Integral Quadratic Separation Framework Dimitri PEAUCELLE LAAS-CNRS - Universit´ e de Toulouse - FRANCE joint work with Denis Arzelier and Fr´ ed´ eric Gouaisbaut French-Israeli Workshop on Delays and Robustness Haifa April 2011
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 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 Workshop on Delays and Robustness 1 April 2011, Haifa
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 ? ... Workshop on Delays and Robustness 2 April 2011, Haifa
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 || ) } ● φ 1 and φ 2 are positive functions. When ¯ w = 0 , ¯ z = 0 separation reads as G I (0) = { ( w, z ) : G 0 ( z, w ) = 0 } ⊂ { ( w, z ) : θ ( w, z ) ≤ 0 } F (0) = { ( w, z ) : F 0 ( w, z ) = 0 } ⊂ { ( w, z ) : θ ( w, z ) > 0 } Workshop on Delays and Robustness 3 April 2011, Haifa
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 Workshop on Delays and Robustness 4 April 2011, Haifa
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 (0) is obtained with � ∞ − ∂V θ ( w = x, z = ˙ x ) = ∂x ( x ( τ )) ˙ x ( τ ) dτ = lim t →∞ − V ( x ( t )) < 0 0 ▲ Topological separation w.r.t. F (0) does hold as well � ∞ − ˙ θ ( w, z = f ( w )) = V ( w ( τ )) dτ > 0 0 Workshop on Delays and Robustness 5 April 2011, Haifa
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 (0) : t →∞ − x T ( t ) Px ( t ) ≤ 0 , i.e. P > 0 lim ▲ Quadratic separation w.r.t. F (0) guaranteed if − 2 w T ( t ) PAw ( t ) > 0 , i.e. A T P + PA < 0 ∀ t > 0 , Workshop on Delays and Robustness 6 April 2011, Haifa
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 Workshop on Delays and Robustness 7 April 2011, Haifa
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 ( ς ) 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( ς ) dτ w 0 Workshop on Delays and Robustness 8 April 2011, Haifa
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 Workshop on Delays and Robustness 9 April 2011, Haifa
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 Workshop on Delays and Robustness 10 April 2011, Haifa
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 ∆ ∆ how to choose Π = Θ ? ( i.e. the quadratic separator) Plenty of results in µ -analysis and IQC theory D-scalings, DG-scalings etc. but still, conservative Workshop on Delays and Robustness 11 April 2011, Haifa
Outline ➊ Topological separation & related theory 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 Workshop on Delays and Robustness 12 April 2011, Haifa
➋ IQS for the descriptor case w w G (z, w)=0 w z z w w F (w, z)=0 z z z ■ LTV implicit application in feedback loop with an uncertain operator E ( ς ) z ( ς ) = A ( ς ) w ( ς ) , w ( ς ) = [ ∇ z ]( ς ) ∇ ∈ ∇ ∇ � �� � � �� � F G ● ∇ is bloc-diagonal contains scalar, full-bloc, LTI and LTV uncertainties and other operators such as integrators, delays... ● ς can be time (continuous/discrete), frequencies. Workshop on Delays and Robustness 13 April 2011, Haifa
➋ IQS for the descriptor case ■ Integral Quadratic Separator ● Well-posedness etc. is defined for bounded signals in L 2 (here for ς = t ) � ∞ z ∈ R p , � z � 2 = Trace z ∗ ( t ) z ( t ) dt < ∞ 0 ▲ With scalar product � ∞ z ∗ ( t ) w ( t ) dt < z | w > = Trace 0 ▲ Notation � T � T � z � 2 z ∗ ( t ) z ( t ) dt , < z | w > T = Trace z ∗ ( t ) w ( t ) dt T = Trace 0 0 Workshop on Delays and Robustness 14 April 2011, Haifa
➋ IQS for the descriptor case ■ Integral Quadratic Separation [Automatica’08, CDC’08] ● For the case of linear application with uncertain operator E ( ς ) z ( ς ) = A ( ς ) w ( ς ) , w = [ ∇ z ] ∇ ∈ ∇ ∇ where E ( ς ) = E 1 ( ς ) E 2 with E 1 ( ς ) strict full column rank, ● Integral Quadratic Separator (IQS) : ∃ Θ( ς ) , matrix, solution of LMI � � ⊥∗ � � ⊥ Θ( ς ) > 0 , ∀ ς E 1 ( ς ) −A ( ς ) E 1 ( ς ) −A ( ς ) and Integral Quadratic Constraint (IQC) ∀∇ ∈ ∇ ∇ � ∗ � � E 2 z E 2 z � � Θ ≤ 0 . � ∇ z ∇ z Workshop on Delays and Robustness 15 April 2011, Haifa
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