LMI results for robust control design of observer-based controllers, - PowerPoint PPT Presentation

LMI results for robust control design of observer-based controllers, the discrete-time case with polytopic uncertainties Dimitri PEAUCELLE Yoshio EBIHARA IFAC World Congress Wednesday August 25, 2014, Cape Town

Introduction ■ Curiosity at the origin of this work: ● Many existing LMI results for robust state-feedback design ● The Luenberger type observer problem is “dual" to state-feedback, but... few LMI results ● Many results for robust output filtering issue, but applicable only to stable plants D. Peaucelle 1 Cape Town, August 2014

Introduction ■ Issues raised by the study of robust observers: x k +1 = A ( θ ) x k + Bu k x k +1 = A o ˆ ˆ x k + Bu k + L ( C ˆ x k − y k ) , y k = Cx k e k = x k − ˆ x k ● What model A o for the observer ? ▲ Usual answer is to decompose a priori A ( θ ) = A o + ∆( θ ) ▲ For example in NL observers, A o models integrators in series ● One cannot expect e k 2 >k 1 = 0 even if e k 1 = 0 ● Separation principle for the design of state-feedback and observer gains?        A ( θ ) + BK − BK  x k +1  x k  =   A ( θ ) − A o A o + LC e k +1 e k D. Peaucelle 2 Cape Town, August 2014

Outline        A ( θ ) + BK − BK  x k +1  x k  =   A ( θ ) − A o A o + LC e k +1 e k ■ S-variable approach to state-feedback design, and analysis of the closed-loop ■ Joint model and gain observer design with minimization of influence on state-feedback ■ Robust analysis of the state-feedback + observer feedback loop ■ Numerical example ■ Conclusions D. Peaucelle 3 Cape Town, August 2014

state-feedback design ■ Discrete-time systems with polytopic uncertainties v ¯ v ¯ θ v A [ v ] : θ v =1 ... ¯ � � A ( θ ) = v ≥ 0 , θ v = 1 v =1 v =1 ■ Example of existing LMI result for state-feedback design (SFdesign) S       ∃ P [ v ] P [ v ] ≻ 0 0 0 F 1 1 1    � �  w − P [ v ] − ( A [ v ] F 1 + B ˆ :  ≺ B w B T .  0 0   K )  0 0 ∃ F 1 I 1    − C [ v ] ∃ ˆ − µ 2   0 0 K ∞ I z F 1   ● If (SFdesign) hold for all vertices then K = ˆ KF − 1 robustly stabilizes the plant and 1 x k +1 = ( A ( θ ) + BK ) x k + B w w k z k = C z ( θ ) x k has H ∞ norm robustly less than µ ∞ , i.e. � z � 2 ≤ µ ∞ � w � 2 . D. Peaucelle 4 Cape Town, August 2014

state-feedback analysis ■ Robust analysis LMI result for the plant with state-feedback (SFanalysis) ∃ P [ v ]  P [ v ]  ≻ 0 0 0 3 3 �� S � � − ( A [ v ] + BK ) Q − P [ v ]   :  ≺ G 3 . ∃ G 3 0 0 I B   3  ∃ Q ≻ 0 0 0 − I ● If K is solution to (SFdesign) then (SFanalysis) is feasible ● If (SFanalysis) hold for all vertices then the following plant is robustly stable and x k +1 = ( A ( θ ) + BK ) x k − BKe k g k = Q 1 / 2 x k has H ∞ norm robustly less than 1, i.e. � Q − 1 / 2 x � 2 ≤ � Ke k � 2 . ● Virtual output g k models the excursions of the state due to perturbations on the control. ● Maximizing Q gives indications on the maximal excusions. D. Peaucelle 5 Cape Town, August 2014

Joint model and gain observer design ■ LMI result for robust observer design (Odesign) 4 p � K T K, ∃ F 4 , ∃ ˆ ∃ P [ v ] 42 ≻ 0 , ∃ P [ v ] A o , ∃ ˆ L : S       P [ v ] 0 0 I 42     � � K T K − P [ v ] − ( ˆ A o + ˆ ˆ  ≺ A o − F 4 A [ v ]     0 0 0 F 4 LC ) 42      − γ 2 0 0 0 2 Q   S       P [ v ] 0 0 I 4 p     � � − P [ v ]  ≺ − ( ˆ A o + ˆ ˆ A o − F 4 A [ v ]     0 0 0 F 4 LC ) 4 p      − γ 2 0 0 p Q 0   ● For all K and Q the LMI problem (Odesign) is feasible. ˆ ˆ ● (Odesign) provides A o = F − 1 A o , L = F − 1 L s.t. the error dynamics are stable 4 4 e k +1 = ( A o + LC ) e k + ( A ( θ ) − A o ) x k and guarantees the following robust properties: � Ke � 2 ≤ γ 2 � Q − 1 x � 2 , � Ke k � ≤ γ p � Q − 1 x � 2 � Ke � p = max k D. Peaucelle 6 Cape Town, August 2014

Closed-loop analysis ■ Small gain theorem guarantees closed-loop robust stability if γ 2 < 1 . ● Minimizing γ 2 improves stability, but tends to give high gain observers with large peak responses. ● Minimizing a linear combination of γ 2 and γ p gives a trade-off between the two effects D. Peaucelle 7 Cape Town, August 2014

Closed-loop analysis ■ Robust analysis LMI result for the closed-loop plant (Oanalysis) ∃ P [ v ] ≻ 0 , ∃ G 6 : 6 P [ v ]   0 0 6 �� S T � �      C [ v ] T  C [ v ] − A [ v ] I 0   − BK − B w z z − P [ v ]  ≺ G 6 .   0 0   6   0 I LC − A o − BK − LC 0 0 0  − ν ∞ 2 I 0 0 ● If (Oanalysis) hold for all vertices then the state-fedback + observer loop robustly stabilizes the plant and x k +1 = A ( θ ) x k + Bu k + B w w k , x k +1 = ( A o + BK + LC )ˆ ˆ x k − Ly k y k = Cx k , z k = C z ( θ ) x k u k = K ˆ x k has H ∞ norm robustly less than ν ∞ , i.e. � z � 2 ≤ ν ∞ � ˆ w � 2 . ● One can expect µ ∞ ≤ ν ∞ , i.e. that the observer-based control degrades the performance compared to the ideal state-feedback. D. Peaucelle 8 Cape Town, August 2014

Numerical example       � � y k = x k 0 1  1  0 . 1  a b  x k +  u k +  w k , x k +1 = � � 1 0 0 0 z k = x k 0 1 ■ a ∈ [ 0 . 9 , 1 . 1 ] and b ∈ [ 0 . 9 , 1 . 1 ] , i.e. ¯ v = 4 vertices. None of the vertices are stable. � � ● (SFdesign) with µ ∞ = 1 gives K = − 1 . 0633 − 1 . 0324 .    0 . 1239 0 . 0527 ● (SFanalysis) with max Tr ( Q ) gives Q =  . 0 . 0527 0 . 5730 ● (Odesign) with min γ 2 + γ p gives γ 2 = 0 . 8797 , γ p = 0 . 8575 and      0 . 9946 0 . 9807  − 2 . 3637  , L =  . A o = 0 . 9946 − 0 . 0191 − 1 . 3565    1 1 ▲ γ 2 < 1 , the closed-loop is robustly stable ▲ A o � =  1 0 ● (Oanalysis) with min ν ∞ gives ν ∞ = 1 . 0268 . D. Peaucelle 9 Cape Town, August 2014

Numerical example ● Impulse responses (for several random values of uncertainties) Impulse Response Impulse Response 0.25 0.2 0.1 0.15 0.08 0.1 Amplitude Amplitude 0.06 0.05 0.04 0 0.02 −0.05 0 −0.1 −0.02 −0.15 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 35 Time (seconds) Time (seconds) ideal state-feedback with observer-based control D. Peaucelle 10 Cape Town, August 2014

Numerical example ● Control inputs for two different choices of observers Impulse Response Impulse Response 0.3 0.1 0.2 0.1 0 Amplitude Amplitude 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 0 5 10 15 20 25 30 35 0 10 20 30 40 50 60 Time (seconds) Time (seconds) (Odesign) with min γ 2 + 10 4 γ p (Odesign) with min γ 2 + γ p ● Allows to reduce the peak response but with slower convergence D. Peaucelle 11 Cape Town, August 2014

Conclusions ■ Robust observer design revisited ■ Proposed heuristic based on LMIs only ■ Observer optimized not to perturb too strongly the given state-feedback Tradeoff between L 2 and peak criteria ■ Discussions about S-variable approach LMIs: primal (observer case) VS dual (state-feedback) analysis ( G S-variable) VS design ( F structured S-variable) The S-Variable Approach to LMI-Based Robust Control Springer, Y. Ebihara, D. Peaucelle, D. Arzelier, 2015 D. Peaucelle 12 Cape Town, August 2014