H Bounded Resilient state-feedback design for linear - PowerPoint PPT Presentation

H ∞ Bounded Resilient state-feedback design for linear continuous-time systems - A robust control approach C.Briat and J.J. Martinez July/August 2011 IFAC World Congress 2011, Milano, Italy C.Briat and J.J. Martinez corentin@briat.info 1/24

Outline Introduction Main Results Examples C.Briat and J.J. Martinez corentin@briat.info 2/24

Introduction C.Briat and J.J. Martinez corentin@briat.info 3/24

Previous Works Resilience of controllers [Keel et al. ’97] Continuous-Time systems Riccatti [Haddad, 97], [Yang et al, 01] LMI [Jadbabaie et al. 97], [Peaucelle et al. 04] Discrete-Time Systems [Briat and Martinez, 09] Bounded controller design (NP-hard, [Blondel et al. 97]) Continuous-time [Peaucelle et al. 08] Discrete-time [Briat and Martinez, 09] C.Briat and J.J. Martinez corentin@briat.info 4/24

Goal Design of a state-feedback for continuous-time LTI systems Resilient (non-fragile) Bounded Achieve minimal performance C.Briat and J.J. Martinez corentin@briat.info 5/24

Considered system LTI continuous-time linear systems x ( t ) ˙ = Ax ( t ) + Bu ( t ) + Ew ( t ) z ( t ) = Cx ( t ) + Du ( t ) + Fw ( t ) state x , control input u , exogenous input w , controlled output z . Matrices supposed known Method easily extends to the uncertain case C.Briat and J.J. Martinez corentin@briat.info 6/24

Control Laws Find a control law of the form u ( t ) = Kx ( t ) which asymptotically stabilizes the system minimizes a performance criterion, e.g. H ∞ . Additionally, the controller must satisfy A resilience (non-fragility) property A boundedness condition on the coefficients C.Briat and J.J. Martinez corentin@briat.info 7/24

Non-fragility property Self-robustness property of controllers Error on the controller implementation preserves closed-loop system stability Error models : Additive error (rounding, uniform discrete valued space) K i = K c + δ K , δ K : error term Additive and multiplicative error (rounding+nonuniform discrete valued space) K i = K c + θ K c + Γ , θ, Γ : error term K i implemented controller, K c computed one C.Briat and J.J. Martinez corentin@briat.info 8/24

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

Additive-Multiplicative Error Form of implemented gain Γ = U ˜ K i = ( 1 + θ ) K c + Γ , ∆ V || ˜ U , V known , 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 and J.J. Martinez corentin@briat.info 10/24

Bounded coefficients Form of implemented gain (with additive error) K i = K 0 + K c + δ K � �� � previous K c K 0 shifting term ( K c centered around 0) Design of a controller centered about 0 such that √ || K c + δ K || 2 ≤ β mn m , n dimensions of input and state resp, β maximal amplitude for controller coefficients. C.Briat and J.J. Martinez corentin@briat.info 11/24

Main Results C.Briat and J.J. Martinez 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 M 11 E M 14 F T ⋆ − γ I 0    ≺ 0   ⋆ ⋆ − Q 0  ⋆ ⋆ ⋆ M 44 holds where He [ AX + BK 0 X + BY ] + α 2 BUQU T B T M 11 = [ CX + DK 0 X + DY + α 2 DUQU T B T ] T M 14 = − γ I + α 2 DUQU T D T M 44 = In such a case, we have K c = YX − 1 and the closed-loop system satisfies || z || ℓ 2 ≤ γ || w || ℓ 2 . C.Briat and J.J. Martinez corentin@briat.info 13/24

Sketch of the proof Write the closed-loop system Substitute into the Bounded Real Lemma (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) to obtain LMIs C.Briat and J.J. Martinez corentin@briat.info 14/24

Adding constraints on the controller coefficients (1) Idea : Add a condition to the previous design Nonconvex constraint on the controller → no exact LMI formulation Relaxation necessary (Cone complementary algorithm or iterative LMI algorithm) C.Briat and J.J. Martinez 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 X + XN XV T N T ⋆    � 0   ⋆ ⋆ − H 0  ⋆ ⋆ ⋆ − I Π 11 = − mn β 2 I + α 2 UHU T This will result in a gain K c satisfying || K c + δ K || 2 ≤ √ mn β . Iteration between X and the slack-variable N Can be proved using the projection lemma. C.Briat and J.J. Martinez corentin@briat.info 16/24

Example C.Briat and J.J. Martinez corentin@briat.info 17/24

Example (1) Let us consider the unstable system ˙ x ( t ) = Ax ( t ) + Bu ( t ) + Ew ( t ) z ( t ) = Cx ( t ) + Du ( t ) + Fw ( t ) with matrices F = 0 and       2 1 7 1 1 0 1 0 0 0 1 7 1 2 0 1 0 1 0 0       A = C = D =       7 1 0 1 0 0 0 0 1 0       2 1 6 1 0 0 0 0 0 1     1 0 1 1 0 1 3 0     B = E =     1 1 1 1     0 0 2 0 C.Briat and J.J. Martinez corentin@briat.info 18/24

Example (2) With no implementation error we get γ ∗ = 3 . 4155 System stabilizable for all α < 1309 For a precision of α = 0 . 5, we find γ a = 3 . 4368 (worst case) Resilient Controller (after rounding) � − 4148 � − 65751 − 10023 − 20577 K a = − 22161 − 352006 − 53613 − 110142 Too large coefficients Setting β = 15 . 5, α = 1 / 2 and K 0 = − 1 / 2 · 1 m × n (integer coefficients in [ − 16 , 15 ] ), we get � − 8 � 1 − 11 − 5 K = − 1 − 14 0 − 5 and γ = 4 . 3924. C.Briat and J.J. Martinez corentin@briat.info 19/24

Example (3) 16 14 12 10 γ 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 α Evolution of the H ∞ -norm w.r.t. α C.Briat and J.J. Martinez corentin@briat.info 20/24

Example (4) 22 20 18 16 14 12 γ 10 8 6 4 2 20 40 60 80 100 120 β Evolution of the H ∞ -norm w.r.t. β Red : worst-case (any controller in the ball of radius α = 1 / 2) Blue : actual H ∞ -norm for implemented controller C.Briat and J.J. Martinez corentin@briat.info 21/24

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

Conclusion and Future Works Characterization of Resilient SF Controllers Two types of error LMI formulation (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 C.Briat and J.J. Martinez corentin@briat.info 23/24

Thank you for your attention C.Briat and J.J. Martinez corentin@briat.info 24/24