Region of attraction approximations for polynomial dynamical systems

Transcription

1 Region of attraction approximations for polynomial dynamical systems Milan Korda EPFL Lausanne Didier Henrion LAAS-CNRS Toulouse & CTU Prague Colin N. Jones EPFL Lausanne

2 Region of Attraction (ROA) ẋ(t) =f (x(t),u(t)), 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 2

3 Region of Attraction (ROA) Long history - typically tackled using non-convex BMIs or gridding 3

4 How is it done? 4

5 Approach Approach ẋ = f (x,u) Common approach for stochastic or chaotic systems The ensembles are modeled using measures.! 5

6 Measures in control x µ 0 µ 0 0 T t 6

7 Measures in control x µ F µ 0 µ F 0 T t 7

8 Measures in control x µ µ 0 µ F µ(d) = R n T 0 I D (t,x(t x 0 )) dt dµ 0 (x 0 ) 0 T t 8

9 Liouville s equation µ 0 µ µ T v v(t,x) dµ F (x) v(0,x) dµ 0 (x) = R n R t + xv f (t,x,u) dµ(t,x,u) n v C 1 ([0,T] X) [0,T ] R n R m Key fact ẋ = f (x,u) Optimization over system trajectories Optimization over measures satisfying Liouville s equation 9

10 ROA characterization 10

11 ROA characterization x x max µ F µ 0 X T x min 0 T t 11

12 ROA characterization µ x µ F x max µ 0 X T µ F 0 T t 12

13 ROA characterization µ 0 x µ µ F x max µ 0 µ F X T x min 1 0 T t 13

14 ROA characterization µ 0 x µ µ F x max µ 0 µ F X T x min 1 0 T t 14

15 ROA characterization µ 0 x µ µ F x max µ 0 µ F X T x min 1 0 T t 15

16 ROA complement characterization µ x µ F x max Finite-dimensional relaxations give outer approximations µ F X T x min 1 0 T t 16

17 ROA complement characterization ẋ = f (x) 17

18 ROA complement characterization x x max µ F µ 0 X T x min 0 T t 18

19 ROA complement characterization µ x µ F x max µ 0 µ F X T x min 0 T t 19

20 ROA complement characterization µ 0 x µ µ F µ 0 µ F X T 1 0 T t 20

21 ROA complement characterization µ 0 x µ µ F µ 0 µ F X T 1 0 T t 21

22 ROA characterization using measures µ 0 x µ µ F µ 0 µ F X T 1 0 T t 22

23 ROA characterization using measures µ x µ F µ F X T 1 0 T t 23

24 Primal LP Infinite dimensional primal LP in the cone of nonnegative measures µ 0 µ 0 1 sup µ 0 (X) s.t. X vdµ F X vdµ 0 = Primal LP [0,T ] X v t + xv fdµ v C 1 µ 0 µ 0 M(X) +, µ F M(X F ) +, µ M([0,T] X) + {T } X T [0,T] X {T } X \ X T 24

25 Dual LP Infinite dimensional dual LP in the cone of nonnegative continuous functions Dual LP Decrease along trajectories inf s.t. X w(x) dx v t + xv f (t,x) 0, (t,x) [0,T] X v(t,x) 0, (t,x) X F w(x) v(0,x)+1, x X w(x) 0, x X, v C 1 ([0,T] X) w C(X) X F = {T } X T X F =[0,T] X {T } X \ X T w I X0 {x w(x) 1} w I X\X0 {x w(x) < 1} 25

26 Finite-dimensional relaxations 26

27 Finite dimensional SDP relaxations SDP! Optimization over measures Optimization over moment sequences Optimization over moment sequences up to degree 2k No gap Optimization over nonnegative continuous functions Optimization over nonnegative polynomials SDP! Optimization over SOS polynomials up to degree 2k No gap 27

28 Convergence w k (x) k w k (x) w k I X0 L 1 vol(k \ ) 0 k := {x w k (x) 1} w k (x) w k I X\X0 L 1 vol( \ k ) 0 k := {x w k (x) < 1} X X 28

29 Convergence w k (x) k w k (x) w k I X0 L 1 vol(k \ ) 0 k := {x w k (x) 1} w k (x) w k I X\X0 L 1 vol( \ k ) 0 k := {x w k (x) < 1} X X 29

30 Convergence w k (x) k w k (x) w k I X0 L 1 vol(k \ ) 0 k := {x w k (x) 1} w k (x) w k I X\X0 L 1 vol( \ k ) 0 k := {x w k (x) < 1} X X 30

31 Convergence w k (x) k w k I X0 L 1 vol(k \ ) 0 k := {x w k (x) 1} w k I X\X0 L 1 vol( \ k ) 0 k := {x w k (x) < 1} 31

32 Convergence µ k 0 µ k 0 X X 32

33 Convergence µ k 0 µ k 0 X X 33

34 Convergence µ k 0 µ k 0 X X 34

35 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 d = 10 d =

36 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 d =

37 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 d =

38 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 d =

39 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 39

40 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X = {x x 2 1.1}, X T = {x x 2 0.5}, T =1 Stable equilibrium at the origin with a bounded region of attraction d =

41 Numerical examples Backward Van der Pol oscillator ẋ 1 = 2x 2 ẋ 2 =0.8x (x )x 2 X = {x x 2 1.1}, X T = {x x 2 0.5}, T =1 Stable equilibrium at the origin with a bounded region of attraction d =

42 Numerical examples Backward Van der Pol oscillator x 1 = 2x2 x 2 = 0.8x1 + 10(x )x2 X = {x!x!2 1.1}, XT = {x!x!2 0.5}, T = 1 Stable equilibrium at the origin with a bounded region of attraction 18 IX\X0 42

43 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 43

44 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 44

45 Extensions Infinite time version Discrete-time version, sampled-data version Controlled inner approximations Robust inner approximations Rational or trigonometric dynamics ROA of optimization-based controllers 45

46 Conclusion Convex characterization of the ROA and its complement SDP relaxations inner & outer approximations Additional properties (e.g. convexity) can be easily enforced Many extensions Easy to use - Gloptipoly, Yalmip, SOSTOOLS, etc. 46

47 Question time 47