Ellipsoidal and Interval Techniques for State Estimation in Linear - PowerPoint PPT Presentation

Ellipsoidal and Interval Techniques for State Estimation in Linear Dynamical Systems under Model Uncertainty Sergey Nazin Institute of Control Science, Moscow, Russia CESAME – UCL, Louvain-la-Neuve, Belgium Une s´ eminaire ` a CESAME, 21 mars 2006 ` a 14.00 0

Research interests Guaranteed state estimation for uncertain dynamic systems Ellipsoidal estimation Interval estimation Interval analysis, Robust linear algebra Interval systems of linear algebraic equations Invariant sets in state estimation and control problems 1

★ ✥ Today: Ellipsoidal Techniques for State Estimation in Linear Dynamical Systems under Model Uncertainty Outline: 1. Introduction to set-membership estimation , application to state estimation problem, classical ellipsoidal method 2. State estimation for dynamic systems under model uncertainty via method of ellipsoids ✧ ✦ 2

★ ✥ I. Set-membership State Estimation: State vector x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) T Discrete-time model x ( k + 1) = f ( x ( k ) , v ( k ) , k ) , x (0) = x 0 y ( k ) = h ( x ( k ) , w ( k ) , k ) x – approximation of state vector Estimation: � Stochastic models: variables are random with known distribution (Kalman filter) Deterministic models: variables are unknown-but-bounded (set-membership estimation) ✧ ✦ 3

★ ✥ Names and References: Bertsekas, D. P., & Rhodes I. B. (1971). Recursive state estimation for a set-membership description of uncertainty. IEEE TAC , 16, 117–128. Schweppe, F. C. (1973). Uncertain Dynamic Systems . Englewood Cliffs, NJ: Prentice Hall. Fogel, E., & Huang, Y. F. (1982). On the value of information in system identification – bounded noise case. Automatica , 18, 229–238. Milanese, M., Norton, J. P., Piet-Lahanier, H., & Walter, E., Eds. (1996). Bounding Approaches to System Identification . New York: Plenum. Boyd, S., El Ghaoui, L., Ferron, E., & Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory , Philadelphia: SIAM. ✧ ✦ 4

★ ✥ State estimation: problem statement = A k x k + B k v k , Linear discrete-time model: x k +1 = C k x k + w k y k x k ∈ R n – state vector, v k ∈ R r – perturbation, y k ∈ R m – measurements, w k ∈ R q – noise. x 0 ∈ X 0 , v k ∈ V k , w k ∈ W k Uncertainty: The problem is to find the guaranteed estimate for x k under given measurements y 1 , ..., y k and initial state x 0 . Techniques: ellipsoidal, interval, polyhedral, etc... ✧ ✦ 5

★ ✥ Classical ellipsoidal method (without model uncertainty) x k +1 = A k x k + B k v k = C k x k + w k y k A k , B k , C k are known; x 0 ∈ E 0 – bounded, � v k � ≤ 1 , � w k � ≤ 1 . Ellipsoidal estimation: Prediction phase — sum of ellipsoids Correction phase — intersection of ellipsoids x : ( x − c ) T P ( x − c ) ≤ 1 , P ≥ 0 Ellipsoid — � � E ( c, P ) = f det = − ln det P Size of ellipsoid: determinant criterion f tr = tr P − 1 trace criterion ✧ ✦ 6

� ★ ✥ Prediction phase x k +1 = A k x k + B k v k ; x 0 ∈ E ( c 0 , P 0 ) , P 0 > 0 , � v k � ≤ 1 . Lemma 1 (Durieu et al., 2001): 2.5 Let S N = � N i =1 E ( c i , P i ) and 2 � � 1.5 � A N = α : α i > 0 , α i = 1 . 1 Then S N ⊆ E ( c α , P α ) , ∀ α ∈ A N with 0.5 0 −0.5 � � P − 1 α − 1 P − 1 c α = c i , = . α i i −1 −1.5 f det ( P α ) = − ln det P α and −2 f tr ( P α ) = tr P − 1 convex on A N . — α −2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 ✧ ✦ 7

� ★ ✥ Correction phase x k ∈ E ( c k , P k ) , P k > 0; y k = C k x k + w k ; � w k � ≤ 1 . Lemma 2 (Durieu et al., 2001): Let I N = � N i =1 E ( c i , P i ) , A N = { α : α i ≥ 0 , � α i = 1 } . 3 2 Then I N ⊆ E ( c α , P α / (1 − δ α )) with 1 � � c α = P − 1 0 P α = α i P i , α i P i c i , α −1 � α i c T i P i c i − c T δ α = α P α c α −2 ∀ α ∈ A N provided that P α > 0 . −3 −4 f det ( P α ) = − ln det P α −3 −2 −1 0 1 2 3 and ✧ f tr ( P α ) = tr P − 1 ✦ convex on A N . — α 8

★ ✥ Set-membership filtering: Prediction step Correction step Boyd et al. (1994), El Ghaoui & Calafiore (1999) − → LMI approach My approach − → reduce the problem to one-dimentional optimization ✧ ✦ 9

★ ✥ II. Ellipsoidal state estimation under model uncertainty = ( A k + ∆ A k ) x k + v k x k +1 = ( C k + ∆ C k ) x k + w k y k � ∆ A k � 2 + � v k � 2 � ∆ C k � 2 + � w k � 2 ≤ 1 , ≤ 1 ε 2 ε 2 δ 2 δ 2 v k w k A k C k x 0 ∈ E ( c 0 , P 0 ) , P 0 > 0 � � � z : � z � 2 ≤ ε 2 � x � 2 + δ 2 � � H � 2 + � w � 2 Hx + w : ≤ 1 = . � Lemma 3 ε 2 δ 2 Ellipsoidal estimation: Prediction phase Correction phase ✧ ✦ 10

Reachability Sets: � ∆ A k � 2 + � v k � 2 x k +1 = ( A k + ∆ A k ) x k + v k , ≤ 1 ε 2 δ 2 v k A k x 0 ∈ E ( c 0 , P 0 ) , P 0 > 0 25 40 D 5 20 15 30 D 5 D 4 10 20 D 3 5 D 4 D 2 D 1 10 0 D 3 D 2 D 1 −5 0 −10 −10 −15 −20 −20 −25 −50 −40 −30 −20 −10 0 10 20 30 40 50 −20 −10 0 10 20 30 40 50 60 70 80 11

★ ✥ Prediction phase (model uncertainty) x = ( A + ∆ A ) x + v, � � ∆ A � 2 + � v � 2 x ∈ E ( c, P ) , ≤ 1 . ε 2 δ 2 Reachability set: � x ∈ E ( c, P ) , � z � 2 ≤ ε 2 � x � 2 + δ 2 � F = x = A x + z : � F ⊂ E ( d, Q ) ✧ ✦ 12

★ ✥ Theorem 1 (prediction) Each ellipsoid in the family E ( d ( τ ) , Q ( τ )) with (1 − δ 2 τ ) A Q − 1 d ( τ ) = τ P c, � � − 1 τ A T + τ − 1 I A Q − 1 Q ( τ ) = , 1 − ξ ( τ ) where (1 − δ 2 τ ) P − τε 2 I, = Q τ (1 − δ 2 τ ) c T P c − (1 − δ 2 τ ) 2 c T P Q − 1 ξ ( τ ) = τ P c, contains F for all τ such that 0 < τ < τ ∗ = λ min δ 2 λ min + ε 2 , where λ min = min eig P . � τ min = arg 0 <τ<τ ∗ f ( Q τ ) min ✧ ✦ 13

★ ✥ x = ( A + ∆ A ) x + v, � � ∆ A � 2 + � v � 2 x ∈ E ( c, P ) , ≤ 1 . ε 2 δ 2 Example: 8 � � � � 1 . 5 1 / 9 0 6 c = , P = , 2 0 1 4 � � 1 0 A = , ε = 1 , δ = 0 . 5 0 1 2 0 −2 −4 −6 −4 −2 0 2 4 6 8 10 ✧ ✦ 14

★ ✥ x = ( A + ∆ A ) x + v, � � ∆ A � 2 + � v � 2 x ∈ E ( c, P ) , ≤ 1 . ε 2 δ 2 Example: Optimal outer ellipsoid in 4 the case when c = 0 . 3 E(0,Q) � � 2 1 / 9 0 F c = 0 , P = , 0 1 1 � � 1 5 A = , ε = δ = 0 . 5 0 0 1 E(0,P) −1 −2 −3 −4 ✧ ✦ −8 −6 −4 −2 0 2 4 6 8 15

★ ✥ Correction phase (model uncertainty) x ∈ E ( c, P ) � ∆ C � 2 + � w � 2 y = ( C + ∆ C ) x + w, ≤ 1 . ε 2 δ 2 y − Cx = ∆ C x + w, � y − Cx � 2 ≤ ε 2 � x � 2 + δ 2 ( x − d ) T M ( x − d ) ≤ 1 , d = R − 1 C T y, R = C T C − ε 2 I. with R M = y T CR − 1 C T y − y T y + δ 2 , ✧ ✦ 16

★ ✥ Theorem 2 (correction) � ∆ C � 2 + � w � 2 If x ∈ E ( c, P ) , and y = ( C + ∆ C ) x + w , ≤ 1 , ε 2 δ 2 then x ∈ E ( g ( τ ) , Q ( τ )) where  (1 − ν ) − 1 Q τ , Q ( τ ) =   = (1 − τ ) P + τM, Q τ    Q − 1 g ( τ ) = τ [(1 − τ ) Pc + τMd ] ,  (1 − τ ) c T Pc + τd T Md − g ( τ ) T Q τ g ( τ ) , = ν � � 1 0 ≤ τ < τ ∗ = min ∀ τ : 1 , λ min = min eig ( M, P ) . � , 1 − λ min τ min = arg 0 ≤ τ<τ ∗ f ( Q τ ) min ✧ ✦ 17

★ ✥ x ∈ E ( c, P ) � ∆ C � 2 + � w � 2 y = ( C + ∆ C ) x + w, ≤ 1 . ε 2 δ 2 Example: 4 3 2 y = 3 , C = ( − 1 . 5 , 3) , 1 ε = 1 . 5 and δ = 0 . 5 . 0 � � 1 0 E ( c, P ) : c = 0 , P = . 0 1 / 9 −1 −2 −3 −4 −3 −2 −1 0 1 2 3 ✧ ✦ 18

★ ✥ Some extensions to 1. other matrix norms that specifies model uncertainty 2. separate uncertainties Conclusions 1. One-step sub-optimal ellipsoidal estimates for prediction and correction. 2. Difficulties: convexity of cost functions, stable calculations, outliers Paper: Polyak, Nazin, Durieu, & Walter (2004). Ellipsoidal parameter or state estimation under model uncertainty. Automatica , 40 (7). ✧ ✦ 19