LMI Methods in Optimal and Robust Control

Transcription

1 LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 14: LMIs for Robust Control in the LF Framework

2 ypes of Uncertainty In this Lecture, we will cover Unstructured, Dynamic, norm-bounded: Structured, Static, norm-bounded: := { L(L 2 ) : H < 1} := {diag(δ 1,, δ K, 1, N ) : δ i < 1, σ( i ) < 1} Structured, Dynamic, norm-bounded: := { 1, 2, L(L 2 ) : i H < 1} Unstructured, Static, norm-bounded: Parametric, Polytopic: := { R n n : 1} := { R n n : = i Parametric, Interval: := { i α i H i, α i 0, i i δ i : δ i [δ i, δ+ i ]} Each of these can be ime-varying or ime-invariant! α i = 1} M. Peet Lecture 14: 2 / 28

3 Back to the Linear Fractional ransformation he interval and polytopic cases rely on Linearity of the uncertain parameters. ẋ(t) = (A 0 + (t))x(t) he Linear-Fractional ransformation, however [ ] [ ] [ ] ẋ1 = z(t) S(P, x1 (t) ) = (P F (t) 22 +P 21 (I P 11 ) 1 x(t) P 12 ) F (t) is an arbitrary rational function. We focus on two results: he S-Procedure for Unstructured Uncertainty Sets he Structured Singular Value for Structured Uncertainty Sets. p M q M. Peet Lecture 14: 3 / 28

4 Robust Stability p M q Questions: Is S(, M) stable for all? Is I M 11 invertible for all? M. Peet Lecture 14: 4 / 28

5 Redefine Robust and Quadratic Stability Suppose we have the system Definition 1. [ ] M11 M M = 12 M 21 M 22 he pair (M, ) is Robustly Stable if (I M 22 ) is invertible for all. [ẋ(t) ] [ ] Alternatively, if = z(t) S(M, x(t) ) w(t) Definition 2. he pair (M, ) is Robustly Stable if for some β > 0, ρ(m 22 + M 21 (I M 11 ) 1 M 12 ) + βi is Hurwitz for all. [ ] [ ] Alternatively, if xk+1 = z S(M, xk ) k w k Definition 3. he pair (M, ) is Robustly Stable if ρ(m 22 + M 21 (I M 11 ) 1 M 12 ) = β < 1 for all. M. Peet Lecture 14: 5 / 28

6 Quadratic Stability - Parametric Uncertainty Focus on the 1,1 block of S(M, ): If ẋ(t) = S(M, )x(t), Definition 4. he pair (M, ) is Quadratically Stable if there exists a P > 0 such that S(M, ) P + P S(M, ) < βi for all Alternatively, if x k+1 = S(M, )x k, Definition 5. he pair (M, ) is Quadratically Stable if there exists a P > 0 such that S(M, ) P S(M, ) P < βi for all for all. M. Peet Lecture 14: 6 / 28

7 Parametric, Norm-Bounded ime-varying Uncertainty Consider the state-space representation: ẋ(t) = Ax(t) + Mp(t), q(t) = Nx(t) + Qp(t), p(t) = (t)q(t), (t) Parametric, Norm-Bounded Uncertainty: := { R n n : 1} M. Peet Lecture 14: 7 / 28

8 Parametric, Norm-Bounded Uncertainty Quadratic Stability: here exists a P > 0 such that P (Ax(t) + Mp) + (Ax(t) + Mp) P < 0 for all p {p : p = q, q = Nx + Qp} heorem 6. he system ẋ(t) = Ax(t) + Mp(t), p(t) = (t)q(t), q(t) = Nx(t) + Qp(t), := { R n n : 1} is quadratically stable if and only if there exists some P > 0 such that [ ] [ ] [ ] x A P + P A P M x p M < 0 P 0 p [ ] {[ ] [ ] [ ] [ ] x x x N for all : N N Q x p p p Q N I Q Q p } 0 M. Peet Lecture 14: 8 / 28

9 Parametric, Norm-Bounded Uncertainty If. If [ ] [ x A P + P A P M p M P 0 then for all x, p such that ] [ x < 0 p] [ ] x for all p {[ ] [ ] [ [ ] } x x N : N N Q x p p Q N I Q Q] 0 p x P (Ax + Mp) + (Ax + Mp) P x < 0 p 2 Nx + Qp 2 herefore, since p = q implies p q, we have quadratic stability. he only if direction is similar. M. Peet Lecture 14: 9 / 28

10 he S-Procedure A Significant LMI for your oolbox Quadratic stability here requires positivity of a matrix on a subset. his is Generally a very hard problem NP-hard to determine if x F x 0 for all x 0. (Matrix Copositivity) S-procedure to the rescue! he S-procedure asks the question: Is z F z 0 for all z {x : x Gx 0}? Corollary 7 (S-Procedure). z F z 0 for all z {x : x Gx 0} if there exists a scalar τ 0 such that F τg 0. he S-procedure is Necessary if {x : x Gx > 0}. M. Peet Lecture 14: 10 / 28

11 Parametric, Norm-Bounded Uncertainty heorem 8 (Dual Version). he system ẋ(t) = Ax(t) + Mp(t), p(t) = (t)q(t), q(t) = Nx(t) + Qp(t), := { R n n : 1} is quadratically stable if and only if there exists some µ 0 and P > 0 such that [ ] [ ] AP + P A P N MM MQ + µ NP 0 QM QQ < 0} I Noting that the LMI can be written as [ ] [ ] [ ] AP + P A P N M M + µ < 0 NP µi Q Q or AP + P A P N M NP µi Q < 0 M Q 1 µ I we see that this condition is simply an H gain condition on the nominal system H < 1. M. Peet Lecture 14: 11 / 28

12 Necessity of the Small-Gain Condition his leads to the interesting result: p M q If := { L(L 2 ) : 1}, then S(P, ) H if and only if P 11 H < 1 he small gain condition is necessary and sufficient for stability. Quadratic Stability is equivalent to stability. Holds for Dynamic and Parametric Uncertainty M. Peet Lecture 14: 12 / 28

13 Quadratic Stability and Equivalence to Robust Stability Consider Quadratic Stability in Discrete-ime: x k+1 = S l (M, )x k. Definition 9. (S l, ) is QS if S l (M, ) P S l (M, ) P < 0 for all heorem 10 (Packard and Doyle). Let M R (n+m) (n+m) be given with ρ(m 11 ) 1 and σ(m 22 ) < 1. hen the following are equivalent. 1. he pair (M, = R m m ) is quadratically stable. 2. he pair (M, = C m m ) is quadratically stable. 3. he pair (M, = C m m ) is robustly stable. M. Peet Lecture 14: 13 / 28

14 Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty However, we can add controllers: heorem 11. he system with u(t) = Kx(t) and ẋ(t) = A 0 x(t) + Bu(t) + Mp(t), p(t) = (t)q(t), q(t) = Nx(t) + Qp(t) + D 12 u(t), := { R n n : 1} is quadratically stable if and only if there exists some µ 0 and P > 0 such that [ ] [ ] (A + BK)P + P (A + BK) P (N + D 12 K) MM MQ +µ (N + D 12 K)P 0 QM QQ < 0} I Of course, this is bilinear in P and K, so we make the change of variables Z = KP. M. Peet Lecture 14: 14 / 28

15 An LMI for Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty heorem 12. here exists a K such that the system with u(t) = Kx(t) ẋ(t) = Ax(t) + Bu(t) + Mp(t), p(t) = (t)q(t), q(t) = Nx(t) + Qp(t) + D 12 u(t), := { R n n : 1} is quadratically stable if and only if there exists some µ 0, Z and P > 0 such that [ ] [ ] AP + BZ + P A + Z B P N + Z D12 MM MQ +µ NP + D 12 Z 0 QM QQ < 0}. I hen K = ZP 1 is a quadratically stabilizing controller. We can also extend this result to optimal control in the H norm. M. Peet Lecture 14: 15 / 28

16 An LMI for H -Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty In this case, we set Q = 0. heorem 13. here exists a K such that the system with u(t) = Kx(t) ẋ(t) = Ax(t) + Bu(t) + Mp(t) + B 2 w(t), p(t) = (t)q(t), q(t) = Nx(t) + D 12 u(t), := { R n n : 1} y(t) = Cx(t) + D 22 u(t) satisfies y L2 γ u L2 if there exists some µ 0, Z and P > 0 such that AP + BZ + P A + Z B + µmm (CP + D 22 Z) P N + Z D12 CP + D 22 Z γ 2 I 0 < 0. NP + D 12 Z 0 µi hen K = ZP 1 is the corresponding controller. M. Peet Lecture 14: 16 / 28

17 Structure, Norm-Bounded Uncertainty For the case of structured parametric uncertainty, we define the structured set = { = diag(δ 1 I n1,, δ s I ns, s+1,, s+f ) : δ i R, R n k n k } δ and represent unknown parameters. s is the number of scalar parameters. f is the number of matrix parameters. M. Peet Lecture 14: 17 / 28

18 he Structured Singular Value For the case of structured parametric uncertainty, we define the structured singular value. Definition 14. Given system M L(L 2 ) and set as above, we define the Structured Singular Value of (M, ) as µ(m, ) = inf 1 I M is singular Of course, S(M, ) is stable if and only if µ(m 11, ) < 1. Obviously, µ(m, ) < M For := { L(L 2 ) : 1}, µ(m, ) = M µ(αm, ) = α µ(m, ) 1 Can increase M by a factor µ(m, ) before losing stability. In general, computing µ is NP-hard unless uncertainty is unstructured or block-diagonal. M. Peet Lecture 14: 18 / 28

19 Scalings and he Structured Singular Value Suppose Θ = {Θ : Θ = Θ for all } hen µ(m, ) = inf Θ Θ ΘMΘ 1. Θ is the set of scalings. M. Peet Lecture 14: 19 / 28

20 Scalings and he Structured Singular Value = { = diag(δ 1 I n1,, δ s I ns, s+1,, s+f ) : δ i R, R n k n k } Define the set of scalings PΘ := {diag(θ 1,, Θ s, θ s+1 I,, θ s+f I : Θ i > 0, θ j > 0} heorem 15. Suppose system M has transfer function ˆM(s) = C(sI A) 1 B + D with ˆM H. he following are equivalent here exists Θ Θ such that ΘMΘ 1 2 < γ. here exists Θ PΘ and X > 0 such that [ ] A X + XA XB B + 1 [ ] C X Θ γ 2 D Θ [ C D ] < 0 Note: o minimize γ, you must use bisection. M. Peet Lecture 14: 20 / 28

21 An LMI for Stability of Structured, Norm-Bounded Uncertainty his allows us to generalize the S-procedure to structured uncertainty heorem 16. he system ẋ(t) = Ax(t) + Mp(t), p(t) = (t)q(t), q(t) = Nx(t) + Qp(t),, 1 is quadratically stable if and only if there exists some Θ PΘ and P > 0 such that [ ] [ ] AP + P A P N MΘM MΘQ + NP 0 QΘM QΘQ < 0} Θ his is an LMI in Θ and P. he constraint Θ PΘ is linear PΘ := {diag(θ 1,, Θ s, θ s+1 I,, θ s+f I) : Θ i > 0, θ j > 0} M. Peet Lecture 14: 21 / 28

22 An LMI for Stability with Structured, Norm-Bounded Uncertainty o prove the theorem, we can take a closer look at the scalings: Since = for PΘ, the system can equivalently be written as ẋ(t) = Ax(t) + M 1 p(t), p(t) = (t)q(t), q(t) = Nx(t) + Q 1 p(t),, 1 for any PΘ. hen [ ] [ ] AP + P A P N MM MQ + NP 0 QM QQ < 0 I becomes [ AP + P A P N NP 0 Pre- and Post-multiplying by the LMI condition. ] [ M + 2 M M 2 Q Q 2 M Q 2 Q I [ I ] < 0} ], and using Θ = 2 PΘ, we recover M. Peet Lecture 14: 22 / 28

23 An LMI for Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty heorem 17. here exists a K such that the system with u(t) = Kx(t) ẋ(t) = Ax(t) + Bu(t) + Mp(t), p(t) = (t)q(t), q(t) = Nx(t) + Qp(t) + D 12 u(t),, 1 is quadratically stable if and only if there exists some Θ PΘ, P > 0 and Z such that [ ] [ ] AP + BZ + P A + Z B P N + Z D12 MΘM MΘQ + NP + D 12 Z 0 QΘM QΘQ < 0. Θ hen K = ZP 1 is a quadratically stabilizing controller. We can also extend this result to optimal control in the H norm. M. Peet Lecture 14: 23 / 28

24 An LMI for Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty In this case, we set Q = 0. heorem 18. here exists a K such that the system with u(t) = Kx(t) ẋ(t) = Ax(t) + Bu(t) + Mp(t) + B 2 w(t), q(t) = Nx(t) + D 12 u(t),, 1 y(t) = Cx(t) + D 22 u(t) p(t) = (t)q(t), satisfies y L2 γ u L2 if there exists some Θ PΘ, Z and P > 0 such that AP + BZ + P A + Z B + MΘM (CP + D 22 Z) P N + Z D12 CP + D 22 Z γ 2 I 0 < 0. NP + D 12 Z 0 Θ hen K = ZP 1 is the corresponding controller. M. Peet Lecture 14: 24 / 28

25 An LMI for Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty Using the scaled system we get ẋ(t) = Ax(t) + Bu(t) + M 1 p(t) + B 2 w(t), q(t) = Nx(t) + D 12 u(t),, 1 y(t) = Cx(t) + D 22 u(t) p(t) = (t)q(t), AP + BZ + P A + Z B + M 2 M (CP + D 22 Z) P N + Z D 12 CP + D 22 Z γ 2 I 0 < 0. NP + D 12 Z 0 I I 0 0 Pre- and Post-multiplying by 0 I 0, and using Θ = 2 PΘ, we recover the LMI condition. M. Peet Lecture 14: 25 / 28

26 Output-Feedback Robust Controller Synthesis How to Solve the Output Feedback Case??? inf K sup S( S(G, ), K) H M. Peet Lecture 14: 26 / 28

27 D-K Iteration A Heuristic for Dynamic Output Feedback Synthesis Finally, we mention a Heuristic for Output-Feedback Controller synthesis. Initialize: Θ = I. Define: A B 1 Θ 1 2 B 2 Ĝ Θ (s) = Θ 1 2 C 1 Θ 1 2 D 11 Θ 1 2 Θ 1 2 D 12 C 2 D 21 Θ Step 1: Fix Θ and solve inf K S(G Θ, K) H Step 2: Fix K and minimize γ such that there exists Θ PΘ ( or Θ PΘ I if you include the regulated output channel.) and X > 0 such that [ ] A cl X + XA cl XB cl Bcl X Θ + 1 [ ] C cl γ 2 Θ [ ] C cl D cl < 0 D cl where A cl, B cl, C cl, D cl define S(G I, K). (Requires Bisection). Step 3: GOO Step 1 M. Peet Lecture 14: 27 / 28

28 A Word on D-K Iteration with Static Uncertainty A Heuristic for Dynamic Output Feedback Synthesis he D-K iteration outlined in this lecture is only valid for Dynamic Uncertainty: (t). Our Scalings Θ are time-invariant. For Static uncertainties, we should search for Dynamic Scaling Factors Θ(s) is a ransfer Function his is much harder to represent as an LMI (Or by any other method!). Matlab has built-in functionality, but it is hard to use. We will return to µ analysis for static uncertainties when we consider more advanced forms of optimization. M. Peet Lecture 14: 28 / 28