Computation of CPA Lyapunov functions

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1 Computation of CPA Lyapunov functions (CPA = continuous and piecewise affine) Sigurdur Hafstein Reykjavik University, Iceland 17. July 2013 Workshop on Algorithms for Dynamical Systems and Lyapunov Functions Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

2 Motivation CPA method: Numerical algorithm to compute a Lyapunov function on a compact domain for a systems with a stable equilibrium. To concretize the method we consider a system: ẋ = f(x), where f C 2 (R 2, R 2 ) f(0) = 0, i.e. equilibrium at the origin D R 2 a compact neighbourhood of the origin We pursue computing a functional V such that 1 V : D R is continuous 2 V (0) = 0 and V (x) x 2 for all x D (minimum at the origin) 3 D + V (x + hf(x)) V (x) f V (x) := lim sup x 2 for all x D h 0+ h (decreasing along solution trajectories) Note: Works exactly the same for f C 2 (R n, R n ) for n 2. Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

make LP problem: fix the domain and triangulation Always start with a simple standard triangulation, where the vertices have integer coordinates.

3 make LP problem: fix the domain and triangulation Always start with a simple standard triangulation, where the vertices have integer coordinates. Then map it to the desired triangulation, here with F(x) = ρ( x ) x x 2 x = 0.01 x 2 x 2 x F = FEM: shape regular triangulation; elsewhere: simplicial complex The vertices are mapped by F, not the simplices co{x 0, x 1, x 2 } = co{f(x 0 ), F(x 1 ), F(x 2 )} Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

make LP problem: variables Variables of the LP problem are V x for every vertex x of every triangle and C ν,1, C ν,2 for every triangle S ν V x is the value of the to be computed CPA Lyapunov

4 make LP problem: variables Variables of the LP problem are V x for every vertex x of every triangle and C ν,1, C ν,2 for every triangle S ν V x is the value of the to be computed CPA Lyapunov function V at the vertex x and C ν,j is an upper bound on the j-th component of its gradient V ν on the triangle S ν Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

5 make LP problem: enforce V (x) x 2 Every x S ν := co{x 0, x 1, x 2 } can be written uniquely as a convex combination of the vertices x = λ x i x i, λ x i 0, λ x i = 1 We define the CPA function V at x as the same convex combination of the values of the variables V x0, V x1, V x2 : V (x) := λ x i V xi To enforce V to have a minimum at the origin we include the constraints: V 0 = 0 and V x x 2 for all vertices x of all triangles Then V (0) := V 0 = 0 and x 2 = λ x i x i 2 λ x i x i 2 = Note: The origin 0 must be a vertex λ x i V xi = V (x) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

6 make LP problem: enforce V (x) f(x) x 2 Orbital derivative of V along the solutions to ẋ = f(x) is enforced to be decreasing by: For every triangle/simplex S ν := co{x 0, x 1, x 2 } and i = 0, 1, 2: x i 2 V ν f(x i ) + E ν,i V ν 1 Here ) E ν,i := B ν x i x 0 2 (max x j x x i x 0 2 j=1,2 where B ν is an upper bound B ν max max 2 f m m,r,s=1,2 z S ν (z) x r x s These constraints are implemented in two steps Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

7 make LP problem: enforce V ν 1 j C ν,j Gradient V ν of V on S ν : V (x) = λ x i V xi = V ν (x x 0 ) + V x0 V ν := X 1 ν ( Vx1 V x0 ), where X ν := V x2 V x0 Constraints (linear in the variables) ( (x1 x 0 ) T ) (x 2 x 0 ) T C ν,j ( V ν ) j C ν,j for all triangles S ν and j = 1, 2 Then V ν 1 j=1 C ν,j Note: X ν depends solely on the geometry of the triangle/simplex S ν. Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

8 make LP problem: enforce V (x) f(x) x 2 Orbital derivative of V along the solutions to ẋ = f(x) is enforced to be decreasing by: For every triangle/simplex S ν := co{x 0, x 1, x 2 } and i = 0, 1, 2: Here x i 2 V ν f(x i ) + E ν,i j=1 C ν,j ) E ν,i := B ν x i x 0 2 (max x j x x i x 0 2 j=1,2 where B ν is a constant fulfilling B ν max max 2 f m m,r,s=1,2 z S ν (z) x r x s The B ν are the only nontrivial inputs to the CPA method The B ν are upper bounds and do not have to be tight Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

9 Implications of the constraints Orbital derivative of V along the solutions to ẋ = f(x) decreasing: For x = 2 i=1 λx i x i S ν = co{x 0, x 1, x 2 } we have ( V ν f(x) = λ x i [ V ν f(x i )] + V ν f(x) i=1 i=1 ) λ x i f(x i ) i=1 λ x i [ V ν f(x i )] + V ν }{{} 1 f(x) C ν,1 +C ν,2 λ x i f(x i ) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

10 Lemma Let S := co{x 0, x 1,..., x n } R n be an n-simplex and g C 2 (S, R). Then, for every x = n λx i x i (convex combination) we have n g(x) λ x n i g(x i ) λ x i E g i, where and E g i ( ) nbg := 2 x i x 0 2 max x j x x i x 0 2 j=1,2,...,n B g := max max 2 g r,s=1,2,...,n z S (z) x r x s Implies: f(x) λ x i f(x i ) λ x i E ν,i Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

11 Implications of the constraints Orbital derivative of V along the solutions to ẋ = f(x) decreasing: For x = 2 λx i x i S ν = co{x 0, x 1, x 2 } we have V ν f(x) λ x i [ V ν f(x i )] + V ν 1 f(x) λ x i V ν f(x i ) + E ν,i λ x i x i 2 j=1 C ν,j λ x i f(x i ) λ x i x 2 i x 2 Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

12 How to select x 0 in co{x 0, x 1, x 2 } For every triange/simplex S ν = co{x 0, x 1, x 2 } the vertex x 0 serves as a kind of a reference point in X ν and ) E ν,i := B ν x i x 0 2 (max x j x x i x 0 2 j=1,2 If 0 / S ν the vertex x 0 is arbitrary. If 0 S ν we must take x 0 = 0 because for x i = 0 the constraints x i 2 V ν f(x i ) + E ν,i 0 E ν,i C ν,j, j=1 j=1 C ν,j reduce to which is true if and only if i = 0. Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

13 Solution to the LP problem = CPA Lyapunov function It now follows that if the LP problem has a solution, then the CPA function V, defined at x = 2 λx i x i S ν := co{x 0, x 1, x 2 }, by V (x) = λ x i V xi fufills: V is continuous and affine on any triangle/simplex S ν V (0) = 0 and V (x) x 2 for all x ν S ν D + f V (x) min V ν f(x) x 2 for all x ( ν S ν ) x S ν V is a Lyapunov function for the system ẋ = f(x) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

14 CPA method: Example n = 2 ) ( (ẋ1 System = ẋ 2 x 2 x 1 + x 3 1 /3 x 2 ), B ν = 2 max x S ν x 1 solution to the LP problem, the V x same triangulation as before Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

15 CPA method: Example n = 2 ) ( (ẋ1 = ẋ 2 x 2 x 1 + x 3 1 /3 x 2 ), B ν = 2 max x S ν x 1 convex interpolation of the values of the V x delivers a CPA Lyapunov function Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

16 CPA method: Example n = 3 ẋ x x 1x x x 2 ẋ 2 = 0.1x 2 x x x 3, B ν = 2.29 ẋ x x 2x x x 3 level set of a computed CPA Lyapunov function Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

17 Sufficiency of the CPA method CPA method without the error term, i.e. x i 2 V ν f(x i ) delivers an approximation to a Lyapunov function (Julian 1999; Julian, Guivant, Desages 1999). Might be a Lyapunov function but a posteriori analysis needed. CPA method with the error term delivers a true Lyapunov function and not an approximation (Marinosson=Hafstein 2002). x i 2 V ν f(x i )+E ν,i j C ν,j Unusual of numerical methods to deliver exact results, usually deliver approximations Takes advantage of V (x) f(x) < 0 being an inequality (first order partial differential inequality) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

18 Does the CPA method always work? (1) What about necessity, i.e. if there exists a Lyapunov function can the CPA method always compute one? Previous results: If an arbitrary small neighbourhood of the origin is excluded from the domain and the equilibrium is exponentially stable (Hafstein 2004) or asymptotically stable (Hafstein 2005) then the LP problem has a feasible solution if the triangles/simplices are regularly shaped and small enough (diam(s ν ) X 1 ν 1 is bounded and diam(s ν ) 0). Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

19 Does the CPA method always work? (2) Proof: Assign values to V x, C ν,i such that the constraints are fulfilled, algorithms find feasible solutions if there are any. Let D be a compact subset of the basin of attraction and let N D be a (small) open neighbourhood of the origin. There exists a Lyapunov function W C (D, R) such that W (x) x 2 and W (x) f(x) 2 x 2 on D \ N. ( ) Assign V x := W (x) and C ν,j = e T j Xν 1 Vx1 V x0 V x2 V x0 V x x 2 and C ν,j ( V ν ) j C ν,j are trivially fulfilled. x i 2 V ν f(x i ) + E ν,i j C ν,j are fulfilled because V ν W (x 0 ) 1 A X 1 ν 1 diam(s ν ) 2, A = bound on the second order derivatives of W on D \ N (compact) X 1 ν 1 diam(s ν ) bounded and diam(s ν ) 0 Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

20 The CPA method always works Newer and better results (Giesl, Hafstein 2010, 2012, 2013?): An arbitrary neighbourhood does not have to be excluded if one uses more advanced triangulations Idea of proof: There exists a Lyapunov function W similar to before, but W (x) = Q 1 2 x 2 close to the origin, where Q > 0 is the solution to the Lyapunov equation J T Q + QJ = I, J := Df(0) Lyapunov functions can be agglutinated in a certain way (Giesl 2007) Use W to assign values to the variables V x, C ν,i of LP problems made for a sequence of ever refined triangulations Several challenges: second order derivatives of W diverge at the origin, diam(s ν ) Xν 1 1 is not bounded close to the origin, etc. Much more difficult to prove Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

21 Z More advanced triangulations More advanced triangulations means fan like triangulations at the origin schematic figures for n = 2 and n = Y X Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

22 CPA method: constructive and decidable? If a system possesses a Lyapunov function V : D R, D R n compact, then the CPA method can compute one in a finite number of steps (constructive) What if there does not exist a Lyapunov function V : D R? LP problem has no feasible solution triangulation has to little structure to support a CPA Lyapunov function CPA is not decidable but: CPA Lyapunov function exists exponential stability D compact exponential stability is a local property local exponential stability can be checked directly (eigenvalues) use CPA method to compute Lyapunov functions with larger domains than the standard quadratic ones Given D and α, M > 0, there is a simpler LP problem that can foreclose φ(t, ξ) Me αt ξ for all ξ D, t > 0 (Marinosson=Hafstein 2002) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

23 CPA method: Evaluation Pros: True Lyapunov functions are computed (exact method) No a posteriori analysis needed Works in n dimensions and for general nonlinear systems Low demands on regularity of f in ẋ = f(x) (piecewise C 2 ) Constructive, i.e. always works Flexible and extendable to different types of systems Neutral: Speed ca. O([ variables] 4 ) with simplex method Contras: Triangulation somewhat cumbersome, particularly local refinements CPA Lyapunov functions not smooth No usual formula for the computed Lyapunov function Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

24 CPA method: Extensions Published: (nonautonomous) Arbitrary switched systems (Hafstein 2007) Differential inclusions (Baier, Grüne, Hafstein 2012) Contraction metrics for periodic orbits (Giesl, Hafstein 2012) semidefinite optimization problem In progress: ISS Lyapunov functions (Baier, Grüne, Hafstein, Li, Wirth) quadratic optimization problem Discrete systems and difference inclusions (Giesl, Hafstein) Finite time systems (Giesl, Hafstein). theoretic preparation for CPA method published talk on this subject in the workshop Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

25 CPA method: Future work Publish basic C++ code for the CPA method (Hafstein, est. end 2013) implemented in Visual Studio Express (Windows), GLPK used to solve the LP problems, use of Scilab and Gnuplot (all freeware) Publish more advanced and user friendly C++/Matlab/Scilab code for the CPA method (Björnsson, Hafstein) Verify computed approximations of complete Lyapunov functions (Björnsson, Giesl, Grüne, Hafstein) Combine the advantages of the CPA method and the RBF method to compute CPA Lyapunov functions fast(er) (Giesl, Hafstein) talk on this subject in the workshop Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

26 References Julian, A high level canonical piecewise linear representation: Theory and applications, Ph.D. Thesis: Universidad Nacional del Sur, Bahia Blanca, Argentina, Julian, Guivant, and Desages, A parametrization of piecewise linear Lyapunov function via linear programming, Int. Journal of Control, 72 (1999), Marinosson, Stability analysis of nonlinear systems with linear programming: A Lyapunov functions based approach, Ph.D. Thesis: Gerhard-Mercator-University, Duisburg, Germany, Marinosson, Lyapunov function construction for ordinary differential equations with linear programming. Dynamical Systems, 17 (2002), Hafstein: A constructive converse Lyapunov theorem on exponential stability. Discrete Contin. Dyn. Syst. 10 (2004), Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations. Dynamical Systems, 20 (2005), Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics 1904, Springer, Hafstein, An algorithm for constructing Lyapunov functions, Electron. J. Differential Equ. Monogr., 8 (2007). Giesl and Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions. J. Math. Anal. Appl., 371 (2010), Baier, Grüne, and Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), Giesl and Hafstein, Construction of Lyapunov functions for nonlinear planar systems by linear programming, J. Math. Anal. Appl., 388 (2012), Giesl and Hafstein, Existence of piecewise linear Lyapunov functions in arbitary dimensions, Discrete Contin. Dyn. Syst., 32 (2012), Giesl and Hafstein: Revised CPA method to compute Lyapunov functions for nonlinear systems, submitted Giesl and Hafstein: Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Analysis 86, (2013), Giesl and Hafstein. Local Lyapunov Functions for periodic and finite-time ODEs. In: Recent Trends in Dynamical Systems, eds. A. Johann, H.Kruse, F. Rupp, and S. Schmitz, Springer Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

27 RBF-CPA: Fast approximation, fast verification (1) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

28 RBF-CPA: Fast approximation, fast verification (2) Sigurdur Hafstein (Reykjavik University) Computation of CPA Lyapunov functions Reykjavik, 17. July / 28

Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction

Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction Jóhann Björnsson 1, Peter Giesl 2, Sigurdur Hafstein 1, Christopher M. Kellett