Modeli v Teorii Robastnogo Upravleni Dmitri i anoviq Konovalov - PowerPoint PPT Presentation

Modeli v Teorii Robastnogo Upravleni� Dmitri i Жanoviq Konovalov Dmitry Peaucelle Dimitri i Poselь Dimitri Peaucelle Tradicionna� Xkola ”Upravlenie, Informaci� i Optimizaci�” Pereslavlь-Zalesski i I�nь 2010

Models in Robust Control Framework Dimitri Peaucelle LAAS-CNRS, Toulouse, FRANCE peaucelle@laas.fr http://homepages.laas.fr/peaucell Traditional School "Control, Information and Optimization" Pereslavl’-Zalesskii June 2010

Outline ➊ What is Robustness? x = f ( x ) − ˙ → ˙ x = A (∆) x + B w (∆) w ➋ Why concentrating on linear systems? [2] Σ Σ(∆) ➌ Affine polytopic-type models [1] Σ [v] Σ K ∆ w z ∆ ∆ Σ w z ➍ LFT-type models u y K ➎ RoMulOC Toolbox » sys = ussmodel( sys, delta ) Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 1

➊ What is Robustness? 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 ) and T ( s ) K ( s ) (for k = 10 ) phase margin > 90 ◦ , gain margin = ∞ Robust !? K Τ +− Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 2

➊ What is Robustness? 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 ) and ˜ 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] . ▲ Stability of extremal values � = Stability of interval Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 3

➊ What is Robustness? k ( s 3 +2 . 1 s 2 +1 . 2 s +0 . 1)( s 2 +0 . 1 s +2 . 252) K Τ ( s 3 +9 s 2 +27 s +27) s 2 ( s 2 +0 . 2 s +1 . 01)+ k ( s 3 +2 . 1 s 2 +1 . 2 s +0 . 1)( s 2 +0 . 1 s +2 . 252) +− k ( s 3 +2 . 1 s 2 +1 . 2 s +0 . 1)( s 2 +0 . 1 s +2 . 252) = s 7 + a 6 s 6 + a 5 ( k ) s 5 + a 4 ( k ) s 4 + a 3 ( k ) s 3 + a 2 ( k ) s 2 + a 1 ( k ) s + a 0 ( k ) ▲ Uncertain characteristic polynomial included in interval polynomial with independent coefficients: a i ( k ) = a i ≤ a i ≤ a i = a i ( k ) s 7 + a 6 s 6 + a 5 ( k ) s 5 + a 4 ( k ) s 4 + a 3 ( k ) s 3 + a 2 ( k ) s 2 + a 1 ( k ) s + a 0 ( k ) = s 7 + a 6 s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 ■ Kharitonov’s Theorem: stability of interval polynomial iff stability of four polynomials s 7 + a 6 s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 s 7 + a 6 s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 s 7 + a 6 s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 s 7 + a 6 s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 ▲ Conservative test when coefficients not independent ● Extendend for feedback loop of interval polynomials: 16/32 polynomials test Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 4

➊ What is Robustness? ● The controller for k = 1 or k = 200 stabilizes both models T ( s ) and ˜ T ( s ) . ▲ Does it hold in between ? s 2 +0 . 1 s +∆ s 2 ( s 2 +0 . 2 s +1 . 01) = y T ( s, ∆) = K Τ +− u K ( s ) = k s 3 +2 . 1 s 2 +1 . 2 s +0 . 1 u = s 3 +9 s 2 +27 s +27 y r − y Define w ∆ = ∆ z ∆ and z ∆ = u , the feedback loop for y r = 0 writes as ∆ s 3 +2 . 1 s 2 +1 . 2 s +0 . 1 Σ k =1 ( s ) = w z ∆ s 7 +9 . 2 s 6 +30 . 81 s 5 +43 . 69 s 4 +34 . 08 s 3 +27 . 49 s 2 +0 . 01 s ∆ 200 s 3 +420 s 2 +240 s +20 Σ Σ k =200 ( s ) = s 7 +9 . 2 s 6 +229 . 8 s 5 +481 . 5 s 4 +314 . 7 s 3 +71 . 27 s 2 +2 s ● Circle criterion for 1 . 002 ≤ ∆( t ) ≤ 2 . 252 Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 5

➊ What is Robustness? Step and frequency responses of closed-loop systems K ⋆ T and K ⋆ ˜ T for k = 200 : ● K ⋆ ˜ T less "robust" to disturbances at frequencies ∈ [1 2] rad/s. ▲ Faulty behavior can be related to narrow peaks in the frequency domain Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 6

➊ What is Robustness? ■ Two ‘definitions’ of robustness for non-linear systems ● Guarantee some characteristics of an output for a given class of input perturbations w z Σ u y z ∈ Z , ∀ w ∈ W K � ∞ � z ∗ ( t ) z ( t ) dt ≤ γ 2 � � z � 2 = ▲ Many results using L 2 norm: Z = 0 u,y ⋆ K � = max || z || ▲ Defines induced L 2 norm of systems: � Σ || w || = || z || / || w ||≤ γ γ min ▲ Invariant set problem (and others ?) can also be defined in this way (∆) Σ for a given set of uncertainties ∆ ∈ ∆ ∆ ● Guarantee stability of K ▲ Parametric uncertainty: ∆ is a vector of scalar, bounded, constant parameters ▲ Time-Varying: ∆( t ) grasps TV and Non-Linear characteristics ▲ E.g. ∆ is a sector bounded operator defined on exogenous signals w ∆ ( t ) = [∆ z ∆ ]( t ) : ρ � z ∆ � ≤ � w ∆ � ≤ ρ � z ∆ � e.g. Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 7

Outline ➊ What is Robustness? x = f ( x ) − ˙ → ˙ x = A (∆) x + B w (∆) w ➋ Why concentrating on linear systems? [2] Σ Σ(∆) ➌ Affine polytopic-type models [1] Σ [v] Σ K ∆ w z ∆ ∆ Σ w z ➍ LFT-type models u y K ➎ RoMulOC Toolbox » sys = ussmodel( sys, delta ) Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 8

➋ Why concentrating on linear systems? ■ 1st Lyapunov Method viewpoint Linearization around equilibrium x e ˙ x = f ( x ) ˙ = ⇒ X = AX ⇐ = Locally asymptotically stable Asymptotically stable ● Properties on approximated model kept true for original model: robustness ▲ Holds because asymptotic stability kept true for small variations ˙ X = ( A + ∆) X , || ∆ || ≪ 1 . ● Need to define precisely ∆ and || ∆ || ≪ 1 Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 9

➋ Why concentrating on linear systems? ■ Including model viewpoint x = A (∆( t )) x + w ( t ) : (∆ , ˙ x = f ( x ) : x ∈ X ˙ = ⇒ ˙ ∆) ∈ ∆ ∆ × ∆ ∆ d , w ∈ W ● Example of a 1 axis rotating mechanical system with input u and perturbation θ ¨ ǫ ( t ) = − a 1 ˙ ǫ ( t ) − a 0 sin( ǫ ( t ) − ǫ 0 ) + b cos θ ( t ) · u ( t ) Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 10

➋ Why concentrating on linear systems? ■ Including model viewpoint (appropriate for Lipschitz function, or at least locally) x = A (∆( t )) x + w ( t ) : (∆ , ˙ x = f ( x ) : x ∈ X ˙ = ⇒ ˙ ∆) ∈ ∆ ∆ × ∆ ∆ d , w ∈ W ● Example of a 1 axis rotating mechanical system with input u and perturbation θ ¨ ǫ ( t ) = − a 1 ˙ ǫ ( t ) − a 0 sin( ǫ ( t ) − ǫ 0 ) + b cos θ ( t ) · u ( t ) ▲ Assumption: Model is valid only for ǫ ∈ [ − 10 ◦ 20 ◦ ] , θ ∈ [ − 45 ◦ 45 ◦ ] . ▲ Errors in identification (parametric uncertainty): a 1 ∈ [0 . 02 0 . 15] , a 0 ∈ [2 5] , ǫ 0 ∈ [ − 5 ◦ 5 ◦ ] , b ∈ [0 . 1 0 . 3] . ● Uncertain time-varying linear model a 0 ( t ) ǫ ( t ) + w ( t ) + ˜ ¨ ǫ ( t ) = − a 1 ˙ ǫ ( t ) − ˜ b ( t ) u ( t ) sin ǫ ( t ) ˜ a 0 ( t ) = a 0 cos ǫ 0 ∈ [1 . 95 5] , w ( t ) = a 0 cos ǫ ( t ) sin ǫ 0 ∈ [ − 0 . 0131 0 . 0131] ǫ ( t ) ˜ b ( t ) = b cos θ ( t ) ∈ [0 . 0707 0 . 3] ▲ May be possible to add information on the derivatives of the uncertainties Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 11

➋ Why concentrating on linear systems? ■ Input-to-output viewpoint x = f ( x, 0) ˙ x = f ( x, w ) , z = g ( x ) ˙ = ⇒ � z � ≤ γ if � w � ≤ α ≪ 1 Locally asymptotically stable Bounded output for small perturbations ▲ How to characterize the performances? ● Many criteria for linear systems - H ∞ , H 2 with interpretations in time-domain, frequency domain, stochastic - Other criteria such as impulse-to-peak (invariant set) - Pole location: information on dynamics of modes ▲ Much attention has been paid to robust performance problems of linear MIMO systems ˙ x = f ( x, w ) , z = g ( x ) ˙ X = A (∆) X + B w (∆) w , z = C (∆) X = ⇒ : || z || / || w || ≤ γ guaranteed performance: max ∆ min γ Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 12

Outline ➊ What is Robustness? x = f ( x ) − ˙ → ˙ x = A (∆) x + B w (∆) w ➋ Why concentrating on linear systems? [2] Σ Σ(∆) ➌ Affine polytopic-type models [1] Σ [v] Σ K ∆ w z ∆ ∆ Σ w z ➍ LFT-type models u y K ➎ RoMulOC Toolbox » sys = ussmodel( sys, delta ) Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 13

➌ Affine polytopic-type models ■ Interval polynomial SISO models (in Kharitonov’s like results) b 0 + b 1 s + b 2 s 2 · · · + b m s m b i ≤ b i ≤ b i , a 0 + a 1 s + a 2 s 2 · · · + a n s n a i ≤ a i ≤ a i ▲ Restricted to SISO systems ▲ Coefficients assumed independent one from the other ▲ Only for parametric uncertainty: Constant uncertainties ▲ Structure not preserved by a change of basis a 0 + a 1 s + a 2 s 2 + a 3 s 3 = a 3 ( s + 1) 3 + ( a 2 − 3 a 3 )( s + 1) 2 + ( a 1 − 2 a 2 + 3 a 3 )( s + 1) + ( a 0 − a 1 + a 2 − a 3 ) Pereslavlь-Zalesski i I�nь 2010 D. Peaucelle 14