Optimized Robust Control Invariant Tubes. & Robust Model Predictive Control

Transcription

1 Optimized Robust Control Invariant Tubes & Robust Model Predictive Control S. V. Raković Imperial College, London Imperial College, 17. November 2004 Imperial College ICM meeting Talk, 17. November 2004 p. 1/36

2 0 Contents & Problem Description 0 Contents & Problem Description Imperial College ICM meeting Talk, 17. November 2004 p. 2/36

3 Contents Robust Control Invariance 1 Set Robust Control Invariance 2 Robust Control Invariant Tubes 3 Robust Model Predictive Control (RMPC) 4 Brief Comments on Extensions 5 Imperfect State Information Case 5 Parametrically Uncertain Linear Discrete Time Systems 5 Piecewise Affine Discrete Time Systems 5 Conclusions 6 Imperial College ICM meeting Talk, 17. November 2004 p. 3/36

4 Problem Description System: x + = Ax + Bu + w, (A, B) controllable, Constraints: (x, u, w) X U W, X, U, W polyhedral and each contains the origin (wlog), in its interior. Control Objectives: Robust Constraint Satisfaction, Robust Asymptotic/Exponential Stability of an appropriate set, Optimization of certain performance criteria. Control Tools: Set Invariance, Model Predictive Control. Imperial College ICM meeting Talk, 17. November 2004 p. 4/36

5 1 Robust Control Invariance 1 Robust Control Invariance Imperial College ICM meeting Talk, 17. November 2004 p. 5/36

6 Set Invariance Definition 1 A set Ω R n is a robust control invariant (RCI) set for system x + = Ax + Bu + w and constraint set (X, U, W) if Ω X and for every x Ω there exists a u U such that Ax + Bu + w Ω, w W. {Ax + Bu 2 + w w W} Ω, u 2 U {Ax + Bu 1 + w w W} Ω, u 1 U x Ω Ω X Imperial College ICM meeting Talk, 17. November 2004 p. 6/36

7 Set Invariance Definition 1 A set Ω R n is a robust control invariant (RCI) set for system x + = Ax + Bu + w and constraint set (X, U, W) if Ω X and for every x Ω there exists a u U such that Ax + Bu + w Ω, w W. u U s.t. {Ax + Bu + w w W} Ω, x Ω Ω X Imperial College ICM meeting Talk, 17. November 2004 p. 6/36

8 Set Invariance Definition 2 A set Ω R n is a robust positively invariant (RPI) set for system x + = Ax + Bν(x) + w and constraint set (X ν, W) if Ω X ν Ax + Bν(x) + w Ω, w W for every x Ω. (X ν X {x ν(x) U}). {Ax + Bν(x) + w w W} Ω x Ω Ω X ν Imperial College ICM meeting Talk, 17. November 2004 p. 6/36

9 Set Invariance Definition 2 A set Ω R n is a robust positively invariant (RPI) set for system x + = Ax + Bν(x) + w and constraint set (X ν, W) if Ω X ν Ax + Bν(x) + w Ω, w W for every x Ω. (X ν X {x ν(x) U}). {Ax + Bν(x) + w x Ω, w W} Ω Ω X ν Imperial College ICM meeting Talk, 17. November 2004 p. 6/36

10 Set Invariance Definition of Control Invariant and Positively Invariant Sets same with W {0} Set Invariance has been studied by many authors, some of them: Bertsekas, Sonnevend, Kolmanovsky, Gilbert, Lasserre, Blanchini, De-Santis, Aubin, Kerrigan, Kouramas & Many Others Set Invariance Issues: Efficient Set Computations, Finite Time Computations. Our contributions: Invariant approximations of RPI sets, Optimized Robust Control Invariance. Imperial College ICM meeting Talk, 17. November 2004 p. 6/36

11 The maximal and the minimal RPI sets Definition 3 A set Ω R n is the maximal RPI (MRPI) set for system x + = Ax + Bν(x) + w and constraint set (X ν, W) if Ω is RPI set and it contains all RPI sets for system x + = Ax + Bν(x) + w and constraint set (X ν, W). Ω Ω 0 Imperial College ICM meeting Talk, 17. November 2004 p. 7/36

12 The maximal and the minimal RPI sets Definition 4 A set Ω 0 R n is the minimal RPI (mrpi) set for system x + = Ax + Bν(x) + w and constraint set (X ν, W) if Ω 0 is RPI set and is contained in all RPI sets for system x + = Ax + Bν(x) + w and constraint set (X ν, W). If ν : X ν U is stabilizing control law Ω 0 Ω Φ i {Ax + Bν(x) + w x Φ i 1, w W}, i 1, Φ 0 Ω Φ i+1 Φ i i 0, each Φ i is RPI, Φ i Ω 0 as i Imperial College ICM meeting Talk, 17. November 2004 p. 7/36

13 Computations of RCI (RPI) sets Maximal RCI set computations Recall x + = Ax + Bu + w, (x, u, w) X U W Standard Set Recursion (Bertsekas, Aubin,... ): Ψ i {x Ψ i 1 u U s.t. Ax + Bu + w Ψ i 1, w W}, i 1 Boundary condition: Ψ 0 X If Ψ k +1 = Ψ k = Ω = Ψ k, else Ω i 0 Ψ i Merely Brute Force Method! Linearity and Convexity should be exploited New Concept Optimized Robust Control Invariance! Imperial College ICM meeting Talk, 17. November 2004 p. 8/36

14 Optimized Robust Control Invariance Characterization of a family of RCI sets M i R m n, i 0, and M k (M 0, M 1,...,M k 1 ), k 0 Family of RCI sets for x + = Ax + Bu + w and (R n, R m, W): R k (M k ) k 1 i=0 D i(m k )W (A B {a + b a A, b B}) D 0 (M k ) I and D i (M k ) A i + i 1 j=0 Ai 1 j BM j, i 1 D k (M k ) = 0 (can be relaxed!) Allows optimization over M k Can incorporate X and U Single Linear Programming Problem! Imperial College ICM meeting Talk, 17. November 2004 p. 9/36

15 Optimized Robust Control Invariance Let U(M k ) k 1 i=0 M iw and Ω {(M k, α, β, δ) M k M k, R k (M k ) αx, U(M k ) βu, Optimization Problem is: P k : (M 0 k, α 0, β 0, δ 0 ) = arg (α, β) [0, 1] [0, 1], q α α + q β β δ} min {δ (M k, α, β, δ) Ω} M k,α,β,δ If Ω the solution to P k yields R k (M 0 k ) and R k(m 0 k ) is RCI for x + = Ax + Bu + w and (α 0 X, β 0 U, W)... Imperial College ICM meeting Talk, 17. November 2004 p. 9/36

16 Optimized Robust Control Invariance Let w = {w 0, w 1,...,w k 1 }, D = [D k 1 (M 0 k )... D 0(M 0 k )] and Set valued good controls: W(x) {w W k Dw = x}, x R k (M 0 k) U(x) M 0 kw, w W(x), x R k (M 0 k) Selection of a control law ν : R k (M 0 k ) β0 U ν(x) U(x), ν(x) M 0 kw 0 (x), w 0 (x) arg min w { w 2 w W(x)} ν : R k (M 0 k ) β0 U piecewise affine function! Imperial College ICM meeting Talk, 17. November 2004 p. 9/36

17 Optimized Robust Control Invariance Comparison to u = Kx (Gilbert & Kolmanovsky) x 2 4 x X 3 2 X 1 0 Ω 0 (K 1 ) 1 0 Ω 0 (K 2 ) x (a) Set for Controller K 1 (b) Set for Controller K 2 4 x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 9/36

18 Optimized Robust Control Invariance Comparison to u = Kx (Gilbert & Kolmanovsky) x X x X R k1 (M k 0 1) 0 R k2 (M k 0 2) x (c) Set for Controller M k1 (d) Set for Controller M k2 4 x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 9/36

19 Optimized Robust Control Invariance Comparison to u = Kx (Gilbert & Kolmanovsky) x 2 4 x X 3 2 X Ω 0 (K 3 ) 0 Ω 0 (K 4 ) x (e) Set for Controller K 3 (f) Set for Controller K 4 4 x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 9/36

20 2 Set Robust Control Invariance 2 Set Robust Control Invariance Imperial College ICM meeting Talk, 17. November 2004 p. 10/36

21 Abstract Set Invariance Ordinary Set Invariance considers States {Ax + Bu 2 + w w W} Ω, u 2 U {Ax + Bu 1 + w w W} Ω, u 1 U x Ω Ω X Imperial College ICM meeting Talk, 17. November 2004 p. 11/36

22 Abstract Set Invariance Abstract Set Invariance considers Sets of States X X, x X, u U s.t. {Ax + Bu + w w W} X + Φ Z f Φ {z R z Z f } X Φ X Φ X, Φ X Z f R Imperial College ICM meeting Talk, 17. November 2004 p. 11/36

23 Abstract Set Invariance Definition 5 A set of sets Φ is set robust control invariant (SRCI) for system x + = Ax + Bu + w and constraint set (X, U, W) if any set X Φ satisfies: (i) X X and, (ii) for all x X, there exists a u U such that Ax + Bu W Y for some Y Φ. X X, x X, u U s.t. {Ax + Bu + w w W} X + Φ Z f Φ {z R z Z f } X Φ X Φ X, Φ X Z f R Imperial College ICM meeting Talk, 17. November 2004 p. 11/36

24 Characterization of Family of SRCI sets Recall x + = Ax + Bu + w, (x, u, w) X U W Introduce z + = Az + Bv, (z, v) Z V A1: (i) The set R is a RCI set for system x + = Ax + Bu + w and constraint set (αx, βu, W) where (α, β) [0, 1) [0, 1), (ii) The control law ν : R βu is such that R is RPI for system x + = Ax + Bν(x) + w and constraint set (X ν, W), where X ν αx {x ν(x) βu}. Let U ν {ν(x) x R}, Z X R V U U ν. A2: (i) The set Z f is a CI set for the nominal system z + = Az + Bv and constraint set (Z, V), (ii) The control law ϕ : Z f V is such that Z f is PI for system z + = Az + Bϕ(z) and constraint set Z ϕ, where Z ϕ Z {z ϕ(z) V}. Imperial College ICM meeting Talk, 17. November 2004 p. 12/36

25 Family of SRCI sets, the MSRCI and the msrci sets SRCI set for x + = Ax + Bu + w and (X, U, W): Φ {z R z Z f } θ : Z f R U defined by θ(x) ϕ(z) + ν(x z), x X, X Φ X For a fixed R: Φ (R) {z R z Ω } the MSRCI set Φ 0 (R) {z R z {0}} the msrci set Φ (R) X Φ (R) {z R z Ω } Φ 0 (R) X Φ 0 (R) {z R z {0}} Imperial College ICM meeting Talk, 17. November 2004 p. 13/36

26 Exploiting Linearity Let X Φ so that X = z R, z Z f, for any x X we have x = z + r, r R. Then: z + = Az + Bϕ(z) Z f and r + = Ar + Bν(r) + w R, w W x + = Ax + θ(x) + w = A(z + r) + B(ϕ(z) + ν(r)) + w = z + + r + Hence, x X, x + X + z + R, w W X + = z + R, z Z f x z r z + r + x + X = z R, z Z f Imperial College ICM meeting Talk, 17. November 2004 p. 14/36

27 Convergence Observation A3: There exists a Lyapunov Function V ( ) : Z f R s.t. ρ 1 z V (z) ρ 2 z, z Z f, V (Az + Bϕ(z)) V (z) ρ 3 z, z Z f \ {0}, 0 < ρ 1 < ρ 2 ρ 3 X 0 Φ (R) X Y i X i Az i 1 + Bϕ(z i 1 ) R Φ (R) X, i, Y i {Ax + Bθ(x) + w x X i 1, w W}, and X i R as i Z X 0 z 0 R, z 0 Z Φ (R) X Imperial College ICM meeting Talk, 17. November 2004 p. 15/36

28 3 Robust Control Invariant Tubes 3 Robust Control Invariant Tubes Imperial College ICM meeting Talk, 17. November 2004 p. 16/36

29 Robust Control Invariant Tubes Translation Invariance Property of RCI sets X 1 = z 1 R X 0 = z 0 R {Ax + Bθ(x) + w x X 0, w W} X 1 Az 0 + Bϕ(z 0 ) R Imperial College ICM meeting Talk, 17. November 2004 p. 17/36

30 Robust Control Invariant Tubes Translation Invariance Property of RCI sets X 2 = z 2 R X 1 = z 1 R X 0 = z 0 R {Ax + Bθ(x) + w x X 1, w W} X 2 Az 1 + Bϕ(z 1 ) R Imperial College ICM meeting Talk, 17. November 2004 p. 17/36

31 Robust Control Invariant Tubes Translation Invariance Property of RCI sets X 3 = z 3 R X 2 = z 2 R X 1 = z 1 R X 0 = z 0 R {Ax + Bθ(x) + w x X 2, w W} X 3 Az 2 + Bϕ(z 2 ) R Imperial College ICM meeting Talk, 17. November 2004 p. 17/36

32 Robust Control Invariant Tubes Translation Invariance Property of RCI sets ; Let X = z R for any x X we have x = z + r, r R. Then: z + = Az + Bϕ(z) and r + = Ar + Bν(r) + w R, w W x + = Ax + θ(x) + w = A(z + r) + B(ϕ(z) + ν(r)) + w = z + + r + Hence, x X, x + X + z + R, w W X + = z + R, z Z f x z r z + r + x + X = z R, z Z f Imperial College ICM meeting Talk, 17. November 2004 p. 17/36

33 Basic Idea of Tubes Basic Idea of Tubes X + i {y y = f(z, µ i (z), w), z X i, w W }, X + i X i+1 X i = z i α i S i x X 0 = z 0 α 0 S 0 X f X i+1 = z i+1 α i+1 S i+1 Tubes studied, mainly in cont. time by Aubin, Frankowska, Kurzhanski, Filippova, Quincampoix, Veliov,... Imperial College ICM meeting Talk, 17. November 2004 p. 18/36

34 Tubes & Robust Constraint Satisfaction Consider Inclusion and a sequence of Sets of states X {X 0, X 1,...,X N } and associated control policy π {µ 0 ( ), µ 1 ( ),...,µ N 1 ( )} satisfying: Important consequence: x X 0 X i X, i I N 1 X N X f X µ i (z) U, z X i, i I N 1 f(z, µ i (z), W) X i+1, z X i, i I N 1 φ(i; x, π,w) X i X, µ i (φ(i; x, π,w); x) U, i I N 1, φ(n; x, π,w) X f Idea: Parametrize tube and corresponding policy! Imperial College ICM meeting Talk, 17. November 2004 p. 19/36

35 Optimized Robust Control Invariant Tubes Linearity & Convexity X i = z i R tube cross section at time i z i center and R a fixed set z i corresponds to the nominal (reference) system Abstract Set Invariance R Optimized RCI set (minimal in some sense) Control Policy Laws µ i ( ) by using θ( ) Appropriate Target Set X f Z f R Use Receding Horizon Control Strategy Imperial College ICM meeting Talk, 17. November 2004 p. 20/36

36 4 Robust Model Predictive Control 4 Robust Model Predictive Control Imperial College ICM meeting Talk, 17. November 2004 p. 21/36

37 RMPC & Optimized RCI Tubes Recall v = {v 0,...,v N 1 } and: x + = Ax + Bu + w, (x, u, w) X U W ( x state at time i φ(i; x, π,w)) z + = Az + Bv, (z, v) Z V ( z state at time i φ(i; z,v,w)) A1: (i) The set R is a RCI set for system x + = Ax + Bu + w and constraint set (αx, βu, W) where (α, β) [0, 1) [0, 1), (ii) The control law ν : R βu is such that R is RPI for system x + = Ax + Bν(x) + w and constraint set (X ν, W), where X ν αx {x ν(x) βu}. Let U ν {ν(x) x R} and Z X R, V U U ν Imperial College ICM meeting Talk, 17. November 2004 p. 22/36

38 Robust Optimal Control Problem Ingredients Define: V N (x) {(v, z) ( φ(k; z,v), v k ) Z V, k N N 1, φ(n; z,v) Zf, x z R} Cost function: z i φ(i; z,v), V N (v, z) N 1 i=0 l(z i, v i ) + V f (z N ), l(x, u) Qx 2 + Ru 2, V f (x) Px 2, A3: V f (Az + Bϕ f (z)) + l(z, ϕ f (z)) V f (z), z Z f, V f (z) c z, c > 0, z Z f, and Z f is s.t. Az + Bϕ f (z) Z f, z Z f. Imperial College ICM meeting Talk, 17. November 2004 p. 23/36

39 Robust Optimal Control Problem Robust Optimal Control Problem (QP) (Mayne, Seron, Raković): P N (x) : V 0 N(x) min v,z {V N(v, z) (v, z) V N (x)} and its unique minimizer is: (v 0 (x), z 0 (x)) arg min v,z {V N(v, z) (v, z) V N (x)} The domain of the value function VN 0 ( ), the controllability set, is: X N {x V N (x) } Imperial College ICM meeting Talk, 17. November 2004 p. 24/36

40 Robust RHC Controller Implicit RMPC law κ 0 N ( ): κ 0 N(x) = v0(x) 0 + ν(x z 0 (x)) Corresponding Set Sequence {X0(x(i))} 0 X0(x(i)) 0 z 0 (x(i)) R, i 0 Let Z N {z z R X N }, then: Φ N {z R z Z N } is SRCI. Imperial College ICM meeting Talk, 17. November 2004 p. 25/36

41 Simple Optimized RCI tube Simple Optimized RCI tube Illustration X 0 = {X 0 i (x)} X0 i (x) φ(i; z 0 (x),v 0 (x)) R x z 0 (x) R z 0 (x) x r 0 (x) = (x z 0 (x)) κ 0 N (x) = v0 0(x) + ν(x z 0 (x)) Imperial College ICM meeting Talk, 17. November 2004 p. 26/36

42 Stability Properties Definition 6 A set R is robustly exponentially stable (Lyapunov stable and exponentially attractive) for x + = Ax + Bκ(x) + w, w W, with a region of attraction X N if there exists a c > 0 and a γ (0, 1) such that any solution x( ) of x + = Ax + Bκ(x) + w with initial state x(0) X N, and admissible disturbance sequence w( ) (w(i) W for all i 0) satisfies d(x(i), R) cγ i d(x(0), R) for all i 0. Imperial College ICM meeting Talk, 17. November 2004 p. 27/36

43 Stability Properties Proposition 1 (i) For all x R, V 0 N (x) = 0, z0 (x) = 0, v 0 (x) = {0, 0,...,0} and κ 0 N (x) = ν(x). (ii) Let x X N and let (v 0 (x), z 0 (x)) be minimizer of P N (x), then for all x + Ax + Bκ 0 N (x) W there exists (v(x+ ), z(x + )) V N (x + ) and V 0 N(x + ) V 0 N(x) l(z 0 (x), v 0 0(x)). Imperial College ICM meeting Talk, 17. November 2004 p. 27/36

44 Stability Properties Theorem 1 Suppose that X N is bounded, then the set R is robustly exponentially stable for controlled uncertain system x + = Ax + Bκ 0 N (x) + w, w W. The region of attraction is X N. Imperial College ICM meeting Talk, 17. November 2004 p. 27/36

45 Optimized RCI Tubes& RMPC examples Example 1 (Double Integrator) RHC tube, random w x z 0 0(x(4)) x(4) X f 5 Z f X 0 0(x(2)) = z 0 0(x(2)) R z0(x(0)) 0 x(0) x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 28/36

46 Optimized RCI Tubes& RMPC examples Example 1 (Double Integrator) RHC tube, random & extreme w x z 0 0(x(4)) x(4) X f 5 z 0 0(x(3)) R Z f z0(x(0)) 0 x(0) x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 28/36

47 Optimized RCI Tubes& RMPC examples Example 1 (Double Integrator) Sets X i x X 0 = X f 5 X x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 28/36

48 Optimized RCI Tubes& RMPC examples Example 2 (Flight Vehicle) RHC tube, random w x z 0 0(x(5)) x(5) z 0 0(x(i)) R, i 10 x(0) z 0 0(x(0)) X f Z f x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 28/36

49 Optimized RCI Tubes& RMPC examples Example 2 (Flight Vehicle) RHC tube, random & extreme w x z 0 0(x(6)) R z 0 0(x(3)) x(3) x(0) z 0 0(x(0)) 0 X f Z f x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 28/36

50 Optimized RCI Tubes& RMPC examples Example 2 (Flight Vehicle) Sets X i x X 0 = X f X x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 28/36

51 5 Extensions 5 Extensions Imperial College ICM meeting Talk, 17. November 2004 p. 29/36

52 Imperfect State Information Case Only Output Measurements are available System: x + = Ax + Bu + w, y = Cx + v Constraints: (x, u, w, v) X U W V A priori Initial State Uncertainty: x X 0 z E Joint Set Membership State Estimation and Control by using Abstract Set Invariance. Proper and simple receding horizon control scheme permitting feedback control based on the observations and set of possible states! Extends and improves the concepts of Bertsekas, Schweppe, Glover, Rhodes, Witsenhausen... Imperial College ICM meeting Talk, 17. November 2004 p. 30/36

53 Imperfect State Information Case Higher Level of Abstract Set Invariance Φ X Z D X z D, z Z X(X, y 1 ) X(X, y 1, u 1 ) X X + (X, u 1, y 1 ) X(X, y 2 ) X + (X, u 1, y 2 ) X(X, y 2, u 2 ) X(X, y i ) {x X Cx y i ( V)}, i = 1, 2 X(X, y i, u i ) X + (X, u i, y i ) z(u i, y i ) D Φ X, i = 1,2 X(X, y i, u i ) A X(X, u i, y i ) + Bu i W, i = 1, 2 Imperial College ICM meeting Talk, 17. November 2004 p. 30/36

54 Imperfect State Information Case Set Set Robust Control Invariant Set (solves output feedback RCI problem) 10 Controlling Set of Sets based on the output measurments Imperial College ICM meeting Talk, 17. November 2004 p. 30/36

55 Imperfect State Information Case Robust Control Invariant Tube of Tubes 50 Robust Control Invariant Tube of Tubes Imperial College ICM meeting Talk, 17. November 2004 p. 30/36

56 Linear Systems Parameter Uncertainty x x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 31/36

57 Linear Systems Parameter Uncertainty x x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 31/36

58 Linear Systems Parameter Uncertainty x x 1 Imperial College ICM meeting Talk, 17. November 2004 p. 31/36

59 RCI Tubes for PWA DTS 4 P 1 P 2 3 X f j = {2, 2, 2, 2, 2, 2, 1, 1} s = {2, 2, 2, 2, 2, 2, 2, 1} Imperial College ICM meeting Talk, 17. November 2004 p. 32/36

60 RCI Tubes for PWA DTS 4 P 1 P 2 3 X f j = {2, 2, 2, 2, 2, 2, 1, 1} s = {2, 2, 2, 2, 2, 2, 2, 1} Imperial College ICM meeting Talk, 17. November 2004 p. 32/36

61 RCI Tubes for PWA DTS 3 2 P 1 P 2 X f j = {2, 1, 1, 1, 1, 1, 1} s = {2, 2, (1, 2), 1, 1, 1, 1} Imperial College ICM meeting Talk, 17. November 2004 p. 32/36

62 RCI Tubes for PWA DTS 3 2 P 1 P 2 X f j = {2, 1, 1, 1, 1, 1, 1} s = {2, 2, (1, 2), 1, 1, 1, 1} Imperial College ICM meeting Talk, 17. November 2004 p. 32/36

63 6 Conclusions 6 Conclusions Imperial College ICM meeting Talk, 17. November 2004 p. 33/36

64 Conclusions Robust MPC of Linear Systems can be achieved by using optimized RCI tubes Tube MPC can be extended to uncertain Linear & PWA Systems Set Invariance is an important concept, but exploit the structure of the considered problem! New concepts Optimized Robust Control Invariance and Abstract Set Invariance Robust MPC connected to these new concepts Robust Output Feedback MPC can be obtained by using RCI Tube of tubes Robust MPC for general non-linear systems is still an open problem Imperial College ICM meeting Talk, 17. November 2004 p. 34/36

65 Acknowledgments This research is supported by the Engineering and Physical Sciences Research Council UK - EPSRC UK Geometric Bounding Toolbox is used for simulations (GBT S.M. Veres) Credits to: Prof. D. Q. Mayne, Prof. R. B. Vinter, Dr. E. C. Kerrigan, Dr. K. I. Kouramas, Dr. M. Seron, Dr W. Langson, Dr. I. Chryssochoos. Imperial College ICM meeting Talk, 17. November 2004 p. 35/36

66 Question Time Thank you Any questions? Imperial College ICM meeting Talk, 17. November 2004 p. 36/36