S- peremennye dl� poluqeni� rezulьtatov v vide matriqnyh neravenstv i neskolьko rezulьtatov dl� robastnogo analiza duskretnyh sistem Dimitri PEAUCELLE / Dmitri i Жanoviq Poselь -Konovalov LAAS-CNRS - Université de Toulouse - FRANCE homepages.laas.fr/peaucell Sankt-Peterburg Okt�brь 2013
Slack variables for deriving LMI results and some examples of such results for robust analysis of discrete-time systems Dimitri Peaucelle Yoshio Ebihara To be submitted to 19th IFAC World Congress / Cape Town
Abstract Since year 2000 many results in the LMI-framework tend to introduce some additional (slack) variables that are apparently unnecessary from the Lyapunov theory. Such vari- ables are related to Finsler lemma that can also be seen as a variant of the S-procedure. The methodology producing these variables is sometimes called the descriptor form ap- proach and results sometimes designated as dilated, or extended, LMIs. In this talk we shall explain the rationale of all these appellations. We shall show that the slack variables are useful for robust analysis and are not if all system parameters are known. Issues of numerical complexity induced by the slack variables will also be discussed. Finally, if we have time, some new results for robust analysis of switching discrete-time systems will be exposed. Sankt-Peterburg, I�nь 2013 D. Peaucelle 1
LMIs ■ Positive definite matrices M ∈ R n × n M = M T ≻ 0 ∀ x � = 0 , x T Mx > 0 ⇔ ⇔ λ ( M ) > 0 + = { M ∈ R n × n : M = M T ≻ 0 } is an open convex cone ● S n Sankt-Peterburg, I�nь 2013 D. Peaucelle 2
LMIs ■ Linear matrix inequality constraints [BGFB94, NN94, EGN00] ▲ Representation with scalar decision variables � y i M i ≻ 0 , y i ∈ R M ( y ) = M 0 + ▲ Representation with matrix decision variables M ( Y s , Y f ) = M 0 + ( � N T si Y si N si ) + ( � N T 2 j Y fjT N 1 j ) 1 j Y fj N 2 j + N T Y si = Y siT ∈ R n si × n si , Y fj ∈ R m fj × p fj Sankt-Peterburg, I�nь 2013 D. Peaucelle 3
LMIs ■ About LMIs ● Convex constraints ● Exist efficient solvers (Semi-Definite Programming) for (polynomial-time) optimization ● Recommended (free) tool in Matlab : YALMIP users.isy.liu.se/johanl/yalmip ● A nice lecture about LMIs homepages.laas.fr/henrion/courses/lmi13/ ■ Any “LMI representable" problem is considered as “solved" ▲ Numerical burden grow very fast with size of problem: O ( n 6 ) ● Example: Global optimization over polynomials is (almost) “solved" ▲ See results on SOS and the moment problem [Las01, Las02, Las06, HL03], [Par03, PPSP04], [SH06] Sankt-Peterburg, I�nь 2013 D. Peaucelle 4
Historical key contributions ■ A.M. Lyapunov - “grand father of LMIs" ● Asymptotic stability of a linear system proved with existence of V s.t. V = x T Px > 0 , x k +1 T Px k +1 − x kT Px k < 0 , ∀ x k +1 � = 0 : x k +1 = Ax k � x T Px > 0 , x T ( A T PA − P ) x < 0 , ∀ x � = 0 � P ≻ 0 , A T PA − P ≺ 0 Sankt-Peterburg, I�nь 2013 D. Peaucelle 5
Historical key contributions ■ V.A. Yakubovich - “father of LMIs" ● S-procedure x T Q 0 x < 0 , ∀ x � = 0 : x T Q 1 x ≤ 0 ⇔ ∃ τ > 0 : Q 0 ≺ τQ 1 ▲ ( τ denoted s in first publication using this technique [E.N. Rozenvasser 1963]) ● KYP lemma ∗ x x < 0 , ∀ u � = 0 , ∀ ω ∈ R : ( e ω I − A ) x = Bu M u u T A B P 0 A B ⇔ ∃ P = P T : M ≺ − P I 0 0 I 0 Sankt-Peterburg, I�nь 2013 D. Peaucelle 6
Historical key contributions ■ P . Finsler ● Finsler’s lemma y T My < 0 , ∀ y � = 0 : By = 0 ⇔ B ⊥ T MB ⊥ ≺ 0 : BB ⊥ = 0 , rank ( B ) = dim ( ker ( B )) : M ≺ τB T B ⇔ ∃ τ : M ≺ FB + B T F ⇔ ∃ F ▲ Example: S-procedure y T My < 0 , ∀ y � = 0 : By = 0 ⇔ y T My < 0 , ∀ y � = 0 : y T B T By ≤ 0 : M ≺ τB T B ⇔ ∃ τ Sankt-Peterburg, I�nь 2013 D. Peaucelle 7
Historical key contributions ■ P . Finsler y T My < 0 , ∀ y � = 0 : By = 0 ⇔ B ⊥ T MB ⊥ ≺ 0 : BB ⊥ = 0 , rank ( B ) = dim ( ker ( B )) : M ≺ τB T B ⇔ ∃ τ : M ≺ FB + B T F ⇔ ∃ F � T � x k +1 T x kT y = ▲ Example: The Lyapunov result P 0 � � y < 0 : ∀ y � = 0 : V k +1 − V k = y T y = 0 − A I − P 0 P 0 A B ⊥ ≺ 0 : B ⊥ = ⇔ A T PA − P = B ⊥ T 0 − P I � � y = 0 . ● Notice the descriptor-like representation of the model: − A I Sankt-Peterburg, I�nь 2013 D. Peaucelle 8
Slack variables result ● Finsler’s lemma y T My < 0 , ∀ y � = 0 : By = 0 ⇔ B ⊥ T MB ⊥ ≺ 0 : BB ⊥ = 0 , rank ( B ) = dim ( ker ( B )) : M ≺ τB T B ⇔ ∃ τ : M ≺ FB + B T F ⇔ ∃ F ■ What if introducing F ? Approach known as: “Finsler based" - “Slack variables" - “dilated LMI" - “extended LMI" - “descriptor" ● For robust analysis results (LTI, LTV, TDS, Periodic...): [PABB00, DOS01, OG05, EPAH05, PDSV09, EPA09, PS09, TPAE13]... ● For robust state-feedback and filter design: [OBG99, OGH99, AP00, GdOB02, FS02, EH02, EH04, PG05, EPA11]... ● For output feedback and anti-windup design: [APT00, AHP02, PA01, ACP06, GKB07, AGPP10, TGGdSJQ11]... Sankt-Peterburg, I�nь 2013 D. Peaucelle 9
Slack variables result ● Finsler’s lemma y T My < 0 , ∀ y � = 0 : By = 0 ⇔ B ⊥ T MB ⊥ ≺ 0 : BB ⊥ = 0 , rank ( B ) = dim ( ker ( B )) : M ≺ τB T B ⇔ ∃ τ : M ≺ FB + B T F ⇔ ∃ F ■ What if introducing F ? ● Example: stability of x k +1 = Ax k : P 0 I � � ≺ F F T ∃ P ≻ 0 , F : + I − A − A T 0 − P ▲ 2 n × 2 n LMIs with n ( n +1) + 2 n 2 variables !! ( n × n and n ( n +1) in original problem) 2 2 ▲ Why using such a numerically expensive condition ? Sankt-Peterburg, I�nь 2013 D. Peaucelle 10
Example of a discrete-time system with uncertainties ■ Discrete-time system with uncertainties a ∈ [ 1 , 2] , b ∈ [ − 0 . 5 , β ] . ay k +2 + b 2 y k +1 + aby k = 0 . ● By hand: Robust stability is guaranteed for β < 1 . ● Can we build an LMI problem that guarantees robust stability for fixed β ? Sankt-Peterburg, I�nь 2013 D. Peaucelle 11
Polytopic embedding system approach ■ Discrete-time system with uncertainties a ∈ [ 1 , 2] , b ∈ [ − 0 . 5 , β ] . ay k +2 + b 2 y k +1 + aby k = 0 . − b 2 /a − b x k = A ( a, b ) x k . ● State-space representation x k +1 = 1 0 ▲ Interval arithmetics: b 2 /a ∈ [ 0 , β 2 ] (assuming β ≥ 0 . 5 ) 0 0 . 5 0 − β a ∈ [ 1 , 2] 1 0 1 0 ⊂ CO A ( a, b ) , − β 2 − β 2 0 . 5 − β b ∈ [ − 0 . 5 , β ] 1 0 1 0 ● Enters the general formulation of polytopic systems x k +1 = A ( θ ) x k � v ] � A [1] , A [2] , . . . A [¯ A ( θ ) ∈ CO ▲ I.e. A ( θ ) = � ¯ v v = { θ v ≥ 0 , � ¯ v v =1 θ v A [ v ] where θ ∈ Ξ ¯ v =1 θ v = 1 } . Sankt-Peterburg, I�nь 2013 D. Peaucelle 12
“Quadratic stability" results [Bar85] ■ Robust stability condition: existence of P ( θ ) s.t. P ( θ ) ≻ 0 , A ( θ ) T P ( θ ) A ( θ ) − P ( θ ) ≺ 0 , ∀ θ ∈ Ξ ¯ v ■ Conservative assumption: P ( θ ) = P unique Lyapunov Matrix ∀ θ P ≻ 0 , A ( θ ) T PA ( θ ) − P ≺ 0 , ∀ A ( θ ) ∈ CO � v ] � A [1] , A [2] , . . . A [¯ ● Convexity of S + allows to conclude that ⇔ P ≻ 0 , A [ v ] T PA [ v ] − P ≺ 0 , ∀ v = 1 . . . ¯ v ▲ For the example, LMIs feasible up to β = 0 . 7057 (far from the actual upper bound) � � − β 2 0 . 5 is unstable as soon as β = 0 . 7071 ▲ Actually, vertex 1 0 ● Need for better representations of the model with uncertainties Sankt-Peterburg, I�nь 2013 D. Peaucelle 13
Descriptor model representation ay k +2 + b 2 y k +1 + aby k = 0 . ▲ Equivalent model, affine in the uncertainties a 0 b 0 0 x k +1 + π k = x k . 0 1 0 1 0 0 0 1 b a ■ General descriptor model to be considered here: E x ( θ ) x k +1 + E π ( θ ) π k = F ( θ ) x k � � ● Assumption: E ( θ ) = square invertible ∀ θ ∈ Ξ ¯ E x ( θ ) E π ( θ ) v ▲ System is causal, without impulsive modes, π k well defined for all k ≥ 0 . Sankt-Peterburg, I�nь 2013 D. Peaucelle 14
Stability of the descriptor model ■ General descriptor model: x k +1 � � M ( θ ) y k = = 0 E x ( θ ) E π ( θ ) − F ( θ ) π k x k ■ Robust stability if exists V ( k, θ ) = x kT P ( θ ) x k such that P ( θ ) 0 0 V ( k + 1 , θ ) = y kT y k < 0 , ∀ y k � = 0 : M ( θ ) y k = 0 0 0 0 − V ( k, θ ) − P ( θ ) 0 0 ● Finsler lemma: equivalent condition (should hold ∀ θ ∈ Ξ ¯ v ). P ( θ ) 0 0 ∃ P ( θ ) ≻ 0 < F ( θ ) M ( θ ) + M T ( θ ) F T ( θ ) : 0 0 0 ∃ F ( θ ) 0 0 − P ( θ ) Sankt-Peterburg, I�nь 2013 D. Peaucelle 15
Recommend
More recommend
