Poluqenie rezulьtatov v vide. line inyx matreqnyh neravenstv dl robastnosti. Tradicionna Xkola Upravlenie, Informaci i Optimizaci

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1 Poluqenie rezulьtatov v vide line inyx matreqnyh neravenstv dl robastnosti Dmitri i Жanoviq Konovalov Dimitri i Poselь Dmitry Peaucelle Dimitri Peaucelle Tradicionna Xkola Upravlenie, Informaci i Optimizaci Pereslavlь-Zalesski i I nь 2010

Obtaining LMI results for robust control problems Dimitri Peaucelle LAAS-CNRS, Toulouse, FRANCE peaucelle@laas.

2 Obtaining LMI results for robust control problems Dimitri Peaucelle LAAS-CNRS, Toulouse, FRANCE Traditional School "Control, Information and Optimization" Pereslavl -Zalesskii June 2010

3 Outline ➊ Robust multi-objective problems Σ1 Π Σ Π K K K Σ Π ➋ Linear Matrix Inequalities & Optimization tools ➌ Manipulating inequalities to obtain LMI results ➍ Some LMI results A T P + P A < 0 Ax = b, x K A > BC 1 B T A B > 0 C > 0 B T C A T (ζ)p + P A(ζ) < 0 ζ A [v] P + P A [v] < 0 v {1... v} ➎ Solving robust multi-objective problems in RoMulOC» quiz= ctrpb( sf, unique )+h2(usys_h2)+hinf(usys_hinf,10); D. Peaucelle 1 Pereslavlь-Zalesski i I nь 2010

4 ➊ Robust multi-objective problems Standard robust analysis problem: w Σ z prove stability of (Σ w,z ) for all Standard robust design problem: w u Σ K z y Find K that guarantees stability of ((Σ w,z ) u,y K) for all contains unknown parameters, scheduling parameters, approximations of non-linearities, delays... Σ is a linear model: crude but simple representation of the system D. Peaucelle 2 Pereslavlь-Zalesski i I nь 2010

5 ➊ Robust multi-objective problems Generalizes to robust performance problems w w Σ z z Guarantee an input/output property for all w Σ w z u y K z Find a controller that guarantees input/output properties for all [2] Σ Σ() Same holds for polytopic-type models Σ [1] Σ [v] K D. Peaucelle 3 Pereslavlь-Zalesski i I nь 2010

6 ➊ Robust multi-objective problems Robust multi-objective problem Find a controller K that guarantees simultaneously several robust specifications Π 1, Π 2,... each of which being defined for a possibly different models (Σ 1 1 ), (Σ 2 2 ),... Σ 1 1 Π Σ 2 3 Π Σ Π K K K Example: Robust H 2 /H problem min ẋ = A()x + B u ()u z = u y = C y ()x + w u = Ky 2 : ẋ f = A f x f + B f w ẋ = A()x + B w ()x f + B u ()u z = C z ()x y = C y ()x u = Ky γ D. Peaucelle 4 Pereslavlь-Zalesski i I nь 2010

7 ➊ Robust multi-objective problems Σ 1 1 Π Σ 2 3 Π Σ Π K K K Naturally defined as existence (feasibility) problem over several constraints While a nominal performance Σ K = γ may be defined by an equality Robust performance (Σ ) K γ, can only be an inequality Finding the best robust controller: optimization problem over inequality constraints What type of optimization problem? Unique optimum? Convex? Convergence time?... LMI: Upper-bounds, Convex optimization, polynomial-time. D. Peaucelle 5 Pereslavlь-Zalesski i I nь 2010

8 Outline ➊ Robust multi-objective problems Σ1 Π Σ Π K K K Σ Π ➋ Linear Matrix Inequalities & Optimization tools ➌ Manipulating inequalities to obtain LMI results ➍ Some LMI results A T P + P A < 0 Ax = b, x K A > BC 1 B T A B > 0 C > 0 B T C A T (ζ)p + P A(ζ) < 0 ζ A [v] P + P A [v] < 0 v {1... v} ➎ Solving robust multi-objective problems in RoMulOC» quiz= ctrpb( sf, unique )+h2(usys_h2)+hinf(usys_hinf,10); D. Peaucelle 6 Pereslavlь-Zalesski i I nь 2010

9 ➋ Linear Matrix Inequalities & Optimization tools Convex cones A set K is a cone if for every x K and λ 0 we have λx K. A set is a convex cone if it is convex and a cone. D. Peaucelle 7 Pereslavlь-Zalesski i I nь 2010

10 ➋ Linear Matrix Inequalities & Optimization tools Convex cones Convex cone of positive reals: x R + { ( Second order (Lorentz) cone: Ksoc n = x = x 1... x n ), x x 2 n 1 x 2 n } K 3 soc : D. Peaucelle 8 Pereslavlь-Zalesski i I nь 2010

11 ➋ Linear Matrix Inequalities & Optimization tools Convex cones Convex cone of positive reals: x R + { ( ) Second order (Lorentz) cone: Ksoc n = x = x 1... x n Positive semi-definite matrices: K n psd = x = ( x 1 x n 2 ), mat(x) = mat(x) T =, x x 2 n 1 x 2 n x 1 x n+1 x n(n 1)+1. x n x 2n x n 2. } 0 K 2 psd : x 1 x 2 x 2 x 3 0 D. Peaucelle 9 Pereslavlь-Zalesski i I nь 2010

12 ➋ Linear Matrix Inequalities & Optimization tools Convex cones Convex cone of positive reals: x R + { ( ) Second order (Lorentz) cone: Ksoc n = x = x 1... x n ( Positive semi-definite matrices: Kpsd {x n = = x 1 x n 2 Unions of such: K = R + K n 1 soc... K n q psd }, x x 2 n 1 x 2 n ) }, mat(x) 0 D. Peaucelle 10 Pereslavlь-Zalesski i I nь 2010

13 ➋ Linear Matrix Inequalities & Optimization tools Optimization over convex cones p = min cx : Ax = b, x K Linear programming: K = R + R +. Semi-definite programming: K = K n 1 psd Kn q psd Dual problem d = max b T y : A T y c T = z, z K Primal feasible Dual infeasible Dual feasible Primal infeasible If primal and dual strictly feasible p = d Polynomial-time algorithms (O(n 6.5 log(1/ɛ))) D. Peaucelle 11 Pereslavlь-Zalesski i I nь 2010

14 ➋ Linear Matrix Inequalities & Optimization tools Optimization over convex cones p = min cx : Ax = b, x K Dual problem d = max b T y : A T y c T = z, z K Possibility to perform convex optimization, primal/dual, interior-point methods, etc. Interior-point methods [Nesterov, Nemirovski 1988] - Matlab Control Toolbox [Gahinet et al.] Primal-dual path-following predictor-corrector algorithms: SeDuMi (Sturm), SDPT3 (Toh, Tütüncü, Todd), CSDP (Borchers), SDPA (Kojima et al.) Primal-dual potential reduction: MAXDET (Wu, Vandenberghe, Boyd) Dual-scaling path-following algorithms: DSDP (Benson, Ye, Zhang) Barrier method and augmented Lagrangian: PENSDP (Kocvara, Stingl) Cutting plane algorithms... D. Peaucelle 12 Pereslavlь-Zalesski i I nь 2010

15 ➋ Linear Matrix Inequalities & Optimization tools Semi-Definite Programming and LMIs SDP formulation LMI formalism p = min cx : Ax = b, x K d = max b T y : A T y c T = z, z K d = min g i y i : F 0 + F i y i 0 p = max Tr(F 0 X) : Tr(F i X) + g i = 0, X 0 In control problems: variables are matrices The H norm computation example for G(s) (A, B, C, D) : G(s) 2 = min γ : P > 0, AT P + P A + Cz T C z B w P + Cz T D zw P Bw T + DzwC T z γ1 + DzwD T zw }{{} C T z D T zw C T z D T zw T 2 γ T +p < 0 Need for a nice parser D. Peaucelle 13 Pereslavlь-Zalesski i I nь 2010

16 ➋ Linear Matrix Inequalities & Optimization tools Parsers: LMIlab, tklmitool, sdpsol, SeDuMiInterface... YALMIP Convert LMIs to SDP solver format (all available solvers!) Simple to use >> P = sdpvar( 3, 3, symmetric ); >> lmiprob = lmi ( A *P+P*A<0 ) + lmi (P>0 ); >> solvesdp( lmiprob ); Works in Matlab - free! Extends to other non-sdp optimization problems (BMI...) SDP dedicated version in Scilab [S. Solovyev] D. Peaucelle 14 Pereslavlь-Zalesski i I nь 2010

17 ➋ Linear Matrix Inequalities & Optimization tools SDP-LMI issues and prospectives Any SDP representable problem is "solved (numerical problems due to size and structure) Find "SDP-ables" problems (linear systems, performances, robustness, LPV, saturations, delays, singular systems...) Equivalent SDP formulations distinguish which are numerically efficient New SDP solvers: faster, precise, robust (need for benchmark examples) D. Peaucelle 15 Pereslavlь-Zalesski i I nь 2010

18 ➋ Linear Matrix Inequalities & Optimization tools SDP-LMI issues and prospectives Any SDP representable problem is "solved (numerical problems due to size and structure) Find "SDP-ables" problems (linear systems, performances, robustness, LPV, saturations, delays, singular systems...) Equivalent SDP formulations distinguish which are numerically efficient New SDP solvers: faster, precise, robust (need for benchmark examples) Any "SDP-able problem has a dual interpretation New theoretical results (worst case) New proofs (Lyapunov functions = Lagrange multipliers; related to SOS) SDP formulas numerically stable (KYP-lemma) D. Peaucelle 16 Pereslavlь-Zalesski i I nь 2010

19 ➋ Linear Matrix Inequalities & Optimization tools SDP-LMI issues and prospectives Any SDP representable problem is "solved (numerical problems due to size and structure) Find "SDP-ables" problems (linear systems, performances, robustness, LPV, saturations, delays, singular systems...) Equivalent SDP formulations distinguish which are numerically efficient New SDP solvers: faster, precise, robust (need for benchmark examples) Any "SDP-able problem has a dual interpretation New theoretical results (worst case) New proofs (Lyapunov functions = Lagrange multipliers; related to SOS) SDP formulas numerically stable (KYP-lemma) Non "SDP-able" : Robustness & Multi-objective & Relaxation of NP-hard problems Optimistic / Pessimistic (conservative) results Reduce the gap (upper/lower bounds) while handling numerical complexity growth. D. Peaucelle 17 Pereslavlь-Zalesski i I nь 2010

20 ➋ Linear Matrix Inequalities & Optimization tools SDP-LMI issues and prospectives Any SDP representable problem is "solved (numerical problems due to size and structure) Find "SDP-ables" problems (linear systems, performances, robustness, LPV, saturations, delays, singular systems...) Equivalent SDP formulations distinguish which are numerically efficient New SDP solvers: faster, precise, robust (need for benchmark examples) Any "SDP-able problem has a dual interpretation New theoretical results (worst case) New proofs (Lyapunov functions = Lagrange multipliers; related to SOS) SDP formulas numerically stable (KYP-lemma) Non "SDP-able" : Robustesse & Multi-objective & Relaxation of NP-hard problems Optimistic / Pessimistic (conservative) results Reduce the gap (upper/lower bounds) while handling numerical complexity growth. Develop software for "industrial" application / adapted to the application field RoMulOC toolbox D. Peaucelle 18 Pereslavlь-Zalesski i I nь 2010

21 Outline ➊ Robust multi-objective problems Σ1 Π Σ Π K K K Σ Π ➋ Linear Matrix Inequalities & Optimization tools ➌ Manipulating inequalities to obtain LMI results ➍ Some LMI results A T P + P A < 0 Ax = b, x K A > BC 1 B T A B > 0 C > 0 B T C A T (ζ)p + P A(ζ) < 0 ζ A [v] P + P A [v] < 0 v {1... v} ➎ Solving robust multi-objective problems in RoMulOC» quiz= ctrpb( sf, unique )+h2(usys_h2)+hinf(usys_hinf,10); D. Peaucelle 19 Pereslavlь-Zalesski i I nь 2010

22 ➌ Manipulating inequalities to obtain LMI results Congruence A > 0 for any non zero vector x: x T Ax > 0. A > 0 for any full column rank matrix B: B T AB > 0. A > 0 for any matrix B: B T AB 0. A > 0 exists a square non-singular matrix B: B T AB > 0. Most LMI results are formulated as (sufficiency) If P... : L(P...) > 0 then the system ẋ = f(x, w...) is such that... To prove these results: perform congruence with vectors x, w... Example (Lyapunov): If P : P > 0, A T P + P A < 0 then the system ẋ = Ax is stable. Proof: V (x) = x T P x > 0, V (x) = x T (A T P + P A)x = 2ẋ T P x < 0 for all x 0. D. Peaucelle 20 Pereslavlь-Zalesski i I nь 2010

23 ➌ Manipulating inequalities to obtain LMI results Examples of nominal performance analysis: (P > 0) Stability (discrete-time) A T P A P < 0 [ ] Regional pole placement 1 A r 11P r 12P r 12 P r 22 P 1 A < 0 H norm AT P + P A + Cz T C z P B w + Cz T D zw BwP T + DzwC T z γ DzwD T zw < 0 H 2 norm Impulse-to-peak A T P + P A + Cz T C z < 0 trace(bwp T B w ) < γ 2 A T P + P A < 0 BwP T B w < γ 2 1 Cz T C z < P DzwD T zw < γ 2 1 D. Peaucelle 21 Pereslavlь-Zalesski i I nь 2010

24 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results Schur complement A > BC 1 B T C > 0 A B B T C > 0 Example: (AX + BS)X 1 (XA T + S T B T ) X < 0 X > 0 X AX + BS XA T + S T B T X < 0 D. Peaucelle 22 Pereslavlь-Zalesski i I nь 2010

25 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results Finsler lemma - Elimination lemma - Creation lemma x T Ax < 0 x : Bx = 0 τ R : A < τb T B X = X T : A < B T XB where B columns generate the null space of B: G : A < B T G T + GB B T AB < 0 B R p m, rank(b) = r < m, BB = 0, B R m (m r), B T B > 0 G is a Slack variable (Lagrange multiplier) D. Peaucelle 23 Pereslavlь-Zalesski i I nь 2010

26 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results Finsler lemma - Elimination lemma - Creation lemma x T Ax < 0 x : Bx = 0 τ R : A < τb T B X = X T : A < B T XB G : A < B T G T + GB B T AB < 0 Example: V (x) = T ẋ 0 P ẋ < 0, x P 0 x G : 0 P + 1 P 0 A T [ 1 A [ G T + G ] ẋ = 0 x ] 1 A < 0 D. Peaucelle 24 Pereslavlь-Zalesski i I nь 2010

27 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results Finsler lemma - Elimination lemma - Creation lemma C T AC < 0 H : A < B T H T C + C T HB B T AB < 0 Example P = [ 0 1 A T P A P = ] P P 1 [ ] A 1 P 0 0 P < 0 A 1 < 0 H : P 0 0 P < 1 A T H T [ 1 0 ] H [ 1 A ] D. Peaucelle 25 Pereslavlь-Zalesski i I nь 2010

28 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results S-procedure [Yakubovich] x T Mx < 0 x : x T Nx 0 τ > 0 : M < τn Example: T 1 M 1 < 0, T 1 z w T M z τ > 0 : M < τ w < 0, z w T z w 0 D. Peaucelle 26 Pereslavlь-Zalesski i I nь 2010

29 Tools to build LMI results S-procedure [Yakubovich] ➌ Manipulating inequalities to obtain LMI results x T Mx < 0 x : x T Nx 0 τ > 0 : M < τn Example: T 1 1 M < 0, T 1 (1 D) 1 C (1 D) 1 C T T x M x < 0, z 1 0 z w w w 0 1 w 0 τ > 0 : M < τ C D 0 1 z = Cx + Dw 1 0 C D T Special case: X + C T T B T + BC < 0, T 1 τ > 0 : X + τc T C + τ 1 BB T < 0 D. Peaucelle 27 Pereslavlь-Zalesski i I nь 2010

30 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results D and DG-scalling T 1 1 M < 0, = δ1 : δ C, δ 1 (1 D) 1 C (1 D) 1 C T Q > 0 : M < C D Q 0 C D Q 0 1 T 1 1 M (1 D) 1 C (1 D) 1 C Q > 0, T = T T : M < C D 0 1 < 0, = δ1 : δ R, δ 1 T Q T T T C D Q 0 1 D. Peaucelle 28 Pereslavlь-Zalesski i I nь 2010

31 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results Kalman-Yakubovich-Popov KYP lemma 1 1 M (jω1 A) 1 B (jω1 A) 1 B T Q : M < C D 0 Q 0 1 Q 0 < 0, ω R C D 0 1 And so on... Full-Block S-procedure [Scherer], Quadratic Separation [Iwasaki] 1 (1 D) 1 C M < C D M < 0, (1 D) 1 C T Θ C D, 1 Θ 1 < Difficulty: build the separator Θ, losslessly... D. Peaucelle 29 Pereslavlь-Zalesski i I nь 2010

32 ➌ Manipulating inequalities to obtain LMI results Tools to build LMI results S-procedure [Yakubovich] x T Mx < 0 x : x T Nx 0 τ > 0 : M < τn x T Mx < 0 x : x T N 1 x 0. x T N p x 0 τ 1 > 0,... τ p > 0 : M < τ 1 N 1 + τ p N p Not lossless except in few special cases: composed of m r scalar real repeated, m c scalar complex repeated, m F full complex non-repeated blocks [Meinsma et al.] DG-scalling lossless if 2(m r + m c ) + m F 3! D. Peaucelle 30 Pereslavlь-Zalesski i I nь 2010

33 Outline ➊ Robust multi-objective problems Σ1 Π Σ Π K K K Σ Π ➋ Linear Matrix Inequalities & Optimization tools ➌ Manipulating inequalities to obtain LMI results ➍ Some LMI results A T P + P A < 0 Ax = b, x K A > BC 1 B T A B > 0 C > 0 B T C A T (ζ)p + P A(ζ) < 0 ζ A [v] P + P A [v] < 0 v {1... v} ➎ Solving robust multi-objective problems in RoMulOC» quiz= ctrpb( sf, unique )+h2(usys_h2)+hinf(usys_hinf,10); D. Peaucelle 31 Pereslavlь-Zalesski i I nь 2010

34 ➍ Some LMI results Examples of nominal performance analysis: (P > 0) Stability (discrete-time) A T P A P < 0 [ ] Regional pole placement 1 A r 11P r 12P r 12 P r 22 P 1 A < 0 H norm AT P + P A + Cz T C z P B w + Cz T D zw BwP T + DzwC T z γ DzwD T zw < 0 H 2 norm Impulse-to-peak A T P + P A + Cz T C z < 0 trace(bwp T B w ) < γ 2 A T P + P A < 0 BwP T B w < γ 2 1 Cz T C z < P DzwD T zw < γ 2 1 D. Peaucelle 32 Pereslavlь-Zalesski i I nь 2010

35 ➍ Some LMI results Robust performance analysis: V (x, ) parameter-dependent Lyapunov function. Nominal analysis (LMI) Robust analysis (NP-hard) P : L Σ (P ) < 0, P () : L Σ() (P ()) < 0 Test over sample values { 1...N } gives optimistic results (some results exist if { 1...N } is uniform distribution of and large N ) D. Peaucelle 33 Pereslavlь-Zalesski i I nь 2010

36 ➍ Some LMI results Robust performance analysis: V (x, ) parameter-dependent Lyapunov function. Nominal analysis (LMI) Robust analysis (NP-hard) P : L Σ (P ) < 0, P () : L Σ() (P ()) < 0 Test over sample values { 1...N } gives optimistic results (some results exist if { 1...N } is uniform distribution of and large N ) Choice of P () for having a finite number of decision variables : Quadratic Stability : P () = P Polytopic PDLF: P () = ξ v P [v] [2] Σ Σ() P () polynomial w.r.t. ξ v 2 h i Quadratic-LFT PDLF: P () = 1 T C ˆP 4 1 C = (1 D ) 1 C C 3 5 w w u Σ [1] K Σ K Σ [v] z z y P () polynomial w.r.t. C D. Peaucelle 34 Pereslavlь-Zalesski i I nь 2010

37 ➍ Some LMI results Robust performance analysis: V (x, ) parameter-dependent Lyapunov function. Nominal analysis (LMI) Robust analysis (NP-hard) P : L Σ (P ) < 0, P () : L Σ() (P ()) < 0 Test over sample values { 1...N } gives optimistic results (some results exist if { 1...N } is uniform distribution of and large N ) Choice of P () for having a finite number of decision variables : Quadratic Stability : P () = P Polytopic PDLF: P () = ξ v P [v] 2 h i Quadratic-LFT PDLF: P () = 1 T C ˆP 4 1 LMIs over infinite number of variables, P () : L Σ() (P ()) < 0 P [v] or ˆP :, L Σ() (P ()) < 0 C 3 5 D. Peaucelle 35 Pereslavlь-Zalesski i I nь 2010

38 ➍ Some LMI results Conservative LMIs for polytopic models (Example of stability analysis) ẋ = A()x with A() = v v=1 ξ va [v] : ξ Ξ = {ξ v 0, v v=1 ξ v = 1} D. Peaucelle 36 Pereslavlь-Zalesski i I nь 2010

39 ➍ Some LMI results Conservative LMIs for polytopic models (Example of stability analysis) ẋ = A()x with A() = v v=1 ξ va [v] : ξ Ξ = {ξ v 0, v v=1 ξ v = 1} Quadratic Stability : P () = P V (x) < 0 A T ()P + P A() < 0 A [v]t P + P A [v] < 0 D. Peaucelle 37 Pereslavlь-Zalesski i I nь 2010

40 ➍ Some LMI results Conservative LMIs for polytopic models (Example of stability analysis) ẋ = A()x with A() = v v=1 ξ va [v] : ξ Ξ = {ξ v 0, Quadratic Stability : P () = P v v=1 ξ v = 1} V (x) < 0 A T ()P + P A() < 0 A [v]t P + P A [v] < 0 Polytopic PDLF: P () = ξ i P [i] T ẋ 0 P () ẋ x P () 0 x < 0 : [ A() 1 ] x ẋ = 0 Finsler Lemma 0 P () P () 0 [ + G() A() 1 ] + AT () 1 G T () < 0 G() = G & convexity 0 P [v] [ + G P [v] 0 A [v] 1 ] + A[v]T 1 G T < 0 D. Peaucelle 38 Pereslavlь-Zalesski i I nь 2010

41 ➍ Some LMI results Conservative LMIs for LFT models (Example of stability analysis) ẋ = Ax + B w with w = z = C x + D w D. Peaucelle 39 Pereslavlь-Zalesski i I nь 2010

42 ➍ Some LMI results Conservative LMIs for LFT models (Example of stability analysis) ẋ = Ax + B w with w = z = C x + D w Quadratic Stability : P () = P V (x) < 0 [ 1 (1 D ) 1 C ] [ A P + P A P B B T P 0 ] [ 1 (1 D ) 1 C ] < 0 Quadratic separation A P + P A P B B T P 0 < C 0 1 D T D Θ C 0 1 T, 1 Θ 1 < 0 D. Peaucelle 40 Pereslavlь-Zalesski i I nь 2010

43 ➍ Some LMI results Conservative LMIs for LFT models (Example of stability analysis) ẋ = Ax + B w with w = z = C x + D w Quadratic Stability : P () = P V (x) < 0 [ 1 (1 D ) 1 C ] [ A P + P A P B B T P 0 ] [ 1 (1 D ) 1 C ] < 0 Quadratic separation A P + P A P B B T P 0 < C 0 1 D T D Θ C 0 1 T, 1 Θ 1 < 0 Quadratic-LFT PDLF - same methodology (yet needs many matrix manipulations). D. Peaucelle 41 Pereslavlь-Zalesski i I nь 2010

44 ➍ Some LMI results State-feedback designs results Quadratic Stability case P () = P - Procedure 1- Write the analysis LMI results for the closed-loop (A() + B()K) T P + P (A() + B()K) < 0 2- Write it for the dual system (or perform congruence by X = P 1 ) (A() + B()K)X + X(A() + B()K) T < 0 3- Perform the change of variables S = KX (A()X + B()S) + (A()X + B()S) T < 0 4- Eliminate with previously explained methodology (A [v] X + B [v] S) + (A [v] X + B [v] S) T < 0 5- If a solution is found K = SX 1 solves the robust design problem. D. Peaucelle 42 Pereslavlь-Zalesski i I nь 2010

45 ➍ Some LMI results State-feedback designs results Quadratic Stability case P () = P - Multi-objective Σ K 1 1 Π Σ K 2 3 L Π1 (X 1, S 1 ) < 0, L Π2 (X 2, S 2 ) < 0, L Π3 (X 3, S 3 ) < 0 Lyapunov Shaping Paradigm : Solve the set of LMI for the same X = X i, S = S i Π K = SX 1 Σ K Π Procedure not fully applicable to Polytopic PDLF (non conservative structure on G?) 0 X[v] X [v] 0 + G T [ A [v]t + K T B [v]t 1 No results available for state-feedback with PDLF in the LFT case. ] + A[v] + B [v] K 1 G < 0 D. Peaucelle 43 Pereslavlь-Zalesski i I nь 2010

46 ➍ Some LMI results Full-order dynamic output-feedback results Only for some LFT models and Quadratic Stability case P () = P Change of variables proposed by [Scherer 97] Other control design problems? Not LMI, or still some unknown changes of variables to be found Many (non LMI) results for static-output feedback u = Ky (includes all control problems) Software based on variational analysis - nonsmooth optimization HIFOO: H-Infinity Fixed Order Optimization [Noll, Apkarian] Sold to Matlab c Other solvers PENBMI in PENOPT [Kocvara,Stingl] D. Peaucelle 44 Pereslavlь-Zalesski i I nь 2010

47 Outline ➊ Robust multi-objective problems Σ1 Π Σ Π K K K Σ Π ➋ Linear Matrix Inequalities & Optimization tools ➌ Manipulating inequalities to obtain LMI results ➍ Some LMI results A T P + P A < 0 Ax = b, x K A > BC 1 B T A B > 0 C > 0 B T C A T (ζ)p + P A(ζ) < 0 ζ A [v] P + P A [v] < 0 v {1... v} ➎ Solving robust multi-objective problems in RoMulOC» quiz= ctrpb( sf, unique )+h2(usys_h2)+hinf(usys_hinf,10); D. Peaucelle 45 Pereslavlь-Zalesski i I nь 2010

➎ Solving robust multi-objective problems in RoMulOC Helicopter example System defined at maximal value of parameters >> sysmax = ssmodel( Helicopter ); >> sysmax.a = [0 1 0 ;0 0 1;0-2.8-0.

48 ➎ Solving robust multi-objective problems in RoMulOC Helicopter example System defined at maximal value of parameters >> sysmax = ssmodel( Helicopter ); >> sysmax.a = [0 1 0 ;0 0 1; ]; >> sysmax.bw = [0;0;-14]; >> sysmax.bu = [0;0;8]; >> sysmax.dzu = 1 name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator ) D. Peaucelle 46 Pereslavlь-Zalesski i I nь 2010

49 ➎ Solving robust multi-objective problems in RoMulOC System defined at maximal value of parameters >> sysmax = ssmodel( Helicopter ); >> sysmax.a = [0 1 0 ;0 0 1; ]; >> sysmax.bw = [0;0;-14]; >> sysmax.bu = [0;0;8]; >> sysmax.dzu = 1; System defined at minimal value of parameters >> sysmin = ssmodel( Helicopter ); >> sysmin.a = [0 1 0 ;0 0 1; ]; >> sysmin.bw = [0;0;-14]; >> sysmin.bu = [0;0;8]; >> sysmin.dzu = 1 name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator ) D. Peaucelle 47 Pereslavlь-Zalesski i I nь 2010

50 ➎ Solving robust multi-objective problems in RoMulOC Uncertain system defined as interval of max and min >> usys = uinter( sysmin, sysmax ) Uncertain model : interval 2 param WITH name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator ) Interval model converted to polytopic model >> usys = u2poly( usys ) Uncertain model : polytope 4 vertices WITH name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator ) D. Peaucelle 48 Pereslavlь-Zalesski i I nь 2010

51 ➎ Solving robust multi-objective problems in RoMulOC Declare a state-feedback design problem >> quiz = ctrpb( state-feedback, Lyap-unique ) control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) No specified performance Add an H performance objective >> quiz = quiz + hinfty( usys, 4 ); Add a pole location performance objective >> r = region( plane, -0.1 ) Half-plane such that: Re(z)<-0.1 >> quiz = quiz + dstability( usys, r ) Add an impulse-to-peak performance minimization objective >> quiz = quiz + i2p( usys ) control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) Specified performances / systems: # Hinfty < 4 / Helicopter # D-stability / Helicopter # minimize I2P / Helicopter D. Peaucelle 49 Pereslavlь-Zalesski i I nь 2010

52 ➎ Solving robust multi-objective problems in RoMulOC The quiz object >> quiz control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) Specified performances / systems: # Hinfty < 4 / Helicopter # D-stability / Helicopter # minimize I2P / Helicopter Contains decision variables >> quiz.vars [3x3 sdpvar] Lyapunov matrix [1x3 sdpvar] S=-K*P [1x1 sdpvar] S-procedure scaling [1x1 sdpvar] g > (I2P cost)^2 D. Peaucelle 50 Pereslavlь-Zalesski i I nь 2010

53 ➎ Solving robust multi-objective problems in RoMulOC Constrained by LMIs >> quiz.lmi ID Constraint Type Tag #1 Numeric value Matrix inequality 3x3 Lyap >0 #2 Numeric value Matrix inequality 4x4 Var Lyap <0 #3 Numeric value Matrix inequality 4x4 Var Lyap <0 #4 Numeric value Matrix inequality 4x4 Var Lyap <0 #5 Numeric value Matrix inequality 4x4 Var Lyap <0 #6 Numeric value Matrix inequality 3x3 Var Lyap <0 #7 Numeric value Matrix inequality 3x3 Var Lyap <0 #8 Numeric value Matrix inequality 3x3 Var Lyap <0 #9 Numeric value Matrix inequality 3x3 Var Lyap <0 #10 Numeric value Matrix inequality 3x3 Constraint 1 #11 Numeric value Matrix inequality 4x4 Constraint 2 #12 Numeric value Matrix inequality 3x3 Constraint 3 #13 Numeric value Element-wise 1x1 Constraint 4 #14 Numeric value Matrix inequality 3x3 Constraint 1 #15 Numeric value Matrix inequality 4x4 Constraint 2 #16 Numeric value Matrix inequality 3x3 Constraint 3 #17 Numeric value Element-wise 1x1 Constraint 4 D. Peaucelle 51 Pereslavlь-Zalesski i I nь 2010

54 ➎ Solving robust multi-objective problems in RoMulOC #18 Numeric value Matrix inequality 3x3 Constraint 1 #19 Numeric value Matrix inequality 4x4 Constraint 2 #20 Numeric value Matrix inequality 3x3 Constraint 3 #21 Numeric value Element-wise 1x1 Constraint 4 #22 Numeric value Matrix inequality 3x3 Constraint 1 #23 Numeric value Matrix inequality 4x4 Constraint 2 #24 Numeric value Matrix inequality 3x3 Constraint 3 #25 Numeric value Element-wise 1x1 Constraint D. Peaucelle 52 Pereslavlь-Zalesski i I nь 2010

55 ➎ Solving robust multi-objective problems in RoMulOC And can be solved (SeDuMi solver by default) >> K = solvesdp( quiz ) SeDuMi 1.1R3 by AdvOL, 2006 and Jos F. Sturm, Alg = 2: xz-corrector, theta = 0.250, beta = eqs m = 11, order n = 76, dim = 250, blocks = 22 nnz(a) = , nnz(ada) = 117, nnz(l) = 64 it : b*y gap delta rate t/tp* t/td* feas cg cg prec 0 : 6.96E : -1.79E E E+02 2 : -1.05E E E+02 3 : -2.56E E E+01 4 : -5.54E E E+00 5 : -1.84E E E-01 6 : -7.08E E E-01 7 : -2.95E E E-02 8 : -2.30E E E-02 9 : -1.97E E E : -1.91E E E : -1.91E E E : -1.91E E E : -1.91E E E : -1.91E E E-08 D. Peaucelle 53 Pereslavlь-Zalesski i I nь 2010

56 ➎ Solving robust multi-objective problems in RoMulOC 15 : -1.91E E E : -1.91E E E-10 iter seconds digits c*x b*y Inf e e-01 Ax-b = 3.6e-10, [Ay-c]_+ = 2.6E-11, x = 5.0e-01, y = 3.7e+02 Detailed timing (sec) Pre IPM Post 1.800E E E-02 Max-norms: b =1, c = 196, Cholesky add =0, skip = 0, L.L = Feasibility is not strictly determined Worst constraint residual is e-11 < (=sqrt(double(ctrpb.vars{4}))) may be a guaranteed I2P norm K = D. Peaucelle 54 Pereslavlь-Zalesski i I nь 2010

57 ➎ Solving robust multi-objective problems in RoMulOC Welcome to new RoMulOC users Version 1 - started in june 2005 Contains robust analysis results presented in this talk. LMIs are coded using YALMIP parser, all available SDP solvers can be used. Version 2 - started in february 2007 Includes design facilities for robust multi-objective state-feedback. Coding LMI results for full order output-feedback design is planned. Version 3 - maybe this year Would include heuristic tools for solving static output-feedback design problems. D. Peaucelle 55 Pereslavlь-Zalesski i I nь 2010

Conclusions Robustnes: important issue in control theory Robustness results have impact for many other problems LMI: central tool for robustness Effective to transfer theory to industrial

58 Conclusions Robustnes: important issue in control theory Robustness results have impact for many other problems LMI: central tool for robustness Effective to transfer theory to industrial applications with software Quadratic Separation framework not yet fully exploited Extensions to non-linearities, time-varying operators... Descriptor version of RoMulOC: Romuald D. Peaucelle 56 Pereslavlь-Zalesski i I nь 2010

Robust Multi-Objective Control for Linear Systems

Robust Multi-Objective Control for Linear Systems Elements of theory and ROMULOC toolbox Dimitri PEAUCELLE & Denis ARZELIER LAAS-CNRS, Toulouse, FRANCE Part of the OLOCEP project (includes GloptiPoly)