Synthèse de correcteurs par retour d état observé robustes. pour les systèmes à temps discret rationnels en les incertitudes

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1 Synthèse de correcteurs par retour d état observé robustes pour les systèmes à temps discret rationnels en les incertitudes Dimitri Peaucelle Yoshio Ebihara & Yohei Hosoe Séminaire MOSAR, 16 mars 2016 Ecole des Mines de Nantes Extends results from two papers presented at 19th IFAC World Congress (Cape Town). Results submitted to Automatica: hal.archives-ouvertes.fr/hal v1

2 Motivation Literature full of robust state-feedback design results, few for robust observer design Output filter Observer (no open loop stability assumption) Observers of states in given state-space + assuming MIMO systems i.e. not restricted to SISO systems in canonical form (integrators in series) ẋ = x (f(x, θ) + g(x, θ)u) D. Peaucelle 1 16 mars 2016

3 Motivation Discrete-time linear system with uncertainties x k+1 = A r (θ)x k + B r (θ)u k, y k = Cx k Luenberger-like observer ˆx k+1 = A oˆx k + B o u k + L(C ˆx k y k ) Observed-state feedback u k = K ˆx k Our goals: Build a separation-like heuristic with first, K design, then, A o, B o, L design Use up-to-date SV-LMI tools For systems rational in the uncertainties θ D. Peaucelle 2 16 mars 2016

4 Motivation Closed-loop dynamics (state x and error e = x ˆx) driven by the state matrix x k+1 = A r(θ) + B r (θ)k B r (θ)k x k e k+1 A (θ) + B (θ)k A o + LC B (θ)k e k where A (θ) = A r (θ) A o and B (θ) = B r (θ) B o. Separation obtained when A (θ) = 0 and B (θ) = 0 x k+1 = A r(θ) + B r (θ)k B r (θ)k e k+1 0 A o + LC x k e k Impossible when θ are uncertainties (Notion of observed-state not quite defined for uncertain systems) D. Peaucelle 3 16 mars 2016

5 Motivation Closed-loop dynamics (state x and error e = x ˆx) driven by the state matrix x k+1 = A r(θ) + B r (θ)k B r (θ)k x k e k+1 A (θ) + B (θ)k A o + LC B (θ)k e k where A (θ) = A r (θ) A o and B (θ) = B r (θ) B o. Choices from the literature: A o = A r (θ nom ), but why? Possible choice min Ao max θ A r (θ) A o, but what properties? Our choice: optimize the input/output performances of e k+1 = (A o + LC B (θ)k)e k + ( A (θ) + B (θ)k)x k, ɛ k = Ke k where x k is treated as the input and ɛ k is the output. D. Peaucelle 4 16 mars 2016

6 Outline ❶ Descriptor multi-affine modeling of rational systems ❷ LMI results for robust design and robust analysis ❸ Observed-state feedback design heuristic ❹ Example D. Peaucelle 5 16 mars 2016

7 ❶ Descriptor multi-affine modeling of rational systems p independent uncertain vectors θ p R m p indexed by p = 1 p θ Θ = {(θ 1,..., θ p ) Θ 1... Θ p }. { } Each θ p in a polytope with v p vertices V p = θ p [1,..., θ [ v p p Θ p = Co(V p ) = { θ p = v p v=1 ξ p,v θ [v p : ξ p,v 0, v p v=1 ξ p,v = 1 Example: scalar uncertainty in an interval: θ p [ θ [1 p, θ [2 p. Example: 2D vector in convex hull of points issued from identification process }. D. Peaucelle 6 16 mars 2016

8 ❶ Descriptor multi-affine modeling of rational systems Multi-affine matrices: affine in each θ p Example for two scalar uncertainties θ 1 [θ [1 1, θ[2 1, θ 2 [θ [1 2, θ[ θ 1 + θ 1 θ 2 = ξ 1,1 ξ 2,1 (1 + θ [1 1 + θ[1 1 θ[1 2 ) +ξ 1,1 ξ 2,2 (1 + θ [1 1 + θ[1 1 θ[2 2 ) +ξ 1,2 ξ 2,1 (1 + θ [2 1 + θ[2 1 θ[1 2 ) +ξ 1,2 ξ 2,2 (1 + θ [2 1 + θ[2 1 θ[2 2 ). Not the same as the convex hull of all possible vertices Example: [ θ 1 θ 1 θ 2 θ 2 with θ 1 [1, 2 and θ 2 [1, [ [ = [ [ D. Peaucelle 7 16 mars 2016

9 ❶ Descriptor multi-affine modeling of rational systems Any matrix rational in θ admits a descriptor multi-affine representation (DMAR) R(θ) = M 1 (θ)m 1 2 (θ)m 3(θ) where M 1 (θ), M 2 (θ), M 3 (θ) are multi-affine in θ. Alternative to linear-fractional representations Usually of smaller size, and easier to build Example: θ 1 1+θ 2 θ 2 1 θ 2 1 θ 1 0 = θ 1 θ θ θ θ 1 θ D. Peaucelle 8 16 mars 2016

10 ❶ Descriptor multi-affine modeling of rational systems Discrete-time linear system, with performance I/O, rational in the uncertainties x k+1 z k = A r (θ)x k + B rw (θ)w k = C rz (θ)x k + D rzw (θ)w k The DMAR A r(θ) C rz (θ) B rw (θ) D rzw (θ) = E x(θ) E z (θ) Eπ 1 (θ) [ A(θ) B w (θ) gives the following descriptor multi-affine representation of the system I 0 E x (θ) I E z (θ) E π (θ) A(θ) B w (θ) x k+1 π k : exogenous vector = E 1 π (θ)(a(θ)x k + B w (θ)w k ) z k π k x k w k = E(θ)η k = 0 D. Peaucelle 9 16 mars 2016

11 ❷ LMI results for robust design and robust analysis If there exists P [v = P [vt 0, S and µ 2 such that for all vertices θ [v V [ diag P [v I 0 P [v µ 2 I (SE(θ [v )) + (SE(θ [v )) T then the system is robustly stable (i.e. θ Θ) with robust H performance µ. Proof - step 1 - By convexity the condition holds for all θ Θ: [ diag P (θ) I 0 P (θ) µ 2 I (SE(θ)) + (SE(θ)) T with multi-affine Lyapunov matrix P (θ) 0. Proof - step 2 - Since E(θ)η k = 0 one gets η T k diag [ P (θ) I 0 P (θ) µ 2 I η k = x T k+1 P (θ)x k+1 + z T k z k x T k P (θ)x k µ 2 w T k w k < 0 D. Peaucelle mars 2016

12 ❷ LMI results for robust design and robust analysis If there exists P [v = P [vt 0, S and µ 2 such that for all vertices θ [v V [ diag P [v I 0 P [v µ 2 I (SE(θ [v )) + (SE(θ [v )) T then the system is robustly stable (i.e. θ Θ) with robust H performance µ. S-variable result Extended in present work to multi-affine representations Exist tools to reduce numerical burden (sometimes lossless) Example: no S if plant is multi-affine in θ & common P = P (θ) Extensions to mixed constant/time-varying uncertainties D. Peaucelle mars 2016

13 ❷ LMI results for robust design and robust analysis SV-LMI for robust state-feedback design If there exist P [v d 0, S dx, S dy, S dπ such that LMIs L sf (θ [v ) hold for all θ [v V then K = S dy T (S dx T ) 1 is a robustly stabilizing state-feedback gain s.t. x k+1 z k = A r (θ)x k + B r (θ)u k + B rw (θ)w k, u k = Kx k = C rz (θ)x k + D rzu (θ)u k + D rzw (θ)w k has an H performance smaller than µ d whatever θ Θ. Linearizing change of variables on S-variables Proof uses equivalence with dual system x d,k+1 = A T r (θ)x d,k Result is new because for rational systems Easy extensions for regional pole location, H 2 performance, etc. D. Peaucelle mars 2016

14 ❷ LMI results for robust design and robust analysis SV-LMI for analysis of state trajectories under fixed state-feedback K = K If there exist P [v 0, Q and S such that LMIs L sf,a (θ [v ) hold for all θ [v V, then x k+1 = A r (θ)x k + B r (θ)u k, u k = Kx k + ɛ k is robustly stable and x k is bounded for bounded control errors ɛ k : W x 2 ɛ 2 where W = Q 1/2. Allow to estimate the state trajectories in case of corrupted state-feedback (inevitable when feedback is with observed-state) D. Peaucelle mars 2016

15 ❷ LMI results for robust design and robust analysis SV-LMI for robust observer design under fixed K = K If there exist P [v and expected state trajectories W = W 0, P [v p K T K, S x, S a, S b, S l, S 2π, S pπ such that LMIs L ob (θ [v ) hold for all θ [v V, then A o = S x 1 S a, B o = S x 1 S b, L = S x 1 S l define an observer that guarantees: ɛ 2 γ 2 W x 2, ɛ p γ p W x 2 where ɛ k = Ke k. The properties hold whatever bounded x and whatever θ Θ. Norm-to-norm perf: asymptotic coupling of observation error on system dynamics Norm-to-peak perf: avoid waterbed effects of transient peaks Small gain theorem: if γ 2 < 1 observed-state feedback robustly stabilizes D. Peaucelle mars 2016

16 ❷ LMI results for robust design and robust analysis SV-LMI for observed-state feedback analysis under fixed K = K, A o = A o etc. If there exist P [v c 0, S c such that LMIs L ob,a (θ [v ) hold for all θ [v V, then x k+1 z k = A r (θ)x k + B r (θ)u k + B rw (θ)w k = C rz (θ)x k + D rzu (θ)u k + D rzw (θ)w k ˆx k+1 = A oˆx k + B o u k + L(C ˆx k y k ), u k = K ˆx k has an H performance smaller than µ c whatever θ Θ. Exists also SV-LMI conditions L ob,da for the dual system: µ dc upper-bound No apriori relation between upper-bounds µ d (ideal state-feedback), µ c and µ dc D. Peaucelle mars 2016

17 ❸ Observed-state feedback design heuristic 4 steps 1- Design stabilizing state-feedback K (for example using LMI L sf ) 2- Get estimate of state trajectories represented by W (using LMIs L sf,a ) (max λ min (W T W ) leads to tight estimates) 3- Design observer (using LMIs L ob ) (min β 2 γ β p γ 2 p to adjust tradeoff between norm and peak performances) 4- Analyze observed-state closed-loop (using LMIs L ob,a or L ob,da ) No guarantee that next step would be feasible No guarantee to find a robustly stabilizing control when exists All step are purely LMI with clear control theory justification Each step based on new LMI conditions D. Peaucelle mars 2016

18 ❹ Example Academic example for illustration x k+1 = z k = θ 1 2 /θ 2 θ [ 1 0 x k + 0 u k + θ 2 [ x k + θ 2 u k, y k = 0 1 θ 1 0 x k w k DMAR E x = A = θ θ 1 θ [, E z = 0 1, E π = θ 2 0, 0 1, B = 0, B w = θ 2. θ 2 0 θ 1 [ 1 δ 1, 1+δ 1, θ 2 [ 1 δ 2, 1+δ 2 at limit of stability when θ 1 = θ 2 = 1 D. Peaucelle mars 2016

19 ❹ Example Results when µ d = 10 at first step (δ 1, δ 2 ) (β 2, β p ) (γ 2, γ p ) µ c µ dc (0, 0) (1, 1) (10 4, 10 4 ) (0.1, 0) (1, 1) (1.0747, ) (0, 0.1) (1, 1) (0.3947, ) (0.1, 0.1) (10, 1) (1.2736, ) (0.1, 0.1) (1, 1) (1.2962, ) (0.1, 0.1) (1, 10) (1.3339, ) (0.2, 0.1) (1, 1) (1.3006, ) (0.1, 0.2) (1, 1) (1.3242, ) (0.2, 0.2) (1, 1) (3.5392, ) D. Peaucelle mars 2016

20 ❹ Example Results when µ d is minimized at first step and (β 2, β p ) = (1, 1). (δ 1, δ 2 ) µ d µ d (γ 2, γ p ) µ c µ dc (0, 0) 1 1 (10 4, 10 4 ) (0.1, 0) (0.2349, ) (0, 0.1) (0.3061, ) (0.1, 0.1) (0.5611, ) (0.2, 0.1) (1.1902, ) (0.1, 0.2) (1.3334, ) (0.2, 0.2) (3.0445, ) µ d computed on SV-LMI with reduced size D. Peaucelle mars 2016

21 ❹ Example Size of LMI conditions (applying non-conservative size reduction procedures) (δ 1, δ 2 ) L sf Lsf L sf,a L ob L ob,a L ob,da (0, 0) {5, 0} {5, 0} {3, 0} {6, 0} {5, 0} {5, 0} (1, 0) {6, 1} {6, 1} {5, 2} {7, 1} {7, 2} {7, 2} (0, 1) {7, 2} {6, 1} {5, 2} {8, 2} {7, 2} {10, 5} (1, 1) {7, 2} {6, 1} {6, 3} {8, 2} {8, 3} {11, 6} {N r, N c } with N r rows in each LMI (to be multiplied by nb of vertices) {N r, N c } with N c columns of S-variables D. Peaucelle mars 2016

22 ❹ Example System model x k+1 = θ 1 2 /θ 2 θ x k + 0 u k + θ 2 θ 1 0 w k Observer model (design for δ 1 = δ 2 = 0.2) A o =, B o = D. Peaucelle mars 2016

23 Conclusions Revisited Luenberger observer design in case of uncertain systems Separation principle replaced by a mixed norm/peak performance measure LMI design of state-feedback and observer gains one after the other Heuristic with no guarantee of success Surprisingly, K design for fixed observer is more complex (prospective work) Extensions for continuous-time systems: raises issues about S-variable conditions for design (tuning parameters) Promising descriptor multi-affine representation combined to SV-LMIs D. Peaucelle mars 2016