Integral Quadratic Separators for performance analysis Dimitri PEAUCELLE , Lucie Baudouin, Fr´ ed´ eric Gouaisbaut LAAS-CNRS - Universit´ e de Toulouse - FRANCE European Control Conference - Budapest August, 24-26 2009
Outline ➊ Topological separation & Integral Quadratic Separation ➋ Norm-to-norm performance in IQS framework ➌ Impulse-to-norm performance in IQS framework ➍ Impulse-to-peak performance in IQS framework ➎ Conclusions & The Romuald toolbox 1 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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 || ) } 2 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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 3 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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 4 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ For linear systems : quadratic Lyapunov function, i.e. quadratic separator w ( z,w ) G ¯ z ( z,w ) 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 , 5 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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 6 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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 7 ECC - Budapest - August, 24-26 2009
➊ Topological separation 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 8 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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 ( ˙ δ ) ▲ G = I is an integrator ● Necessary and sufficient to have − P ( δ ) 0 Θ( δ ) = − P ( δ ) 0 9 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ 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) 10 ECC - Budapest - August, 24-26 2009
➊ Topological separation w G (z, w)=0 w z w F (w, z)=0 z z ■ Linear implicit application in feedback loop with an uncertain operator E z ( t ) = A w ( t ) , w ( t ) = [ ∇ z ]( t ) ∇ ∈ ∇ ∇ � �� � � �� � F G ● ∇ is bloc-diagonal contains scalar, full-bloc, LTI and LTV uncertainties and other operators such as integrator etc. 11 ECC - Budapest - August, 24-26 2009
➊ Topological separation ■ Integral Quadratic Separation [Automatica’08, CDC’08, ROCOND’09] ● For the case of linear application with uncertain operator E z ( t ) = A w ( t ) , w ( t ) = [ ∇ z ]( t ) ∇ ∈ ∇ ∇ where E = E 1 E 2 with E 1 full column rank, ● Integral Quadratic Separator (IQS) : ∃ Θ , matrix, solution of LMI � � ⊥∗ � � ⊥ Θ > 0 E 1 −A E 1 −A and Integral Quadratic Constraint (IQC) ∀∇ ∈ ∇ ∇ ∗ � ∞ E 2 z ( t ) E 2 z ( t ) dt ≤ 0 Θ [ ∇ z ]( t ) [ ∇ z ]( t ) 0 12 ECC - Budapest - August, 24-26 2009
Outline ➊ Topological separation & Integral Quadratic Separation ● Rich framework for robust stability analysis ▲ Can input-output performances be treated in the same framework ? ➋ Norm-to-norm performance in IQS framework ➌ Impulse-to-norm performance in IQS framework ➍ Impulse-to-peak performance in IQS framework ➎ Conclusions & The Romuald toolbox 13 ECC - Budapest - August, 24-26 2009
Outline ■ Integral Quadratic Separator : all signals are assumed L 2 : � z � 2 < ∞ � ∞ � ∞ � z � 2 = Trace z ∗ ( t ) z ( t ) dt , z ∗ ( t ) w ( t ) dt < z | w > = Trace 0 0 ▲ Notation � T � T � z � 2 z ∗ ( t ) z ( t ) dt , < z | w > T = Trace z ∗ ( t ) w ( t ) dt T = Trace 0 0 14 ECC - Budapest - August, 24-26 2009
➌ Norm-to-norm performance in quadratic separation framework ■ Induced L 2 norm ( H ∞ in the LTI case) E ˙ x = Ax + Bv , g = Cx + Dv ▲ Prove that system is asymptotically stable ▲ and � g � < γ � v � for zero initial conditions x (0) = 0 (strict upper bound on the L 2 gain attenuation) ● Equivalent to well-posedness with respect to � t Integrator with zero initial conditions x ( t ) = [ I 1 ˙ x ]( t ) = 0 ˙ x ( τ ) dτ and signals such that � v � ≤ 1 γ � g � 15 ECC - Budapest - August, 24-26 2009
➌ Norm-to-norm performance in quadratic separation framework ■ Induced L 2 norm E ˙ x = Ax + Bv , g = Cx + Dv ▲ Define ∇ n 2 n the fictitious non-causal uncertain operator such that iff � v � ≤ 1 v = ∇ n 2 n g γ � g � ● Induced L 2 norm problem is equivalent to well-posedness of E ˙ x A B x I 1 0 0 = , ∇ = g C D v ∇ n 2 n 0 1 0 � �� � � �� � � �� � � �� � E z A w 16 ECC - Budapest - August, 24-26 2009
➌ Norm-to-norm performance in quadratic separation framework ■ Induced L 2 norm E ˙ x A B x I 1 0 0 = , ∇ = g C D v ∇ n 2 n 0 1 0 � �� � � �� � � �� � � �� � E z A w ● Elementary IQS for bloc I 1 is − P 0 Θ I 1 = : P > 0 − P 0 � t Indeed (recall x ( t ) = [ I 1 ˙ x ]( t ) = 0 ˙ x ( τ ) dτ and x (0) = 0 ) � � � x ˙ x ˙ � = − x ∗ ( T ) Px ( T ) ≤ 0 � Θ I 1 � I 1 ˙ x I 1 ˙ x T 17 ECC - Budapest - August, 24-26 2009
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