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

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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

SLIDE 1

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

SLIDE 2

Plan

Introduction Main Results Examples

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 2/24

SLIDE 3

Introduction

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

Systems

Discrete-time linear systems x(k + 1) z(k)

A B E C D F   x(k) u(k) 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

SLIDE 7

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

SLIDE 8

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) Ki = Kc + δK Additive and multiplicative error (rounding+nonuniform discrete valued space) Ki = Kc + θKc + Γ

Ki implemented controller, Kc computed one, δK, θ, Γ error terms

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 8/24

SLIDE 9

Additive Error

Form of implemented gain Ki = Kc + δK δK = U∆V ∆ diagonal, ||∆||2 ≤ α Coefficients of δK inside [−α, α] ✻ ✲ K δK ♦ −α ✰ ✶ +α possible values for δK

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 9/24

SLIDE 10

Additive-Multiplicative Error

Form of implemented gain Ki = (1 + θ)Kc + Γ Γ = U ˜ ∆V θ ∈ [−µ, µ] || ˜ ∆||2 ≤ ˜ α Illustration of possible error behavior : ✻ ✲ K δK ✸ possible values for δK ❦ q Maximal error ✠ Linear approximation ☛ δK total implementation error

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 10/24

SLIDE 11

Bounded coefficients

Form of implemented gain (with additive error) Ki = M1(K0 + Kc)M2

previous Kc

+δK M1, M2 scaling terms, K0 shifting term K0 allows for looking for a controller centered around 0 such that ||Kc + M−1

1 δKM−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

SLIDE 12

Main Results

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

SLIDE 13

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       −X XV T M14 M15 ⋆ −γI F T ET ⋆ ⋆ −Q ⋆ ⋆ ⋆ −γI + α2DUQUTDT α2DUQUTBT ⋆ ⋆ ⋆ ⋆ −X + α2BUQUTBT       ≺ 0 holds where M15 = [AX + BM1K0M2X + BM1Y]T M14 = [CX + DM1K0M2X + DM1Y]T In such a case, we have Kc = Y(M2X)−1 and the closed-loop system satisfies ||z||ℓ2 ≤ γ||w||ℓ2.

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 13/24

SLIDE 14

Sketch of the proof

Write the closed-loop system Substitute into the BRL Rewrite the BRL into the form Ψ + UT∆V + VT∆TU ≺ 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

SLIDE 15

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

SLIDE 16

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     Π11 Y ⋆ NTM2X + XMT

2 N

XM−T

V T NT ⋆ ⋆ −H ⋆ ⋆ ⋆ −I     0 Π11 = −s2mnβ2I + α2M−1

1 UHUTM−T 1

This will result in a gain Kc satisfying ||Kc + M−1

1 δKM−1 2 ||2 ≤ s√mnβ.

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

SLIDE 17

Example

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

SLIDE 18

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 A =     9.3547 0.5789 1.3889 2.7219 9.1690 3.5287 2.0277 1.9881 4.1027 8.1317 1.9872 0.1527 8.9365 0.0986 6.0379 7.4679     C = 0.1     1 1 1     D =     1 1     B =     1 1 2 −1     E = 0.1     1 1 2 1 1 1    

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 18/24

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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 γa = 132.9090 (worst case) After rounding and verification, we get γr = 88.4425

C.Briat, J.J. Martinez speaker A. Seuret corentin@briat.info 19/24

SLIDE 20

Optimal Controller K ∗ =

−18.6097

−3.5441 −5.8235 −7.7459 −2.3537 −3.0544 −0.8132 −0.0420

Resilient Controller

Ka =

−53.4820

−16.3820 −20.9800 −24.6260 42.0900 13.3060 18.5000 21.4660

C.Briat, J.J. Martinez speaker A. Seuret

corentin@briat.info 20/24

SLIDE 21

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

SLIDE 22

Conclusion and Future Works

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

SLIDE 23

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

SLIDE 24

Thank you for your attention

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