Computation of Local ISS Lyapunov Function Via Linear Programming

Transcription

1 Computation of Local ISS Lyapunov Function Via Linear Programming Huijuan Li Joint work with Robert Baier, Lars Grüne, Sigurdur F. Hafstein and Fabian Wirth Institut für Mathematik, Universität Würzburg

2 Outline 1 Introduction 2 Relation between ISS and Robust Lyapunov Functions 3 Computing Local Robust Lyapunov Functions by Linear Programming 4 Computing Local ISS Lyapunov Functions by Linear Programming 5 Example 6 Conclusion and Future Works

3 Motivation Estimate the domain of attraction of interconnected systems x 1 x 2 x n 1 ẋ 1 = f 1 (x 1,x 2,,x n ), x n. x i R n i, ẋ n = f n (x 1,x 2,,x n ), f i is Lipschitz continuous and f i (0,0,,0) = 0. n n i = N, i=1

4 Motivation Domain of attraction ẋ = f (x). (1) f (0) = 0, f is Lipschitz continuous. If 0 is locally asymptotically stable, then the domain of attraction of 0 is defined as D(0) = { x(0) R N : x(t,x(0)) 0 as t }. Here x(t,x(0)) denotes the solution of system (1).. 0 x(t,x(0))

5 Motivation Estimate the domain of attraction of interconnected systems x 1 x 2 x n 1 ẋ 1 = f 1 (x 1,x 2,,x n ), x n. x i R n i, ẋ n = f n (x 1,x 2,,x n ), n n i = N, i=1 f i is Lipschitz continuous and f i (0,0,,0) = 0. If subsystems are input to state stable(iss), an effective way to estimate the domain of attraction using small gain theorems with subsystems input to state stability (ISS) Lyapunov functions. Question: How to compute local ISS Lyapunov functions for subsystems?

6 Subsystems x j x j x i x j Treating effect of x j (j = i) as perturbation. System with perturbation Assumptions: j = i S i : ẋ i = f i (x 1,x 2,...,x i 1,x i,x i+1,...,x n ). ẋ = f (x,u), x R n,u U R, (2) f : R n R m R n is Lipschitz continuous and f (0,0) = 0. System (2) is locally input to state stable(iss). Notations: U R := B(0,R) R m, admissible input values u U R and the control input functions are u U R := {u : R R m measurable u L R}, u L = ess.sup{ u(t) 2,t 0}.

7 ISS Lyapunov function Definition (ISS Lyapunov function) A smooth function V : D R 0, with D R n open, is a local ISS Lyapunov function of system (2) if there exist ρ 0 > 0, ρ u > 0, K functions ϕ 1, ϕ 2, α and β such that B(0,ρ 0 ) D and ϕ 1 ( x 2 ) V (x) ϕ 2 ( x 2 ), V =< V (x),f (x,u) > α( x 2 ) + β ( u 2 ), for all x 2 ρ 0, u 2 ρ u. If ρ 0 = ρ u = then V is called a global ISS Lyapunov function. K function α( ) β ( ) 0

8 Relation Between ISS and Robust Lyapunov Functions Introduce two Lipschitz continuous functions η 1 : R n R n and η 2 : R m R 0, then consider the auxiliary System ẋ = f η (x,u) := f (x,u) η 1 (x)η 2 (u), u R. x(0) = x 0, f η (0,u) = 0. Theorem If there exists a local robust Lyapunov function V (x) for the auxiliary system and α( ) K such that < V (x),f η (x,u) > α( x 2 ), for all u U R, K > 0 and β ( ) K such that < V (x),η 1 (x) > K (K > 0), and η 2 (u) β ( u 2 ), then V (x) is a local ISS Lyapunov function for the original system. Question: How to compute such a robust Lyapunov function V (x) with η 1 (x) and η 2 (u)?

9 Relation Between ISS and Robust Lyapunov Functions Introduce two Lipschitz continuous functions η 1 : R n R n and η 2 : R m R 0, then consider the auxiliary System ẋ = f η (x,u) := f (x,u) η 1 (x)η 2 (u), u R. x(0) = x 0, f η (0,u) = 0. Theorem If there exists a local robust Lyapunov function V (x) for the auxiliary system and α( ) K such that < V (x),f η (x,u) > α( x 2 ), for all u U R, K > 0 and β ( ) K such that < V (x),η 1 (x) > K (K > 0), and η 2 (u) β ( u 2 ), then V (x) is a local ISS Lyapunov function for the original system. Question: How to compute such a robust Lyapunov function V (x) with η 1 (x) and η 2 (u)?

10 Computing Local Robust Lyapunov Functions by Linear Programming Grids Aim: Compute a robust Lyapunov function V (x) which is linear affine on each simplex. Such a function V (x) is Lipschitz continuous. We divide a compact set Ω R n into N n simplices T = {Γ ν ν = 1,...,N}, Γ ν := co{x 0,x 1,,x n }, h x := diam(γ ν ), diam(γ ν ) := max x y 2. x,y Γ ν We divide a compact set Ω u U R into N u m simplices T u = {Γν u ν = 1,...,N u }, Γν u := co{u 0,u 1,,u m }, h u := diam(γν u ), n or m = 2

11 Clarke s subdifferential for Lipschitz continuous functions How to extend the definition of smooth robust Lyapunov function to definition of piecewise affine robust Lyapunov function i.e. nonsmooth robust Lyapunov function? Proposition (Clarke,1998) For a Lipschitz function V : R n R Clarke s subdifferential satisfies Cl V (x) = co{lim i V (x i ) x i x, V (x i )exists and lim i V (x i ) exists}. V (x) = x,x R Cl V (0) = [ 1,1] 0 x

12 Nonsmooth robust Lyapunov function System ẋ = f (x,u), x R n,u R m, f : R n R m R n is Lipschitz continuous and f (0,u) = 0. Definition (Nonsmooth robust Lyapunov function) A Lipschitz continuous function V : D R 0, with D R n open is said to be a local nonsmooth robust Lyapunov function of system if there exist ρ 0 > 0, ρ u > 0, K functions ϕ 1 and ϕ 2, and a positive definite function α such that B(0,ρ 0 ) D and ϕ 1 ( x 2 ) V (x) ϕ 2 ( x 2 ), sup < ξ,f (x,u) > α( x 2 ), ξ cl V (x), u U R for all x 2 ρ 0, u 2 ρ u. If ρ 0 = ρ u = then V is called a global nonsmooth robust Lyapunov function.

13 Computing local robust Lyapunov functions by linear programming System ẋ = f (x,u), x R n,u R m, Assumptions: f : R n R m R n is Lipschitz continuous and f (0,u) = 0. System is locally asymptotically stable at the origin uniformly in u. Problem How to compute a local robust Lyapunov function V (x) for the system?

14 Linear programming based algorithm for computing piecewise affine Lyapunov functions Previous results of Linear programming based algorithm for computing piecewise affine Lyapunov functions 1 Linear programming based algorithm for computing piecewise affine Lyapunov function was first presented in [Marinósson,2002] for ordinary differential equations. 2 Further developed in [Hafstein,2007] for systems with switching time. 3 Extended to nonlinear differential inclusions in [Baier, Grüne, Hafstein,2012]. * Gives a true Lyapunov function, i.e., not an approximation of a Lyapunov function.

15 Computing local robust Lyapunov functions by linear programming Notation: PL(Ω): the space of continuous functions V : Ω R which are linear affine on each simplex, i.e. V ν := V intγν const, for all Γ ν Ω. System: ẋ = f (x,u), f (0,u) = 0. Assumption: System is locally asymptotically stable at the origin uniformly in u. Aim: compute a local robust Lyapunov function V (x) PL(Ω\B(0,ε))(ε > 0 small enough) satisfying by linear programming. < V (x),f (x,u) > x 2. x(0) x(t,x(0)) B(0, ε) Reason for excluding B(0, ε) < V (x),f (x,u) > x 2.

16 Computing robust Lyapunov functions by linear programming Linear programming based algorithm 1 Let T ε := {Γ ν Γ ν B(0,ε) = /0} T, ε > 0 is small enough. 1. For all vertices x i of Γ ν T ε, introduce V (x i ) as the variables and demand V (x i ) x i 2 = V (x) x 2,x Γ ν T ε. 2. For every Γ ν T ε, introduce the variables C ν,i (i = 1,2,...,n), G and require V ν,i C ν,i G, V ν,i is the i-th component of the vector V ν. 3. For every Γ ν T ε, and every Γ u ν, demand < V ν,f (x i,u j ) > + n k=1 C νk (A x (u,h x ) + A u (x,h u )) }{{} x i 2, compensate for interpolation errors for the points (x,u) with x = x i,u = u j, i = 0,1,...,n, j = 0,1,...,m.

17 Theorem (Result of linear programming based algorithm 1) If x f (x,u) is Lipschitz continuous in x uniformly in u and u f (x,u) is similar for u, and the linear programming problem with the constraints in has a feasible solution, then the values V (x i ) at all the vertices x i of all the simplices Γ ν T ε and the condition V PL(T ε ) uniquely define the function V : Γ ν R, Γ ν T ε satisfying < V ν,f (x,u) > x 2 for x Γ ν and u Γ u ν. Furthermore V is a robust Lyapunov function for system. Problem How to compute a local ISS Lyapunov function V (x) with η 1 (x) and η 2 (u) for the original system by linear programming?

18 Nonsmooth ISS Lyapunov function System ẋ = f (x,u), x R n,u R m, f : R n R m R n is Lipschitz continuous and f (0,0) = 0. Definition (Nonsmooth ISS Lyapunov function) A Lipschitz continuous function V : D R +, with D R n open is said to be a local nonsmooth ISS-Lyapunov function of system if there exist ρ 0 > 0, ρ u > 0, K functions ϕ 1, ϕ 2, α and β such that B(0,ρ 0 ) D and ϕ 1 ( x 2 ) V (x) ϕ 2 ( x 2 ), sup < ξ,f (x,u) > α( x 2 ) + β ( u 2 ), ξ cl V (x), u U R for all x 2 ρ 0, u 2 ρ u. If ρ 0 = ρ u = then V is called a global nonsmooth ISS Lyapunov function.

19 Computing Local ISS Lyapunov Functions by Linear Programming Auxiliary system Ansatz for the original system two Lipschitz continuous functions η 1 : R n R n and η 2 : R m R 0, η 2 (u) = r η 1 (x) = m j=0 Auxiliary system: n i=0 λ i η 1 (x i ), for x = µ j u j 2, for u = m j=0 n i=0 λ i x i Γ ν, n i=0 µ j u j Γ u ν and r 0, ẋ = f η (x,u) := f (x,u) η 1 (x)η 2 (u). λ i = 1, m j=0 µ j = 1.

20 Computing Local ISS Lyapunov Functions by Linear Programming Auxiliary system: ẋ = f η (x,u) := f (x,u) η 1 (x)η 2 (u). Aim: compute a local robust Lyapunov function V (x) PL(Ω\B(0,ε)) for x Γ ν Ω\B(0,ε) satisfying by linear programming K < V (x),η 1 (x) > 1 (K > 0). < V (x),f η (x,u) > x 2. Reasons for excluding B(0, ε) (i) 0 cl V (0) may hold which is contradictory with < V (x),η 1 (x) > 1. (ii) < V (x),f η (x,u) > x 2.

21 Computing Local ISS Lyapunov Functions by Linear Programming Linear programming based algorithm 2 Linear programming problem 1: 1. V (x) is positive definite. The constraints are the same as 1. in linear programming based algorithm V ν,i C ν,i G. The requirements are the same as 2. in linear programming based algorithm For every Γ ν T ε, and every Γ u ν, introduce a nonnegative variable r and demand < V ν,f (x i,u j ) > r u j 2 + Objective function: min{r + G}. n k=1 C νk (A x (u,h x )+A u (x,h u )) x i 2. After getting V PL(T ε ) defined by values V (x i ) at all the vertices x i of all the simplices Γ ν T ε, consider another linear programming problem.

22 Computing Local ISS Lyapunov Functions by Linear Programming Linear programming based algorithm 2 Linear programming problem 2: 4. Introduce a nonnegative variable K and variables η 1,k (x i ) (k = 1,2,...,n). For every Γ ν T ε, demand K < V ν,η 1 (x i ) > 1,i = 1,2,...,n, η 1 (x i ) = (η 1,1 (x i ),η 1,2 (x i ),...,η 1,n (x i )). Objective function: min{k}

23 Result of linear programming based algorithm 2 Consider ẋ = f (x,u), x R n,u U R. (3) Assumptions: f (0,0) = 0, x f (x,u) is Lipschitz continuous in x uniformly in u, and u f (x,u) is similar. Theorem If the assumptions hold and the linear programming problems constructed by the algorithm 2 have feasible solutions, then the values V (x i ) at all the vertices x i of all the simplices Γ ν T ε and the condition V PL(T ε ) uniquely define the function V : Γ ν R. Furthermore Γ ν T ε V (x) is a local ISS Lyapunov function and satisfies < V ν,f (x,u) > x 2 + rk u 2. Remark: If the assumption x u f (x,u) is two times continuously differentiable with respect to x for every fixed u and with respect to u for every fixed x hold, then we get better computation errors.

24 Existence of Solutions of Linear Programming Problems Consider ẋ = f (x,u), x R n,u U R. (4) Assumptions: f (0,0) = 0, x f (x,u) is Lipschitz continuous in x uniformly in u, and u f (x,u) is similar. Theorem If the assumptions hold, and system (4) has a local C 2 ISS Lyapunov function V : Ω R and let ε > 0, then there exist triangulations T ε and T u such that the linear programming problems constructed by the algorithm 2 have feasible solutions and deliver a local ISS Lyapunov function V PL(T ε ).

25 Example Consider system adapted from [Michel, Sarabudla, Miller, 1982] is described by S 1 : { ẋ1 = x 1 [4 (x x 2 2 )] + 0.1u 1, ẋ 2 = x 2 [4 (x x 2 2 )] 0.1u 2, on x = (x 1,x 2 ) [ 1.5,1.5] 2, u = (u 1,u 2 ) [ 0.6,0.6] 2. ε = r = k = V (x) obtained by the algorithm ε = Theoretical V 1 (x) based on V = x 2

26 Difference between V (x) and V 1 (x) Difference between contours of V (x) and V 1 (x) ε = 0.012

27 Conclusion and Future Works Conclusion A new way of computing local ISS Lyapunov functions is given. Future works Consider two linear programming problems of algorithm 2 as one quadratic problem. Consider the problem of computing local ISS Lyapunov functions without introducing auxiliary systems. Estimate the domain of attraction of interconnected systems by small gain theorems with subsystems local ISS Lyapunov functions obtained by linear programming.

28 Thanks for your attention!