Design of H Bounded Non-Fragile Controllers for Discrete-Time - PowerPoint PPT Presentation

Design of H ∞ Bounded Non-Fragile Controllers for Discrete-Time Systems C.Briat, J.J. Martinez speaker A. Seuret December 2009 CDC 09 - Shanghai, China C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 1/24

Plan Introduction Main Results Examples C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 2/24

Introduction C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 3/24

Previous Works Resilience of controllers [Keel et al. ’97] Continuous-Time systems Ricatti [Haddad, 97], [Yang et al, 01] LMI [Jadbabaie et al. 97], [Peaucelle et al. 04] Discrete-Time Systems ? Bounded controller design (NP-hard, [Blondel et al. 97]) Sporadic results, e.g. [Peaucelle et al. 08] in continuous-time Discrete-time ? C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 4/24

Goals Resilient Controllers (SF) Synthesis for DT systems Bounded Controllers Synthesis Efficient characterization of solutions (LMIs) Include performance optimization C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 5/24

Systems Discrete-time linear systems �   x ( k ) � x ( k + 1 ) � � A B E = u ( k )   z ( k ) C D F w ( k ) state x , control input u , exogenous input w , controlled output z . Matrices supposed known Can be extended easily to the uncertain case C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 6/24

Control Laws Find a control law of the form u ( k ) = Kx ( k ) such that it stabilizes the system minimizes a performance criterium, e.g. H ∞ . Moreover, the controller must also satisfy A resilience (non-fragility) property A boundedness condition for the coefficients C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 7/24

Non-fragility property Self-robustness property of the controller Error on the controller implementation gain maintain closed-loop stability Model of the implementation error Two type of errors : Additive error (rounding, uniform discrete valued space) K i = K c + δ K Additive and multiplicative error (rounding+nonuniform discrete valued space) K i = K c + θ K c + Γ K i implemented controller, K c computed one, δ K , θ, Γ error terms C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 8/24

Additive Error Form of implemented gain K i = K c + δ K δ K = U ∆ V ∆ diagonal , || ∆ || 2 ≤ α Coefficients of δ K inside [ − α, α ] ✻ δ K + α ✰ K ✲ ✶ ♦ possible values for δ K − α C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 9/24

Additive-Multiplicative Error Form of implemented gain Γ = U ˜ K i = ( 1 + θ ) K c + Γ ∆ V || ˜ θ ∈ [ − µ, µ ] ∆ || 2 ≤ ˜ α Illustration of possible error behavior : δ K ✻ Linear approximation Maximal error q ✠ ☛ ✲ K ❦ ✸ possible values for δ K δ K total implementation error C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 10/24

Bounded coefficients Form of implemented gain (with additive error) K i = M 1 ( K 0 + K c ) M 2 + δ K � �� � previous K c M 1 , M 2 scaling terms, K 0 shifting term K 0 allows for looking for a controller centered around 0 such that √ || K c + M − 1 1 δ KM − 1 2 || 2 ≤ β mn m , n dimensions of input and state resp. C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 11/24

Main Results C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 12/24

Resilient state-feedback (additive) Theorem There exists a quadratically stabilizing resilient state-feedback if there exist a matrix X = X T ≻ 0 , a diagonal matrix Q ≻ 0 and a scalar γ > 0 such that the following LMI   XV T − X M 14 M 15 0 F T E T ⋆ − γ I 0     ⋆ ⋆ − Q 0 0 ≺ 0     − γ I + α 2 DUQU T D T α 2 DUQU T B T ⋆ ⋆ ⋆   − X + α 2 BUQU T B T ⋆ ⋆ ⋆ ⋆ holds where [ AX + BM 1 K 0 M 2 X + BM 1 Y ] T M 15 = [ CX + DM 1 K 0 M 2 X + DM 1 Y ] T M 14 = In such a case, we have K c = Y ( M 2 X ) − 1 and the closed-loop system satisfies || z || ℓ 2 ≤ γ || w || ℓ 2 . C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 13/24

Sketch of the proof Write the closed-loop system Substitute into the BRL Rewrite the BRL into the form Ψ + U T ∆ V + V T ∆ T U ≺ 0 Apply the Petersen’s lemma (or Scaled-bounded real lemma), congruence transformations, Schur complement and change of variables (standard) C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 14/24

Adding constraints on the controller coefficients (1) Idea : Add a condition to the previous design → add-on Nonlinear constraint on the controller (proved NP-hard, nonconvex) → no exact LMI formulation Relaxation necessary (Cone complementary algorithm or iterative LMI algorithm) C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 15/24

Adding constraints on the controller coefficients (2) Iterative LMI based result (no additional optimization cost) Theorem Find N , Y and X ≻ 0 of appropriate dimension such that   Π 11 Y 0 0 N T M 2 X + XM T XM − T V T N T ⋆ 2 N   2  � 0   ⋆ ⋆ − H 0  ⋆ ⋆ ⋆ − I Π 11 = − s 2 mn β 2 I + α 2 M − 1 1 UHU T M − T 1 2 || 2 ≤ s √ mn β . This will result in a gain K c satisfying || K c + M − 1 1 δ KM − 1 Iteration between X and the slack-variable N Can be proved using the projection lemma. C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 16/24

Example C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 17/24

Example (1) Let us consider the unstable system x ( k + 1 ) = Ax ( k ) + Bu ( k ) + Ew ( k ) z ( k ) = Cx ( k ) + Du ( k ) + Fw ( k ) with matrices F = 0   9 . 3547 0 . 5789 1 . 3889 2 . 7219 9 . 1690 3 . 5287 2 . 0277 1 . 9881   A =   4 . 1027 8 . 1317 1 . 9872 0 . 1527   8 . 9365 0 . 0986 6 . 0379 7 . 4679     1 0 0 0 0 0 0 1 0 1 0 0     C = 0 . 1 D =     0 0 0 0 1 0     0 0 0 0 0 1     1 0 1 0 1 2 1 2     B = E = 0 . 1     − 1 0 1 0     0 0 1 1 C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 18/24

Example (2) With no implementation error we get γ ∗ = 14 . 78 System stabilizable for all α < 0 . 0032 (need quite large precision) For a precision of α = 0 . 0020, we find γ a = 132 . 9090 (worst case) After rounding and verification, we get γ r = 88 . 4425 C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 19/24

Optimal Controller � � − 18 . 6097 − 3 . 5441 − 5 . 8235 − 7 . 7459 K ∗ = − 2 . 3537 − 3 . 0544 − 0 . 8132 − 0 . 0420 Resilient Controller � � − 53 . 4820 − 16 . 3820 − 20 . 9800 − 24 . 6260 K a = 42 . 0900 13 . 3060 18 . 5000 21 . 4660 C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 20/24

Example (3) Location of eigenvalues of the closed-loop system for random implementation error lower than 0.0025. Left : optimal controller, Right : Memory resilient controller unstable behavior on the left C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 21/24

Conclusion and Future Works C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 22/24

Conclusion and Future Works Characterization of Resilient SF Controllers Two types of error LMI form (optimization) Additional nonlinear constraint for the boundedness of controllers (relaxation) Characterize more general class of errors Dynamic Output Feedback case Other formulations for boundedness of controllers (more relevant in continuous time) C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 23/24

Thank you for your attention C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 24/24