Integralьnoe kvadratiqnoe razdelenie: opisanie podhoda, sv�zь s funkci�mi L�punova i S- proceduro i �kuboviqa, priloжenie k analizu usto iqivosti sputnika s ograniqeniem po vhodu Dimitri PEAUCELLE / Dmitri i Жanoviq Poselь -Konovalov LAAS-CNRS - Université de Toulouse - FRANCE Sankt-Peterburg Ma i 2013
Evaluating regions of attraction of LTI systems with saturation in IQS framework Dimitri Peaucelle Sophie Tarbouriech Martine Ganet-Schoeller Samir Bennani Presented first at 7th IFAC Symposium on Robust Control Design / Aalborg
Introduction ■ Saturated control of a linear system x = Ax + Bu , u = sat ( Ky ) , y = Cx ˙ ● Assume K designed for the linear system (no saturation) ● System with saturation: Stability is (in general) only local ● Problem: find (largest possible) set of x (0) such that x ( ∞ ) = 0 ■ Goal of this presentation : formalize the problem in the IQS framework ● Can "system augmentation" relaxations provide less conservative results ? Sankt-Peterburg, I�nь 2013 D. Peaucelle 1
Topological separation - [Safonov 80] w G(z, w) = w z ■ Well-posedness of a feedback loop w F(w, z) = z z ● Uniqueness and boundedness of internal signals for all bounded disturbances � � � � w − w 0 ¯ � � � w � � � � � ∃ γ : ∀ ( ¯ w, ¯ z ) ∈ L 2 × L 2 , ≤ γ , � � � � z − z 0 ¯ � � � z � � � � � G ( z 0 , w 0 ) = 0 ● with solution to the system without perturbations F ( w 0 , z 0 ) = 0 Sankt-Peterburg, I�nь 2013 D. Peaucelle 2
Topological separation - [Safonov 80] w G(z, w) = w z ■ Well-posedness of a feedback loop w F(w, z) = z z ■ Theorem: Well-posed iff exists a topological separator θ ● ‘Negative’ on the inverse graph of one component G I ( ¯ w ) = { ( w, z ) : G ( z, w ) = ¯ w } ⊂ { ( w, z ) : θ ( w, z ) ≤ φ 2 ( || ¯ w || ) } ● ‘Positive definite’ on the graph of the other component of the loop F (¯ z ) = { ( w, z ) : F ( w, z ) = ¯ z } ⊂ { ( w, z ) : θ ( w, z ) > − φ 1 ( || ¯ z || ) } ▲ Issue 1: How to choose θ ? Answer: S-procedure. ▲ Issue 2: How to test the separation inequalities ? Answer: LMIs. Sankt-Peterburg, I�nь 2013 D. Peaucelle 3
Example : the small gain theorem w G(z, w) = w z ■ Well-posedness of a feedback loop w F(w, z) = z z ● In case of causal G ( z, w ) : w = ∆ z , ∆ ∈ RH m × l ∞ and stable proper LTI F ( w, z ) : z = H ( s ) w ● Necessary and sufficient (lossless) choice of separator θ ( w, z ) = � w � 2 − γ 2 � z � 2 ● Separation inequalities: θ ( w, z ) = � w � 2 − γ 2 � z � 2 ≤ 0 , ∀ w = ∆ z ⇔ � ∆ � 2 ∞ ≤ γ 2 ∞ < 1 θ ( w, z ) = � w � 2 − γ 2 � z � 2 > 0 , ∀ z = H ( s ) w ⇔ � H � 2 γ 2 Sankt-Peterburg, I�nь 2013 D. Peaucelle 4
Example : stability of passive interconnected systems w G(z, w) = w z ■ Well-posedness of a feedback loop w F(w, z) = z z ● In case of passive G ( z, w ) : w = ∆ z and stable LTI F ( w, z ) : z = H ( s ) w ● Necessary and sufficient (lossless) choice of separator θ ( w, z ) = − < w | z > ● Separation inequalities: � ∞ w T ( t ) z ( t ) dt ≥ 0 θ ( w, z ) = − < w | z > ≤ 0 , ∀ w = ∆ z ⇔ 0 θ ( w, z ) = − < w | z > 0 , ∀ z = H ( s ) w ⇔ H ∗ ( jω ) + H ( jω ) < 0 , ∀ ω Sankt-Peterburg, I�nь 2013 D. Peaucelle 5
Integral Quadratic Separation (IQS) ■ From topological separation to IQS: Choice of an Integral Quadratic Separator � ∞ �� � � � �� � � z z z ( t ) � � � z T ( t ) w T ( t ) θ ( w, z ) = � Θ = Θ( t ) dt � � w w w ( t ) 0 ● Identical choice to IQC framework [Megretski, Rantzer, Jönsson] � + ∞ � � z ( jω ) � � z T ( jω ) w T ( jω ) θ ( w, z ) = Π( jω ) dω w ( jω ) −∞ ▲ Π is called a multiplier. θ ( w, z ) ≤ 0 is called an IQC. ▲ Conservatism reduction in IQC framework : ω -dependent multipliers: 1 Ψ 1 ( jω ) � � ˆ Ψ 1 ( jω ) ∗ Ψ r ( jω ) ∗ Π( jω ) = Π · · · 1 . . . Ψ r ( jω ) Sankt-Peterburg, I�nь 2013 D. Peaucelle 6
Integral Quadratic Separation (IQS) ■ Main IQS result (both for ω or t or k dependent signals) ■ IQS is necessary and sufficient under assumptions (proof based on [Iwasaki 2001]) ● One component is a linear application, can be descriptor form F ( w, z ) = A w − E z ▲ can be time-varying A ( t ) w ( t ) −E ( t ) z ( t ) or frequency dep. ˆ w ( ω ) − ˆ A ( ω ) ˆ E ( ω )ˆ z ( ω ) ▲ A ( t ) , E ( t ) are bounded and E ( t ) = E 1 ( t ) E 2 where E 1 ( t ) is full column rank ● The other component can be defined in a set G ( z, w ) = ∇ ( z ) − w , ∇ ∈ ∇ ∇ ▲ ∇ ∇ must have a linear-like property ∇ , ∃ ˜ ∇ : ∇ ( z 1 ) − ∇ ( z 2 ) = ˜ ∀ ( z 1 , z 2 ) , ∀∇ ∈ ∇ ∇ ∈ ∇ ∇ ( z 1 − z 2 ) ▲ ∇ ∇ does not need to be causal ■ The matrix Θ must satisfy an IQC over ∇ ∇ + an LMI involving ( E , A ) Sankt-Peterburg, I�nь 2013 D. Peaucelle 7
Examples - Topological Separation and Lyapunov ■ Global stability of a non-linear system ˙ x = f ( x, t ) w � t G(z, w) = w G ( z = ˙ x, w = x ) = 0 z ( τ ) dτ − w ( t ) , z w F ( w, z, t ) = f ( w, t ) − z ( t ) F(w, z) = z z ● ¯ w plays the role of the initial conditions, ¯ z are external disturbances ● Well-posedness: for all bounded initial conditions and all bounded disturbances, the state remains bounded around the equilibrium ≡ global stability Sankt-Peterburg, I�nь 2013 D. Peaucelle 8
Examples - Topological Separation and Lyapunov ■ Global stability of a linear TV system ˙ x = A ( t ) x w � t G(z, w) = w 0 z ( τ ) dτ − w ( t ) = s − 1 z − w, G ( z = ˙ x, w = x ) = z w F ( w, z, t ) = A ( t ) w ( t ) − z ( t ) F(w, z) = z z − P ( t ) z ( t ) 0 � ∞ � � z T ( t ) w T ( t ) dt ● IQS: θ ( w, z ) = 0 − ˙ − P ( t ) P ( t ) w ( t ) ▲ θ ( w, z ) ≤ 0 for all G ( z, w ) = 0 iff P ( t ) ≥ 0 � t � � x T Px + x T ˙ x ) dτ = x T ( t ) P ( t ) x ( t ) Px + x T P ˙ x (0) = 0 , ( ˙ 0 ▲ θ ( w, z ) > 0 for all F ( w, z ) = 0 iff A T ( t ) P ( t ) + P ( t ) A ( t ) + ˙ P ( t ) < 0 z T Pw + w T ˙ Pw + w T Pw = w T ( A T P + PA + ˙ � � P ) w Sankt-Peterburg, I�nь 2013 D. Peaucelle 9
Examples - Topological Separation and Lyapunov ■ Global stability of a system with a dead-zone � t G 1 ( ˙ x, x ) = 0 ˙ x ( τ ) dτ − x ( t ) , w w G 2 ( g, v ) = dz ( g ( t )) − v ( t ) , G(z, w) = w z −1 z w F 1 ( x, v, ˙ x, t ) = f 1 ( x, v, t ) − ˙ x ( t ) , F(w, z) = z 1 z F 2 ( x, v, g, t ) = f 2 ( x, v, t ) − g ( t ) ● Dead-zone embedded in a sector uncertainty ∇ ∇ ∞ = {∇ ∞ : 0 ≤ ∇ ∞ ( g ) ≤ g } G I 2 = { ( v, g ) : G 2 ( g, v ) = 0 } ⊂ { ( v, g ) : v = ∇ ∞ ( g ) , ∇ ∞ ∈ ∇ ∇ ∞ } ∇ ∞ rather than w.r.t G I ▲ Choosing θ IQS w.r.t. ∇ 2 , is a source of conservatism Sankt-Peterburg, I�nь 2013 D. Peaucelle 10
Examples - Topological Separation and Lyapunov ■ IQS applies for linear f 1 , f 2 ■ Global stability of a system with a dead-zone � t G 1 ( ˙ x, x ) = 0 ˙ x ( τ ) dτ − x ( t ) , w G 2 ( g, v ) = dz ( g ( t )) − v ( t ) , G(z, w) = w z w F 1 ( x, v, ˙ x, t ) = Ax ( t ) + Bv ( t ) − ˙ x ( t ) , F(w, z) = z z F 2 ( x, v, g, t ) = Cx ( t ) + Dv ( t ) − g ( t ) ● LMI conditions obtained for the IQS defined by − P 0 0 0 P > 0 , − p 1 0 0 0 Θ = , − P 0 0 0 p 1 > 0 . − p 1 2 p 1 0 0 ● Result is exactly identical to circle theorem result Sankt-Peterburg, I�nь 2013 D. Peaucelle 11
Launcher model ■ Launcher in ballistic phase : attitude control ● Neglected atmospheric friction, sloshing modes, ext. perturbation, axes coupling : I ¨ θ = T T ( u ) = u − ¯ T dz ( 1 ● Saturated actuator: T = sat ¯ T u ) ¯ ● PD control u = − K P θ − K D ˙ θ � t G 1 ( ˙ x, x ) = 0 ˙ x ( τ ) dτ − x ( t ) , G 2 ( g, v ) = dz ( g ( t )) − v ( t ) , 0 1 0 x ( t ) + v ( t ) − ˙ F 1 ( x, v, ˙ x, t ) = x ( t ) , − ¯ − K P − K D T � � − K P − K D F 2 ( x, v, g, t ) = x ( t ) − g ( t ) ¯ ¯ T T Sankt-Peterburg, I�nь 2013 D. Peaucelle 12
Launcher model ■ Global stability LMI test fails ∇ ∞ includes ∇ ∞ = 1 for which the system is I ¨ ▲ Sector uncertainty ∇ θ = 0 (unstable) ● LMI test succeeds (whatever ¯ g < ∞ ) if dead-zone is restricted to belong to w 1 ! z ! 1 g ( g ) ≤ 1 − ¯ g ∇ ∇ ¯ g = {∇ ¯ g : 0 ≤ ∇ ¯ g g } z ¯ 1 z ▲ Useful if one can prove for constrained x (0) that | g ( θ ) | ≤ ¯ g holds ∀ θ ≥ 0 . ■ How can one prove local properties in IQS framework ? Sankt-Peterburg, I�nь 2013 D. Peaucelle 13
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