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-01225068v1
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) 0 1 0 0 . . ... . . . . x = ˙ x + ( f ( x, θ ) + g ( x, θ ) u ) 1 0 0 · · · 0 1 D. Peaucelle 1 16 mars 2016
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 ˆ x k +1 = A o ˆ x k + B o u k + L ( C ˆ x k − y k ) ■ Observed-state feedback u k = K ˆ x 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
Motivation ■ Closed-loop dynamics (state x and error e = x − ˆ x ) 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 x k = e k +1 0 A o + LC e k ▲ Impossible when θ are uncertainties (Notion of observed-state not quite defined for uncertain systems) D. Peaucelle 3 16 mars 2016
Motivation ■ Closed-loop dynamics (state x and error e = x − ˆ x ) 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 A o 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 , ǫ k = Ke k where x k is treated as the input and ǫ k is the output. D. Peaucelle 4 16 mars 2016
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
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 } . � � p , . . . , θ [¯ v p ] θ [1] ■ Each θ p in a polytope with ¯ v p vertices V p = p v p ¯ v p ¯ � � � � ξ p,v θ [ v ] Θ p = Co ( V p ) = θ p = : ξ p,v ≥ 0 , ξ p,v = 1 . p v =1 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
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 [2] 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 ) 1 + θ 1 + θ 1 θ 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 � � with θ 1 ∈ [1 , 2] and θ 2 ∈ [1 , 2] . ● Example: θ 1 θ 1 θ 2 θ 2 1 + 1 � � � � � � � � 3 5 3 3 9 3 = � = 1 1 1 2 4 2 2 2 2 2 4 2 2 2 D. Peaucelle 7 16 mars 2016
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: − 1 1 + θ 2 0 0 1 0 θ 1 θ 12 θ 2 θ 1 θ 1 0 1+ θ 2 = . 0 1 0 0 θ 1 θ 2 1 0 0 0 1 θ 1 0 0 θ 1 0 1 D. Peaucelle 8 16 mars 2016
❶ Descriptor multi-affine modeling of rational systems ■ Discrete-time linear system, with performance I/O, rational in the uncertainties x k +1 = A r ( θ ) x k + B rw ( θ ) w k z k = C rz ( θ ) x k + D rzw ( θ ) w k ● The DMAR A r ( θ ) B rw ( θ ) E x ( θ ) � � = E − 1 π ( θ ) A ( θ ) B w ( θ ) C rz ( θ ) D rzw ( θ ) E z ( θ ) ■ gives the following descriptor multi-affine representation of the system x k +1 I 0 E x ( θ ) 0 0 z k = E ( θ ) η k = 0 0 I E z ( θ ) 0 0 π k 0 0 E π ( θ ) A ( θ ) B w ( θ ) x k w k ▲ π k : exogenous vector = E − 1 π ( θ )( A ( θ ) x k + B w ( θ ) w k ) D. Peaucelle 9 16 mars 2016
❷ LMI results for robust design and robust analysis ■ If there exists P [ v ] = P [ v ] T ≻ 0 , S and µ 2 such that for all vertices θ [ v ] ∈ V � � ≺ ( SE ( θ [ v ] )) + ( SE ( θ [ v ] )) T P [ v ] − P [ v ] − µ 2 I I 0 diag 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 : � � ≺ ( SE ( θ )) + ( SE ( θ )) T − µ 2 I P ( θ ) I 0 − P ( θ ) diag with multi-affine Lyapunov matrix P ( θ ) ≻ 0 . ● Proof - step 2 - Since E ( θ ) η k = 0 one gets � � η T − µ 2 I η k P ( θ ) I 0 − P ( θ ) k diag = 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 10 16 mars 2016
❷ LMI results for robust design and robust analysis ■ If there exists P [ v ] = P [ v ] T ≻ 0 , S and µ 2 such that for all vertices θ [ v ] ∈ V � � ≺ ( SE ( θ [ v ] )) + ( SE ( θ [ v ] )) T P [ v ] − P [ v ] − µ 2 I I 0 diag 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 11 16 mars 2016
❷ LMI results for robust design and robust analysis ■ SV-LMI for robust state-feedback design ≻ 0 , S dx , S dy , S dπ such that LMIs L sf ( θ [ v ] ) hold for all θ [ v ] ∈ V If there exist P [ v ] d then K = S dyT ( S dxT ) − 1 is a robustly stabilizing state-feedback gain s.t. x k +1 = A r ( θ ) x k + B r ( θ ) u k + B rw ( θ ) w k , u k = Kx k z 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 12 16 mars 2016
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 + ǫ k is robustly stable and x k is bounded for bounded control errors ǫ k : � Wx � 2 ≤ � ǫ � 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 13 16 mars 2016
❷ LMI results for robust design and robust analysis ■ SV-LMI for robust observer design under fixed K = K and expected state trajectories W = W � K T K , S x , S a , S b , S l , S 2 π , S pπ such that LMIs If there exist P [ v ] ∞ ≻ 0 , P [ v ] p 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: � ǫ � 2 ≤ γ 2 � Wx � 2 , � ǫ � p ≤ γ p � Wx � 2 where ǫ 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 14 16 mars 2016
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