Evaluating regions of attraction of LTI systems with saturation in - PowerPoint PPT Presentation

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 Seminar SKIMAC 2013 - jan 22-24

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 ? 1 SKIMAC / Jan 22-24, 2013

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 ¯ G ( z 0 , w 0 ) = 0 � � � � ∃ γ : ∀ ( ¯ w, ¯ z ) ∈ L 2 × L 2 , � ≤ γ � , � � � � � � � � z − z 0 z ¯ F ( w 0 , z 0 ) = 0 � � ■ iff exists a topological separator θ ● Negative on the inverse graph of one component ● Positive definite on the graph of the other component of the loop G I ( ¯ w ) = { ( w, z ) : G ( z, w ) = ¯ w } ⊂ { ( w, z ) : θ ( w, z ) ≤ φ 2 ( || ¯ w || ) } F (¯ z ) = { ( w, z ) : F ( w, z ) = ¯ z } ⊂ { ( w, z ) : θ ( w, z ) > − φ 1 ( || ¯ z || ) } ▲ Issues: How to choose θ ? How to test the separation inequalities ? 2 SKIMAC / Jan 22-24, 2013

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 3 SKIMAC / Jan 22-24, 2013

Integral Quadratic Separation (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, J¨ onsson] � + ∞ � � 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 ) 4 SKIMAC / Jan 22-24, 2013

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 ) ▲ ∇ ∇ need not to be causal ■ The matrix Θ must satisfy an IQC over ∇ ∇ + an LMI involving ( E , A ) 5 SKIMAC / Jan 22-24, 2013

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 x ( t ) = A ( t ) x ( t ) , ∇ = s − 1 1 ■ For linear systems ˙     − 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 ▲ θ ( w, z ) > 0 for all F ( w, z ) = 0 iff A T ( t ) P ( t ) + P ( t ) A ( t ) + ˙ P ( t ) < 0 6 SKIMAC / Jan 22-24, 2013

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(z, w) = w G 2 ( g, v ) = dz ( g ( t )) − v ( t ) , 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 ) ■ IQS applies for linear f 1 , f 2 ● Dead-zone embedded in a sector uncertainty ∇ ∇ ∞ = {∇ ∞ : 0 ≤ ∇ ∞ ( g ) ≤ g } G I 2 = { ( v, g ) : G 2 ( g, v ) = 0 } ⊂ { ( v, g ) : v = ∇ ∞ ( g ) , ∇ ∞ ∈ ∇ ∇ ∞ } ▲ This is the only source of conservatism ● 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 7 SKIMAC / Jan 22-24, 2013

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 ■ Global stability LMI test fails ▲ Sector uncertainty includes ∇ ∞ = 1 for which the system is I ¨ 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 ? 8 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS 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 ¯ G ( z 0 , w 0 ) = 0 w � � � � ∃ γ : ∀ ( ¯ w, ¯ z ) ∈ L 2 × L 2 , � ≤ γ � , � � � � � � � � z − z 0 ¯ F ( w 0 , z 0 ) = 0 z � � ▲ How to introduce initial conditions x (0) and “final” conditions g ( θ ) in IQS framework? ■ Square-root of the Dirac operator: linear operator such that � ∞ 0 ϕ θ x T ( t ) Mϕ θ x ( t ) dt = x T ( θ ) Mx ( θ ) < ϕ θ x | Mϕ θ x > = x �→ ϕ θ x : < ϕ θ 1 x | Mϕ θ 2 x > = 0 if θ 1 � = θ 2 ● Such operator is also used for PDE to describe states on the boundary 9 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS ■ System with initial and final conditions writes as       T θ x ϕ 0 x 0 0 0 1       T θ ˙  x   A B   ϕ θ x  0 0       =       T θ g C T θ v       0 0 0             ϕ θ g C ϕ 0 x 0 0 0 ▲ T θ x is the truncated signal such that T θ x ( t ) = x ( t ) for t ≤ θ and = 0 for t > θ . ● The integration operator is redefined as a mapping      T θ x  ϕ 0 x  = I  T θ ˙ ϕ θ x x 10 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS       ϕ 0 x T θ x 0 0 0 1       T θ ˙ x A B ϕ θ x       0 0       =       T θ g T θ v C    0 0 0                ϕ θ g C ϕ 0 x 0 0 0 ● Restricted sector constraint assumed to hold up to t = θ : w 1 ! z ! 1 T θ v = ∇ ¯ g T θ g z 1 z 11 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS       T θ x ϕ 0 x 0 0 0 1       T θ ˙  x   A B   ϕ θ x  0 0       =       T θ g C T θ v       0 0 0             ϕ θ g C ϕ 0 x 0 0 0 ● Goal is to prove the restricted sector condition holds strictly at time θ (whatever θ ) ▲ i.e. find sets 1 ≥ x T (0) Qx (0) = < ϕ 0 x | Qϕ 0 x > s.t. | g ( θ ) | = � ϕ θ g � < ¯ g ▲ reformulated as well posedness problem where ϕ 0 x = ∇ ci ϕ θ g defined by g 2 < w ci | Qw ci > ≤ � z ci � 2 w ci = ∇ ci z zi : ¯ 12 SKIMAC / Jan 22-24, 2013