Integral Quadratic Separation Framework Dimitri PEAUCELLE LAAS-CNRS - Universit´ e de Toulouse - FRANCE joint work with Denis Arzelier and Fr´ ed´ eric Gouaisbaut Seminar at UFSC, Florian´ opolis October 2009
Outline ➊ Topological separation & related theory ● Well-posedness definition and main result ● Relations with Lyapunov theory ● The case of linear uncertain systems : quadratic separation ● The Lur’e problem ● Relations with IQC framework & µ -theory ● A S -procedure like result ➋ Integral Quadratic Separation (IQS) for the descriptor case ➌ Performances in the IQS framework ➍ System augmentation : a way towards conservatism reduction ➎ The Romuald toolbox 1 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Well-posedness w G (z, w)=0 w Well-Posedness: z w Bounded ( ¯ w, ¯ z ) ⇒ unique bounded ( w, z ) F (w, z)=0 z z F G � �� � � �� � z = Aw + ¯ z and w = ∆ z + ¯ w are linear applications ● In case Well-posedness : ( 1 − A ∆) non-singular ▲ What if ∆ = ∆ ∈ ∆ ∆ is uncertain ? ▲ If A = T ( jω ) is an LTI system ? ▲ If G is non-linear ? ... 2 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Well-posedness & topological separation Well-Posedness: w G (z, w)=0 w Bounded ( ¯ w, ¯ z ) z � � � � w � � � � w w ¯ � � � � F (w, z)=0 ⇒ ∃ !( w, z ) , ∃ γ : ≤ γ � � � � z � � � � z z ¯ z � � � � ● [Safonov 80] ∃ θ topological separator: G I ( ¯ w ) = { ( w, z ) : G ¯ w ( z, w ) = 0 } ⊂ { ( w, z ) : θ ( w, z ) ≤ φ 2 ( || ¯ w || ) } F (¯ z ) = { ( w, z ) : F ¯ z ( w, z ) = 0 } ⊂ { ( w, z ) : θ ( w, z ) > − φ 1 ( || ¯ z || ) } 3 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ For dynamic systems ˙ x = f ( x ) , topological separation ≡ Lyapunov theory G � �� � F � t � �� � z ( t ) = f ( w ( t )) + ¯ z ( t ) , w ( t ) = z ( τ ) dτ + ¯ w ( t ) ���� ���� 0 x ( t ) x ( t ) ˙ ▲ ¯ w : contains information on initial conditions ( x (0) = 0 by convention) ● Well-posedness ⇒ for zero initial conditions and zero perturbations : w = z = 0 (equilibrium point). ● Well-posedness (global stability) ⇒ whatever bounded perturbations the state remains close to equilibrium 4 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ For dynamic systems ˙ x = f ( x ) , topological separation ≡ Lyapunov theory G F � �� � � t � �� � z ( t ) , z ( t ) = f ( w ( t )) + ¯ w ( t ) = z ( τ ) dτ + ¯ w ( t ) ���� ���� 0 x ( t ) x ( t ) ˙ ● Assume a Lyapunov function V (0) = 0 , V ( x ) > 0 , ˙ V ( x ) < 0 ▲ Topological separation w.r.t. G I ( ¯ w ) is obtained with � ∞ − ∂V θ ( w = x, z = ˙ x ) = ∂x ( x ( τ )) ˙ x ( τ ) dτ = lim t →∞ − V ( x ( t )) < γ 1 � ¯ w � 0 ▲ Topological separation w.r.t. F (¯ z ) does hold as well � ∞ − ˙ θ ( w, z = f ( w )) = V ( w ( τ )) dτ > − γ 2 � ¯ z � 0 5 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ For linear systems : quadratic Lyapunov function, i.e. quadratic separator G F � �� � � t � �� � z ( t ) , z ( t ) = Aw ( t ) + ¯ w ( t ) = z ( τ ) dτ + ¯ w ( t ) ���� ���� 0 x ( t ) x ( t ) ˙ ● A possible separator based on quadratic Lyapunov function V ( x ) = x T Px � � � � � � ∞ � − P z ( τ ) 0 z T ( τ ) w T ( τ ) θ ( w, z ) = dτ − P w ( τ ) 0 0 ▲ Quadratic separation w.r.t. G I ( ¯ w ) : t →∞ − x T ( t ) Px ( t ) ≤ γ 1 � ¯ lim w � , i.e. P > 0 ▲ Quadratic separation w.r.t. F (¯ z ) guaranteed if − 2 w T ( t ) PAw ( t ) > − γ 2 � ¯ z ( t ) � , i.e. A T P + PA < 0 ∀ t > 0 , 6 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Topological separation : alternative to Lyapunov theory ▲ Needs to manipulate systems in a new form ● Suited for systems described as feedback connected blocs Any linear system with rational dependence w.r.t. parameters writes as such x = Ax + B ∆ w ∆ ˙ LFT z ∆ = C ∆ x + D ∆ w ∆ x = ( A + B ∆ ∆( 1 − D ∆ ∆) − 1 C ∆ ) x ˙ ← → w ∆ = ∆ z ∆ ▲ Finding a topological separator is a priori as complicated as finding a Lyapunov function ● Allows to deal with several features simultaneously in a unified way 7 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Quadratic separation [Iwasaki & Hara 1998] ● If F ( w ) = Aw is a linear transformation and G = ∆ is an uncertain operator defined as ∆ ∈ ∆ ∆ convex set it is necessary and sufficient to look for a quadratic separator � � � ∞ � � z θ ( z, w ) = Θ dτ z T w T w 0 ● If F ( w ) = A ( ω ) w is a linear parameter dependent transformation and G = ∆ is an uncertain operator defined as ∆ ∈ ∆ ∆ convex set necessary and sufficient to look for a parameter-dependent quadratic separator � � � ∞ � � z θ ( z, w ) = z T w T Θ( ω ) dτ w 0 8 October 2009, Florian´ opolis
➊ Topological separation & related theory w G (z, w)=0 w z ■ A well-known example : the Lur’e problem w F (w, z)=0 z z ▲ F = T ( jω ) is a transfer function ▲ G ( z ) /z ∈ [ − k 1 , − k 2 ] is a sector-bounded gain ( i.e. the inverse graph of G is in [ − 1 /k 1 , − 1 /k 2 ] ) ● Circle criterion : exists a quadratic separator (circle) for all ω 9 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Another example : parameter-dependent Lyapunov function w G (z, w)=0 w z w F (w, z)=0 z z ▲ F = A ( δ ) parameter-dependent LTI state-space model ▲ G = I is an integrator ● Necessary and sufficient to have − P ( δ ) 0 Θ( δ ) = − P ( δ ) 0 10 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Direct relation with the IQC framework ▲ F = T ( jω ) is a transfer matrix ▲ G = ∆ is an operator known to satisfy an Integral Quadratic Constraint (IQC) � + ∞ 1 � � dω ≤ 0 ∆ ∗ ( jω ) Π( ω ) 1 ∆( jω ) −∞ ● Stability of the closed-loop is guaranteed if for all ω T ( jω ) � � > 0 T ∗ ( jω ) Π( ω ) 1 1 ▲ Knowing ∆ ∆ the set of ∆ how to choose Π = Θ ? ( i.e. the quadratic separator) 11 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Some choices of quadratic separators Θ ∆ = { ∆ ∗ ∆ ≤ ¯ k 2 1 } ▲ ∆ is full-bloc complex norm-bounded : ∆ Θ should be such that − ¯ k 2 1 0 1 1 � � � � ≤ 0 ⇒ ≤ 0 ∆ ∗ ∆ ∗ Θ 1 1 ∆ ∆ 0 1 ● S -procedure ! [Yakubovitch 70’s] − ¯ k 2 1 0 ∃ τ > 0 : Θ ≤ τ 0 1 12 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Some choices of quadratic separators Θ ∆ = { δ 1 : | δ | ≤ ¯ ▲ ∆ = δ 1 is scalar complex norm-bounded : ∆ k } Θ should be such that − ¯ k 2 0 1 � � � � 1 ≤ 0 ⇒ ≤ 0 δ ∗ δ ∗ 1 Θ 1 1 0 1 δ δ 1 ● D -scaling − ¯ k 2 D 0 ∃ D > 0 : Θ ≤ D 0 13 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Some choices of quadratic separators Θ k , δ = δ ∗ } ∆ = { δ 1 : | δ | ≤ ¯ ▲ ∆ = δ 1 is scalar real norm-bounded : ∆ ● DG -scaling − ¯ k 2 D G ∃ D > 0 , G = − G ∗ : Θ ≤ G ∗ D 14 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Some choices of quadratic separators Θ ▲ ∆ = jω 1 with ω ∈ R ● K-Y-P lemma P 0 ∃ P = P ∗ : Θ ≤ P 0 15 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ Some choices of quadratic separators Θ ∆ 1 0 ▲ ∆ = δ 2 1 0 ∆ 1 if full-bloc complex norm-bounded in { ∆ 1 ∗ ∆ 1 ≤ ¯ k 2 1 1 } k 2 , δ 2 = δ 2 ∗ } δ 2 is scalar real norm-bounded in { δ 2 1 : | δ 2 | ≤ ¯ ● One can take (full-block S -procedure [Scherer], etc.) − τ ¯ k 2 1 1 0 0 0 − ¯ k 2 2 D G 0 0 Θ = τ 1 0 0 0 G ∗ D 0 0 16 October 2009, Florian´ opolis
➊ Topological separation & related theory ■ µ -theory is a special case of IQC framework ▲ F = T ( jω ) is a transfer matrix ▲ ∆ is bloc-diagonal composed of m F full-bloc complex norm-bounded uncertainties m c scalar complex norm-bounded uncertainties m r scalar real norm-bounded uncertainties ▲ All uncertainties bounded by same ¯ k (at the expense of modifying T ( jω ) ) ▲ Goal : k m = max ¯ k 1 ● If µ = k m < 1 stability is proved. ● Convex problem with DG -scalings (LMI for fixed ¯ k , else Generalized Eigenvalue Problem) 17 October 2009, Florian´ opolis
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