Lyapunov-based methods in control

Transcription

1 Dr. Alexander Schaum Lyapunov-based methods in control Selected topics of control engineering Seminar Notes Stand: Summer term 2018 c Lehrstuhl für Regelungstechnik Christian Albrechts Universität zu Kiel

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3 Contents 1 Lyapunov stability and Lyapunov s direct method Stability in the sense of Lyapunov Lyapunov s direct method Lyapunov-based design methods Integrator backstepping Passivity-based feedback control References iii

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5 Preface These notes provide the basis theory about Lyapunov s direct method and its applications in control engineering including the integrator backstepping and passivity-based approaches. They are the product of seminars and courses on this subject held at Kiel University during the last semesters. I did my best to present the results in a correct manner but know that, anyway always some errors are possible. Thus I appreciate constructive criticism and indications where something should and could be improved. Alexander Schaum Kiel, summer term

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7 Chapter 1 Lyapunov stability and Lyapunov s direct method 1.1 Stability in the sense of Lyapunov Consider the nonlinear dynamical system ẋ = f(x), x(0) = x 0, (1.1) with solutions x(t) = x(t; x 0 ). A fundamental property of linear and nonlinear systems (or flows) is the stability of equilibrium points, i.e. solutions x of the algebraic equation 0 = f(x, p). (1.2) Throughout this text we will be concerned with the stability and need a formal definition. Here, we employ the common definitions associated to A. Lyapunov [1, 2]. Definition An equilibrium point x of (1.1) is said to be stable, if for any ɛ > 0 there exists a constant δ > 0 such that for any initial deviation from equilibrium within a δ-neighborhood, the trajectory is comprised within an ɛ-neighborhood, i.e. x 0 : x 0 x δ x(t) x ɛ t 0. (1.3) If x is not stable, it is referred to as unstable. This concept is illustrated in Figure 1.1 (left), and is sometimes referred to stability in the sense of Lyapunov. Stability implies that the solutions stay arbitrarily close to the equilibrium whenever the initial condition is chosen sufficiently close to the equilibrium point. Note that stability implies boundedness of solutions, but not that these converge to an equilibrium point. Convergence in turn is ensured by the concept of attractivity. Definition The equilibrium point x is called an attractor for the set S, if x 0 S : lim t x(t) x = 0. (1.4) The set S is called the domain of attraction. Note that an equilibrium point may be attractive without being stable, i.e. that trajectories always have a large transient, so that for small ɛ no trajectory will stay for all times within the ɛ-neighborhood, but will return to it and converge to the equilibrium point. An example of such a behavior is given by Vinograd s system [3] 3

8 4 1 Lyapunov stability and Lyapunov s direct method δ δ ɛ ɛ Fig. 1.1: Qualitative illustration of the concepts of stability (left) and asymptotic stabilty (right). x 2 ẋ 1 = 1(x 2 x 1 ) + x 5 2 (x x2 )(1 + (x2 1 + x2 2 )2 ) x 2 ẋ 2 = 2(x 2 2x 1 ) (x x2 2 )(1 + (x2 1 + x2 2 )2 ) (1.5) with phase portrait shown in Figure 1.2, illustrating a butterfly-shaped behavior where, for very small the initial deviation from equilibrium will be, there is a large transient returning asymptotically to the equilibrium point x = 0. Fig. 1.2: Phase portrait of the Vinograd system (1.5), with an unstable but attractive equilibrium point at x = 0. It should be mentioned, that if one can demonstrate convergence within a domain S 1, this does not necessarily imply that S 1 is the maximal domain of attraction. The determination of the maximal domain of attraction for an attractive equilibrium of a nonlinear system is, in general, a non-trivial task, due to

9 1.1 Stability in the sense of Lyapunov 5 the fact that, in contrast to linear systems, nonlinear systems can have multiple attractors, each one with its own domain of attraction. This issue will be analyzed later with more detail. A common, similar concept, including a statement for the transient behavior, is called asymptotic stability and is defined next. Definition An equilibrium point x of (1.1) is said to be asymptotically stable if it is stable and attractive. This concept is illustrated in Figure 1.1 (right). Note that in contrast to pure attractivity, the concept of asymptotic stability does not allow for large transients associated to small initial deviations, and is thus much stronger, and more of practical interest. The asymptotic stability does not state anything about convergence speed. It only establishes that after an infinite time period the solution x(t), if starting in the domain of attraction, will approach x without ever reaching it actually. A concept which allows to overcome this issue is the stronger one of exponential stability. Definition An equilibrium point x of (1.1) is said to be exponentially stable in a set S, if it is stable, and there are constants a, λ > 0 so that x 0 S : x(t) x a x 0 x e λt. (1.6) The constant a is known as the amplitude, and λ as the convergence rate. Clearly, exponential stability implies asymptotic stability. It is quite noteworthy that, unless the convergence is still asymptotic, for exponentially stable equilibria it is possible to determine exactly the time needed to approach the equilibrium up to a given distance. Note that all the above stability and attractivity concepts can also be applied to sets. To examplify this, and for later reference, consider the definition of an attractive compact set. Definition A compact set M is called attractive for the domain S, if x 0 S : lim x(t) M. t Convergence to a set is of practical interest because in many situations it may be used to obtain a reduced model for analysis and design purposes, or may even be part of the design as in the sliding-mode control approach (see e.g. [4]). In many situations it is an essential part of the stability assessment of a dynamical system [5]. Furthermore, in many applications it is more important to show the convergence to a set rather than to an equilibrium point. This is due to the fact that in application situations the exact system parameters are often not known exactly, but within some tolerance interval, defining on the other hand a tolerance interval for the required convergence. This requirement is conceptually defined via the practical stability notion introduced by Lefschetz [6]. Consider a nominal parameter p for which the operation point x of (1.1) is defined. Then consider a (possibly time varying) deviated paramter p(t) with which the system is actually operating. Then practical stability is defined as follows. Definition The operation point x is said to be practically stable if for given parameter and initial condition deviation sizes δ p and δ 0, respectively, it holds that t 0 : p(t) p δ p, x 0 x δ x x(t) x α( x 0 x, t) + β(δ p ), (1.7) with α being a increasing-decreasing (i.e. class KL) and β an increasing (i.e. class K) function a. a Class KL are monotonically increasing in the first and decreasing in the second argument, while class K functions are monotonically increasing

10 6 1 Lyapunov stability and Lyapunov s direct method Geometrically speaking, practical stability ensures the boundedness of solutions in presence of bounded perturbations, and that the initial condition deviation asymptotically vanishes. In fact, for δ p = 0 the concept of asymptotic stability (see Definition 1.1.3) is recovered. Finally, it will be helpfull in the sequel to understand the concept of (positively) invariant sets M for a dynamical system. Definition A set M R n is called positively invariant, if for all x 0 M it holds that x(t; x 0 ) M for all t 0. The distinction positive invariant is used to distinguish the concept from negative invariance, referring to a reversion of time (i.e., letting time tending to minus infinity). 1.2 Lyapunov s direct method A very usefull way to establish the stability of an equilibrium point for a nonlinear dynamical systems consists in Lyapunov s direct method. Motivated by studies on energy dissipation in physical processes, in particular in astronomy, Aleksandr Mikhailovich Lyapunov, generalized these considerations to functions which are positive for any non-zero argument [1]. In the sequel consider that the equilibrium point under consideration is the origin x = 0. If other equilibria have to analyzed a linear coordinate shift x = x x can be employed to move the equilibrim to the origin in the coordinate x. To summarize the results of Lyapunov and generalizations of it some definitions are in order. Definition A continuous functional V : D R n R is called positive semi-definite if x : V (x) 0. positive definite if x 0 : V (x) > 0 and V (x) = 0 only for x = 0. negative semi-definite if x : V (x) 0. negative definite if x 0 : V (x) < 0 and V (x) = 0 only for x = 0. Theorem Let V : D R n, V (x) > 0 be positive definite. If x D : then x = 0 is stable in the sense of Lyapunov. dv V (x) (x) = x ẋ 0, Proof: Given that V (x) > 0 and its continuity, there exists a function W (x) > 0 such that W (x) V (x), x D. (1.8) Let ɛ > 0 and set m := min W (x) > 0. x =ɛ (1.9) Choose δ > 0 such that max V (x) m. 0 x δ Given that m > 0, V (x) > 0 and the continuity of V such a positive δ always exists. It follows from the fact that V is non-increasing over time ( V (x) < 0) that x 0 : x 0 δ V (x(t; x 0 )) m.

11 1.2 Lyapunov s direct method 7 By (1.8) this implies that W (x(t; x 0 )) m. By virtue of the definition of m in equation (1.9) it follows that x 0 : x 0 δ x(t; x 0 ) ɛ showing that the origin is stable in the sense of Lyapunov. A function V > 0 that satisfies the conditions of Theorem is called a Lyapunov function. Note that if V > 0 is continuously differentiable but it is not clear if dv 0 or the sign depends on some system parameters, then it is called a Lyapunov function candidate. As can be seen from the proof, an essential part consists in that the sets Γ c = {x R n V (x) = c} (1.10) defined by level curves of V (x) are the boundaries of compact subsets D c of the state space. In virtue of the non-increasing nature of V these sets are positively invariant. The geomtric idea of the proof of Theorem is quite beautiful and will be shortly discussed. See Figure 1.3 for an illustration. The conditions of the theorem ensure that for a given ɛ there exists a value c > 0 such that the set D c with the boundary Γ c defined in (1.10) is completely contained in the ɛ-neigborhood N ɛ of the origin, i.e. it holds that D c N ɛ. Choosing δ > 0 such that the δ-neighborhood N δ is completely contained in D c one obtains that N δ D c N ɛ with D c being positively invariant. Thus it holds that for all x 0 with x 0 δ, i.e. x 0 N δ the solution x(t; x 0 ) is contained in D c N ɛ, implying that x(t) ɛ for all t 0. x 2 ɛ N ɛ x 1 δ D c N δ Fig. 1.3: Geometrical idea behind the proof of Lyapunov s direct method in a two-dimensional state space. The above only holds locally, unless V (x) is strictly growing with x. Thus the result is only local. The maximum compact set implied by the particular Lyapunov function can be explicitely determined. In the case that lim x V (x) = the function is called radially unbounded. For a radially unbounded Lyapunov function the above result becomes global, i.e. it holds with D = R n. By evaluating explicitely the inequality dv (x) = 0 which holds over the set { } X 0 = x R n dv (x) = 0 (1.11)

12 8 1 Lyapunov stability and Lyapunov s direct method one can apply the following result going to back to Nikolay Nikolayevich Krasovsky and Joseph Pierre LaSalle and is known as the invariance theorem. Theorem (Krasovsky-LaSalle) Let D R n be a postively invariant compact set and V C 1 (D R) positive defininte function with dv (x) 0 for all x D. Then the trajectories x(t) converge to the largest positively invariant set M X 0 with X 0 defined in (1.11). If the conditions of this theorem are satisfied, an additional condition implies the asymptotic stability of the origin as stated next. Theorem If the conditions of Theorem are satisfied and it holds that M = {0}, then the origin x = 0 is asymptotically stable. A typical system where these results can be illustrated is given by the following Lienard oscillator ẍ + dẋ + f(x) = 0 (1.12) with d > 0 and f(x) > 0 for x > 0, f(x) = 0 for x = 0 and f( x) = f(x). The oscillator (1.12) can be written equivalently in state-space form with x 1 = x and x 2 = ẋ as ẋ 1 = x 2 ẋ 2 = f(x 1 ) dx 2. (1.13a) (1.13b) Consider the following Lyapunov function candidate V (x) = x1 0 f(ξ)dξ x2 2 motivated by the energy contained in the motion of x in form of potential and kinetic energy. The change in time of V is governed by dv (x) = f(x 1)ẋ 1 + x 2 ẋ 2 = f(x 1 )x 2 + x 2 ( f(x 1 ) dx 2 ) = dx implying stability of the origin x = 0 in virtue of Theorem From Theorem it is additionally known that x converges into the set X 0 = {x R 2 x 2 = 0}, and more specifically into the largest positively invariant subset of M X 0. This set in turn contains only trajectories for which x 2 (t) = 0 for all times, given that it is positively invariant. This means that ẋ 2 (t) = 0 for all times. Substituting x 2 = 0, ẋ 2 = 0 into (1.13b) this means that f(x 1 (t)) = 0 for all times, showing that M = {0} given that f(x 1 ) = 0 only for x 1 = 0. Corollary implies that the origin x = 0 is asymptotically stable. The asymptotic stability can also be concluded using Lyapunov s direct method if dv (x) is negative definite. This is stated in the next theorem. Theorem Let V : D R n R n, V > 0. If x D : asymptotically stable in D. dv (x) < 0, then x = 0 is locally

13 1.2 Lyapunov s direct method 9 Proof: In virtue of Theorem we have that x = 0 is stable in the sense of Lyapunov. It thus remains to show that lim t V (x) = 0 to conclude, by taking into account the positive definiteness and the continuity of V (x), that lim t x(t) = 0. Assume that V does not converge to zero. Then there exists a positive constant c > 0 such that lim t V (x(t)) = c > 0. Let S = {x D V (x) c} By assumption, for x 0 / S, i.e. V (x 0 ) > c it holds that t 0 : x(t) / S. Let Γ S be the boundary of the set S, i.e. Γ S = {x D V (x) = c}. We have that V (x) x ΓS < 0. Introduce γ := max x Γ S V (x) < 0. Now, let x 0 / S, i.e. V (x 0 ) > c and let t > t := V (x0) c γ. Observe that t V (x(t; x 0 )) = V (x 0 ) + V (x(τ; x 0 )dτ 0 V (x 0 ) γt < V (x 0 ) γt = c implying that t > t it holds that x(t; x 0 ) S. This contradicts the initial assumption that t 0 : x(t) / S, and thus c cannot be positive and it must hold that c = 0. This, in turn, implies that lim t V (x(t)) = 0, and thus x 0 D : lim t x(t) = 0. At this place it is noteworthy that using Lyapunov functions one can establish a domain for which the equilibrium point is an attractor. This domain will always be included in the domain of attraction of the equilibrium point. Note that eventhough it is not possible to conclude if the domain of attraction established in this way is the complete domain of attraction or only a subset of it, unless the result is global. As discussed above, in may cases it is not sufficient to conclude only the asymptotic stability and it becomes important to have a quantitative value for the convergence speed towards an equilibrium. This can be established if the equilibrium is exponentially stable (see Definition 1.1.4). Exponential stability can be concluded using Lyapunov functions if some additional properties are given. These are stated in the next theorem. Theorem Let V : D R n R, V > 0 be a positive definite functional. If there exist constants α, β, γ > 0 so that (i) α x 2 V (x) β x 2 (1.14a) (ii) dv (x) γv (x) (1.14b) then x = 0 is exponentially stable and (1.6) holds with a = β/α and λ = γ/2. Proof: In virtue of (1.14b) it holds that V (x(t)) V (x 0 )e γt. From (1.14a) this implies that x(t) 2 1 α V (x(t)) 1 α V (x 0)e γt β α e γt x 0 2

14 10 1 Lyapunov stability and Lyapunov s direct method and finally β x(t) α x 0 e γ 2 t. Exponentially stability as defined in (1.6) follows with a and λ stated above. Note that, in particular, the fact that the value of V (x) monotonically decreases over time rules out the possibility of closed trajectories, given that they could only exist on level cures of V. Thus the use of Lyapunov functions is also an effective means for the preclusion of limit cycles. Finally, it is possible to show that the existence of a Lyapunov function is intrinsically related to the stability properties of the equilibrium point as stated in the next theorem for the case of an exponentially stable equilibrium origin. Theorem Let x = 0 be exponentially stable in D R n. Then there exists a Lyapunov function V : D R, V (x) > 0 and a constant γ > 0 such that (1.14b) holds true. Proof: By assumption there are constant a, λ such that x(t) a x 0 e λt. Consider the functional V (x(t)) = It holds that 0 x(t + τ) 2 dτ. V (x(t)) = 0, x(t + τ) = 0, τ 0, showing that V is positive definite. On the other hand, in virtue of the exponential stability of the origin it holds that V (x(t)) 0 a 2 x(t) 2 e 2λτ dτ = a2 x(t) 2 2λ showing that for finite x 0 the function V (x) is quadratically bounded from above by the norm of x(t). Consider the rate of change of V at time t evaluated at the point x(t) given by ( dv 1 ) (x(t)) = lim x(t + s) 2 ds x(t + s) 2 ds τ 0 + τ τ 0 1 τ = lim x(t + s) 2 ds τ 0 + τ 0 = x(t) 2 2λ a 2 V (x(t)) showing that inequality (1.14b) holds with γ = 2λ a 2.

15 Chapter 2 Lyapunov-based design methods In this chapter, two important control design approaches are presented that are closely related to or directly based on Lyapunov s direct method to achieve asymptotically (or exponentially) stable operation points for the closed-loop system. The first one is the integrator backstepping, that is presented in the most basic set-up, and the second one is passivity theory and the passivation by feedback control. 2.1 Integrator backstepping Consider the following system ẋ 1 = f 1 (x 1 ) + g(x 1 )x 2, x 1 (0) = x 10 (2.1a) ẋ 2 = u, x 2 (0) = x 20, (2.1b) with smooth vector fields f 1 (x 1 ) and g(x 1 ) 0, x 1 D 1 R. Classical examples for such dynamics are motivated for systems where the actual actuator dynamics have to be taken into account and can be summarized in form of an integrator (e.g. a pump with supplied electric current). In the following a way to exploit the particular structure of this system dynamics for stabilization of the origin by state-feedback is discussed. Consider in a first, auxiliary step the state x 2 as virtual control input. By the condition that g(x 1 ) 0 in D 1 it follows that a simple (linearizing) control can be assigned of the form x 2 = v f(x 1) g(x 1 ) = µ(x 1 ) with closed-loop dynamics (for x 1 ) given by ẋ 1 = v, x 1 (0) = x 10. A desired behavior for x 1 can be introduced by adequately choosing v, e.g. as v = kx 1 to obtain exponential convergence with rate k. Another, more general approach could be to consider the Lyapunov function V 1 (x 1 ) = 1 2 x2 1, dv 1 (x 1 ) = x 1 v! = Q 1 (x 1 ) (2.2) for some desired Q 1 (x 1 ) = x 1 q 1 (x 1 ) > 0 what is achieved using µ(x 1 ) = q 1(x 1 ) f(x 1 ). g(x 1 ) (2.3) 11

16 12 2 Lyapunov-based design methods Choosing W 1 (x 1 ) = kx 2 1, the aforementioned control v = kx 1 is recovered. So, consider for the moment that v is chosen such that (2.2) holds. Clearly, this control is not implementable, as it was assumed that x 2 would be the control input. Thus, actually the difference betweeen x 2 and µ(x 1 ) has to be considered: z = x 2 µ(x 1 ), ż = u µ(x ( 1) x 1 Introducing the Lyapunov function candidate f 1 (x 1 ) + g(x 1 ) [z + µ(x 1 )] }{{} =x 2 ). (2.4) V z (z) = 1 2 z2 (2.5) it follows that dv z (z) = z ( u µ(x ( 1) f 1 (x 1 ) + g(x 1 )[z + µ(x 1 )]) ). x 1 Clearly, one can use this equation to find u in dependence of z and x 1 so that dvz(z) < 0. Nevertheless, this would neglect the dynamics of x 1 for the transient during which x 2 µ(x 1 ). Thus, consider the system dynamics in (x 1, z) coordinates ẋ 1 = f 1 (x 1 ) + g(x 1 )[z + µ(x 1 )], x 1 (0) = x 10 (2.6a) ż = u µ(x ( ) 1) f 1 (x 1 ) + g(x 1 )[z + µ(x 1 )], x 1 z(0) = z 0 (2.6b) and the joint Lyapunov function candidate W (x 1, z) = V 1 (x 1 ) + V z (z). (2.7) The rate of change of W (x 1, z) over time is given by dw (x 1, z) ( = x 1 f 1 (x 1 ) + x 1 g(x 1 )[z + µ(x 1 )] + z u µ(x ( ) 1) ) f 1 (x 1 ) + g(x 1 )[z + µ(x 1 )] x 1 ( = Q 1 (x 1 ) + z x 1 g(x 1 ) + u µ(x ( ) 1) ) f 1 (x 1 ) + g(x 1 )[z + µ(x 1 )] x 1 Thus, to achieve the condition dw (x 1, z)! = Q 1 (x 1 ) Q 2 (z) < 0 for some Q 2 (z) = zq 2 (z) so that Q 2 (z) > 0 it is sufficient to choose the control input u as u = q 2 (z) x 1 g 1 (x 1 ) + µ(x ( ) 1) f 1 (x 1 ) + g(x 1 )[z + µ(x 1 )] = ϖ(x 1, z) (2.8) x 1 with µ(x 1 ) given in (2.3). In terms of (x 1, x 2 ) this stabilizing controller can be written as u = α(x 1, x 2 ) := ϕ(x 1, z) z=x2 µ(x 1). This approach is known as integrator backstepping and can be summarized in the following two steps: a) Design x 2 as auxiliary (virtual) input variable to obtain the relationship x 2 = µ(x 1 ) for which asymptotic (or exponential) stability is ensured b) Design the actual controller input u = ϖ(x 1, x 2 ) so that the difference z = x 2 µ(x 1 ) asymptotically (or exponentially) converges to zero, taking into account the (possibly open-loop unstable) of x 1 during the transient for which x 2 µ(x 1 ).

17 2.1 Integrator backstepping 13 This is put in the Lyapunov-framework in the way discussed above and can directly be generalized to the case where x 1 is a vector in R n and for the case that an integrator chain of n integrators separate the dynamics of x 1 and the input u (so that the relative degree is at least n). Theorem Consider the system ẋ 1 = f(x 1 ) + g(x 1 )x 2, x 1 (0) = x 10 ẋ 2 = u, x 2 (0) = x 20 with x 1 R n, x 2 R and smooth vector fields f, g with f(0) = 0. Let µ(x 1 ) be a continuously differentiable function with µ(0) = 0 and V 1 (x 1 ) a differentiable, positive definite (radially unbounded) function so that V 1 (x 1 ) x 1 (f(x 1 ) + g(x 1 )µ(x 1 )) Q 1 (x 1 ) 0. (2.9) Then the following holds true: a) If Q(x 1 ) > 0 then the feedback u = α(x 1, x 2 ) = µ(x ( ) 1) f(x 1 ) + g(x 1 )x 2 V 1(x 1 ) g(x 1 ) k(x 2 µ(x 1 )) (2.10) x 1 x 1 with k > 0 asymptotically stabilizes the origin [x T 1, x 2 ] T = 0 T and W (x 1, x 2 ) = V 1 (x 1 ) (x 2 µ(x 1 )) 2 is an associated Lyapunov function. b) If Q 1 (x 1 ) 0 then for the closed-loop system with the feedback control (2.10) the trajectories converge into the largest positively invariant subset of {[ ] } x1 W = R n+1 Q x 1 (x 1 ) = 0, x 2 = µ(x 1 ). (2.11) 2 Proof: a) According to the assumptions it holds that dw (x 1, x 2 ) = V 1(x 1 ) (f(x 1 ) + g(x 1 )[µ(x 1 ) + x 2 µ(x 1 )]) + x 1 ( + (x 2 µ(x 1 )) u µ(x 1) x 1 and taking into account (2.9) and (2.10) it follows that dw (x 1, x 2 ) ) (f(x 1 ) + g(x 1 )x 2 ) = Q 1 (x 1 ) k(x 2 µ(x 1 )) 2 < 0. (2.12) Given that Q 1 (x 1 ) > 0 by assumption, the asymptotic stability follows from Theorem b) For Q 1 (x 1 ) 0 it follows from Theorem that the trajectories converge into the largest positively invariant subset of {[ ] } x1 W 0 = R n+1 dw (x 1, x 2 ) = 0. x 2 According to (2.12) this set is identical with (2.11).

18 14 2 Lyapunov-based design methods Beyond this result, one can directly consider chains of integrators, like in the system ẋ 1 = f(x 1 ) + g(x 1 )x 2, x 1 (0) = x 10 ẋ 2 = x 3, x 2 (0) = x 20. ẋ n = u, x n (0) = x n0 or the more general set-up where ẋ 1 = f 1 (x 1, x 2 ), x 1 (0) = x 10 ẋ 2 = f 2 (x 1, x 2 ) + u, x 2 (0) = x 20 under the assumption that vector field α(x 1 ) and a Lyapunov function V (x 1 ) is known which proofs the asymptotic stability of the system ẋ 1 = f 1 (x 1, α(x 1 )), x 1 (0) = x 10. For more information, the interested reader is referred to the literature [7, 8]. 2.2 Passivity-based feedback control Consider the SISO control system ẋ = f(x) + g(x)u, x(0) = x 0 (2.13a) y = h(x) with state x R n and smooth vector fields f(x), g(x) and differentiable output map h(x). (2.13b) The notion of passivity in systems theory is motivated by the notion of passivity in electrical engineering, where this concept refers to an electrical circuit where the electric power consumption P = U I is always positive, in the understanding that this means that the circuit dos not produce energy by itself and the net flow of energy is always into the circuit. This has been extended to a general set-up for (open) dynamical (control) systems [9, 10, 11, 12, 13, 14, 15, 16, 17]. For the purpose at hand the notion of passivity can be defined in the following way. Definition The system (2.13) is called passive, if there exists a (positive semi-definite) storage function S(x) 0 so that ds(x) = S(x) x (f(x) + g(x)u) uy. Clearly, this restricts the class of systems (2.13) in a considerable manner. The advantage of considering passive systems lies in the important fact, that the simple output feedback control u = ky, k > 0 yields the condition ds(x) ky 2 0 so that in the case that the storage function S(x) > 0 is positive definite, it acts as a Lyapunov function and in virtue of Theorem the state converges into the largest positively invariant subset of

19 2.2 Passivity-based feedback control 15 Y 0 = {x R n y = h(x) = 0}. (2.14) The positive invariance goes at hand with the condition that y 0, or equivalently, that y (k) = 0 for all 0 k N 0. Using the notion of the Lie-derivative L f h(x) = h(x) x f(x), Lk f h(x) = Lk 1 f h(x) f(x), x This in turn means that L 0 f h(x) = h(x) y = h(x) = 0 ẏ = L f h(x) = 0 y (2) = L 2 f h(x) = 0. y (n) = L n f h(x) = 0. In vector form this becomes h(x) L f h(x) O(x) =. = 0. L n f h(x) (2.15) The map O(x) is the nonlinear observability map 1 [18, 19], so that in case that the system is completely observable in the sense that this map is invertible it turns out that only the zero vector x = 0 is a solution of (2.15). In linear systems, if the map can be inverted along one trajectory, it can be inverted along any trajectory 2. In nonlinear systems this is not true and it is worth introducing a new concept which corresponds to the invertibility along the solution (and thus uniqueness of this solution) for which y 0. Definition The system (2.13) is called a) zero-state observable if y 0 implies x = 0, i.e. if the solution of O(x) = 0 is uniquely given by x = 0. b) zero-state detectable if y 0 implies x 0. Note that if the system is not zero-state observable but zero-state detectable, then the map O(x) is not invertible, but all solutions x(t) which are mapped by O to the zero vector 0 converge asymptotically to zero. Furthermore, it should be clear, that zero-state observability implies zero-state detectability but not vice versa. On the other hand, looking on the constraint y 0 the notion of the zero dynamics comes into play. The zero dynamics is given by (2.13) with the constraint y 0 and u chosen so that this constraint holds true. To make this point clear, note that if the relative degree [18, 7] of (2.13) is equal to one at x = 0, i.e. there exists a neighborhood N 0 of x = 0 so that L g h(x) 0, x N 0, 1 It can be quickly shown, that for a linear system this map corresponds to the Kalman observability map K o = C CA. CA n 1 2 For linear systems the map (2.15) can actually be written as the Kalman observability matrix K o times the state vector x..

20 16 2 Lyapunov-based design methods there exists a state transformation (in N 0 ) of the form [ [ ] z h(x) = Ψ(x) =, z R, ζ R ζ] n 1, (2.16) Φ(x) that is a diffeomorphism 3, and the dynamics in the new coordinates reads ż = L f h(x) + L g h(x)u, z(0) = z 0 (2.17a) ζ = ϕ(z, ζ, u), ζ(0) = ζ 0 (2.17b) y = z. (2.17c) Actually, it is possible to choose the map Φ(x) so that the vector field ϕ does not depend on the input u, but this does not make a difference at this stage. The zero dynamics is given by ζ = ϕ(0, ζ, u) =: ϕ 0 (ζ), ζ(0) = ζ 0 (2.18a) u = L f h(x) L g h(x) x=ψ 1 0 (2.18b) ζ With these notions and results at hand the following result is a direct consequence of the passivity property and the application of Lyapunov s direct method. Proposition Let (2.13) be passive with positive definite storage function S(x) > 0. Then the following holds true: a) The zero dynamics (2.18) is Lyapunov stable. b) If (2.13) is zero-state detectable then the output-feedback control u = ky asymptotically stabilizes the origin x = 0. Proof: a) Under the assumptions of the proposition, the storage function S(x) > 0 is a Lyapunov function. From the passivity property it follows that for any state trajectory which is a solution of the zero dynamics (2.18) it holds that ds(x) yu = 0 implying Lyapunov stability in virtue of Theorem b) From Theorem it follows that with u = ky the state trajectories x(t) converge into the largest positively invariant subset M Y 0, with Y 0 defined in (2.14). By the zero-state detectability assumption this subset is given by M = {0}. In the following the question is addressed if it is possible to passivate a system using feedback control. For this purpose, recall the state transformation (2.16) along with the dynamics in the new coordinates (2.17). According to the relative degree one property, it follows that using the control u = v L f h(x) L g h(x) (2.19) and introducing the positive semi-definite storage function S(x) = 1 2 z2, the system is passive with respect to the new input v and it holds that ds(x) = zv = yv. 3 A map is a diffeomorphism if it is continuously differentiable and invertible, with continuously differentiable inverse. Such maps conserve geometric and topological properties in state space and are frequently used in the theory of dynamical systems, in particular for control design purposes.

21 2.2 Passivity-based feedback control 17 From this relation alone, it is nevertheless not possible to conclude about the zero dynamics, given that here S(x) 0 is only positive semi-definite. The theory of Lyapunov has been extended to positive semi-definite Lyapunov functions, and is related to the concept of conditional stability, but treating this subject goes beyond the scope of the present notes. The reader can explore this interesting subject e.g. in the seminal work [7] or the related literature. For the purpose at hand, we focus on positive definite storage functions S(x) > 0. The following concept further characterizes systems in dependence of the properties of the zero dynamics (2.18) [13, 7]. Definition Let L g h(0) 0. Then (2.13) is said to be minimum phase if ζ = 0 is a locally asymptotically stable equilibrium point of the zero dynamics (2.18) weakly minimum phase if there exists a positive definite Lyapunov function V 0 (ζ) > 0 defined in a neighborhood N ζ,0 of ζ = 0 that is at least 2 times continuously differentiable and satisfies L ϕ0 V 0 (ζ) 0 for all ζ N ζ,0. Note that the weakly minimum phase property implies the Lyapunov stability of ζ = 0 (see also the discussion in [13]). Having these concepts at hand, the following result is stated without its proof which can be found in the related literature [13, 7] (and directly extends to the MIMO case). Theorem The system (2.13) is locally feedback equivalent to a passive system (i.e. there exists a feedback such that the closed-loop system is passive) with a C 2 positive definite storage function S(x) > 0 if and only if it has relative degree r = 1 at x = 0 and is weakly minimum phase. Accordingly, a system which is feedback eequivalent to a passive system can be stabilized by the feedback control u = ky L f h(x). L g h(x) A direct extension of this result holds for the case that the system is minimum phase. In this case the origin can be asymptotically stabilized using the above control law.

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