j/j 1,b), s l with i,j {1,...,k 1 1}, a {1,...,ρ i }, b {1,...,µ i }, l {1,...,n n k1 1}. These are constant vector fields. From the expression (3.50) for dφ 0 (p 0 ) it follows that ( )] ad k i 1 f g j (p 0 ) = [(Φ s (p 0 )), 1 i j ρ 0, s=0 so that s (k i 1/k i 2, j) s (k i 1/k i 2, j) = [ (Φ s (p 0 )) 1 ( ad k i 1 f g j (x) )], 1 i j ρ x=p 0. 0

146 Chapter 3. Local Transverse Feedback Linearization 131 In light of this and the definition of λ i, i {1,...,ρ 0 }, given by (3.51), in s-coordinates the values of the entries of (3.52), along and below the diagonal, at p 0 are ( ) ds (k i 1/k i 2, i)(0) = δ ij, 1 i j ρ 0. s (k i 1/k i 2, j) Thus, at 0 = (Φ s (p 0 )) 1 (p 0 ) the matrix (3.52), in s-coordinates, has ones along its diagonal and zeros below. Therefore it is non-singular at s = 0, which is equivalent to being non-singular at p 0 = Φ 0 (p 0 ) Example: Synchronization To illustrate these ideas, we present an example of centralized control of two Lorenz oscillators. The equations of motion are ẋ 1 = σ(x 2 x 1 ) + u 1 ẏ 1 = σ(y 2 y 1 ) + u 2 ẋ 2 = rx 1 x 2 x 1 x 3 ẏ 2 = ry 1 y 2 y 1 y 3 (3.53) ẋ 3 = bx 3 + x 1 x 2 + u 1 ẏ 3 = by 3 + y 1 y 2 + u 2. For simplicity, we assume that σ = r = b = 1. Here col(x,y) = col(x 1,x 2,x 3,y 1,y 2,y 3 ) is the state of the system, u = col(u 1,u 2 ) is the control input and σ(x 2 x 1 ) 1 0 rx 1 x 2 x 1 x bx f(x,y) = 3 + x 1 x 2 1 0, G 0 (x,y) = span{g 1,g 2 } = span. σ(y 2 y 1 ) 0 1 ry 1 y 2 y 1 y by 3 + y 1 y We consider two separate problems using centralized control laws: (a) a partial synchronization problem and (b) the problem of full state synchronization. We will show that, when using transverse feedback linearization, the former is solvable while the latter is not. These types of problems are common and have appeared in the literature [90]. We begin with the partial synchronization problem.

147 Chapter 3. Local Transverse Feedback Linearization 132 Partial synchronization Suppose we are interested in forcing the variables x 1 and y 1 to lie on a unit circle Γ = {(x,y) R 3 R 3 : x y1 2 = 1}. In this case, it is clear that u = col( σ(x 2 x 1 ) + y 1, σ(y 2 y 1 ) x 1 ) is a suitable, though not unique, element of F(f,g, Γ ) for (3.53). The constraint h = x 2 1+y1 2 1 defining Γ trivially satisfies condition (i) of Theorem since Γ = h 1 (0). It turns out that h also yields a relative degree of n n = 1, so that it can be used as the virtual output y in (3.25). In other words, in this example, and with this choice of virtual output, transverse feedback linearization amounts to input-output feedback linearization for non-square systems, Theorem Nevertheless, it is instructive to also check the conditions of Theorem In this example we have that for all (x,y) Γ, T (x,y) Γ = Im y x Simple calculations reveal that for all (x,y) Γ, dim(t (x,y) Γ + G 0 (x,y)) = 6, i.e. condition (i) of Theorem is satisfied. In the special case when n = n 1, condition (ii) disappears and condition (i) becomes both necessary and sufficient. Since dim (T (x,y) Γ + G 0 (x,y)) = 6, we obtain ρ 0 = 1, ρ 1 = 0. Therefore, there is one controllability index, k 1 1. Following the procedure from Lemmas 3.5.4, 3.5.5, we find that, the following regular feedback transformation on R 6 \{(x,y) : x 1 = y 1 = 0}, [ ] [ g 1 g 2 = [ g 1 g 2 ]β(x,y) = ] g 1 g 2 x 1 y 1 y 1 x 1

148 Chapter 3. Local Transverse Feedback Linearization 133 yields, for every (x,y) Γ, T (x,y) Γ + G 0 (x,y) = T (x,y) Γ span{ g 1 }(x,y), T (x,y) Γ G 0 (x,y) = span{ g 2 }(x,y). Following the procedure in the proof of Theorem and letting (x 0,y 0 ) = col(1, 0, 0, 0, 0, 0) Γ, one obtains the maps S := (s 1,...,s 5) Φ S (x 0,y 0 ) and S 0 := (s 1) Φ 0 S 0 (x,y), defined as Φ S (x 0,y 0 ) := φ v 5 φ v 4 φ v 3 φ v 2 s 5 s 4 s 3 s 2 Φ 0 S 0 (x,y) := φ g 1 (x,y). s 1 φ v 1(x s 0,y 0 ), 1 Composing these two maps we get Φ s (x 0,y 0 ) := Φ 0 S 0 Φ S (x 0,y 0 ) = e s 1 cos (s 5) s 1 e s 1 + s 2 e s 1 sin (s 5) s 3 s 4. The desired output function is given by inverting the Φ s (x 0,y 0 ) to obtain ( ) λ(x,y) = s 1(x,y) = ln x y1 2. Now, using the output function λ(x,y), we can easily derive the normal form (3.2) using Theorem Let ξ = λ(x,y) and note that η 1 = x 2, η 2 = x 3, η 3 = y 2 η 4 = y 3, η 5 = arg (x 1 + iy 1 ) are functions that are independent of λ(x,y) in a neighbourhood of (x 0,y 0 ). Set Ξ(x,y) = col(η 1,η 2,η 3,η 4,η 5,ξ) so that in (η,ξ) coordinates (3.53) is

149 Chapter 3. Local Transverse Feedback Linearization 134 represented as η 1 = e 2ξ cos (η 5 ) (r η 2 ) η 1 η 2 = bη 2 + e 2ξ η 1 cos (η 5 ) + u 1 η 3 = e 2ξ sin (η 5 ) (r η 4 ) η 3 η 4 = bη 4 + e 2ξ η 3 sin (η 5 ) + u 2 η 5 = σ (η 1 sin (η 5 ) η 3 cos (η 5 )) + e ξ sin (η 5 )u 1 + e ξ cos (η 5 )u 2 ξ = σ ( η 1 cos (η 5 ) + η 3 sin (η 5 ) e 4ξ) + e ξ cos (η 5 )u 1 + e ξ sin (η 5 )u 2. Finally, we apply the feedback transformation u = a(x,y)+β(x,y)v outlined in the proof of Theorem In the expression for u we have [ ] β(x,y) = M(x,y) N(x,y) = e ξ cos (η 5) sin (η 5 ) sin (η 5 ) cos (η 5 ) where M(x,y) is the right inverse of the decoupling matrix L g λ, while N(x,y) spans its kernel. Furthermore a(x,y) = β(x,y) col(l f λ(x,y), 0) = β(x,y) σ ( η 1 cos (η 5 ) + η 3 sin (η 5 ) e 4ξ) 0 and v = (v,v ). With this construction, we obtain the desired normal form η = f 0 (η,ξ) + g 1 (η,ξ)v + g 2 (η,ξ)v ξ = v. The problem of stabilizing Γ is now reduced to the problem of stabilizing the state ξ and can be done by choosing a linear feedback v = kξ with gain k > 0. As for the remaining degree of freedom in the control, v, this input can be used to control the motion on the set Γ. We will say more about this in Chapter 5. For now, set v = σ(η 1 sin(η 5 ) η 3 cos(η 5 )) + ω where ω R is the desired angular velocity on the circle. Figure 3.5 shows the phase curves of the closed loop system projected onto the (x 1,y 1 )-plane while Figure 3.6 shows the one of these phase curves in three dimensional view.,

150 Chapter 3. Local Transverse Feedback Linearization y x 1 Figure 3.5: Partial synchronization of a circle for various initial conditions in the (x 1,y 1 ) plane of the Lorenz oscillator (3.53). Full synchronization Next, we pursue the question of whether or not transverse feedback linearization can be used to fully synchronize system (3.53), i.e, we want to know if the diagonal Γ = {(x,y) R 3 R 3 : x 1 = y 1,x 2 = y 2,x 3 = y 3 } can be stabilized using transverse feedback linearization. The set Γ is invariant for any choice of u so long as u 1 = u 2. In particular, we choose u = 0. Thus n = 3 and for

151 Chapter 3. Local Transverse Feedback Linearization 136 x(t) y(t) x 3, y x2, y x 1, y Figure 3.6: One of the phase curves from Figure 3.5 of the Lorenz oscillator in three dimensions. any (x,y) Γ we have that T (x,y) Γ = Im To check the conditions of Theorem hold we note that G 0 = Ḡ0 and that, for all (x,y) Γ dim(t (x,y) Γ + G 0 (x,y)) = 4 which means that ρ 0 = 1 everywhere on Γ. Next we find the distribution G 1 by calculating ad f g 1 (x,y) = col(1,x 1 + x 3 1, 1 x 2, 0, 0, 0) and ad f g 2 (x,y) = col(0, 0, 0, 1,y 1 + y 3 1, 1 y 2 ). Simple calculations reveal dim(t (x,y) Γ + G 1 (x,y)) = 5, so that ρ 1 = 1 and so for condition (ii) of Theorem to

152 Chapter 3. Local Transverse Feedback Linearization 137 hold we require that for all (x,y) Γ, dim(t (x,y) Γ +G 1 (x,y)) = dim(t (x,y) Γ +Ḡ1(x,y)). One can easily check that this condition fails because [g 1, ad f g 1 ] = col (0, 2, 0, 0, 0, 0), [g 2, ad f g 2 ] = col (0, 0, 0, 0, 2, 0) and dim(t (x,y) Γ + G 1 (x,y) + span{[g 1, ad f g 1 ], [g 2, ad f g 2 ]}(x,y)) = 6. Therefore, the conditions of Theorem do not hold. We conclude that transverse feedback linearization cannot be used to synchronize the Lorenz oscillators (3.53). 3.8 Transverse feedback linearization with partial information In this section we turn our attention to the set stabilization problem with partial information, restricting ourselves to the case of single-input systems, for simplicity. In the single-input setting, Theorem asserts that LTFLP is solvable if and only if there exists a virtual output function y = λ(x), λ C (U), with Γ λ 1 (0) yielding a welldefined relative degree of n n. If this is the case, following the proof of Theorem , we let ξ = col (λ(x),l f λ(x),...,l n n 1 f λ(x)) and choose n n functions η i = φ i (x), i {1,...,n n }, to complete the coordinate transformation. In the single-input case, such function can always be chosen (see [52]) so that its time derivative along the control system does not depend on u. This way, in (η,ξ) coordinates the system reads as η = f 0 (η,ξ) ξ = Aξ + b(a 1 (η,ξ) + a 2 (η,ξ)u), (3.54) where the pair (A,b) is in Brunovský normal form (one chain of integrators). The feedback transformation u = a 1 /a 2 + v/a 2 yields the desired normal form (3.2). As before, stabilizing the subspace ξ = 0 in (η, ξ) coordinates corresponds to stabilizing the set Γ in original coordinates (if the trajectories of the closed loop system are

153 Chapter 3. Local Transverse Feedback Linearization 138 bounded). Suppose that the state x is not available for feedback but, rather, the only available information is given by a vector output y = h(x), h : R n R p. Then, (η,ξ) is not available for feedback, the feedback transformation above cannot be implemented, and it may be impossible to stabilize the ξ subsystem. Suppose, however, that the linearizing output λ(x) is measured, that is, there exists a function λ such that λ = λ h, and consider the ξ subsystem before feedback transformation ξ 1 = ξ 2 ξ n n 1 = ξ n n (3.55) ξ n n = a 1 (η,ξ) + a 2 (η,ξ)u. Since ξ 1 = λ(x) = λ(h(x)), it is available for feedback. For system (3.55) with ξ 1 measured, a theory exists [73] that asserts the existence of a dynamic feedback ζ = ϕ(ζ,ξ 1 ) u = (ζ,ξ 1 ) capable of stabilizing the origin of (3.55), provided (3.54) satisfies a weak minimum phase assumption. Motivated by these observations, we seek to find structural conditions guaranteeing the existence of a measured output function λ(x) satisfying Theorem LTFLP with Partial Information (LTFLPI): Given a smooth single-input system with output ẋ = f(x) + g(x)u (3.56) y = h(x), h : R n R p, where g(x) 0 and the components of h are linearly independent, a set Γ I (f,g, R n ), a feedback u F(f,g, Γ ), and a point x 0 Γ, find, if possible, a virtual output function y = λ(x) = λ(h(x)) satisfying the conditions of Theorem 3.4.1, i.e., on a neighbourhood U of x 0 in R n,

154 Chapter 3. Local Transverse Feedback Linearization 139 (i) Γ U {x U : λ(h(x)) = 0}, and (ii) the system ẋ = f(x) + g(x)u y = λ(h(x)) has relative degree n n at x 0. To the measurement output h(x) = col (h 1 (x),...,h p (x)) we associate the nonsingular distribution W := ann (span {dh 1,...,dh p }), (3.57) which, by Proposition and Proposition , is involutive. We have the following result Theorem Suppose that ( G n n 2 + W ) is regular at x 0 Γ. Then LTFLPI is solvable at x 0 for system (3.56) if and only if (i) T x0 Γ G n n 1(x 0 ) = T x0 R n and, (ii) there exists an open neighbourhood U of x 0 in R n such that, ( x Γ U), dim (T x Γ G n n 2(x)) = dim ( T x Γ ( G n n 2 + W ) (x) ) = constant. Proof. Suppose that LTFLPI is solvable at x 0 Γ. Since, LTFLPI solvable implies that LTFLP is solvable at x 0, by Theorem 3.4.4, condition (i), we have T x0 Γ + G n n 1(x 0 ) = T x0 R n. Furthermore, since dim (G n n 1) n n, the subspaces T x0 Γ and G n n 1(x 0 ) are independent which proves that condition (i) is necessary. We are left to show that condition (ii) is necessary. Since LTFLPI is solvable and ξ 1 (x) = λ(x), we have Γ U = {x R n : ξ(x) = 0} {x R n : λ(x) = 0}

155 Chapter 3. Local Transverse Feedback Linearization 140 so that, for all x Γ and for any v T x Γ, L v λ(x) = 0. This implies that dλ ann (TΓ ). Furthermore, since λ(x) yields a well-defined relative degree of n n at x 0, we have that, by Definition , for any x in an open neighbourhood of x 0, without loss of generality U, and L g λ(x) = L adf gλ(x) = = L ad n n 2 f g λ(x) = 0 L ad n n 1 f gλ(x) 0. This means that, in a neighbourhood of x 0, without loss of generality U, dλ ann (G n n 2), dλ ann (G n n 1). By the chain rule, ) dλ x = d ( λ h = d λ h(x) dh x, x so that, for any vector field w W, and all x U, dλ x (w(x)) = d λ h(x) dh x (w(x)) = 0. In other words, dλ ann (W) = span{dh 1,...,dh p }. This shows that, in U, dλ ann (G n n 2) ann (W) G n n 2 + W ann (dλ) but, by Proposition , ann (dλ) is an involutive distribution so that G n n 2 + W ann (dλ) and by Proposition , dλ ann ( G n n 2 + W ).

156 Chapter 3. Local Transverse Feedback Linearization 141 This shows that, on Γ U, dλ ann (TΓ ) ann ( G n n 2 + W ). Thus, by Proposition , dλ ann ( TΓ + G n n 2 + W ) which implies that on U Γ, dim ( ann ( TΓ + G n n 2 + W )) 1. (3.58) Therefore, by (3.58) and Proposition , at any point on x U Γ dim ( T x Γ + G n n 2 + W(x) ) < n. We have already proven that condition (i) is necessary and therefore on U Γ, dim (T x Γ G n n 2(x)) = n 1. (3.59) Therefore, by (3.58) and (3.59), we have that for any point in U Γ n 1 = dim (T x Γ + G n n 2) (x) dim (T x Γ + G n n 2(x) + W(x)) dim ( T x Γ + ( G n n 2 + W ) (x) ) < n. which proves the necessity of condition (ii). We now turn to the proof of sufficiency. Conditions (i) and (ii), and the regularity of G n n 2 + W at x 0 imply that TΓ ( G n n 2 + W ) is a smooth nonsingular distribution near x 0. Using an argument identical to that in the proof of Lemma 3.5.5, it can be shown that, by taking U sufficiently small, there exists a smooth nonsingular distribution G G n n 2 + W on U enjoying the two properties ( Gn n 2 + W ) = G G n n 2 on U, where V := Γ U. G V = TV ( G n n 2 + W ),

157 Chapter 3. Local Transverse Feedback Linearization 142 Let w 1,...,w µ be a set of local generators for G on U. Similarly, there exist n µ vector fields {v 1,...,v n µ} on V, such that TV = G V span{v 1,...,v n µ}. Now we have a collection of n linearly independent vector fields, TV/G V G n n 2 G {}}{{}} {{}}{ v 1,...,v n µ; w 1,...,w }{{ µ ; g,...,ad n n 2 f g,ad n n 1 f g } } TV {{ } TR n (3.60) which we use to generate a coordinate transformation analogous to that used in Theorem We work our way from left to right in the list (3.60) starting with the group of vector fields spanning TV/ G V. Define the map S := (s 1,...,s n µ) Φ S (x 0 ) as Next define S := Φ S (x 0 ) = φ v n µ s n µ φ v 1 s 1 (x 0 ). ( ) s 1,...,s µ Φ (x), as S Φ S (x) := φ wµ and the map S := ( s 0,...,s n n 2) Φ S (x), as s µ φ w 1(x), s 1 Φ S (x) := φ g s 0 φ adn n f s n n 2 2 g (x). Finally, let s := (S,s n n 1,S,S ) Φ s (x 0 ), with domain a neighbourhood U of s = 0, be defined as Φ s (x 0 ) := Φ Φ S S φ adn n f s n n 1 1 g Φ S (x 0 ). (3.61) Since the vector fields in the list (3.60) are linearly independent near x 0, it follows from the inverse function theorem (Theorem 2.4.6) that there exists a neighbourhood U of s = 0 such that (3.61) is a diffeomorphism onto its image. Let λ(x) = s n n 1(x). (3.62)

158 Chapter 3. Local Transverse Feedback Linearization 143 We will not show that (3.62) yields a well-defined relative degree of n n at x 0 because the arguments are similar to the proof of Theorem Instead, we focus on showing that there exists a function λ such that λ = λ(h(x)). The function (3.62) yields relative degree n n on near x 0, so in particular, by Definition , L g λ(x) = L adf gλ(x) = = L ad n n 2 f gλ(x) = 0. Furthermore, since Γ λ 1 (0), these facts imply that, dλ ann (TΓ ) ann (G n n 2) = ann (TΓ + G n n 2) = ann (TΓ + G n n 2 + W) ann (TΓ + G n n 2 + W) = ann (TΓ ) ann (G n n 2 + W). Therefore, dλ ann (G n n 2) ann (W) so = ann (G n n 2) span{dh 1,...,dh p } dλ = p σ i (x)dh i (x) i=1 which implies that λ = λ(h(x)) and σ i (x) = λ y i. Example Consider the kinematic unicycle χ 1 cos (χ 3 ) 0 χ 2 = sin (χ 3 ) + 0 u + χ w (3.63) with disturbance w generated by the exosystem ẇ1 ẇ 2 = w 2 w 1. (3.64)

159 Chapter 3. Local Transverse Feedback Linearization 144 We wish to make the unicycle converge to the unit circle in the (χ 1,χ 2 )-plane. The goal set Γ = {(χ,w) R 5 : χ χ = 0} is not controlled invariant. The maximal controlled invariant subset of Γ is given by Γ = {(χ,w) R 5 : χ χ = χ 1 cos (χ 3 ) + χ 2 sin (χ 3 ) + χ 2 w 1 = 0}. Here, n = dim (Γ ) = 3. Set x := col (χ,w), cos (x 3 ) sin (x 3 ) f(x) := 0, g(x) := x 5 x , and suppose the measured states are χ = col (χ 1,χ 2,χ 3 ), i.e. h(x) = col (x 1,x 2,x 3 ) = χ. The question we pose is: can transverse feedback linearization be used to stabilize Γ using only χ for feedback? The answer, of course, is yes. For, one can check that the function λ(x) = x x yields a well-defined relative degree of 2 = n n at each x Γ. If we set ξ 1 = λ(x), ξ 2 = L f λ and let η 1,η 2,η 3 be any 4 three additional linearly independent functions, then, the local diffeomorphism Ξ = (η,ξ) solves LTFLPI. Next we confirm this observation using Theorem In this case, conditions (i) and (ii) of the theorem become (i) dim (T x0 Γ + G 1 (x 0 )) = 5 and, (ii) there exists an open neighbourhood U of x 0 in R 5 such that ( x Γ U) dim(t x Γ + G 0 (x)) = dim(t x Γ + (G 0 + W)(x)) = constant, 4 If the functions η 1, η 2, η 3 are chosen so that dη 1,dη 2,dη 3 ann (G 0 ), then we obtain the normal form (3.54). It is always possible to do this. If, on the other hand, dη 1,dη 2,dη 3 ann (G 0 ), then the control u will appear in the η subsystem.

160 Chapter 3. Local Transverse Feedback Linearization 145 where W, in this case, is given by W = ann (span{dh 1,dh 2,dh 3 }) = ann (span{e 1,e 2,e 3}) = span{e 4,e 5 }. Since G 0 and W are constant distributions, condition (b) is automatically satisfied. It is also easy to check that condition (a) holds.

161 Chapter 4 Global Transverse Feedback Linearization Building on the results of Chapter 3, in this chapter we pose the global transverse feedback linearization problem (GTFLP) in which, roughly speaking, one seeks a single coordinate and feedback transformation such that (2.8) is feedback equivalent to the normal form (3.2) in a tubular neighbourhood of Γ. The geometry of Γ plays an increased role in characterizing the solution. In [80], we provided sufficient conditions for the solvability of GTFLP in the single-input case. In this chapter we present sufficient conditions for global (in a tubular neighbourhood) feedback equivalence to a system whose dynamics transversal to the submanifold are linear and controllable. The conditions are restrictive and GTFLP for multi-input systems remains an essentially open problem. 146

162 Chapter 4. Global Transverse Feedback Linearization Global transverse feedback linearization problem Consider a control system of the form (1.1) m ẋ = f(x) + g i (x)u i. (4.1) Once again x R n is the state, u = (u 1,...,u m ) R m is the control input, and the vector fields f,g 1,...,g m : R n TR n are smooth (C ). We assume throughout this chapter that g 1,...,g m are linearly independent. Suppose we are given Γ I (f,g, R n ) and u F(f,g, Γ ), we can define the global transverse feedback linearization problem. Global Transverse Feedback Linearization Problem (GTFLP): Given a pair Γ I (f,g, R n ), u F(f,g, Γ ) with n = dim (Γ ), find, if possible, a tubular neighbourhood Γ ǫ of Γ in R n, a transformation Ξ Diff(Γ ǫ), Ξ : Γ ǫ Γ R n n, x (η,ξ), and a regular static feedback transformation (α, β), such that (4.1) is feedback equivalent i=1 on Γ ǫ to η = f 0 (η,ξ) + g 1 (η,ξ)v + g 2 (η,ξ)v ξ = Aξ + Bv, (4.2) where v = col(v,v ) R m, B is full rank, the pair (A,B) is controllable, and Ξ(Γ ) = {(η,ξ) Ξ(Γ ǫ) : ξ = 0}. Let the integers {ρ 0 (p),...,ρ i (p),...} and the transverse controllability indices of (4.1) with respect to Γ at p Γ, {k 1 (p),...,k ρ0 (p)}, be defined by (3.30) and (3.31) respectively. The next result does not yield a constructive solution to GTFLP, but is interesting for comparison with Theorem Theorem Suppose that Γ is contractible. Then GTFLP is solvable if and only if there exist ρ 0 smooth R-valued functions λ 1,...,λ ρ0 defined on an open neighbourhood U of Γ in R n, such that (i) U Γ {x U : λ i (x) = 0, i = 1,...,ρ 0 }

163 Chapter 4. Global Transverse Feedback Linearization 148 (ii) The system m ẋ = f(x) + u i g i (x) i=1 y = λ(x) = col (λ 1 (x),...,λ ρ0 (x)) (4.3) has uniform vector relative degree {k 1,...,k ρ0 } over Γ. Proof. Suppose that the conditions of Theorem hold. Let D(x) be the decoupling matrix associated to the output y, i.e., the ρ 0 m matrix with components D ij (x) = L gj L k i 1 λ i (x) which, by property (ii) and Definition , has rank ρ 0 on Γ and therefore also in a tubular neighbourhood Γ ǫ of Γ. Since D(x) has constant rank on Γ ǫ, for each point p Γ ǫ there exists a smooth matrix-valued function N(x), defined in a neighbourhood U Γ ǫ containing p, spanning the kernel of D(x) at each x U. This isn t enough, as we need to define N(x) over a neighbourhood of Γ, without loss of generality, Γ ǫ. Since Γ is a contractible set, so too is the tubular neighbourhood Γ ǫ. Let E be the vector bundle over Γ ǫ whose fibre at each point x Γ ǫ is the kernel of D(x). The bundle E is smooth because the components D ij (x) of D(x) are smooth functions. By Theorem E is trivializable, i.e., E Γ ǫ ker (D). Therefore, by Theorem , there exist m ρ 0 smooth sections n i : Γ ǫ E, i {1,...,m ρ 0 }, such that at each x Γ ǫ, ker (D(x)) = span{n 1 (x),...,n m ρ0 (x)}. These sections can be used to obtain the columns of the matrix valued function N(x). Now let β(x) = [M(x) N(x)], where M(x) := D (x)(d(x)d (x)) 1 is a m ρ 0 rightinverse of D(x), and N(x) is, as discussed above, a m (m ρ 0 ) smooth matrix-valued map whose columns span the kernel of D(x) in Γ ǫ. Notice that β(x) is non-singular on Γ ǫ. Let α(x) = β(x) col(l k 1 f λ 1,...,L kρ 0 f λ ρ0, 0 (m ρ0 ) 1). Consider the feedback transformation u = α(x) + β(x)v, where v = col(v,v ), with

164 Chapter 4. Global Transverse Feedback Linearization 149 v R ρ 0 and v R m ρ 0. This feedback transformation and property (ii) give d k 1λ 1 dt k 1. d kρ 0 λ ρ0 dt kρ 0 Defining the map x ξ as = L k 1 f λ 1. L kρ 0 f λ ρ0 + D(x)(α(x) + β(x)v) = v. ξ := col(λ 1,...,L k 1 1 f λ 1,...,λ ρ0,...,l kρ 0 1 f λ ρ0 )(x), the above implies that ξ = Aξ + Bv, where (A,B) is in Brunovský normal form with controllability indices {k 1,...,k ρ0 }. Next, following the ideas in the proof of Proposition in [53], one finds that there exists a map η = ϕ(x) mapping a neighbourhood of Γ, without loss of generality Γ ǫ, onto Γ such that the transformation Ξ : x (η,ξ) is a diffeomorphism of Γ ǫ onto a neighbourhood of Γ R n n. In transformed coordinates, the dynamics have precisely the form (4.2). The remainder of the proof of sufficiency, as well as the proof of necessity, are almost identical to the arguments found in the proof of Theorem and are omitted. Please refer to the proof of Theorem given on page 115. Definition Let λ 1,...,λ ρ0 be ρ 0 smooth R-valued functions satisfying the conditions of Theorem The map λ(x) := col (λ 1 (x),...,λ ρ0 (x)) is called a transverse output of (4.1) with respect to Γ. 4.2 Necessary conditions In light of Theorem and the other results from Chapter 3 we have the following obvious lemma. Lemma Suppose that GTFLP is solvable. Then,

165 Chapter 4. Global Transverse Feedback Linearization 150 (a) ( p Γ ) T p Γ + G n n 1(p) = T p R n, and (b) ( i n n 1) ( p Γ ) dim(t p Γ + G i (p)) = dim(t p Γ + Ḡi(p)) = constant. Condition (b) of Lemma implies that the controllability indices of (4.1) with respect to Γ are constant over Γ. Our goal is to extend the proof technique used in solving LTFLP, the proof of Theorem 3.4.4, to GTFLP. Unfortunately, using that technique, the conditions of Lemma are not sufficient. 4.3 On the interaction between system and set The guiding theoretical basis for this chapter is the generalized inverse function theorem (Theorem 2.4.7). As we progress through the remainder of this chapter we must constantly keep in mind the fact that in order to extend the proof of Theorem to the global case, we hope to find a diffeomorphism Φ s (p 0 ) : W Φ W (p 0 ) analogous to the map (3.49) (page 128), where Φ W (p 0 ) is a tubular neighbourhood of Γ. For the reader s convenience we restate Theorem Generalized inverse function theorem. Suppose that f : M N is a smooth map between manifolds. Let P M be a submanifold of M and assume that (i) df p is an isomorphism for every p P and (ii) f P maps P diffeomorphically onto f(p). Then, f maps a neighbourhood of P diffeomorphically onto a neighbourhood of f(p). The second hypothesis of this theorem is one of the main obstacles to globalizing Theorem This obstacle is manifested as conditions imposed on the local generators for the distributions G i := TΓ Ḡi, i n n 1.

166 Chapter 4. Global Transverse Feedback Linearization 151 These distributions characterize the interaction between the system s dynamics and the geometry of the set Γ. In Chapter 3, we found a set of n linearly independent vector fields that we used to generate the local diffeomorphism (3.49). These vector fields were identified through Lemma and Lemma These lemmas led to the array (3.47) of n linearly independent vector fields. The reason these lemmas are needed is that they allow us to choose the appropriate set of n vector fields with which to generate s-coordinates and in turn find the output functions that satisfy Theorem In the case of LTI systems, this is not an obstacle because all of the vectors involved are constant. In the global case, in order to apply the generalized inverse function theorem, we require that the part of the flow map generated by vector fields that are tangent to Γ be onto Γ. One way to ensure this is to impose that G i = TΓ Ḡi = 0, i n n 1, (4.4) and that there exist n linearly independent vector fields v i : Γ TΓ, i {1,...,n }, such that, for every p Γ, T p Γ = span{v 1,...,v n } and, furthermore [v i, v j ] = 0, i,j {1,...,n }, (4.5) i.e., the vector fields commute. Condition (4.4) is a generalization of one of the sufficient conditions for the global solution to the single-input case [80, Theorem 4.4., condition (3)]. Condition (4.4), although checkable, is restrictive. For example, condition (4.4) will not hold in the situation illustrated in Figure 4.1. In Figure 4.1 the set Γ is a line in the plane. Suppose that system (4.1) has a single input and that at the points x Γ and x Γ condition (4.4) holds in the manner depicted in Figure 4.1. By continuity of the vector field g, there exists a point on Γ on which condition (4.4) fails to hold. We now discuss why conditions (4.4) and (4.5) allow us to apply the generalized inverse function theorem. Assuming, for a moment, that conditions (4.4) and (4.5) hold, let s examine their effect on the map Φ s (p 0 ) in (3.49). Recall that the last row of

167 Chapter 4. Global Transverse Feedback Linearization 152 g(x) x x Γ g(x ) Figure 4.1: The vector field g is continuous so that there is at least one point on Γ at which g belongs to the tangent space of Γ. vector fields in the array (3.47) span, at each point p Γ, the subspace T p Γ /Ḡk 1 1(p). However, if (4.4) holds, then at each p Γ, T p Γ /Ḡk 1 1(p) = T p Γ. As a result, the array (3.47) becomes 1 g 1,...,g m ;...; ad km 1 f g 1,...,ad km 1 f g m ; ad km f g 1,...,ad km f g m 1;...;ad k m 1 1 f g 1,...,ad k m 1 1 f g m 1 ; ad k 3 f g 1,ad k 3 f g 2;...; ad k 2 1 f ad k 2 f g 1;...; ad k 1 1 f g 1 ; v 1,...,v n. g 1,ad k 2 1 f g 2 ; (4.6) Following the proof in Section 3.6, we introduce the flow maps used therein to generate s-coordinates, but using the array (4.6) instead of (3.47). Fix p 0 Γ and define the map S := (s 1,...,s n ) Φ S (p 0 ), R n Γ, as Φ (p S 0 ) := φ v n φ v 1(p s 0 ). (4.7) n This map is the global version of the map Φ S (p 0 ) in Section 3.6. It consists of the flows generated by vector fields that are tangent to Γ and not contained in any Ḡi, i k 1. Next, set s := col (S,S ) = col(s 1,...,s n,s 1,...,s n n ) and define the map s 1 1 Condition (4.4) implies that the integer ρ 0, defined as ρ 0 (p) = dim(t p Γ + G 0 (p)) dim(t p Γ ), is equal to m.

168 Chapter 4. Global Transverse Feedback Linearization 153 s Φ s (p 0 ), R n W Φ R n W (p 0), as Φ s (p 0 ) := φ g 1 s 1 φ gm s m φ adk 2 f g 1 s n n k 1 +k 2 +1 adk 1 1 f g 1 φ Φ (p s n n S 0 ). (4.8) In order to apply the generalized inverse function theorem to (4.8), we require that (i) The differential (dφ s (p 0 )) s of the map (4.8) is an isomorphism for every s R n {0}, i.e., at every s for which Φ s (p 0 ) Γ. (ii) The mapping (4.7) maps R n diffeomorphically onto Γ. The enforcement of condition (ii) above is the subject of the next section. Remark In lieu of conditions (4.4) and (4.5), we could instead use the following, milder, condition: for each distribution G i, i k 1 1, there exist local generators {v1,...,v i µ i i } in a tubular neighbourhood of Γ, such that, fixing i,j k 1 1, [ ] v i a, v j Γ b = 0, a {1,...,µ i }, b {1,...,µ j }. This, however, introduces difficulties in showing that the differential of the map (4.8) is an isomorphism at each p Γ. Furthermore, this condition is not checkable. 4.4 On the geometry of Γ If we impose condition (4.4) then we have essentially decoupled the role of system s dynamics from the geometry of the set Γ in finding a solution to GTFLP. However, in order to apply the generalized inverse function theorem, condition (4.4) does not guarantee that the map (4.7) is onto Γ. This is where condition (4.5) comes into play. Here we investigate the ramifications of this condition on the geometry of the set Γ and show that this condition makes the map (4.7) onto, which is crucial to satisfying condition (ii) of the generalized inverse function theorem. Suppose then, that Γ is parallelizable, i.e., TΓ Γ R n, so that by Theorem there exist vector fields on Γ that globally generate TΓ. If the generators v 1,...,v n

169 Chapter 4. Global Transverse Feedback Linearization 154 are commutative (i.e., [v i, v j ] = 0) and complete, then the map (4.7) is onto, as shown next. Lemma ([52], Lemma 2.18). Let be a nonsingular involutive distribution and let S be a maximal integral submanifold of. Then, given any two points p and q in S, there exist vector fields τ 1,...,τ k in and real numbers t 1,...,t k such that q = φ τ 1 t 1 φ τ k tk (p). Corollary If there exist n complete and linearly independent generators v 1,...,v n : Γ TΓ of TΓ that commute, i.e., [v i, v j ] = 0, then for any p 0 Γ, Φ (p S 0 ) : R n Γ defined as maps R n onto Γ. Φ (p 0) = φ v n φ v 1(p (s 1,...,s n ) s 0 ) n s 1 Proof. Let D = span{v 1,...,v n }. Then, by assumption, D is non-singular and, since [v i, v j ] = 0, by Lemma D is involutive on Γ. Therefore, by the Frobenius theorem, D is integrable and through every p Γ there passes an n -dimensional maximal integral submanifold of D. Since, at every p Γ, D(p) = T p Γ, it follows that Γ is the maximal integral submanifold of D through every p Γ. By Lemma 4.4.1, given any two points p,q Γ, there exist k real numbers t 1,...t k such that q = φ v j 1 t 1 φ v j k t k (p), where j l {1,...,n }, l {1,...,k}. By Lemma and completeness, [v i, v j ] = 0 implies that φ v i t φ v j s (p) = φ v j s φ v i t (p) for all p Γ and all s,t R. Therefore, by re-arranging the order of the flows in the above map, we can write q = φ v 1 τ 1 φ v n (p), τ k

170 Chapter 4. Global Transverse Feedback Linearization 155 where τ i, i {1,...,n } is the sum of all the times t j, j {1,...,k}, spent flowing along the integral curve generated by the vector v i. Having shown that condition (4.5) guarantees that Φ S (p 0 ) is onto Γ, we now identify precisely which class of submanifolds Γ satisfies it. Theorem Let M be a connected m-dimensional manifold. Then, M is diffeomorphic to a generalized cylinder T k R m k, k {1,...m} if and only if there exist m complete, commutative and linearly independent vector fields on M. The proof of sufficiency relies on the fact that there exist m linearly independent generators for the tangent bundle of T k R m k that are constant and, therefore, commutative and complete. The necessity of the statement follows from [7, Lemma 49.2] (see, in particular, Problem 10), where it is shown that if TM admits m commutative linearly independent vector fields, and the flows s φ v i s (p) are defined for all s R, then M is diffeomorphic to T k R m k. We have thus seen that condition (4.5), used to guarantee that Φ S (p 0 ) is onto, restricts the allowable submanifolds Γ to the class of submanifolds diffeomorphic to a generalized cylinder. On the other hand, Theorem 4.1.1, a tool needed in the generalization of the main result of the previous chapter, restricts Γ to be contractible. The only n -dimensional generalized cylinder to be contractible is R n, so in order to solve GTFLP we will assume that Γ R n. Theorem Suppose that Γ is diffeomorphic to R n. Then there exist n complete, linearly independent and commutative vector fields v 1,...,v n : Γ TΓ such that Φ S (p 0 ) in (4.7) maps R n diffeomorphically onto Γ. Proof. Let T : R n Γ be a diffeomorphism, and for i {1,...,n } define v i := T e i, where {e 1,...,e n } is the natural basis of R n. Then v 1,...,v n are complete, commutative and linearly independent vector fields on Γ. Without loss of generality

171 Chapter 4. Global Transverse Feedback Linearization 156 assume that T(0) = p 0. To show that Φ (p 0) = φ v n φ v 1(p (s 1,...,s n ) s 0 ) n is a diffeomorphism R n Γ it suffices to show that Φ (s 1,...,s n ) (p 0) = T(s 1e 1 + s n e n ). To this end, we use the result in [114, Problem 10.14] stating that if v 1, v 2 are smooth and commutative vector fields on Γ, then φ v 2 t φ v 1 t (p 0 ) = φ v 1+v 2 t (p 0 ) for all t R. On one hand, we have s 1 T (s 1e 1 + s n e n ) = s 1T (e 1 ) + + s n T (e n ) = s 1v s n v n. (4.9) On the other hand, Φ (p 0) = φ v n φ v 1(p (s 1,...,s n ) s 0 ) = φ s n v n 1 φ s 1 v 1 1 (p 0 ) n s 1 = φ s 1 v 1+ +s n v n 1 (p 0 ) (by [114, Problem 10.14]). Since T is a diffeomorphism, by (4.9) and Lemma we have ( ) φ s 1 v 1+ +s n v n 1 (p 0 ) = T φ s 1 e 1+ +s n e n 1 (0) = T(s 1e 1 + s n e n ) proving that Φ (s 1,...,s n ) (p 0) = T(s 1e 1 + s n e n ). 4.5 Feedback transformations in a tubular neighbourhood of a manifold As mentioned earlier, in Chapter 3 we found a set of n linearly independent vector fields that we used to generate the local diffeomorphism (3.49). These vector fields were identified through Lemma and Lemma Specifically, recall that Lemma generates a feedback transformation (α,β) such that in a neighbourhood of p Γ, T p Γ + Ḡi(p) = T p Γ ( i j=0 ) span {ad j f g } k : 1 k ρ j (p), i n n.

172 Chapter 4. Global Transverse Feedback Linearization 157 A related global requirement on a feedback transformation (α, β) has appeared in the literature [28] but there it only involves the distribution G 0. Lemma can be extended to the global case if we restrict our attention to sets Γ that are contractible. Lemma is not needed under the condition (4.4). Lemma Suppose Γ is a contractible submanifold of R n. Let Γ ǫ be a tubular neighbourhood of Γ. Assume that, for all i n n, ( p Γ ) dim(t p Γ + G i (p)) = dim(t p Γ + Ḡi(p)) = constant ( p Γ ǫ) dim(ḡi(p)) = ν i = constant. Then, ρ 0 ρ 1 ρ n n 1, and there exist a tubular neighbourhood Γ ǫ Γ ǫ, and a regular static feedback (α,β) on Γ ǫ such that, for all p Γ, and for all i n n, the following holds ( i ) T p Γ + Ḡi(p) = T p Γ span{ad j f g k : 1 k ρ j }(p). (4.10) j=0 The proof of this Lemma is almost identical to the proof of Lemma and is omitted. The only differences are that (i) In the global case we use the retraction r : Γ ǫ Γ associated with the tubular neighbourhood of Γ instead of a local retraction. (ii) Since Γ is contractible, we use the fact that this implies that Γ is parallelizable to find n sections of TΓ that span T p Γ at each point p of Γ. (iii) We utilize Theorem in order to find generators for suitable constant dimensional distributions on the entire set Γ. 4.6 Sufficient conditions Theorem Suppose that Γ is diffeomorphic to R n. Then GTFLP is solvable if

173 Chapter 4. Global Transverse Feedback Linearization 158 (i) ( p Γ ) dim (T p Γ + G n n 1(p)) = n (ii) ( i n n 1) Ḡi is nonsingular in a neighbourhood of Γ (iii) ( p Γ )( i n n 1) dim (T p Γ + G i (p)) = dim (T p Γ + Ḡi(p)) = constant. (iv) ( i n n 1) TΓ Ḡi = {0}. Remark When the system (4.1) has a single input, then the conditions of Theorem essentially reduce to the conditions of [80, Theorem 4.4] with the notable exception that there we allow Γ to be a generalized cylinder, whereas in Theorem we have the stronger condition Γ R n. Proof. The manifold Γ is diffeomorphic to R n so by Theorem there exist n complete, linearly independent and commuting vector fields {v 1,...,v n } such that for each p Γ, T p Γ = span{v 1,...,v n } and Φ in (4.7) maps R n diffeomorphically onto Γ. S By conditions (ii) and (iii) we have that the transverse controllability indices of (4.1) are constant on the entire set Γ and we can apply Lemma to obtain a feedback transformation such that (4.10) holds for i n n 1. Hereafter, we will drop the tilde over the vector fields f,g 1,...,g m with the understanding that the feedback transformation of Lemma has been applied. Therefore, since conditions (i) and (iv) hold, we have that the n vector fields in the array (4.6) are linearly independent on Γ. Next, consider the flow maps (4.7) and (4.8) from Section 4.3. We want to apply the generalized inverse function theorem to (4.8). By Theorem 4.4.4, for all p Γ there exists S R n such that p = Φ (S,0)(p 0 ). The differential (dφ s (p 0 )) (S,0) has the matrix representation (dφ s (p 0 )) (S,0) = [ where }{{} n n ad k1 1 f g 1 (p) ad k 2 f g 1(p) g 1 (p) g m (p) = ( dφ S (p 0 ) ) S. ] p=φ (S,0) (p 0 ) (4.11)

174 Chapter 4. Global Transverse Feedback Linearization 159 By Theorem 4.4.4, Φ R n {0}(p 0 ) maps R n {0} diffeomorphically onto Γ. In order to apply the generalized inverse function theorem, it remains to be shown that (dφ s (p 0 )) (S,0) is an isomorphism at each p Γ. By Lemma 4.5.1, the last n n columns of (4.11), i.e., the vectors ad k 1 1 f g 1 (p),...,g m (p), are linearly independent and at each p Γ span a subspace that is independent of T p Γ. The first n columns of (4.11) span ( )S. ( ) the image of dφ (p S 0 ) By definition dφ (p S 0 ) : T S S T Rn p Γ. Hence, span{ } span{ad k 1 1 f g 1 (p),...,g m (p)} = 0. Moreover, since Φ (p S 0 ) is a diffeomorphism R n Γ, dφ (p ( ) S 0 ) is an isomorphism onto T pγ, and hence has rank n, S proving that (dφ s (p 0 )) (S,0) is an isomorphism at every p = Φ (S,0)(p 0 ) Γ. By the generalized inverse function theorem (Theorem 2.4.7), Φ s (p 0 ) is a diffeomorphism of a tubular neighbourhood of R n {0} onto a tubular neighbourhood of Γ. The remainder of the proof is exactly the same as the proof of Theorem from Section 3.6 and we omit it for brevity.

175 Chapter 5 The Path Following Problem In this chapter we apply the results of Chapter 3 to the path following problem. We look at two specific applications: (1) a magnetically levitated positioning system and (2) a planar vertical/short take-off and landing aircraft (PVTOL). The work on the magnetically actuated positioning system includes experimental verification and was done in collaboration with Cameron Fulford, a former M.ASc student in the System Control Group, University of Toronto. The results concerning the PVTOL were obtained in collaboration with Luca Consolini and Mario Tosques at the University of Parma, Italy. The transverse feedback linearization normal form is particularly well-suited to the path following problem. The reason is that the normal form decomposes the system dynamics into two subsystems: a transverse subsystem that determines whether or not the system is on the path, and a tangential subsystem that determines the motion on the path itself. A further decomposition of the control inputs allows one to separately design a transversal control law to stabilize the desired path and then independently design a tangential controller to meet the specifications on the path itself. 160

176 Chapter 5. The Path Following Problem Introduction The path following control problem (PFP) is chiefly concerned with providing a stable motion along a given path with no a priori time parameterization associated with the movement on the path. More specifically, the control objective is to drive the output of a control system to the path so as to traverse it in a desired direction. Usually, specific applications impose additional requirements, such as speed regulation on the path and internal stability. PFP has some affinity to the tracking control problem in which it is desired that the system output asymptotically matches a reference signal, but there are fundamental differences. Consider, for instance, the simple problem in Example of making a planar kinematic point-mass vehicle follow a path on the plane. A path following controller should render the path an invariant set for the closed-loop system. In other words, if the vehicle is initialized on the path, then the vehicle should remain on it at all time. A tracking controller would make the vehicle follow a reference point moving along the path and therefore it would not guarantee invariance of the path. As a matter of fact, if the vehicle is initialized on the path, but its position does not coincide with that of the reference point, then the vehicle will leave the path and then asymptotically approach it. It may be argued that it is sufficient to initialize the reference point at the same location as that of the vehicle, but in practice this is not a good solution because of its intrinsic lack of robustness: if the vehicle were subject to a sudden force that temporarily stops its motion, then once again the reference point would lose synchrony with the vehicle which would then leave the path. More generally, tracking controllers stabilize a specific system trajectory, while path following controllers should stabilize a family of trajectories, all those whose associated output signals lie on the desired path. We call the collection of all such trajectories the path following manifold. Its precise definition is given in Section 5.2. The point of view taken in this chapter is to convert PFP into a set stabilization problem: the stabilization of the path following manifold. This guarantees, among other

177 Chapter 5. The Path Following Problem 162 things, the invariance property mentioned earlier: if the state is appropriately initialized, then the resulting output signal lies on the desired path at all time. We will utilize the technique of transverse feedback linearization introduced in Chapter 3 to stabilize the path following manifold. As discussed throughout this thesis, transverse feedback linearization, when feasible, simplifies the design of control laws that stabilize sets. Besides converting set stabilization problems into the problem of stabilizing the origin of an LTI system, transverse feedback linearization has other benefits in the context of PFP. The dynamical decomposition achieved in the transverse feedback linearization normal form naturally decomposes the system into two subsystems. One subsystem, the transverse subsystem, is the LTI part that determines whether or not the system in on the path. The tangential subsystem, is the remaining, possibly nonlinear, part of the system dynamics that determines the motion along the path. Additionally, the normal form also achieves a decomposition of the control inputs into two distinct groups, v and v. The control v can be designed to stabilize the path following manifold, while v is designed to meet additional application-specific requirements on the path following manifold in order to control the motion along the path. Together, these decompositions make it possible to divide the control design into two steps: the stabilization of the path following manifold (transversal control design) and the control of the motion on the manifold (tangential control design). 5.2 Path following methodology In this section we formulate the path following control design problem and we present a methodology for its solution. We consider smooth control-affine systems with m inputs and p outputs, m ẋ = f(x) + g i (x)u i i=1 y = h(x) = col (h 1 (x),...,h p (x)). (5.1)

178 Chapter 5. The Path Following Problem 163 Given a smooth embedded path 1 in the output space, γ := {y : s(y) = 0}, we want to design a smooth feedback that makes the output of the system (5.1) approach and traverse γ in a desired direction with a desired speed. Moreover, it is required that γ be output invariant for the closed-loop system, which intuitively means that if the state x is appropriately initialized, the resulting output signal lies on the path at all time. In order to give a precise definition of output invariance, let Γ := {x : s(h(x)) = 0}. Stabilizing the set Γ corresponds to sending the output of the plant to the desired path. However, generally Γ is not a controlled invariant set so one should instead stabilize the maximal controlled-invariant subset of Γ, which we denote by Γ. Intuitively, the set Γ is the collection of all those motions of the control system (5.1) whose associated output signals can be made to lie in γ at all time by a suitable choice of input signal. Throughout the rest of this section we consider (5.1) and a smooth embedded path in the output space γ = {y : s(y) = 0}. Assumption The maximal controlled-invariant subset of Γ = {x : s(h(x)) = 0}, Γ, is a non-empty, closed embedded submanifold of the state space. Let n be its dimension. Assumption is a basic feasibility requirement for the path following problem. With this assumption, Γ is precisely the zero dynamics manifold of the control system ẋ = f(x) + g(x)u with output ŷ = s(h(x)) as defined in Definition Definition The path following manifold Γ of γ with respect to (5.1) is the maximal controlled invariant submanifold contained in (s h) 1 (0). Definition Let ū(x) be a smooth feedback and let Γ be the path following manifold of γ with respect to (5.1). The path γ is output invariant under the closed loop vector field f := f + gū if Γ is invariant under f. 1 By smooth we mean that s is a smooth function; by embedded we mean that the path has no self-intersections and it is a closed subset of R p. This is equivalent to requiring that one can choose s : R p R p 1 so that its Jacobian has full rank p 1 everywhere on γ.

179 Chapter 5. The Path Following Problem 164 As discussed in the introduction, output invariance is a fundamental property of path following controllers. The path following control design problem entails finding a feedback ensuring that three objective are met. P1 For each initial condition in a suitable set, the corresponding solution x(t) is defined for all t 0 and h(x(t)) γ 0 as t +. P2 The path γ is output invariant for the closed-loop system in the sense of Definition P3 The motion on γ meets additional application-specific requirements such as direction and speed of traversal of the path, and boundedness of the internal dynamics. Our approach to the solution of the problem above is summarized below. Step 1 Find the path following manifold Γ. Step 2 Transverse feedback linearization (see Chapter 3). Find, if possible, a coordinate transformation Ξ : x (η,ξ), defined in a neighbourhood U of Γ, and a regular feedback transformation u = α(x) + β(x)v such that Ξ(Γ ) = {(η,ξ) : ξ = 0} and, in new coordinates, η = f 0 (η,ξ) + g (η,ξ)v + g (η,ξ)v v = col(v,v ), (5.2) ξ = Aξ + Bv with (A,B) a controllable pair. We refer to the ξ subsystem as the transversal subsystem. On the other hand, the system ż = f 0 (η, 0)+g (η, 0)v is the tangential subsystem. Step 3 Transversal control design. Design a transversal feedback v (ξ) stabilizing the origin of the transversal subsystem. In theory, one can use a linear static feedback. For practical reasons, however, it may be useful to include integral action in the stabilizer. Depending on the application, one may use a nonlinear finite-time stabilizer (see [14]).

180 Chapter 5. The Path Following Problem 165 Step 4 Tangential control design. Design a tangential feedback v (η,ξ) such that, when ξ = 0, the tangential subsystem meets the application-specific goals in P3 and, moreover, the closed-loop system has no finite escape times. The approach outlined above relies on the stabilization of the path following manifold Γ. Other set stabilization approaches may be used to stabilize Γ, but transverse feedback linearization is particularly well suited to path following in that it allows one to separately address the stabilization of Γ (objectives P1 and P2) and the control of the dynamics on Γ (objective P3). More specifically, the tangential subsystem, with state η, describes the motion on Γ, that is, when the plant output lies in γ. The tangential controller is designed to prevent finite escape times from occurring and to meet goal P3 guaranteeing, among other things, that the path is traversed in a desired direction with a desired speed. The transversal subsystem, with state ξ, describes the motion off the set Γ. Due to the absence of finite escape times, the transversal controller stabilizes Ξ(Γ ). If the trajectories of the closed-loop system are bounded 2, then the stabilization of Ξ(Γ ) implies that of Γ, and therefore the transversal controller meets goal P1. It also meets goal P2 because the origin of the ξ subsystem is an equilibrium of the closed-loop system, and thus Γ is an invariant set of the closed-loop system. The computation, in Step 1, of Γ can be performed using the zero dynamics algorithm discussed in Section 2.5, provided some mild regularity conditions hold. In general, however, the algorithm only provides a characterization of Γ in the neighbourhood of a point. One may, if needed, apply the algorithm in a neighbourhood of various points to piece together Γ. 2 It may happen in some applications that the trajectories of the closed-loop system aren t bounded because the path itself is unbounded. In this case, in order to be able to state that the stabilization of Ξ(Γ ) implies that of Γ, it is necessary that there exist a class-k function α such that ξ(x) α( x Γ ).

181 Chapter 5. The Path Following Problem Comparison of path following approaches in the literature Early investigations of the path following problem in [93], [99], [100] focus on specific applications and rely on coordinate transformations decomposing the system dynamics into tangential and transversal components. In this sense, these approaches are similar to ours. The influential work in [9] gives insight on how to convert a tracking control law into a path following control law using a computationally feasible projection operator. The same idea was later used in [4]. Another approach is to parameterize the path, use the parameterization as a reference trajectory, and treat the velocity of the reference point as an extra control input. This approach, and variations on its theme, is popular and is the subject of a considerable amount of work, see [2], [3], [27], [50] among others. See also the related work in [109], [110]. An interesting feature in these papers is that they divide the control design into two subproblems; the geometric task achieves convergence to the path, while the dynamic task assigns a speed profile on the path. In our framework, the transversal control design addresses the geometric task, while the tangential control design addresses the dynamic task. On the other hand, relying on a parameterization of the path, the above mentioned approaches share the drawback of all tracking controllers mentioned in the introduction in that they fail to make the path output invariant. Research has also been devoted to the study of the exact (or perfect) path following problem for specific applications. The exact path following problem entails finding a trajectory of the control system and an open-loop control such that the corresponding output signal traces the required path in its entirety. We refer the reader to [25], [42] and [85] for three representative approaches to this problem. The relationship to our approach is that any trajectory solving the perfect path following problem must necessarily lie on the path following manifold Γ. In general, there might be more than one

182 Chapter 5. The Path Following Problem 167 trajectory solving the exact path following problem (for instance, a vehicle may traverse the path at different speeds) and therefore, generally, Γ has dimension greater than one. The idea of stabilizing the whole Γ, rather than just one trajectory on Γ allows one to change the motion along the path, by means of the tangential control design in step 4, without having to repeat the control design process. This concept is illustrated experimentally in Section When feasible, our approach allows a degree of flexibility in choosing the tangential dynamics (the dynamics on the path) which is not always found in other path-following approaches. There are times when the motion on the path is completely determined by the requirement of making the path following manifold invariant. In this case, all the control effort goes towards the stabilization of Γ [103]; there are no tangential controls. This may be the case, for example, in trying to stabilize a Jordan curve in the configuration space of an underactuated controlled Euler-Lagrange system. In [103], the authors are interested in designing control laws forcing an Euler-Lagrange system to exhibit orbitally stable limit cycles for systems with n-dof. The desired limit cycle is given by n 1 constraints in the configuration space called virtual holonomic constraints (see also [117]) and it is a controlled-invariant set which is stabilized. This approach has some affinity to ours in that the path following problem is solved within the set stabilization framework. In [31], El-Hawwary and Maggiore employed the results from [30] to globally solve the path following problem for the kinematic unicycle using passivity based controllers. 5.4 Path following for a maglev positioning system We now apply our path following methodology to the design of a path following controller for a 5-DOF magnetically levitated (maglev) positioning system at the University of Toronto Systems Control research lab. The current incarnation of the maglev positioning system is the culmination of efforts of many researchers. Maggiore and Becerril

183 Chapter 5. The Path Following Problem 168 in [71] developed a ordinary differential equation model of the system. This model forms the basis of control design. In [87] Owen, Maggiore and Apkarian carried out the implementation of the 2-DOF and 3-DOF versions of the positioning system. Fulford [34] expanded the system to the current 5-DOF model as well as the actual implementation of the path following controller that we present in this chapter. The material in this section has appeared, in different form, in Cameron Fulford s Masters thesis. The control objective is to make a platen move along a closed Jordan curve defined within the system s range of operation. Besides stabilizing the curve, the controller should be able to make the platen traverse it in a desired direction with a desired angular velocity. At the same time, the controller should regulate the rotational dynamics of the platform in such a way that it remains level with the ground. This control system finds application in manufacturing, whereby the platen hosts parts that need to be positioned with high-accuracy, or moved along profiles. There are several advantages in using magnetically levitated positioning systems over traditional mechanically actuated ones; the reader is referred to [87] for a detailed discussion. The path invariance property (see Definition 5.2.3) induced by our control method is particularly beneficial in this application, because it effectively creates virtual mechanical constraints in the system that make it act as if it were being guided by (re-configurable) mechanical guides. This distinguishes our path following methodology from some others in the literature Experimental apparatus and model In this section we give a brief description of the physical experiment and a simplified model used as the basis for controller design. The detailed derivation of the system model is found in [34].

184 Chapter 5. The Path Following Problem 169 Hardware setup The 5-DOF maglev positioning system was developed in collaboration with Quanser and is the evolution of two previous setups, a 2-DOF and a 3-DOF, described in [87]. The apparatus is shown in Figure 5.1. Figure 5.1: The magnetic levitation system The setup used in this research consists of four symmetrically placed iron-cored permanent magnet linear synchronous motors, or PMLSMs, shown from the top view in Figure 5.2. Each PMLSM is labeled Motor 1 to Motor 4, and consists of a stator and a mover. The stators are housed in a heavy stationary frame and each mover is positioned beneath its corresponding stator and affixed to an aluminium platen. Each stator exerts two orthogonal forces on the mover: a horizontal translational force and a vertical normal force. The aluminium platen is positioned below a stationary frame and rests on sets of linear guides that allow the platen to move along two horizontal axes, one vertical axis, as well as rotate about the two horizontal axes (pitch and roll). Figure 5.2 illustrates the platen and the five degrees-of-freedom including three translations and two rotations. The linear guides do not provide any actuation force to the platen other than friction (a disturbance) and are currently required to maintain proper alignment of the platen and, most importantly, facilitate the placement of sensors used to measure displacements and rotations of the platen. The system has a horizontal displacement range of ±50 mm along the X-axis and Z-axis, a vertical range of approximately 13 mm, and rotations about the X-axis and