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Evaluating regions of attraction of LTI systems with saturation in IQS framework Dimitri Peaucelle Sophie Tarbouriech Martine Ganet-Schoeller Samir Bennani 7th IFAC Symposium on Robust Control Design / Aalborg / June 20, 2011 Introduction

  1. Evaluating regions of attraction of LTI systems with saturation in IQS framework Dimitri Peaucelle Sophie Tarbouriech Martine Ganet-Schoeller Samir Bennani 7th IFAC Symposium on Robust Control Design / Aalborg / June 20, 2011 Introduction ■ Integral Quadratic Separation framework ■ Launcher attitude control: local stability of linear system with saturation ■ IQS methodology for the given problem and results 1 IFAC ROCOND / Aalborg / June 20-22, 2012

  2. 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 ¯ G ( z 0 , w 0 ) = 0 w � � � � ∃ 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 the other component ● Positive definite on the graph of one 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 || ) } L Issues: How to choose θ ? How to test the separation inequalities ? 2 IFAC ROCOND / Aalborg / June 20-22, 2012 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 ( t ) w w 0 ● Identical choice to IQC framework [Megretski, Rantzer, J¨ onsson] ■ 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 L 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 ) L 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 , ∇ ∈ ∇ ∇ L ∇ ∇ must have a linear-like property ∇ , ∃ ˜ ∇ : ∇ ( z 1 ) − ∇ ( z 2 ) = ˜ ∀ ( z 1 , z 2 ) , ∀∇ ∈ ∇ ∇ ∈ ∇ ∇ ( z 1 − z 2 ) ■ The matrix Θ must satisfy an IQC over ∇ ∇ + an LMI involving ( E , A ) 3 IFAC ROCOND / Aalborg / June 20-22, 2012

  3. 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 rle 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 , s − 1 ∈ C + ■ 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 ) L θ ( w, z ) ≤ 0 for all G ( z, w ) = 0 iff P ( t ) ≥ 0 L θ ( w, z ) > 0 for all F ( w, z ) = 0 iff A T ( t ) P ( t ) + P ( t ) A ( t ) + ˙ P ( t ) < 0 4 IFAC ROCOND / Aalborg / June 20-22, 2012 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 ) , ∇ ∞ ∈ ∇ ∇ ∞ } L 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 0 0 2 p 1 5 IFAC ROCOND / Aalborg / June 20-22, 2012

  4. 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 L Sector uncertainty includes ∇ ∞ = 1 for which the system is I ¨ θ = 0 (unstable) ● LMIs succeeds (whatever ¯ g < ∞ ) if dead-zone is restricted to belong to � � � � � � g ( g ) ≤ 1 − ¯ g ∇ ∇ ¯ g = {∇ ¯ g : 0 ≤ ∇ ¯ g g } � ¯ � � L 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 ? 6 IFAC ROCOND / Aalborg / June 20-22, 2012 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 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 � � L 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 7 IFAC ROCOND / Aalborg / June 20-22, 2012

  5. 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 T θ v ⎜ ⎟ ⎢ C ⎥ ⎜ ⎟ 0 0 0 ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ϕ θ g C ϕ 0 x 0 0 0 L 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 of ( ϕ 0 x, T θ ˙ x ) to ( T θ x, ϕ θ x ) . ● Restricted sector constraint assumed to hold up to t = θ (i.e. T θ v = ∇ ¯ g T θ g ) ● Goal is to find sets 1 ≥ x T (0) Qx (0) = < ϕ 0 x | Qϕ 0 x > s.t. g ( θ ) = � ϕ θ g � < ¯ g . ■ Problem defined in this way is a well-posedness problem with ∇ composed of 3 blocs ● IQS framework applies and gives conservative LMI conditions ● Equivalent to LaSalle invariance principle with V ( x ) = x T Qx (ellipsoidal sets of IC) 8 IFAC ROCOND / Aalborg / June 20-22, 2012 System augmentation with derivatives ■ How to reduce conservatism ? ● Needed a description of the dead-zone better than sector uncertainty ● Needed to have dead-zone dependent sets of initial conditions ■ Both features derived via descriptor modeling of system augmented with ˙ v and ˙ g ⎧ if g > 1 v = g − 1 v = ˙ ˙ g ⎪ ⎪ ⎪ ⎨ v = dz ( g ) : if | g | ≥ 1 v = 0 v = 0 ˙ ⎪ ⎪ if g < − 1 v = g + 1 v = ˙ ˙ g ⎪  ● For IQS, link between ˙ v and ˙ g is embedded in ˙ v = ∇ { 0 , 1 } ˙ g , with ∇ { 0 , 1 } ∈ { 0 , 1 } . ● Also needed to specify that v is the integral of ˙ v (thus descriptor form) 9 IFAC ROCOND / Aalborg / June 20-22, 2012

  6. System augmentation with derivatives ⎡ ⎤ ⎡ ⎤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎛ ⎞ T θ x ⎛ ⎞ ϕ 0 x 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 T θ v ⎢ ⎥ ⎢ ⎥ ϕ 0 v A B 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ϕ θ x ⎜ ⎟ ⎜ ⎟ T θ ˙ x ⎢ 0 0 0 1 0 0 0 0 ⎥ ⎢ 0 0 0 0 0 0 1 0 0 ⎥ ⎜ ⎟ ⎜ ⎟ ϕ θ v ⎢ ⎥ ⎢ ⎥ T θ ˙ v C 0 0 0 0 1 0 0 0 ⎜ ⎟ 0 0 0 0 0 0 0 0 ⎜ ⎟ = ⎢ ⎥ ⎢ ⎥ T θ v ⎜ ⎟ ⎜ ⎟ T θ g ⎢ 0 0 0 0 0 1 0 0 ⎥ ⎢ 0 0 C 0 0 0 0 0 0 ⎥ ⎜ ⎟ ⎜ ⎟ ϕ θ v ⎢ ⎥ ⎢ ⎥ − C ⎜ ϕ θ g ⎟ ⎜ ⎟ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎢ ⎥ ⎢ ⎥ T θ ˙ v ⎜ ⎟ ⎝ ⎠ ⎢ 0 0 0 0 0 0 0 1 ⎥ T θ ˙ g ⎢ 0 0 C 0 0 0 0 0 0 ⎥ ⎝ ⎠ ϕ 0 x ⎣ ⎦ ⎣ ⎦ − 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ϕ θ g ϕ 0 v 0 0 0 0 0 0 0 0 0 0 0 1 0 − 1 0 0 0 ■ Problem defined in this way is a well-posedness problem with ∇ composed of 5 blocs ● IQS framework applies and gives less conservativeLMI conditions ● Equivalent to LaSalle invariance principle with T T ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎝ x ⎝ x x x ⎠ = V ( x ) = Q a Q a ⎠ ⎝ ⎠ ⎝ ⎠ dz ( Cx ) dz ( Cx ) v v 10 IFAC ROCOND / Aalborg / June 20-22, 2012 Application to the launcher model ■ LMIs tested on the launcher example ���� ��� ���� � � ���� � ��� � ���� � � � ��� � ��� � ● Sets of initial conditions for which | g ( θ ) | ≤ 8 is guaranteed ● Improvement thanks to piecewise quadratic sets of initial conditions 11 IFAC ROCOND / Aalborg / June 20-22, 2012

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