Ellipsoidal Sets for Static Output-Feedback by Dimitri Peaucelle - PowerPoint PPT Presentation

IFAC World Congress July 21-26 2002, Barcelona Ellipsoidal Sets for Static Output-Feedback by Dimitri Peaucelle & Denis Arzelier & Regis Bertrand L aboratoire d’ A nalyse et d’ A rchitecture des S yst` emes du C.N.R.S. Toulouse, FRANCE

Motivation 1 � � � � h i Θ � ∆ � ∆ � ∆ ∆ � � � � � � � � � ∆ � � � � � Θ : � � � Σ � j ω � Σ � � h i Θ � ω Σ � j ω � � � � R � � � � � � � � � � Topological separation for robust analysis Ellipsoidal Sets for Static Output-Feedback

Motivation 1-a � : for synthesis � � � � Σ � j ω � h i Θ � ω � Σ � j ω Σ � � � � R � � � � � � � � � � � Θ : � � � � � h i K Θ � � � � non-empty set � K � � � � � K � Ellipsoidal Sets for Static Output-Feedback

Matrix ellipsoids 2 R n Ellipsoids of the vector space R n � n s.t. Z R n , radius r and geometry Z � � k Z k � 1. Centre x o � , � R n � � Z � � � x � x o � � x � x o � � r x : R m � p Matrix ellipsoids of R p � p and geometry Z R m � m s.t. Z R m � p , radius R � � � k Z k � 1. Centre K o � , � R m � p � � Z � � � K � K o � � K � K o � � R : K f X , Y , Z g -ellipsoid : definition � � � � � � � X Y � � h i � R m � p � � � K : � � Z � � K � � � � � � Y Z K � Ellipsoidal Sets for Static Output-Feedback

Matrix ellipsoids 3 � , R � 1 Y ✪ Algebraic rules: K o � � Z � K o � ZK o � X . � 1 Y ✪ Non-emptiness condition : R � � � � YZ � X � � � YK � K � Y � � Z � X K ✪ LMI description : � � � � Z ZK ✪ f X , Y , Z g -ellipsoid : a compact convex set. � R � m q det ✪ VOL � f X , Y , Z g -ellipsoid � � � VOL � f� � , � g -ellipsoid � . � , � p � Z det Ellipsoidal Sets for Static Output-Feedback

Static output-feedback 4 � � t � � Ax � t � � Bu � t � Σ x ˙ � � � � : Σ y � t � � Cx � t � � Du � t � Notations � K : � K � � u � t � � Ky � t � � Σ is stabilisable via static output-feedback Σ � K s.t. iff � K is stable. f X , Y , Z g -ellipsoid s.t.: iff there exist a Lyapunov matrix P and a non-empty � � � � � � � � � � � � � � � � � P � � � C D � X Y � C D � � � � � � � � � � � A B P � A B � � Y Z � � Ellipsoidal Sets for Static Output-Feedback

Remarks 5 A set of control laws: � Px proves Σ � x � � x f X , Y , Z g -ellipsoid. V � K stability for any gain K in the The non-convex constraint � 1 Y � � LMIs � a non-linear constraint ( X � YZ � ). SOF stabilisability y R 1 � 1 : z � y 2 � � 1 et x Example for K x Ellipsoidal Sets for Static Output-Feedback

Fragility and resilience 6 Definition : Let K o be a stabilising gain and ∆ � K an additive uncertainty. � ∆ K � ∆ � K s.t. Σ � ∆ K ✪ Fragile : � K o � is unstable. � � ∆ K � ∆ Σ � ∆ K ✪ Resilient : � K � K o � is stable. : � ✪ Quadratically resilient : � Px ) proves the resiliency. � x � � x A unique quadratic Lyapunov function ( V Ellipsoidal Sets for Static Output-Feedback

Fragility and resilience 7 Corollaries ✪ ∆ � K : ellipsoidal matrix set centred at the origin. � 1 Y � non-linear constraint X � YZ � � � LMI constraint g -ellipsoid is quadratically resilient to ∆ K Z ∆ K � f X , Y , Z � R . centre K o of the ✪ ∆ � K : norm-bounded uncertainty. � ρ � Z � � non-linear constraint � YY � � X � � � � � LMI constraint g -ellipsoid is quadratically resilient to ∆ K ∆ K � ρ f X , Y , Z � centre K o of the � . ✪ ∆ δ � K : multiplicative uncertainty with radius ¯ � ¯ δ 2 � 1 Y � non-linear constraint X � � 1 � YZ � � � LMI constraint g -ellipsoid is quadratically resilient to ∆ � δ K o with j δ δ . � ¯ � f X , Y , Z j centre K o of the K Ellipsoidal Sets for Static Output-Feedback

Bounded or dissipative specifications on K 8 Definition f X g -ellipsoid. Design a stabilising control law K that belongs to a given K , Y K , Z K � ρ K � K Example 1 : Find a control law with bounded gain ( K � ) � � K � Exemple 2 : Find a passive control law ( K � ) LMI constraint ν � � 0 � � � � � � � � X K Y K � X Y ν � � � � � � � Y K Z K Y Z � Ellipsoidal Sets for Static Output-Feedback

Pole location 9 f X R , Y R , Z R g -stability The poles of Σ � K belong to an ellipsoidal region of the complex plane: � Y R � � Z R f s � � sY R � s � ss g C : X R � � Examples: half-planes, discs, sectors, parabolas... f X R , Y R , Z R g -stabilisability Static output-feedback � � � � � � � � � P � P � X � Y � C � D � X R Y R � � � � � � � � h i h i � � � � � � � � � � � � P � P � A � B � Y � � Z � � � � � � Y R Z R � non-linear constraint Ellipsoidal Sets for Static Output-Feedback

Extensions to multi-objective synthesis 10 ∆ ∆ Λ ∋ Σ <γ {X ,Y ,Z }−stable <γ 3 R R R 2 <γ 1 ∋ K {X ,Y ,Z }−ellip. K K K ✪ An LMI constraint for each specification. ✪ A Lyapunov function for each specification. � 1 Y ✪ A unique non-linear constraint ( X � YZ � ) Ellipsoidal Sets for Static Output-Feedback

Algorithms to solve non-linear matrix inequalities 11 ✪ SOF design has no convex formulation in the general case. ✪ Elimination based approach � LMIs � ( XY � � ) proved to be NP-hard [Fu & Luo 1997]. ✪ Heuristic algorithms such as coordinated-descent iterative resolutions of BMIs. ✪ Efficient sub-optimal first order algorithms: ✫ Cone complementarity algorithm [El Ghaoui & al 1997]. ✫ Alternation projection algorithm [Grigoriadis & Skelton 1996]. Ellipsoidal Sets for Static Output-Feedback

Algorithms to solve non-linear matrix inequalities 12 Numerical experiments with cone complementarity algorithm δ 2 � ¯ � 1 Y ✪ Considered non-linear constraint: X � . � � 1 � YZ δ 2 � ¯ � ˆ � 1 Y � ˆ ✪ Linear relaxation: � � 1 � X X YZ X ✪ Algorithm designed for: ˆ � 1 Y � � YZ � . X � ¯ δ 2 � 1 Y ✪ Stopping criterium: X � � 1 � YZ � . Ellipsoidal Sets for Static Output-Feedback

Algorithms to solve non-linear matrix inequalities 13 � h i R 2 � 1 . � SOF design such that K k 1 k 2 � Poles location ( Re � pˆ � � 0 � 15) Stability oles � � h i and resiliency bounded K (radius 10, center ). 10 10 and resiliency 18 1.3 1.2 1.2 16 16 1.1 14 1 1 δ = 0.5 k 12 12 δ=0.5 2 0.9 δ = 0.25 0.8 0.8 k 2 10 δ = 0 0.7 8 8 0.6 0.6 δ = 0.25 6 0.5 4 0.4 0.4 δ = 0 4 0.3 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.4 −0.3 −0.2 2 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 1 2 3 0 k 1 k 1 Ellipsoidal Sets for Static Output-Feedback

Conclusions and prospectives 14 ✪ New static output-feedback design based on the topological separation theory. ✪ No conservatism when compared to LMI analysis techniques. ✪ Encouraging numerical results. ✫ Develop new adapted algorithms. ✫ Extensions to other design problems. Ellipsoidal Sets for Static Output-Feedback