Proceeings of the 4th IEEE Conference on Decision & Control Manchester Gran Hyatt Hotel San Diego, CA, USA, December 3-, 6 FrIP.7 Approximate Reuction of Dynamical Systems Paulo Tabuaa, Aaron D. Ames, Agung Julius an George Pappas Abstract The reuction of ynamical systems has a rich history, with many important applications relate to stability, control an verification. Reuction is typically performe in an exact manner as is the case with mechanical systems with symmetry which, unfortunately, limits the type of systems to which it can be applie. The goal of this paper is to consier a more general form of reuction, terme approximate reuction, in orer to exten the class of systems that can be reuce. Using notions relate to incremental stability, we give conitions on when a ynamical system can be projecte to a lower imensional space while proviing har bouns on the inuce errors, i.e., when it is behaviorally similar to a ynamical system on a lower imensional space. These concepts are illustrate on a series of examples. I. INTRODUCTION Moeling is an essential part of many engineering isciplines an often a key ingreient for successful esigns. Although it is wiely recognize that moels are only approximate escriptions of reality, their value lies precisely on the ability to escribe, within certain bouns, the moele phenomena. In this paper we consier moeling of close loop control systems, i.e., ifferential equations, with the purpose of simplifying the analysis of these systems. The goal of this paper is to reuce the imension of the ifferential equations being analyze while proviing har bouns on the introuce errors. One promising application of these techniques is to the verification of hybri systems, which is currently constraine by the complexity of high imensional ifferential equations. Reucing ifferential equations an in particular mechanical systems is a subject with a long an rich history. The first form of reuction was iscovere by Routh in the 89 s; over the years, geometric reuction has become an acaemic fiel in it self. One begins with a ifferential equation with certain symmetries, i.e., it is invariant uner the action of a Lie group on the phase space. Using these symmetries, one can reuce the imensionality of the phase space by iviing out by the symmetry group) an efine a corresponing ifferential equation on this reuce phase This research was partially supporte by the National Science Founation, EHS awar 933 an CCR awar 6. P. Tabuaa is with the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 99. E-mail: tabuaa@ee.ucla.eu Web: http://www.ee.ucla.eu/ tabuaa A. D. Ames is with the Control an Dynamical Systems Department, California Institute of Technology, Pasaena, CA 9. E-mail: ames@cs.caltech.eu Web: http://www.cs.caltech.eu/ ames Agung Julius an George Pappas are with the Department of Electrical an Systems Engineering, University of Pennsylvania, Philaelphia, PA 94. E-mail: { agung@seas.upenn.eu, pappasg@ee.upenn.eu} space. The main result of geometric reuction is that one can unerstan the behavior of the full-orer system in terms of the behavior of the reuce system an vice versa [MW74], [vs8], [BKMM96]. While this form of exact reuction is very elegant, the class of systems for which this proceure can be applie is actually quite small. This inicates the nee for a form of reuction that is applicable to a wier class of systems an, while not being exact, is close enough. In systems theory, reuce orer moeling has also been extensively stuie uner the name of moel reuction [BDG96], [ASG]. Contrary to moel reuction where approximation is measure using L norms we are intereste in L norms. The guarantees provie by L norms are more natural when applications to safety verification are of interest. We evelop our results in the framework of incremental stability an our main result is in the spirit of existing stability results for cascae systems that proliferate the Inputto-State Stability ISS) literature. II. PRELIMINARIES A continuous function γ : R + R+, is sai to belong to class K if it is strictly increasing, γ) = an γr) as r. A continuous function β : R + R+ R+ is sai to belong to class KL if, for each fixe s, the map βr, s) belongs to class K with respect to r an, for each fixe r, the map βr, s) is ecreasing with respect to s an βr, s) as s. For a smooth function ϕ : R n R m we enote by Tϕ the tangent map to ϕ an by T x ϕ the tangent map to ϕ at x R n. We will say that ϕ is a submersion at x R n if T x ϕ is surjective an that Tϕ is a submersion if it is a submersion at every x R n.whenϕ is a submersion we will also use the notation kertϕ) to enote the istribution: kertϕ)={x : R n R n T x ϕ Xx) =}. Given a point x R n, x will enote the usual Eucliean norm while f will enote ess sup t [,τ] ft) for any given function f :[,τ] R n, τ>. A. Dynamical an control systems In this paper we shall restrict our attention to ynamical an control systems efine on Eucliean spaces. Definition : A vector fiel is a pair R n,x) where X is a smooth map X : R n R n. A smooth curve x,x): I R n, efine on an open subset I of R incluing the origin, is sai to be a trajectory of R n,x) if the following two conitions hol: ) x,x)=x; -444-7-/6/$. 6 IEEE. 648
4th IEEE CDC, San Diego, USA, Dec. 3-, 6 FrIP.7 ) txt, x) =Xxt, x)) for all t I. A control system can be seen as an uner-etermine vector fiel. Definition : A control system is a triple R n, R m,f) where F is a smooth map F : R n R m R n. A smooth curve x u,x):i R n, efine on an open subset I of R incluing the origin, is sai to be a trajectory of R n, R m,f) if there exists a smooth curve u : I R m such that the following two conitions hol: ) x u,x)=x; ) t x ut, x) =F x u t, x), ut)) for almost all t I. III. EXACT REDUCTION For some ynamical systems escribe by a vector fiel X on R n it is possible to replace X by a vector fiel Y escribing the ynamics of the system on a lower imensional space, R m, while retaining much of the information in X. When this is the case we say that X can be reuce to Y. This iea of exact) reuction is capture by the notion of ϕ-relate vector fiels. Definition 3: Let ϕ : R n R m be a smooth map. The vector fiel R n,x) is sai to be ϕ-relate to the vector fiel R m,y) if: Tϕ X = Y ϕ. ) The following proposition, prove in [AMR88], characterizes ϕ-relate vector fiels in terms of their trajectories. Proposition : The vector fiel R n,x) is ϕ-relate to the vector fiel R m,y), for some smooth map ϕ : R n R m,iff: ϕ xt, x) =yt, ϕx)), ) where xt, x) an yt, y) are the trajectories of vector fiels X an Y, respectively. For ϕ-relate vector fiels, we can replace the stuy of trajectories x,x) with the stuy of trajectories y,ϕx)) living on the lower imension space R m. In particular, formal verification of X can be performe on Y whenever the relevant sets escribing the verification problem can also be reuce to R m. If a vector fiel an a submersion ϕ are given we can use the following result, prove in [MSVS8], to etermine the existence of ϕ-relate vector fiels. Proposition : Let R n,x) be a vector fiel an ϕ : R n R m a smooth submersion. There exists a vector fiel R m,y) that is ϕ-relate to R n,x) iff: [kertϕ),x] kertϕ). In an attempt to enlarge the class of vector fiels that can be reuce we introuce, in the next section, an approximate notion of reuction. IV. APPROXIMATE REDUCTION The generalization of Definition 3 propose in this section requires a ecomposition of R n of the form R n = R m R k. Associate with this ecomposition are the canonical projections π m : R n R m an π k : R n R k taking R n x =y, z) R m R k to π m x) =y an π k x) =z, respectively. Definition 4: The vector fiel R n,x) is sai to be approximately π m -relate to the vector fiel R m,y) if there exists a class K function γ such that the following estimate hols: π m xt, x) yt, π m x)) γ π k x) ). 3) Note that when X an Y are π m -relate we have: π m xt, x) yt, π m x)) =, which implies 3). Definition 4 can thus be seen as a generalization of exact reuction capture by Definition 3. Although the boun on the gap between the projection of the original trajectory x an the trajectory y of the approximate reuce system is a function of x, in concrete applications the initial conitions are typically restricte to a boune set of interest. The following result has interesting implications in these situations. Proposition 3: If R n,x) is approximately π m -relate to R m,y) then for any compact set C R n there exists a δ> such that for all x C the following estimate hols: π m xt, x) yt, π m x)) δ. 4) Proof: Let δ =max x C γ π k x) ). The scalar δ is well efine since γ π k ) ) is a continuous map an C is compact. From a practical point of view, approximate reuction is only a useful concept if it amits characterizations that are simple to check. In orer to erive such characterizations we nee to review several notions of incremental stability. A. Incremental stability In this subsection we review two notions of incremental stability which will be funamental in proving the main contribution of this paper. We follow [BM] an [Ang]. Definition : A control system R n, R m,f) is sai to be incrementally uniformly boune-input-boune-state stable IUBIBSS) if there exist two class K functions γ an γ such that for each x,x R n an for each pair of smooth curves u, u : I R m the following estimate hols: x u t, x ) x u t, x ) γ x x )+γ u u ) ) for all t I. In general it is ifficult to establish IUBIBSS irectly. A sufficient conition is given by the existence of an IUBIBSS Lyapunov function. Note however that IUBIBSS only implies the existence of a IUBIBSS Lyapunov function with very weak regularity conitions [BM]. Definition 6: A C function V : R n R n R + is sai to be an IUBIBSS Lyapunov function for control system R n, R m,f) if there exist a ξ> an class K functions α, α, an μ such that: ) For x x ξ, α x x ) V x,x ) α x x ); ) μr) r + ξ for r R + ; 3) x x μ u u ) = V. 649
4th IEEE CDC, San Diego, USA, Dec. 3-, 6 FrIP.7 A stronger notion than IUBIBSS is incremental input-tostate stability. Definition 7: A control system R n, R m,f) is sai to be incrementally input-to-state stable IISS) if there exist a class KL function β an a class K function γ such that for each x,x R n an for each pair of smooth curves u, u : I R m the following estimate hols: x u t, x ) x u t, x ) β x x,t)+γ u u ) 6) Since β is a ecreasing function of t we immeiately see that 6) implies ) with γ r) =βr, ) an γ r) =γr), r R +. Once again, IISS is implie by the existence of an IISS Lyapunov function. See [Ang] for a converse result when the inputs take values in a compact set. Definition 8: A C function V : R n R n R + is sai to be an IISS Lyapunov function for the control system R n, R m,f) if there exist class K functions α, α, α, an μ such that: ) α x x ) V x,x ) α x x ); ) x x μ u u ) = V α x x ). B. Fiberwise stability In aition to incremental stability we will also nee a notion of partial stability. Definition 9: A vector fiel R n,x) is sai to be fiberwise stable with respect to R k if there exists a class K function γ such that the following estimate hols: π k x,x)) γ π k x) ). Fiberwise stability can be checke with the help of the following result. Lemma : A vector fiel R n,x) is fiberwise stable with respect to R k if there exist two K functions, α an α, an a function V : R n R such that: ) α π k x) ) V x) α π k x) ), ) V. C. Existence of approximate reuctions In this subsection we prove the main result proviing sufficient conitions for the existence of approximate reuctions. Theorem : Let R n,x) be a fiberwise stable vector fiel with respect to R k an let F = Tπ m X : R m R k R m, viewe as a control system with state space R m, be IUBIBSS. Then, the vector fiel R m,y) efine by: Y y) =T y,z) π m Xy, ) = F y, ) is approximately π m -relate to R n,x). Proof: By assumption Y y) =T y,z) π m Xy, ) is IUBIBSS with respect to R k so that we have: y v t, y ) y v t, y ) γ y y )+γ v v ). In particular we can take y = y = π m x), v = π k x,x), v =, to get: π m xt, x) yt, π m x)) = y πk xt,x)t, π m x)) y t, π m x)) = y v t, π m x)) y t, π m x)) γ v )=γ π k x,x) ). But it follows from fiber stability of X with respect to R k that π k x,x) γ π k x) ). We thus have: π m xt, x) yt, π m x)) γ γ π k x) ), which conclues the proof since γ γ is a class K function. Theorem shows that sufficient conitions for approximate reuction can be given in terms of ISS-like Lyapunov functions an how reuce system can be constructe. Before illustrating Theorem with several examples in the next section we present an important corollary. Corollary : Let R n,x) an R m,y) be vector fiels satisfying the assumptions of Theorem. Then, for any compact set C R n there exists a δ > such that for any x C an y π m C) the following estimate hols: π m xt, x) yt, y) δ Proof: Using the same proof as for Theorem, except picking y = π m x) an y = y, it follows that: π m xt, x) yt, y) γ π m x) y )+γ π k x) ). The boun δ is now given by: δ = max γ π m x) y )+γ π k x) )) x,y) C π mc) which is well efine since C π m C) is compact. This result has important consequences for verification. Given a set of initial conitions S R m, the set of points reachable uner trajectories of Y from initial conitions in S can be over-approximate by enclosing a single trajectory of Y, starting at any point in S, by a tube of raius δ. V. EXAMPLES In this section, we consier examples that illustrate the usefulness of approximate reuction. Example : As a first example we consier the ball in a rotating hoop with friction, as escribe in Chapter of [MR99]. For this example, there are the following parameters: m = mass of the ball, R = Raius of the hoop, g = acceleration ue to gravity, μ = friction constant for the ball. The equations of motion are given by: ω = μ m ω + ξ sin θ cos θ g R sin θ θ = ω 7) 64
4th IEEE CDC, San Diego, USA, Dec. 3-, 6 FrIP.7 4 6 8 where θ is the angular position of the ball an ω is its angular velocity. If π ω : R R is the projection π ω ω, θ) =ω, then accoring to Proposition there exists no vector fiel Y on R which is π ω -relate to X as efine by 7)). However, we will show that Y ω) =T ω,θ) π ω Xω, ) is approximate π ω -relate to X. First, we use: V = mr ω + mgr cos θ) mr ξ sin θ as a Lyapunov function to show that 7) is stable. Note that V ω, θ) =for ω, θ) =, ) an V ω, θ) > for ω, θ), ) provie that Rξ <g. Computing the time erivative of V we obtain: V = μr ω, thus showing stability of 7). We now consier a compact set C invariant uner the ynamics an restrict our analysis to initial conitions in this set. Such a set can be constructe, for example, by taking {x R V x) c} for some positive constant c. Note that stability of 7) implies fiberwise stability on C since π m C) is compact. To apply Theorem we only nee to show that: T ω,θ) π ω Xω, θ) = μ m ω + ξ sin θ cos θ g R sin θ 4 6 8 4 6 8 is IUBIBSS on C with θ seen as an input. We will conclue IUBIBSS by proving the stronger property of IISS. Consier the function: U = ω ω ). Its time erivative is given by: [ U = ω ω ) μ m ω ω )+ξ sin θ cos θ g R sin θ ξ sin θ cos θ + g ] R sin θ μ m ω ω ) + ω ω ξ sin θ cos θ g R sin θ ξ sin θ cos θ + g R sin θ 4 6 8 Fig.. A trajectory of the full orer system re) vs. a trajectory for the reuce system blue) for R =,,, 4 from top to bottom, respectively). μ m ω ω ) + ω ω L θ θ = μ m ω ω ) 8) + μ ) m ω ω ) + ω ω L θ θ, where the secon inequality follows from the fact that ξ sin θ cos θ g R sin θ is a smooth function efine on the convex compact set π θ C) an is thus globally Lipschitz on π θ C) since its erivative is continuous an thus boune on π θ C)) with Lipschitz constant L. We now note that the conition: ω ω > ml μ θ θ makes the secon term in 8) negative from which we conclue the following implication: ω ω > ml μ θ θ = U μ m ω ω ) 64
4th IEEE CDC, San Diego, USA, Dec. 3-, 6 FrIP.7 Fig.. A graphical representation of the penulum on a cart mounte to a spring. showing that U is an IISS Lyapunov function. We can thus reuce 7) to: ω = μ m ω. Projecte trajectories of the full-orer system as compare with trajectories of the reuce system can be seen in Figure ; here μ = m =an ξ =.. Note that as R, the reuce system converges to the full-orer system or the full-orer system effectively becomes ecouple). Example : We now consier a penulum attache to a cart with is mounte to a spring see Figure ). For this example, there are the following parameters: M = mass of the cart, m = mass of the penulum, R = length of the ro, k = spring stiffness, g = acceleration ue to gravity, = friction constant for the cart, b = friction constant for the penulum. The equations of motion are given by: ẋ = v θ = ω v = ω = M + m sin θ kx v + b ) R cos θ mrω sin θ + mg sin θ cos θ RM + m sin mrω sin θ cos θ θ) m + M)g sin θ + kx cos θ + v cos θ + M m ) b ) R ω where x is the position of the cart, v its velocity, θ is the angular position of the penulum an ω its angular velocity. If π x,v) : R 4 R is the projection π x,v) x, θ, v, ω) = x, v) an X is the vector fiel as efine in 9), the goal is 9) to reuce X to R by eliminating the θ an ω variables an thus obtaining Y efine by: ) ẋ = Y x, v) v = T x,θ,v,ω) π x,v) Xx,,v,) ) v =. M v + kx) The objective is now to show that X an Y are approximately π x,v) -relate. In particular, note that the reuce system, Y, is linear while the full-orer system, X, is very nonlinear. This will be iscusse in more etail after proving that they are in fact approximately relate. Stability of X, an in particular fiberwise stability, can be proven as in the previous example by noting that X is Hamiltonian for = b =an using the Hamiltonian as a Lyapunov function V. Consier now the control system: F x, v), θ, ω)) = Tπ x,v) Xx, θ, v, ω) ) = M + m sin mrω sin θ kx θ +mg sin θ cos θ v + b ) R cos θ with θ an ω regare as inputs. To show that F is IUBIBSS we first rewrite ) in the form: F x, v), θ, ω)) = ) mrω sin θ kx v mr ω cos θ M + m an consier the following IISS caniate Lyapunov function: U = m + M) x x ) + v v ). Its time erivative is given by: U = m + M v v ) + mr ω sin θ m + M ω cos θ ω sin θ + ω cos θ )v v ). Using an argument similar to the one use for the previous example, we conclue that: v v mrl θ,ω, ω ) θ,ω, ω ), with L the Lipschitz constant of the function ω sin θ ω cos θ, implying: U m + M) v v ), thus showing that X is IISS an in particular also IUBIBSS. That is, we have establishe that X an Y are approximately π x,v) -relate. In orer to illustrate some of the interesting implications of approximate reuction, we will compare the reuce system, Y, an the full-orer system, X, in the case when R = m = 64
4th IEEE CDC, San Diego, USA, Dec. 3-, 6 FrIP.7.... -. -. - - -. -. - - - -.... -. -. - - -. -. - - - - Fig. 3. A projecte trajectory of the full-orer system re) an a trajectory for the reuce system blue) for =.,.,., from left to right an top to bottom, respectively). k = b =an M =. It follows that the equations of motion for the reuce system are given by the linear system: ) ) ) ẋ x = v, v so we can completely characterize the ynamics of the reuce system: every solution spirals into the origin. This is in stark contrast to the ynamics of X see 9)) which are very complex. The fact that X an Y are approximately relate, an more specifically Theorem, allows us to unerstan the ynamics of X through the simple ynamics of Y. To be more specific, because the istance between the projecte trajectories of X an the trajectories of Y is boune, we know that the projecte trajectories of X will essentially be spirals. Moreover, the friction constant will irectly affect the rate of convergence of these spirals. Examples of this can be seen in Figure 3 where is varie to affect the convergence of the reuce system, an hence the full orer system. REFERENCES [AMR88] R. Abraham, J. Marsen, an T. Ratiu. Manifols, Tensor Analysis an Applications. Applie Mathematical Sciences. Springer-Verlag, 988. [Ang] [ASG] D. Angeli. A lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control, 473):4 4,. A. C. Antoulas, D. C. Sorensen, an S. Gugercin. A survey of moel reuction methos for large-scale systems. Contemporary Mathematics, 8:93 9,. [BDG96] C. L. Beck, J. Doyle, an K. Glover. Moel reuction of multiimensional an uncertain systems. IEEE Transactions on Automatic Control, 4):466 477, 996. [BKMM96] A.M. Bloch, P.S. Krishnaprasa, J.E. Marsen, an R. Murray. Nonholonomic mechanical systems with symmetry. Archive for Rational Mechanichs an Analysis, 36): 99, 996. [BM] A. Bacciotti an L. Mazzi. A necessary an sufficient conition for boune-input boune-state stability of nonlinear systems. SIAM Journal on Control an Optimization, 39):478 49,. [MR99] Jerrol E. Marsen an Tuor S. Ratiu. Introuction to Mechanics an Symmetry. Number 7 in Texts in Applie Mathematics. Springer-Verlag, n eition, 999. [MSVS8] Giuseppe Marmo, Alberto Simoni, Bruno Vitale, an Eugene J. Saletan. Dynamical Systems. John Wiley & Sons, 98. [MW74] J. E. Marsen an A. Weinstein. Reuction of symplectic manifols with symmetry. Reports on Mathematical Physics, :, 974. [vs8] A. van er Schaft. Symmetries an conservation laws for Hamiltonian systems with inputs an outputs: A generalization of Noether s theorem. Systems an Control Letters, :8, 98. 643