Outline Input to state Stability Motivation for Input to State Stability (ISS) ISS Lyapunov function. Stability theorems. M. Sami Fadali Professor EBME University of Nevada, Reno 1 2 Recall: _ Space _ Space: Space of all piecewise continuous functions satisfying _ _ Nonlinear Realization Nonlinear realization Locally Lipschitz in Assumptions guarantee local existence and uniqueness of a solution. 3 4
Questions LTI Realizations Given unforced system asymptotically stable equilibrium Q 1: _ Å _ Å Q2: Is the system state bounded for any bounded input (BIBS)? _ For asymptotically stable, 5 6 Answers for LTI Realizations Example _ Å 2. BIBS system Å (LTI system) Unstable equilibrium State diverges for a bounded input. 30 20 10 0-10 -20-30 -3-2 -1 0 1 2 3 _ Not true for nonlinear systems, in general. 7 8
Functions of Class Class : continuous function with I. II. strictly increasing. Class : continuous function with I. II. III. strictly increasing. as Class Continuous function s.t. I. For fixed, is in class w.r.t.. II. For fixed, is strictly decreasing w.r.t.. III. as 9 10 Local Input to State stability (ISS) is locally ISS if a function, a class function, and constants s.t. _ Global Input to State stability (ISS) is globally ISS if a function, a class function, and constants s.t., _, 11 12
Implications of ISS: Unforced System Unforced system (class ) _ as (class ) is asymptotically stable. Interpretation of ISS Bounded input _, (class ) _ as (class )and = ultimate bound of the system System is ultimately bounded (or globally ultimately bounded) 13 14 Alternative ISS Definition For Useful definition for some proofs. _ ISS Lyapunov Function A continuously differentiable function is an ISS Lypunov function on if if class functions s.t. If then it is an ISS Lypunov function 15 16
Properties of ISS Lyapunov Function Lemma 3.1, p. 80, pos. def. in if & only if Outside of the ball is negative definite along the trajectories of Local ISS Theorem 7.1 If an ISS Lypunov function on, for then it is ISS with _ 17 18 Global ISS Theorem 7.2 If an ISS Lyapunov function on if for then it is ISS with Example Check ISS for the system ISS Lyapunov function candidate _ for / then the system is globally ISS 19 20
Example Check ISS for the system ISS Lyapunov function candidate Example Check ISS for the system ISS Lyapunov function candidate for then the system is globally ISS for 21 then the system is locally ISS 22 ISS Theorems 7.3, 7.4 Theorem 7.3: The system is locally (globally) ISS if and only if an ISS Lyapunov function satisfying the conditions of Theorem 7.1 (Theorem 7.2). Theorem 7.4: The system is locally ISS if (i) is continuously differentiable, and (ii) the autonomous system has an asymptotically stable equilibrium Recall: continuously differential implies locally Lipschitz. Theorem 7.5 The system is locally ISS if (i) is continuously differentiable and globally Lipschitz, and (ii) the autonomous system has an exponentially stable equilibrium 23 24
Theorem 7.6: ISS Lyapunov A continuous function is an ISS Lypunov function on iff class functions s.t. (differential dissipation inequality, storage function ) If and then it is an ISS Lyapunov function Definition 7.2 : Same condition for but alternative Condition Proof: Theorem 7.6 Definition 7.2 As in Definition 7.2 25 26 Proof: Definition 7.2 Theorem 7.6 Case 1: Case 2: define satisfies (i) (ii) not monotone as in Theorem 7.6) 27 ISS and ISS Lyapunov function Dissipation inequality: shows how ISS and ISS Lyapunov function are related. Given s.t. s.t. region bounded by the contour all trajectories that enter never leave it. 28
ISS Pair depends on the composition ISS pair for the system : determines the relationship between the bound on and the bound on ISS pair is not unique. Also called a supply pair. Given the functions Big O Notation Ä Å Ä Å if if 29 30 Theorems: Supply Pair Theorem 7.7: If is a supply pair for the globally ISS system, then a supply pair for the system with Theorem 7.8: If a supply pair for the globally ISS system, then a supply pair for the system with Proof shows how to construct new ISS pairs using and (bounds on and not itself) Cascade Connection Show that two the ISS systems in cascade form an ISS system 31 32
Lemma 7.1 Given the ISS systems and with ISS pairs and, respectively. (i) Define (i) Define is an ISS pair of ISS of Cascade Theorem 7.9: The cascade interconnection of two ISS systems is the ISS system Theorem 7.10: The cascade interconnection of two locally ISS systems is the locally ISS system is an ISS pair of 33 34 Using Lemma 7.1: Proof Define the ISS Lyapunov function for Therefore, as is ISS + 35 Asymptotic Stability Consider Corollary 7.1: If is locally ISS and the equilibrium of is asymptotically stable, then the equilibrium of their cascade,,is locally asymptotically stable. Corollary 7.2: If is locally ISS and the equilibrium of is globally asymptotically stable, then the equilibrium of their cascade,, is globally asymptotically stable. 36