Input-to-state stability and interconnected Systems

Transcription

1 10th Elgersburg School Day 1 Input-to-state stability and interconnected Systems Sergey Dashkovskiy Universität Würzburg Elgersburg, March 5, /20

2 Introduction Consider Solution: ẋ := dx dt = ax, x, a R, x(0) = x 0 x(t) = x 0 e at x a > 0 x a < t t If a = 0 then any solution is constant: x(t) = x 0 2/20

3 ẋ 1 = f 1 (t, x 1,..., x n )... ẋ n = f n (t, x 1,..., x n ) Definition ẋ = f (t, x), t [t 0, ) f continuous and locally Lipschitz in x! A solution η is called stable, if ε > 0 δ = δ(ε, t 0 ) s.t. Each solution ξ with ξ(t 0 ) η(t 0 ) < δ is defined on [t 0, ] and ξ(t) η(t) < ε t t 0 Otherwise η is called unstable. x ε ε η ξ 3/20 t t 0

4 Definition η is called asymptotically stable for ẋ = f (t, x), if η is stable and η is attractive, that is = (t 0 ) : ξ(t 0 ) η(t 0 ) < lim t ξ(t) η(t) = 0. If =, then η is called globally asymptotically stable (GAS). For linear systems ẋ(t) = A(t)x(t) + f (t) Stability properties can be checked by linear algebra tools. Attractivity implies stability Stability implies global stability Attractivity implies global attractivity 4/20

5 Let f, a ij : [t 0, ) R be continuous and A = (a ij ) i,j=1,...,n Theorem η of ẋ = Ax + f is stable x 0 0 of ẋ = Ax is stable Theorem η of ẋ = Ax + f is asymptotically stable (AS) x 0 0 of ẋ = Ax is asymptotically stable. Remark Here global and local properties coincide: AS GAS 5/20

6 ẋ = t x, x(0) = x 0 ẋ = x, x(0) = x 0 x(t) = t 1 + (x 0 + 1)e t x(t) = x 0 e t x x ẋ = t x ẋ = x t t 6/20

7 Corollary 1: One solution of ẋ = Ax + f is (asymptotically) stable all solutions are (asymptotically) stable. Definition ẋ = Ax + f is called (asymptotically) stable, if all its solutions are (asymptoically) stable. Corollary 2: ẋ = Ax + f (asympt.) stable ẋ = Ax (asympt.) stable Remark: Stability properties depend on A : [t 0, ) R n n only. 7/20

8 Consider ẋ = Ax, with A : [t 0, ) R n n continuous. Let X (t) = (x jk ) j,k=1,...,n be the fundamental matrix-solution, i.e. X (t 0 ) = I, columns = linear independent solutions Theorem ẋ = Ax is stable all solutions are bounded on [t 0, ). Theorem ẋ = Ax is AS all solutions satisfy lim t x(t) = 0 Remark: This is not true for nonlinear systems. 8/20

9 Nonlinear example { ẋ 1 = x 1 t t 2 x 1 x 2 2 ẋ 2 = x, t 2 0 = 1 t { x 1 = x 1 (t 0 ) t e x2 2 (t 0)(t 1) General solution x 2 = x 2(t 0 ) t Let δ > 0, x 1 (t 0 ) = δ 2, x 2 (t 0 ) = δ, then lim t x(t) = 0 x 1 (1 + 1 δ 2 ) > 1 e x ( ) 0 is not stable 0 9/20

10 Let A R n n be constant, with eigenvalues λ 1,..., λ n and Jordan-NF CAC 1 = diag(j 1 (λ 1 ),..., J m (λ m )), m n λ 1,..., λ m eigenvalues of Jordan-blocks J 1 (λ 1 ),..., J m (λ m ) Any solution of ẋ = Ax can be written in the form m x(t) = e λ k P k (t), deg(p k ) < Rank J k (λ k ) k=1 Theorem ẋ = Ax stable { Reλj 0, j = 1,..., n Reλ k = 0 alg. mult = geom. mult 10/20

11 Motivation For mathematical pendulum modeled by ϕ + g l sin ϕ = 0 x 1 :=ϕ { ẋ1 = x 2 ẋ 2 = g l sin x 1 we do not have explicit solutions. Eigenvalues make no sense. The energy is E = E E(x 1, x 2 ) = m(lx 2) 2 + mg(l l cos x 1 ) 2 ) ( (ẋ1 mgl sin x1 = ẋ 2 ml 2 x 2 ) ( x2 ) g = 0 l sin x 1 If x(0) < δ 1 then E = const is small and the values x 1, x 2 are small, hence x(t) is small for all t x 0 is stable. 11/20

12 Intuition Stability: The energy cannot grow Asymptotic stability: The energy decays Why energy? Lyapunov function energy 1880 ies In case of systems with inputs: dissipativity and storage function (J.C. Willems 1972) A particular case: input-to-state stability (ISS) by E. Sontag /20

13 Autonomous nonlinear system Consider ẋ = f (x), f : D R n continuous and f (0) = 0 so that x(t) 0 is a solution (equilibrium). Definition x 0 is called stable, if ε > 0 δ = δ(ε) s.t. for any solution x with x(0) < δ it follows x(t) < ε t 0 if additionally > 0 : x(0) < lim t x(t) = 0, then x(t) 0 is called locally asymptotically stable (LAS) if = then x(t) 0 is called globally AS (GAS) How to check these properties? 13/20

14 Lyapunov-Function (LF) Definition V : D R, V C 1 (D) is called LF for ẋ = f (x), if V (0) = 0, V (x) > 0 for x 0 and V (x) := V f (x) 0 in D Theorem (Lyapunov 1892) Let f C(D), f (0) = 0. Let V be an LF for ẋ = f (x). Then 1. V 0 in D = x = 0 is stable 2. V < 0 in D \ {0} = x = 0 is LAS Remark: Instead of smooth V one can use also Lipschitz V. 14/20

15 Remarks on LF LF is not uniquely defined. If V is a LF then 2V and V 2 are LFs A converse theorem can be proved: If x = 0 is a (globally) stable equilibrium for ẋ = f (x) then an (unbounded) LF For linear systems ẋ = Ax there is a method to construct a quadratic LF explicitely Another characterization of GAS in terms of comparison function is possible Let A be Hurwitz (all eigenvalues in C ). Take any positive definite matrix Q. Then there always P such that A T P + PA = Q and V (x) = x T Px is a LF for ẋ = Ax. 15/20

16 Suspension shock absorber F = ma ẍ = µẋ kx = 0 (m = 1) Denote x 1 = x and x 2 = ẋ then we get { ( ẋ1 = x 2 0 k ẋ = Ax, A = ẋ 2 = kx 1 µx 2 1 µ Energy: V (x) = V (x 1, x 2 ) = x2 2/2 + kx 1 2/2 V (x) = (kx 1, x 2 ) V (x) = V Ax = kx 1 x 2 kx 1 x 2 µx 2 2 = µx stability (but not GAS). GAS to be proved in the exercises. ) 16/20

17 For short denote R + := [0, ) and consider classes of continuous functions: α : R + R + is positive definite if α(r) = 0 r = 0 α : R + R + is of class K if it is pos.def. and stric. increasing α : R + R + is of klass K if α K and lim r α(r) = α : R + R + is of class L if it is decreasing and lim r α(r) = 0 β : R + R + R + if β(, t) K and β(r, ) L K K L-function pdf 17/20

18 Theorem ẋ = f (t, x) is GAS β KL such that for any ϕ 0 R N for the corresponding solution ϕ = ϕ(t, t 0, ϕ 0 ) it holds that x(t, t 0, x 0 ) β( x 0, t t 0 ), t t 0. Remark: For ẋ = f (x) we write x(t, x 0 ) β( x 0, t), t 0. Definition (Sontag 1989) ẋ(t) = f (x(t), u(t)) is ISS : β KL γ K x(0) R n u L x(t) β( x(0), t) + γ( u ), t 0 18/20

19 Useful properties for proving ISS and derivations of gains γ K γ 1 K γ 1, γ 2 K γ 1 +γ 2, γ 1 γ 2, γ 1 γ 2, max{γ 1, γ 2, } K The next two are weak triangle type inequalities: γ K, a, b R + and ρ K such that ρ id K it holds γ(a + b) γ ρ(a) + γ ρ (ρ id) 1 (b) γ(a + b) γ (id + ρ)(a) + γ (id + ρ 1 )(b) for any ρ K. a + b max{(id + ρ)(a), (id + ρ 1 )(b)} for any ρ K. γ 1 γ 2 (r) < r r > 0 γ 2 γ 1 (r) < r r > 0 γ 1, γ 2 K, β KL γ 1 (β(γ 2 ( ), )) KL β KL α 1, α 2 K β(r, t) α 2 (α 1 (r)e t ), s, t 0 γ K α 1, α 2 K γ(rs) α 1 (r)α 2 (s), r, s 0 γ K α 1, α 2 K with id + γ = (id + α 1 ) (id + α 2 ) γ K γ K with (id + γ) 1 = id γ 19/20

20 And some more properties The set K is in fact the set of all homeomorphisms of [0, ). The pair (K, ) is a non-commutative group. For example α 1 (r) = r 2 and α 2 (r) = e r 1 do not commute. Cyclic permutations: γ 1 γ 2 γ n = id γ n γ 1 γ n 1 = id For any n N we denote γ n = γ γ γ n times. For any α K and any k N γ K such that α = γ k (this γ is not always unique.) We can continue with these properties (also for KL-functions). See more in A compendium of comparison function results by C.M. Kellett /20