Minkowski Sums and Aumann s Integrals in Set Invariance Theory for Autonomous Linear Time Invariant Systems

Transcription

1 Minkowski Sums and Aumann s Integrals in Set Invariance Theory for Autonomous Linear Time Invariant Systems Saša V. Raković (sasa.rakovic@imperial.ac.uk) Imperial College, London Imperial College, 2. February 26 Imperial College ICM meeting Talk, 2. February 26 p. 1/4

2 Contents & Problem Description Contents & Preliminaries Imperial College ICM meeting Talk, 2. February 26 p. 2/4

3 Contents Set Invariance for Linear Time Invariant Systems 1 Minkowski Sums & Discrete Time Case 2 Aumann s Integrals & Continuous Time Case 3 Simple Robust Control Invariant Tubes 4 Conclusions 5 Imperial College ICM meeting Talk, 2. February 26 p. 3/4

4 Problem Description Strictly Stable Linear Time Invariant Discrete Time System: x + = Ax + w, ρ(a) < 1, Strictly Stable Linear Time Invariant Continuous Time System: ẋ = Ax + w, σ(a) in the interior of left half plane Bounded Disturbances (functions): w W, Disturbance Set: W convex, compact and contains the origin (wlog), in its (non empty) interior i.e. W is a C set. Problem of Interest: Characterization of a family of sets (set of sets) that are robust positively invariant and approximate arbitrarily close in sense of Hausdroff Metric the minimal robust positively invariant set and more importantly are finite time computable. Imperial College ICM meeting Talk, 2. February 26 p. 4/4

5 Preliminary Notation and Facts Necessary Preliminary Notation: C(C) (C(R n )) is a family of all compact subsets of a compact subset C of R n (of R n ) Hausdorff Distance for A C(R n ) and B C(R n ): H(A, B) = max{max a A min b B a b, max b B min a A a b } Important Consequences: The Hausdorff Distance induces a complete metric space on C(C) for any compact subset C of R n as well as for C(R n ). Cauchy (Fundamental) Sequences are convergent in C(R n ) (with H( )) and Banach Contraction Principle (Fixed Point Theorem) is valid in C(R n ) (with H( )). Imperial College ICM meeting Talk, 2. February 26 p. 5/4

6 Brunn Minkowski Algebra Necessary Preliminaries Let X C C (R n ), Y C C (R n ) and Z C C (R n ) (for simplicity), A 1 R n n, A 2 R n n, (α, β) R 2 +, α β ( hereafter C C (R n ) {X C(R n ) X is a C set }) Minkowski Set Addition: X Y {x + y x X, y Y } Minkowski Set Difference: X Y {p R n p Y X} Imperial College ICM meeting Talk, 2. February 26 p. 6/4

7 Brunn Minkowski Algebra Necessary Preliminaries Y X X Y Imperial College ICM meeting Talk, 2. February 26 p. 6/4

8 Brunn Minkowski Algebra Necessary Preliminaries X Y Imperial College ICM meeting Talk, 2. February 26 p. 6/4

9 Brunn Minkowski Algebra Necessary Preliminaries X Y Imperial College ICM meeting Talk, 2. February 26 p. 6/4

10 Brunn Minkowski Algebra Necessary Preliminaries Y X X Y X Y Imperial College ICM meeting Talk, 2. February 26 p. 6/4

11 Brunn Minkowski Algebra Necessary Preliminaries X Y X X Y Imperial College ICM meeting Talk, 2. February 26 p. 6/4

12 Brunn Minkowski Algebra Necessary Preliminaries X Y X (X Y ) Y (X Y ) Y X Imperial College ICM meeting Talk, 2. February 26 p. 6/4

13 Brunn Minkowski Algebra Rules & Words of Caution Let X C C (R n ), Y C C (R n ) and Z C C (R n ) (for simplicity), A 1 R n n, A 2 R n n, (α, β) R 2 +, α β Necessary Algebraic Rules & Words of Caution: X Y = Y X, X (Y Z) = (X Y ) Z (X Y ) Y = X (not true if X and Y not convex) (X Y ) Y X and generally not (X Y ) Y = X αx βx = (α + β)x, αx βx = (α β)x (A 1 + A 2 )X A 1 X A 2 X and generally not (A 1 + A 2 )X = A 1 X A 2 X, take A 1 = A 2 = I, X = B(1), (A 1 + A 2 )X = {}, A 1 X A 2 X = B(2) A(X Y ) = AX AY Y Z X Y X Z Y Z X X Z X Y (Rolf Schneider Convex Bodies: The Brunn Minkowski Theory ) Imperial College ICM meeting Talk, 2. February 26 p. 7/4

14 Explicit Form of Reach Set Operators One Step Reach Set Operator For Discrete Time Case: x + = Ax + w, w W, W C(R n ) R(S) = AS W, R( ) : C(R n ) C(R n ) Reach Set Operator For Continuous Time Case: ẋ = Ax + w, w W, W C(R n ), (hereafter a.e., etc., assumed and omitted) A(t) e ta W, A : [, δ] C(R n ) measurable and integrably bounded, hence P integrable (McShane 1969 P integral as a limit of Riemann type Sums) R(S, δ) = e δa S δ A(σ)dσ For P integrable A : [, δ] C(R n ), P integral = Aumann Integral (Artstein & Burns 1975) Imperial College ICM meeting Talk, 2. February 26 p. 8/4

15 1 Set Invariance for Linear Time Invariant Systems 1 Set Invariance for Linear Time Invariant Systems Imperial College ICM meeting Talk, 2. February 26 p. 9/4

16 Discrete Time Case Robust Positive Invariance (RPI) Condition: AS W S, R(S) S Family of all RPI sets in C(R n ): S D RPI (R n ) {S C(R n ) R(S) S} Special RPI set is the minimal RPI set S: R( S) = S, set S is a Fixed Point and much more! There is one and only one S S D RPI (R n ) such that R( S) = S. Imperial College ICM meeting Talk, 2. February 26 p. 1/4

17 Continuous Time Case Robust Positive Invariance (RPI) Condition: R(S, δ) S, δ R + Family of all RPI sets in C(R n ): S C RPI (R n ) {S C(R n ) R(S, δ) S, δ R + } Special RPI set is the minimal RPI set S: R( S, δ) = S, set S is a Fixed Point and much more! There is one and only one S S C RPI (R n ) such that R(S, δ) = S, δ R +. Imperial College ICM meeting Talk, 2. February 26 p. 11/4

18 2 Minkowski Sums & Discrete Time Case 2 Minkowski Sums & Discrete Time Case Imperial College ICM meeting Talk, 2. February 26 p. 12/4

19 Banach Contraction Principle Approach Simpler and Mathematically more elegant method, define: S i+1 R(S i ), S i+1 AS i W, i N with S C(R n ) Relevant Facts: S i C(R n ), i S i+1 = R(S i ) is a standard Fixed Point Iteration Since ρ(a) < 1, Banach Contraction Principle says that There exists a unique set S S D RPI (R n ) C(R n ) such that R( S) = S (Artstein & Raković). Set S C(R n ) is a solution of S = R( S) Imperial College ICM meeting Talk, 2. February 26 p. 13/4

20 Cauchy Sequences Approach Previous techniques (Kolmanovsky & Gilbert) consider {S i }: S i+1 R(S i ), S i+1 AS i W, i N with S {} or alternatively: S i+1 = i k= Ak W Relevant Facts: S i S i+1, S i C(R n ) and S i for all i Since ρ(a) < 1, {S i } is a Cauchy Sequence in C(R n ) There exists a unique set S C(R n ) that is limit (in the Hausdorff Metric) of {S i }, i.e. H( S, S i ) as i. Set S C(R n ) satisfies S = i= Ai W Imperial College ICM meeting Talk, 2. February 26 p. 14/4

21 Minkowski Sums & RPI Approximations Arbitrarily Close RPI Approximations of S (Raković, Kerrigan, Kouramas & Mayne): Recall S i+1 = i k= Ak W Definition of Cauchy Sequences, W C C (R n ) and ρ(a) < 1 suggest: There exists (i, α) N [, 1) such that A i W αw (Kouramas) Let I {(i, α) N [, 1) A i W αw } Let S(i, α) (1 α) 1 S i and S (D RPI,I) {S(i, α) (i, α) I}. Important Fact S (D RPI,I) S D RPI C(R n ). Imperial College ICM meeting Talk, 2. February 26 p. 15/4

22 Minkowski Sums & RPI Approximations Arbitrarily Close RPI Approximations of S (Raković, Kerrigan, Kouramas & Mayne): Definition of Cauchy Sequences, W C C (R n ) set and ρ(a) < 1 further imply: There exists (i, α) I such that α(1 α) 1 S i B(ε) for any arbitrarily small ε > (Raković, Kerrigan, Kouramas & Mayne) Let I ε {(i, α) I αs(i, α) B(ε)} Let S (D RPI,Iε ) {S(i, α) (i, α) I ε }. Important Fact S (D RPI,Iε ) S (D RPI,I) S D RPI C(R n ). Imperial College ICM meeting Talk, 2. February 26 p. 16/4

23 Minkowski Sums & RPI Approximations Let S S (D RPI,Iε ), so that S = (1 α) 1 S i, (i, α) I ε, then: R(S) = AS W = A(1 α) 1 S i W i 1 = (1 α) 1 A i W (1 α) 1 Å i 1 (1 α) 1 Å k=1 i 1 = (1 α) 1 Å k=1 since S C(R n ), we have S S D RPI. k=1 A k W W A k W α(1 α) 1 W W A k W (1 α) 1 W = S. Also, S i S (1 α) 1 S i and (1 α) 1 S i = α(1 α) 1 S i S i S i B(ε) so that S S S B(ε). Imperial College ICM meeting Talk, 2. February 26 p. 17/4

24 Minkowski Sums & Discrete Time Case Case when W C(R n ) is not a C set (but W ) treated in (Raković, Kerrigan, Kouramas & Mayne). Another simple alternative is: W W B(η) (with η > arbitrarily small) so that W CC (R n ) Use Previously Outlined Methods (W replaced with W ) Simple but Relevant Observation AS W S AS W S. Further Observations Possible and Trivial! Imperial College ICM meeting Talk, 2. February 26 p. 18/4

25 3 Aumann s Integrals & Continuous Time Case 3 Aumann s Integrals & Continuous Time Case Imperial College ICM meeting Talk, 2. February 26 p. 19/4

26 Aumann s Integrals Motivating Example Motivating Example Scalar Case: ẋ = λx + w, λ R, λ < and W = [ b, b] C C (R) R({}, t) = t eλt Wdt = {x R b(e λt 1)/λ x b(e λt 1)/λ} S = e λt Wdt = {x R b/λ x b/λ} Imperial College ICM meeting Talk, 2. February 26 p. 2/4

27 Aumann s Integrals Motivating Example R({}, t) R({}, t) 1 8 x 1 ( ) S 1 8 S x 3 ( ) x 4 ( ) 6 8 x 2 ( ) (a) t esa Wds t (b) e sa Wds t Imperial College ICM meeting Talk, 2. February 26 p. 2/4

28 Reach Set Operator R(S, δ) Properties R(S, δ) is a semi group, i.e. R(S, ) = S and R(S, t + δ) = R(R(S, t), δ) X Y R(X, t) R(Y, t), t R + t 1 t 2 R({}, t 1 ) R({}, t 2 ) R({}, δ) C(R n ) and convex (does not require convexity of W Lyapunov 194, Artstein 1976)! lim h 1 h t+h t e σa Wdσ = e ta W Simple but Relevant Facts: W C C (R n ) is Hasudorff Continuous in time ( also A stable ) R({}, t) is Hasudorff Continuous in time. Imperial College ICM meeting Talk, 2. February 26 p. 21/4

29 What about R(S,kδ), k N? R(S, t) semi group says that R(S, (k 1)δ + δ) = R(R(S, (k 1)δ), δ) Since R({}, δ) = δ eσa Wdσ it follows R({}, 2δ) = e δa δ eσa Wdσ δ eσa Wdσ R({}, kδ) = k 1 i= eiδa δ eσa Wdσ A familiar set sequence {R({}, kδ)}? Of course YES! Set Sequence {R({}, kδ)} is a Cauchy Set Sequence in C(R n ) Converges (in Hausdorff Metric) to unique S defined as S = i= eiδa δ eσa Wdσ (any < δ < same set). Imperial College ICM meeting Talk, 2. February 26 p. 22/4

30 The minimal RPI Set e σa Wdσ It appears that S = e σa Wdσ, why? e σa Wdσ = = = δ δ 2 δ+2η 2 δ δ e σa Wdσ e σa Wdσ 2δ δ δ+η δ e σa Wdσ e σa Wdσ 2 δ δ e σa Wdσ e σa Wdσ 3 δ 2 δ+2η e σa Wdσ 3δ 2δ 2 δ δ+η e σa Wdσ... 3 δ 2 δ e σa Wdσ... e σa Wdσ... e σa Wdσ... Relevant Fact S = S(δ) for any < δ < so that S = e σa Wdσ. Imperial College ICM meeting Talk, 2. February 26 p. 23/4

31 RPI Property of e σa Wdσ Claim R( S, δ) = S, δ R + : ( R ) e σa Wdσ, δ = e δa = = = δ e σa Wdσ e (δ+σ)a Wdσ e σa Wdσ e σa Wdσ δ δ δ e σa Wdσ e σa Wdσ e σa Wdσ Set S = e σa Wdσ is a fixed point and the mrpi set; in particular it satisfies S = e δa S δ eσa Wdσ, δ R +. Imperial College ICM meeting Talk, 2. February 26 p. 24/4

32 What about finite time sets? Arbitrarily Close RPI Approximations of S : Recall R({}, δ) = δ eσa Wdσ Stability of A, W C C (R n ) suggest: There exists (δ, α) R + [, 1) such that e δa W αw Let I {(δ, α) R + [, 1) e δa W αw } Let S(δ, α) (1 α) 1 δ eσa Wdσ and S (C RPI,I) {S(δ, α) (δ, α) I}. Important Fact S (C RPI,I) S C RPI C(R n ). Imperial College ICM meeting Talk, 2. February 26 p. 25/4

33 Aumann s Integrals & RPI Approximations Arbitrarily Close RPI Approximations of S Stability of A, W C C (R n ) further suggest: There exists (δ, α) I such that α(1 α) 1 δ eσa Wdσ B(ε) for any arbitrarily small ε > Let I ε {(δ, α) R + [, 1) I αs(δ, α) B(ε)} Let S (C RPI,Iε ) {S(δ, α) (δ, α) I ε }. Important Fact S (C RPI,Iε ) S (C RPI,I) S C RPI C(R n ). Imperial College ICM meeting Talk, 2. February 26 p. 26/4

34 RPI Property of S(δ,α) Consider (1 α) 1 δ eσa Wdσ, (δ, α) I ε, then: δ R((1 δ α) 1 e σa Wdσ, t) = e ta (1 α) 1 δ = (1 α) 1 e (t+σ)a Wdσ t+δ = (1 α) 1 e σa Wdσ t t t e σa Wdσ e σa Wdσ e σa Wdσ t e σa Wdσ Since W C C (R n ), we have: t e t t σa Wdσ = (1 α) 1 e σa Wdσ α(1 α) 1 e σa Wdσ Imperial College ICM meeting Talk, 2. February 26 p. 27/4

35 RPI Property of S(δ,α) Proof Continued: δ t+δ t R((1 α) 1 e σa Wdσ, t) = (1 α) 1 e σa Wdσ α(1 α) 1 e σa Wdσ δ t = t (1 α) 1 e σa Wdσ (1 α) 1 e σa e δa Wdσ α(1 α) 1 e σa Wdσ Since (δ, α) I ε, we have e δa W αw and: t (1 t α) 1 e σa e δa Wdσ α(1 α) 1 e σa Wdσ and finally: δ R((1 α) 1 e σa Wdσ, t) δ t t (1 α) 1 e σa Wdσ α(1 α) 1 e σa Wdσ α(1 α) 1 e σa Wdσ δ (1 α) 1 e σa Wdσ Imperial College ICM meeting Talk, 2. February 26 p. 28/4

36 RPI Property of S(δ,α) Summary We have proven that: R((1 α) 1 δ eσa Wdσ, t) (1 α) 1 δ eσa Wdσ for arbitrary t R +, hence t R +. Set Inclusion (δ, α) I ε. S (1 α) 1 δ eσa Wdσ S B(ε) is trivial for Imperial College ICM meeting Talk, 2. February 26 p. 29/4

37 Computing δ eσa Wdσ Level sets of appropriately defined value functions (Kurzhanski, Varaya,...) x δ eσa Wdσ x {y V (t, y) = } where V : R R n R: V (t, x) min {x() x() x = x(t), x(t) = e ta t x()+ e sa w(s)ds} (x(),w( )) R n W R[,t] and W R[,t] set of Lebesgue Measurable Functions from interval [, t] to W. Simplifications possible in certain cases. Imperial College ICM meeting Talk, 2. February 26 p. 3/4

38 RPI Property of S(δ,α) Example 5 x L 1 S x 1 Imperial College ICM meeting Talk, 2. February 26 p. 31/4

39 RPI Property of S(δ,α) Example 5 x L x() x() 4 S 1 x() x() x 1 Imperial College ICM meeting Talk, 2. February 26 p. 31/4

40 4 Simple Robust Control Invariant Tubes 4 Simple Robust Control Invariant Tubes Imperial College ICM meeting Talk, 2. February 26 p. 32/4

41 Simple RCI Tubes Underlaying Principle Underlying Principle is to construct an object TUBE T D (T C ) that is itself an interesting RPI set, i.e. T D = {X, X 1,...} and T C (X(), t) = δ [,t] X(δ) T D S D RPI for discrete time case generally non convex and disconnected RPI set! T C (t) S C RPI for continuous time case generally non convex but connected RPI set! Simple tube obtained by considering tube that have RPI cross section clearly objects we have constructed. Simplifies the corresponding robust control design and it allows for control synthesis in a class of parametrized closed loop controls. Imperial College ICM meeting Talk, 2. February 26 p. 33/4

42 Simple Robust Tube Controllers Examples 4 Double Integrator MPC tube, random w x 2 5 z (x(4)) x(4) X f 5 Z f X (x(2)) = z (x(2)) R z(x()) x() x 1 Imperial College ICM meeting Talk, 2. February 26 p. 34/4

43 Simple Robust Tube Controllers Examples 4 Double Integrator MPC tube, random & extreme w x 2 5 z (x(4)) x(4) X f 5 z (x(3)) R Z f z(x()) x() x 1 Imperial College ICM meeting Talk, 2. February 26 p. 34/4

44 Simple Robust Tube Controllers Examples 4 Double Integrator MPC tube Controllability Sets X i x 2 5 X = X f 5 X x 1 Imperial College ICM meeting Talk, 2. February 26 p. 34/4

45 Simple Robust Tube Controllers Examples 4 Output Feedback Case RCI Tube of Tubes (joint work with D. Q. Mayne, R. Findeisen & F. Allgöwer) X i = z (ˆx i ) S S Z f Z f S {x i } {ˆx i } {z i (ˆx i )} Z f S S X = z (ˆx ) S S x ˆx S Imperial College ICM meeting Talk, 2. February 26 p. 34/4

46 Simple Robust Tube Controllers Examples 4 Continuous Time 2-D Simple RCI Tube 5 x 2 S 5 X(t) = z(t) S x 1 Imperial College ICM meeting Talk, 2. February 26 p. 34/4

47 5 Conclusions 5 Conclusions Imperial College ICM meeting Talk, 2. February 26 p. 35/4

48 Conclusions Brun Minkowski Algebra of Convex Bodies Plays a Fundamental Role in Set Invariance Theory for Linear Time Invariant Systems: Minkowski Sums in Discrete Time Case and Aumann s Integrals in Continuous Time Case, State Feedback Case rather straight forward, possibly computationally expensive, Novel Family of Robust Positively Invariant Sets for Continuous Time Systems, The mrpi sets extremely powerful objects, stable attractors for abstract dynamical systems! Robust Positively Invariant Sets allow for construction of Simple Robust Control/Positively Invariant Tubes and Tube of Tubes, General non-linear case is still an open and difficult problem. Imperial College ICM meeting Talk, 2. February 26 p. 36/4

49 Acknowledgments This research is supported by the Engineering and Physical Sciences Research Council UK - EPSRC UK Special Thanks to: Dr. K. I. Kouramas, Prof. Z. Artstein, Prof. A. B. Kurzhanski, Prof. D. Q. Mayne, Prof. R. B. Vinter, Dr. E. C. Kerrigan. Imperial College ICM meeting Talk, 2. February 26 p. 37/4

50 The Source or The Book H(X, Y ) Felix Hausdorff Imperial College ICM meeting Talk, 2. February 26 p. 38/4

51 The Source or The Book (C(R n ), H(X, Y )) Felix Hausdorff Imperial College ICM meeting Talk, 2. February 26 p. 38/4

52 The Source or The Book X Y Hermann Minkowski Imperial College ICM meeting Talk, 2. February 26 p. 38/4

53 The Source or The Book X Y Hermann Minkowski Imperial College ICM meeting Talk, 2. February 26 p. 38/4

54 The Source or The Book {S i } Augustin-Louis Cauchy Imperial College ICM meeting Talk, 2. February 26 p. 38/4

55 The Source or The Book S i S, i Augustin-Louis Cauchy Imperial College ICM meeting Talk, 2. February 26 p. 38/4

56 The Source or The Book A(t)dt Robert Aumann Imperial College ICM meeting Talk, 2. February 26 p. 38/4

57 The Source or The Book S = R(S) Stefan Banach Imperial College ICM meeting Talk, 2. February 26 p. 38/4

58 The Source or The Book S = e δa S δ eσa Wdσ, δ R + Stefan Banach Imperial College ICM meeting Talk, 2. February 26 p. 38/4

59 Question Time Thank you for patience! Any questions? Imperial College ICM meeting Talk, 2. February 26 p. 39/4

60 Board Discussion Slide Imperial College ICM meeting Talk, 2. February 26 p. 4/4