Convex computation of the region of attraction for polynomial control systems
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1 Convex computation of the region of attraction for polynomial control systems Didier Henrion LAAS-CNRS Toulouse & CTU Prague Milan Korda EPFL Lausanne
2 Region of Attraction (ROA) ẋ = f (x,u), x(t) X, u(t) U t [0,T] x(t ) X T Region of attraction (ROA) a.k.a. Backward reachable set The set of all initial states that can be admissibly steered to the target set at a given time X X T X 0 2
3 Region of Attraction (ROA) ẋ = f (x,u), x(t) X, u(t) U t [0,T] x(t ) X T 3
4 Region of Attraction (ROA) Fundamentally difficult to determine!! Long history - typically tackled using non-convex BMIs or gridding Convex formulation Infinite dimensional LP formulation for ROA computation Converging hierarchy of SDP relaxations providing outer approximations! Readily modeled using freely available tools (Gloptipoly, Yalmip, etc.)! No initialization data required! 4
5 How is it done? 5
6 Approach Approach Study how ensembles of initial conditions evolve, not single trajectories Common approach for stochastic or chaotic systems How to model these ensembles? Using measures.! 6
7 Measures Mappings from sets to nonnegative real numbers Measures µ R + Integration A g(x) dµ(x) g(x) dµ(x) = A A g(x) ρ(x)dx (x) 1 ρ = µ 7
8 Measures in control µ 0 µ T T µ (t,x(t),u(t)) [0,T] X U x µ T µ 0 0 T t 8
9 Liouville s equation µ 0 µ µ T X v(t,x) dµ T (x) v C 1 ([0,T] X) X v(0,x) dµ 0 (x) = [0,T ] X U v t + xv f Lv(t,x,u) dµ(t,x,u) Key fact ẋ = f (x,u) Optimization over system trajectories Optimization over measures satisfying Liouville s equation 9
10 Characterization of ROA using measures 10
11 Dynamics Dynamics Liouville s equation 11
12 Constraints Constraints Support constraints Support of a measure Smallest closed set whose complement has zero measure spt µ x(0) X (t,x(t),u(t)) [0,T] X U x(t ) X T spt µ 0 X spt µ [0,T] X U spt µ T X T 12
13 Characterization of ROA using measures Need to µ 0 Non-convex 13
14 Convex characterization of ROA Key idea µ 0 µ 0 X 0 1 X 0 X 14
15 Convex characterization of ROA Key idea µ 0 µ 0 X 0 1 X 0 X 15
16 Convex characterization of ROA Key idea µ 0 µ 0 X 0 1 X 0 X 16
17 Convex characterization of ROA Key idea µ 0 µ 0 X 0 1 X 0 X 17
18 Convex characterization of ROA Key idea µ 0 µ 0 X 0 1 X 0 X 18
19 Primal LP The ROA is characterized by the optimization problem µ 0 sup µ 0 (X) Primal LP s.t. X T vdµ T X vdµ 0 = [0,T ] X U Lvdµ v C 1 µ 0 spt µ [0,T] X U, spt µ 0 X, spt µ T X T Infinite dimensional linear program in the cone of nonnegative measures 19
20 Dual LP on continuous functions Decrease along trajectories inf X w(x) dx Dual LP s.t. Lv(t,x,u) 0, (t,x,u) [0,T] X U v(t,x) 0, x X T w(x) v(0,x)+1, x X w(x) 0, x X, v C 1 ([0,T] X) w C(X) Key observation w I X0 {x w(x) 1} X 0 w 20
21 Finite-dimensional relaxations 21
22 Big picture Infinite dimensional Infinite dimensional 2k 2k k k 22
23 Convergence results Convergence of primal and dual relaxations Optimal values of the primal and dual SDPs converge to the optimal value of the two infinite dimensional LPs, which is equal to the volume of the ROA X 0 w k (x) k Functional convergence w k I X0 L 1 min i k w i I X0 k X 0k := {x w k (x) 1} Set-wise convergence X 0k X 0 volume(x 0k \ X 0 ) 0 k 23
24 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X =[ 1.2, 1.2] 2 X T = {x x }, T = 100 Stable equilibrium at the origin with a bounded region of attraction X d = 10 d =
25 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X =[ 1.2, 1.2] 2 X T = {x x }, T = 100 Stable equilibrium at the origin with a bounded region of attraction X d =
26 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X =[ 1.2, 1.2] 2 X T = {x x }, T = 100 Stable equilibrium at the origin with a bounded region of attraction X d =
27 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X =[ 1.2, 1.2] 2 X T = {x x }, T = 100 Stable equilibrium at the origin with a bounded region of attraction X d =
28 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X =[ 1.2, 1.2] 2 X T = {x x }, T = 100 Stable equilibrium at the origin with a bounded region of attraction 18 I X0 28
29 Numerical examples Brockett integrator ẋ 1 = u 1 ẋ 2 = u 2 ẋ 3 = u 1 x 2 u 2 x 1 X = {x x 1} U = {u u 2 1} X T = {0}, T =1 ROA known semi-analytically d =6 29
30 Numerical examples Brockett integrator ẋ 1 = u 1 ẋ 2 = u 2 ẋ 3 = u 1 x 2 u 2 x 1 X = {x x 1} U = {u u 2 1} X T = {0}, T =1 ROA known semi-analytically d = 10 30
31 More examples Examples from robotics + control law extraction: [A. Majumdar, et al. Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures, 2013] 31
32 Computational issues Need to solve large SDPs Conditioning has secondary effect ``Medium scale only Conditioning is very important Large scale Monomial basis Chebyshev basis bad conditioning better conditioning 32
33 Conclusion Convex characterization of the ROA SDP relaxations converging outer approximations Additional properties (e.g. convexity) can be easily enforced Covers a broad class of systems Easy modeling using Gloptipoly, Yalmip, SOSTOOLS, etc. Extremely simple to use! 33
34 Question time 34
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