Poluqenie rezulьtatov v vide line inyx matreqnyh neravenstv dl� robastnosti Dmitri i Жanoviq Konovalov Dmitry Peaucelle Dimitri i Poselь 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.fr http://homepages.laas.fr/peaucell Traditional School "Control, Information and Optimization" Pereslavl’-Zalesskii June 2010
Outline ∆ ∆ ∆ 1 2 3 Σ Σ Σ ➊ Robust multi-objective problems Π Π Π ✛ ✛ 1 1 2 3 2 3 K K K A T P + PA < 0 ↔ Ax = b , x ∈ K ➋ Linear Matrix Inequalities & Optimization tools A > BC − 1 B T A B > 0 ⇔ ➌ Manipulating inequalities to obtain LMI results B T C C > 0 A T ( ζ ) P + PA ( ζ ) < 0 ∀ ζ ∈ ∆ ∆ ➍ Some LMI results � A [ v ] ′ P + PA [ v ] < 0 ∀ v ∈ { 1 . . . ¯ v } ➎ Solving robust multi-objective problems in RoMulOC » quiz= ctrpb(’sf’,’unique’)+h2(usys_h2)+hinf(usys_hinf,10); Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 1
➊ Robust multi-objective problems ■ Standard robust analysis problem: ∆ w z ∆ w ∆ ,z ∆ ∆ ∆) for all ∆ ∈ ∆ prove stability of (Σ ⋆ ∆ Σ ■ Standard robust design problem: ∆ w z ∆ ∆ Σ w ∆ ,z ∆ u,y Find K that guarantees stability of ((Σ ⋆ ∆) ⋆ K ) for all ∆ ∈ ∆ ∆ u y K ● ∆ ∆ contains unknown parameters, scheduling parameters, approximations of non-linearities, delays... ● Σ is a linear model: crude but simple representation of the system Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 2
➊ Robust multi-objective problems ■ Generalizes to robust performance problems ∆ w z ∆ ∆ Guarantee an input/output property for all ∆ ∈ ∆ ∆ ● Σ z w ∆ w z ∆ ∆ Σ w z Find a controller that guarantees input/output properties for all ∆ ∈ ∆ ∆ ● u y K [2] Σ Σ(∆) ● Same holds for polytopic-type models [1] Σ [v] Σ K Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 3
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 ) , . . . ∆ ∆ ∆ 1 2 3 Σ Σ Σ Π Π Π ✛ ✛ 1 1 2 3 2 3 K K K ● Example: Robust H 2 /H ∞ problem � � � � x f = A f x f + B f w ˙ � � � � � � x = A (∆) x + B u (∆) u ˙ � � � � � � x = A (∆) x + B w (∆) x f + B u (∆) u ˙ � � � � � z = u � � � � � min : ≤ γ ∞ z = C z (∆) x � � � � � � y = C y (∆) x + w � � � � � � y = C y (∆) x � � � � � � u = Ky � � � � 2 u = Ky � � ∞ Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 4
➊ Robust multi-objective problems ∆ ∆ ∆ 1 2 3 Σ Π Σ Σ Π Π ✛ ✛ 1 1 2 3 2 3 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. Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 5
Outline ∆ ∆ ∆ 1 2 3 Σ Σ Σ ➊ Robust multi-objective problems Π Π Π ✛ ✛ 1 1 2 3 2 3 K K K A T P + PA < 0 ↔ Ax = b , x ∈ K ➋ Linear Matrix Inequalities & Optimization tools A > BC − 1 B T A B > 0 ⇔ ➌ Manipulating inequalities to obtain LMI results B T C C > 0 A T ( ζ ) P + PA ( ζ ) < 0 ∀ ζ ∈ ∆ ∆ ➍ Some LMI results � A [ v ] ′ P + PA [ v ] < 0 ∀ v ∈ { 1 . . . ¯ v } ➎ Solving robust multi-objective problems in RoMulOC » quiz= ctrpb(’sf’,’unique’)+h2(usys_h2)+hinf(usys_hinf,10); Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 6
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. Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 7
➋ Linear Matrix Inequalities & Optimization tools ■ Convex cones ▲ Convex cone of positive reals: x ∈ R + � � � � ▲ Second order (Lorentz) cone: K n , x 2 1 + . . . x 2 n − 1 ≤ x 2 soc = x = x 1 . . . x n n K 3 : soc Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 8
➋ Linear Matrix Inequalities & Optimization tools ■ Convex cones ▲ Convex cone of positive reals: x ∈ R + � � � � , x 2 1 + . . . x 2 n − 1 ≤ x 2 ▲ Second order (Lorentz) cone: K n soc = x = x 1 . . . x n n ▲ Positive semi-definite matrices: · · · x 1 x n +1 x n ( n − 1)+1 x = mat ( x ) . . � , K n psd = = mat ( x ) T = . . ≥ 0 � . . x 1 · · · x n 2 x n x 2 n · · · x n 2 x 1 x 2 ≥ 0 K 2 : psd x 2 x 3 Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 9
➋ Linear Matrix Inequalities & Optimization tools ■ Convex cones ▲ Convex cone of positive reals: x ∈ R + � � � � ▲ Second order (Lorentz) cone: K n , x 2 1 + . . . x 2 n − 1 ≤ x 2 soc = x = x 1 . . . x n n � � � � ▲ Positive semi-definite matrices: K n psd = x = , mat ( x ) ≥ 0 · · · x 1 x n 2 soc × . . . × K n q ▲ Unions of such: K = R + × · · · × K n 1 psd × · · · Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 10
➋ Linear Matrix Inequalities & Optimization tools ■ Optimization over convex cones p ⋆ = min cx x ∈ K : Ax = b , ▲ Linear programming: K = R + × · · · R + . psd × · · · K n q ▲ Semi-definite programming: K = K n 1 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 /ǫ )) ) Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 11
➋ Linear Matrix Inequalities & Optimization tools ■ Optimization over convex cones p ⋆ = min cx x ∈ K : Ax = b , ● 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 ... Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 12
➋ Linear Matrix Inequalities & Optimization tools ■ Semi-Definite Programming and LMIs p ⋆ = min cx : Ax = b , x ∈ K ● SDP formulation 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 : ● LMI formalism 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 ) : A T P + PA + C T B w P + C T z C z z D zw � G ( s ) � 2 = min γ : P > 0 , < 0 ∞ PB T w + D T − γ 1 + D T zw C z zw D zw � �� � T T 2 3 2 3 2 3 2 3 C T C T 0 0 z z 6 7 6 7 6 7 6 7 + p 11 ... − γ 6 7 6 7 6 7 6 7 D T D T 4 5 4 5 4 5 4 5 1 1 zw zw ■ Need for a nice parser Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 13
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! http://users.isy.liu.se/johanl/yalmip ▲ Extends to other non-SDP optimization problems (BMI...) ▲ SDP dedicated version in Scilab [S. Solovyev] http://www.laas.fr/OLOCEP/SciYalmip Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 14
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