Input to state Stability

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1 Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part IV: Applications

2 ISS Consider with solutions ϕ(t, x, w) ẋ(t) = f(x(t), w(t)) The system is called ISS, if there exist β KL and γ K such that for all initial values x, all perturbation functions w and all times t 0 the following inequality holds: ϕ(t, x, w) max{β( x, t), γ( w )} In this part of the course, we will investigate applications and some aspects of ISS controller design Applications: stability of interconnected and discretized systems

3 GAS Cascades Consider ẋ 1 (t) = f 1 (x 1 (t)) ẋ 2 (t) = f 2 (x 2 (t), w(t)) x 1 x 1 w x 2 x 2

4 GAS Cascades Coupling via w = x 1 yields ẋ 1 (t) = f 1 (x 1 (t)) ẋ 2 (t) = f 2 (x 2 (t), x 1 (t)) x 1 x 1 x 2 x 2 Theorem: x 1 GAS + x 2 ISS coupled system GAS

5 GAS Cascades Proof: The proof is easier with ISDS formulation ϕ 1 (t, x 0 1 ) µ 1(σ 1 ( x 0 1 ), t)) ϕ 2 (t, x 0 2, w) max{µ 2(σ 2 ( x 0 2 ), t), ν 2(w, t)} ν 2 (w, t) := ess sup τ [0,t] µ 2 (γ 2 ( w(τ) ), t τ) For w = ϕ 1 we obtain ν 2 (ϕ 1, t) max µ 2(γ 2 (µ 1 (σ 1 ( x 0 1 ), τ), t τ) =: β 2 ( x 0 1, t) KL τ [0,t] Thus ϕ = (ϕ 1, ϕ 2 ) with ϕ = max{ ϕ 1, ϕ 2 } satisfies ϕ(t, x 0 ) max{µ 1 (σ 1 ( x 0 1 ), t)), µ 2(σ 2 ( x 0 2 ), t), β 2( x 0 1, t)}

6 GAS Cascades Note: The function β 2 ( x 0 1, t) := max µ 2(γ 2 (µ 1 (σ 1 ( x 0 1 ), τ), t τ) τ [0,t] takes care of the coupling For it is bounded by with η 2 KLD given by µ 1 = g 1 (µ 1 ) and µ 2 = g 2 (µ 2 ) β 2 (r, t) η 2 (γ 2 σ 1 (r), t) η 2 = max{ g 2 (η 2 ), γ 2 (γ 1 2 (η 2)) g 1 (γ 1 2 (η 2))}

7 Small Gain Theorem Consider ẋ 1 (t) = f 1 (x 1 (t), w 1 (t)) ẋ 2 (t) = f 2 (x 2 (t), w 2 (t)) x 2 w 1 x 1 x 1 x w 2 2

8 Small Gain Theorem Coupling via w 1 = x 2, w 2 = x 1 yields ẋ 1 (t) = f 1 (x 1 (t), x 2 (t)) ẋ 2 (t) = f 2 (x 2 (t), x 1 (t)) x 1 x 1 x 2 x 2 Theorem: x 1 ISS + x 2 ISS + γ 1 (γ 2 (r)) < r coupled system GAS

9 Small Gain Theorem Proof: For θ < 1 we use ISDS in order to analyze ẋ 1 (t) = f 1 (x 1 (t), θx 2 (t)), ẋ 2 (t) = f 2 (x 2 (t), θx 1 (t)) Induction over a suitable time sequence yields GAS with where with η i KLD determined by i = 1, 2, j = 2, 1 ϕ i (t, x 0 ) η i (α i ( x 0 ), t) α i (x) max{σ i ( x i ), γ i σ j ( x j )} η i = max{ g i (η i ), γ i (γ 1 i (η i )) g j (γi 1 (η i ))} By continuity, the estimate also holds for θ = 1

10 Example ẋ 1 = x 1 + w 3 1 /2 and ẋ 2 = x w 2 Using V 1 (x) = V 2 (x) = x we obtain µ 1 (r, t) = e t/4 r, γ 1 (r) = 2r 3 /2 2t+4/r 2 µ 2 (r, t) = t+2/r 2, γ 2(r) = 3 4r/3 the coupled system is asymptotically stable with ϕ i (t, x 0 ) η i (α i (x 0 ), t), where α 1 (x) = max { x 1, 2 3 x 2 3}, α 2 (x) = max { x 2, 5 η 1 = max{ c 1 η 1, c 2 η 3 1 } 3 43 x 1 }, η 2 = max{ c 3 η 2, c 4 η2 3}

11 Small Gain Theorem ISS version ([Jiang/Teel/Praly 94, Teel 95]) Consider ẋ 1 (t) = f 1 (x 1 (t), w 1 (t), v 1 (t)) ẋ 2 (t) = f 2 (x 2 (t), w 2 (t), v 2 (t)) v1 w 1 x 1 x 1 x v 2 w 2 x 2 2 write ISS as ϕ i (t, x, w i, v i ) max{β i ( x, t), γ wi ( w i ), γ vi ( v i )}

12 Small Gain Theorem ISS version ([Jiang/Teel/Praly 94, Teel 95]) Coupling via w 1 = x 2, w 2 = x 1 yields ẋ 1 (t) = f 1 (x 1 (t), x 2 (t), v 1 (t)) ẋ 2 (t) = f 2 (x 2 (t), x 1 (t), v 2 (t)) v 1 v 2 x x 1 1 x 2 x 2 Theorem: x 1 & x 2 ISS + γ w1 (γ w2 (r)) < r coupled system ISS

13 Small Gain Theorems Recall the notion of input-to-output stability (IOS) for systems with output y = h(x): y(t) max{β( x, t), γ( w )} Using IOS, the small gain results are easily extended to systems with output

14 Numerical Discretization All stability concepts are easily extended to compact sets A R n Denoting the Euclidean distance of x R n to A by x A we can, e.g., define A is called (locally) asymptotically stable with neighborhood B and attraction rate β KL, if for all x B ϕ(t, x) A β( x A, t), t 0 Similarly, all the ISS concepts can be generalized

15 Numerical Discretization Goal: find asymptotically stable sets by numerical simulations, e.g., using a one step method x(t + h) = ϕ h (x(t)), with solution trajectories ϕ h (t, x 0 )

16 Numerical Discretization Theorem [Kloeden/Lorenz 86] Let A be an asymptotically stable set for ϕ(t, x) and let be ϕ h an approximation of ϕ by a numerical one step method with ϕ h (h, x) ϕ(h, x) ch q+1 Then ϕ h has numerical asymptotically stable sets Ãh with Hausdorff limit Lim h 0Ãh = A. But: For arbitrary numerical as. stable sets Ãh the limit A =Lim h 0Ãh is not asymptotically stable for ϕ Question: When is A =Lim h 0Ãh as. stable for ϕ? Idea: Interpret ϕ h as perturbed system ẋ = f(x) + w

17 Numerical Discretization Theorem: A = Lim h 0Ãh asymptotically stable the sets Ãh are locally ISS with β h KL, γ h K such that β h β KL and γ h γ K for h 0 the sets Ãh have attraction rates β h KL with β h β KL for h 0

18 Numerical Discretization: Example For x = ( x1 x 2 ) R 2 consider ẋ = ( ) x max{ x 1, 0}x Euler approximation suggests that S 1 = {x R 2 x = 1} 2 y x 2 1 is asymptotically stable 2 In fact, 2 y 1 D 1 = {x R 2 x 1} x 2 1 is the only asymptotically stable set 2

19 Construction of ISS Feedbacks Consider ẋ(t) = f(x(t), u(t), w(t)) w u x x

20 Construction of ISS Feedbacks Find a feedback law u such that is ISS ẋ(t) = f(x(t), u(x(t)), w(t)) w x x u

21 Construction of ISS Feedbacks This is a special case of stabilization via feedback, hence the same obstructions arise: continuous static state feedbacks might not exist (Brockett s condition) no universal design method for the general nonlinear case Here, we will focus on an abstract existence result a design procedure based on an ISS Lyapunov function

22 Sampled solutions How to define solutions for a discontinuous feedback map u : R n U? by sampling: Consider a sampling sequence π = (t i ) i N0, 0 = t 0 < t 1 < t 2 <... with maximal sampling rate (π) := sup i N t i t i 1 < Define sampled solution ϕ π (t, x) recursively for i = 0, 1, 2,... via x i := ϕ π (t i, x), ϕ π (t, x) := ϕ(t t i, x i, u(x i )), t [t i, t i+1 ]

23 Stabilization by discontinuous feedback This framework allows for a general abstract stabilization result Theorem [Clarke/Ledyaev/Sontag/Subbotin 97]: If the system is asymptotically controllable to 0 then there exists a feedback u : R n U such that the sampled system is semiglobally practically asymptotically stable, i.e.: there exists β KL such that for all R, ε > 0 there is δ > 0 with ϕ π (t, x) β( x, t) + ε if x R and (π) δ

24 ISS Stabilization by discontinuous feedback Consider the input affine system ẋ = f(x) + G(x)u + G(x)w Theorem: If the system is asymptotically controllable to 0 for w 0 then there exists a feedback u : R n U such that the sampled system is semiglobally practically ISS, i.e.: there exists β KL, γ K such that for all R, ε > 0 there is δ > 0 with ϕ π (t, x, w) max{β( x, t), γ( w )} + ε if x R, γ( w ) R and (π) δ Idea of Proof: Use control Lyapunov function V and nonsmooth analysis techniques to make V an ISS Lyapunov function

25 ISS Stabilization via universal formula A more constructive approach is based on ISS control Lyapunov functions (ISS clf) for control affine systems ẋ = f(x, w)+g(x)u: A smooth function V : R n R is called an ISS clf (in dissipation form), if there exist α i K, i = 1, 2, 3, 4, such that and α 1 ( x ) V (x) α 2 ( x ) inf DV (x)(f(x, w) + G(x)u) α 3( x ) + α 4 ( w ) u U hold for all x R n, w R m

26 ISS Stabilization via universal formula Given: inf DV (x)(f(x, w) + G(x)u) α 3( x ) + α 4 ( w ) u U Theorem: Consider u(x) = K( ω(x), DV (x)g(x) T ) with K(a, b) := a + a 2 + b 4 b 2 b, b 0 0, b = 0 and ω being a continuous and outside 0 smooth function with ω(x) ω(x) := max w {DV (x)f(x, w) α 4( w )} and assume the small control property, i.e., for small x there exists small u with ω(x) + DV (x)g(x)u α 3 ( x ) Then u is continuous and smooth outside 0 and the closed loop system ẋ = f(x, w) + G(x)u(x) is ISS (in the classical sense)

27 ISS Stabilization via universal formula A similar result is available for integral ISS Proof: Show that V is an ISS Lyapunov function for the closed loop system

28 Summary of Part IV ISS can be used for the stability analysis of cascades and fully interconnected systems ISS can be used for the analysis of numerical discretizations ISS controller design: abstract result and universal formula