Robust Control Using Convex Optimization Kyoto University March 12, 2010 Dimitri Peaucelle LAAS-CNRS, Toulouse, FRANCE peaucelle@laas.fr http://homepages.laas.fr/peaucell
Introduction ➊ Robustness in control theory ● Robustness properties of the feedback loop ● Some classical measures of robustness ● Disturbance rejection and robustness to parametric uncertainty - a modeling issue ● Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models ➋ Optimization based tools ● Linear Matrix Inequalities (LMI) framework [1990’s] ● Efficient fast solvers and nice parser for Matlab [2000’s] ● Adapted tool for robustness issues - examples of Lyapunov-based results ➌ Robust control theory at LAAS ● RoMulOC - A Matlab toolbox ● Applications in aerospace ● Integral Quadratic Separation D. Peaucelle 1 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory Let the following double integrator with a flexible mode (attitude of satellite with solar pannel) s 2 + 0 . 1 s + 1 . 002 T ( s ) = 1 s 2 + 0 . 2 s + 1 . 01 s 2 for which the following controller is designed K ( s ) = k s 3 + 2 . 1 s 2 + 1 . 2 s + 0 . 1 s 3 + 9 s 2 + 27 s + 27 The Nichols plots of T ( s ) K ( s ) (for k = 10 ) phase margin > 90 ◦ , gain margin = ∞ Robust !? D. Peaucelle 2 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory Assume uncertainty on the flexible mode (damaged solar panel) s 2 + 0 . 1 s + 1 . 002 s 2 + 0 . 1 s + 2 . 252 s 2 ( s 2 + 0 . 2 s + 1 . 01) → ˜ T ( s ) = T ( s ) = s 2 ( s 2 + 0 . 2 s + 1 . 01) for the same control ( k = 10 ) the Nichols plots of ˜ T ( s ) K ( s ) indicates that the closed-loop is unstable for k = 10 but it is stable for k = 1 and k = 200 . ▲ The controller stabilizes ˜ T ( s ) for k = 1 and k = 200 but not for k ∈ [4 100] . ▲ The controller for k = 1 or k = 200 stabilizes both models T ( s ) and ˜ T ( s ) . D. Peaucelle 3 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory Step and frequency responses of closed-loop systems for k = 200 : ▲ Small variations in the parametric space can have large impact. ▲ Faulty behavior can be related to narrow peaks in the frequency domain D. Peaucelle 4 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ Two ‘definitions’ of robustness ● Guarantee some characteristics for a given class of perturbations w z Σ u y � z � ≤ γ � w � , ∀ w : � w � < ∞ e.g. K � ∞ ▲ Many results using L 2 norm: � w � 2 = w ∗ ( t ) w ( t ) dt 0 u,y ▲ Defines induced L 2 norm of the system: � Σ ⋆ K � = min γ (also known as H ∞ norm for linear systems) D. Peaucelle 5 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ Two ‘definitions’ of robustness ● Guarantee some characteristics of an output for a given class of input perturbations w z Σ u y � z � ≤ γ � w � , ∀ w : � w � < ∞ e.g. K � ∞ ▲ Many results using L 2 norm: � w � 2 = w ∗ ( t ) w ( t ) dt 0 u,y ▲ Defines induced L 2 norm of the system: � Σ ⋆ K � = min γ (also known as H ∞ norm for linear systems) (∆) Σ for a given set of uncertainties ∆ ∈ ∆ ∆ ● Guarantee stability of K ▲ Parametric uncertainty: ∆ is a vector of scalar, bounded, constant parameters ▲ Linear Time-Varying (LTV): Σ is linear , ∆( t ) grasps TV and Non-Linear characteristics ▲ Uncertain dynamic uncertainties: ∆ is a constrained operator w ∆ ( t ) = [∆ z ∆ ]( t ) : � w ∆ � ≤ ρ � z ∆ � e.g. D. Peaucelle 6 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ● Example of uncertain modeling ▲ Non-linear system with unknown parameter x ( t ) = − αx ( t ) + sin( x ( t )) , α ∈ [ − 2 . 5 − 1 . 5] ˙ ▲ Included in the following uncertain model x ( t ) = − 2 x ( t ) + δ 1 x ( t ) + δ 2 ( t ) x ( t ) , δ 1 ∈ [ − 0 . 5 0 . 5] , δ 2 ( t )(= sin x ˙ x ) ∈ [ − 0 . 2 1] ▲ Included itself in � � x ( t ) = − 2 x ( t ) + ˙ ∆( t ) x ( t ) 1 1 δ 1 ∆( t ) = δ 2 ( t ) � � T � � − 0 . 7576 0 − 0 . 6061 1 1 ≤ 0 0 1 . 8182 0 ∆( t ) ∆( t ) − 0 . 6061 0 1 . 5152 D. Peaucelle 7 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ● Example of uncertain modeling (followed) ▲ Non-linear system with unknown parameter x ( t ) = − αx ( t ) + sin( x ( t )) , α ∈ [ − 2 . 5 − 1 . 5] ˙ ▲ Included in the following uncertain model x ( t ) = − 1 . 6 x ( t )+0 . 5˜ δ 1 x ( t )+0 . 6˜ δ 2 ( t ) x ( t ) , ˜ δ 1 ∈ [ − 1 1] , ˜ δ 2 ( t )(= sin x ˙ x ) ∈ [ − 1 1] ▲ Included itself in � � ˜ x ( t ) = − 1 . 6 x ( t ) + ˙ ∆( t ) x ( t ) 0 . 7416 0 . 8124 ˜ δ 1 : ∆( t ) T ∆( t ) ≤ 1 ˜ ∆( t ) = ˜ δ 2 ( t ) D. Peaucelle 8 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ The case of norm-bounded uncertainties ● Non-structured uncertain norm-bounded operator - enters affinelly in the model x = A (∆) x = ( A + B ∆ ∆ C ∆ ) x : ∆ T ∆ ≤ ρ 1 ˙ ● More general: the uncertain operator enters rationally in the model x = A (∆) x = ( A + B ∆ ∆( 1 − D ∆∆ ∆) − 1 C ∆ ) x : ∆ T ∆ ≤ ρ 1 ˙ ● Linear-Fractional-Transform: equivalent modeling using exogenous signals x = Ax + B ∆ w ∆ ˙ , w ∆ = ∆ z ∆ : ∆ T ∆ ≤ ρ 1 z ∆ = C ∆ x + D ∆∆ w ∆ ∆ w z ∆ w ∆ ,z ∆ ∆ Σ ⋆ ∆ Σ ● Generalizes to the case of bounded operators w ∆ ( t ) = [∆ z ∆ ]( t ) : � w ∆ � ≤ ρ � z ∆ � D. Peaucelle 9 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ Robustness w.r.t. uncertain parameters v.s. disturbance rejection w ∆ ,z ∆ ● Stability of Σ ⋆ ∆ for all w ∆ ( t ) = [∆ z ∆ ]( t ) : � w ∆ � ≤ ρ � z ∆ � equivalent to ● L 2 induced norm � Σ � ≤ γ = 1 ρ ∆ z=z w=w w z ∆ Σ ∆ ∆ ∆ stable for � w ∆ � ≤ ρ � z ∆ � ⇔ � z � ≤ γ � w � Σ ▲ Robust stability optimization problem: find maximal ρ ≥ 1 such that the closed loop is stable for all admissible uncertainties ▲ L 2 -Performance optimization problem: find minimal γ ≤ 1 that guarantees performance rejection D. Peaucelle 10 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ Modulus margin example K Τ +− L 2 -gain given by maximal gain of the closed-loop for all frequencies (see Bode plot) ∆ K Τ − + Robustness to norm-bounded uncertainties ■ Extends to MIMO case: Bode plot replaced by sigma plot σ ( K ( jω ) T ( jω )) = maximal singular value of matrix K ( jω ) T ( jω ) for each frequency ω D. Peaucelle 11 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ∆ ■ µ theory µ ( ω ) = 1 /k m ( ω ) < 1 , ∀ ω structured singular value : M(j ) ω k m ( ω ) = min { k : ∃ ∆ det( 1 − k ∆ M ( jω )) = 0 } δ 1 1 r 1 0 ... δ p R 1 r pR ˆ δ 1 1 ˆ r 1 ... ∆ = ˆ δ p C 1 ˆ r pC ∆ 1 ... ∆ p F 0 δ p ∈ C : | ˆ ˆ ∆ p ∈ C m p × l p δ p ∈ R : | δ p | ≤ 1 δ p | ≤ 1 : � ∆ p � ≤ 1 , , D. Peaucelle 12 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ● Example of structured singular value [Skogestad 96] a a M ( jω ) = b b ▲ If ∆ = δ 1 then µ ( ω ) = max | λ ( M ( jω )) | = | a + b | δ 1 0 then µ ( ω ) = | a | + | b | ▲ If ∆ = 0 δ 2 √ 2 a 2 + 2 b 2 ▲ If ∆ is full block then µ ( ω ) = σ ( M ( jω )) = D. Peaucelle 13 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ Standard robust analysis problem: ∆ w ∆ ,z ∆ prove stability of (Σ ⋆ ∆) w z ∆ ∆ Σ for all ∆ ∈ ∆ ∆ (block diagonal operator with LTI and/or TV uncertainties) ■ Standard robust design problem: ∆ w z ∆ ∆ w ∆ ,z ∆ u,y Find K that guarantees stability of ((Σ ⋆ ∆) ⋆ K ) Σ for all ∆ ∈ ∆ ∆ (block diagonal operator with LTI and/or TV uncertainties) u y K ● ∆ contains unknown parameters, scheduling parameters, approximations of non-linearities, delays... ● Σ is a linear model: crude but simple representation of the system D. Peaucelle 14 Seminar at Kyoto Univ. - March 12, 2010
➊ Robustness in control theory ■ Generalizes to robust performance problems ∆ w z ∆ ∆ ● Guarantee an input/output property for all uncertainties Σ w z ∆ w z ∆ ∆ Σ w z Find a controller that guarantees input/output properties for all ∆ ∈ ∆ ∆ ● u y K ● Find a controller that guarantees simultaneously several robust specifications (possibly defined on different models) ∆ ∆ ∆ 1 2 3 Σ Σ Σ Π Π Π ✛ ✛ 1 1 2 3 2 3 K K K D. Peaucelle 15 Seminar at Kyoto Univ. - March 12, 2010
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