A discrete-time Sliding Window Observer for Markovian Switching System Abdelfettah Hocine, Mohammed Chadli, Didier Maquin and Jos´e Ragot Presented by Benoˆıt Marx Institut National Polytechnique de Lorraine Centre de Recherche en Automatique de Nancy CRAN, UMR 7039, CNRS-Nancy Universit´e INPL, 8 d´ecembre 2006

Plan

Goals & Means Goals : ◮

Fault detection.

◮

State estimation.

◮

Linear systems.

◮

Subjected to noises.

Faults to be detected are a priori indexed and modelled. Means : ◮

◮

Markovian switching system.

◮

Model associated to a particular operating mode.

◮

Random commutations between operating modes.

◮

Linear matrix inequalities & quadratic Lyapunov function.

Finite memory observer Consider the system (

xk+1 = Axk + Buk + Gwk yk = Cxk + vk

On the time horizon [k, k + m − 1], where the observation window is of size m y k = Lm xk + Bm u k + Gm w k + v k with the following definitions h iT T T z k = zkT zk+1 . . . zk+m−1 , z ∈ {y , u, w , v } h iT Lm = C T (CA)T . . . (CAm )T 2

6 6 6 6 6 6 6 Bm = 6 6 6 6 6 6 4 2

6 6 6 6 6 6 6 Gm = 6 6 6 6 6

0

0

... .

CB

0 .

. .

... .

. . .

. .

. .

CAB

CB

. . . m−2 CA B

. . . m−3 CA B

...

CB

0

0

...

...

.

. CG

0

CAG

CG

.

.

. .

.

. .

. .

. . . . .

.

. .

.

. .

03 7 7 7 07 7 .7 7 .7 .7 7 7 7 7 05 0 03 7 7 7 07 7 7 .7 .7 .7 7 7 7

State estimate

by minimizing the criterion Jk = (Lm xˆk + Bm u ¯k − y¯k )T (Lm xˆk + Bm u ¯k − y¯k ) subject to xk . We obtain xˆk = αy k − βu k where ´−1 T ` Lm , α = LTm Lm

´−1 T ` Lm Bm β = LTm Lm

ek = xˆk − xk = αv k + γw k

`

´−1 LTm Lm

LTm Gm .

with γ = Express the state vector

xk = αy k − βu k − αv k − γw k

Sliding window observer structure Expression of the system state xk+1 =τ (αy k+1 − βu k+1 − αv k+1 − γw k+1 ) +(1 − τ )(Axk + Buk + wk ) where τ is a weighting scalar the following observer structure is proposed: 8 > < xˆk+1 = τ (αy k+1 − βu k+1 ) + (1 − τ )(Aˆ xk + Buk − K (ˆ yk − yk )) > : yˆk = C xˆk where K is the observer gain to be computed.

Stability conditions Estimation error ek = xˆk − xk concatenation of vectors noise g k = [vk , v k+1 , wk , w k+1 ]T is the vector of noise and F = [(1 − τ )K , τ α, (τ − 1)G , τ γ] Quadratic Lyapunov function Vk (ek ) = ekT Pek and its variation: T ∆Vk (ek ) = ek+1 Pek+1 − ekT Pek

Stability conditions

Proposition Suppose that there exists a gain matrix K , two positive definite matrices P > 0 and Q > 0, and a given positive scalar τ , satisfying the following matrix inequality: Q − P + (1 − τ )2 (A − KC )T P (A − KC ) + (1 − τ )2 (A − KC )T P 2 (A − KC ) < 0 then the observer (??) has a bounded estimation error, i.e. there exists a positive constant r 2 = k F k2 k g k k2 λmax (P + I )/λmin (Q) such that ∆Vk (ek ) < 0 for kek k > r , where I denotes the identity matrix.

Linearization To solve the nonlinear matrix inequality (??), the following sufficient LMI condition in the variables P > 0, Q > 0 and X is proposed: 3 2 (1 − τ )−2 (P − Q) (PA − XC )T (PA − XC )T 5>0 4 PA − XC P 0 PA − XC 0 I

The gain observer is given by K = P −1 X .

Development of the method Markovian process governed by an a priori known Markov transition matrix Π given by: 3 2 p11 · · · p1r 6 .. 7 .. Π = 4 ... . . 5 pr1 · · · prr where pij is the mode transition probability from the model Mi to the model Mj ; we note µkj the probability that the j th model is active at time k. Consider the j th model described by: xk+1 = Aj xk + Bj uk + Gj wk Mj : yk = Cj xk + vk The state estimation of this model is carried out using the SWO described in section ?? to give the following equations: 8 j xˆ = τj (αj y k−m+1 − βj u k−m+1 ) > > < k−m+1 +(1 − τj )(Aj xˆk−m + Bj uk−m − Kj (ˆ yk−m − yk−m )) > > : j yˆk−m = Cj xˆk−m

The state estimate at the final time k of the observation window is obtained by integrating the system (??): j xˆkj = Am−1 xˆk−m+1 + Tj,m u k−m+1 j h

i

The state estimate xˆk of the switching system is then computed as a weighted sum of the states of the “local” models: xˆk =

r X

xˆkj µjk

j=1

the following recurrence on the probability that the system operates according to the model j at the moment k can be established: P Lj (k) ri=1 pij µjk−1 j µk = Pr (1) Pr j l =1 Ll (k) i =1 pil µk−1

Fault models An actuator fault can be modelled by ”modifying” an appropriate column of the control input matrix B. Thus, a fault on the i th actuator is described by writing the following equation: xk+1 = Axk + (B + ∆Bi ) uk + wk On a same way, a sensor fault is described by: yk = (C + ∆Ci ) xk + vk

Simulation example we consider a model of normal operating (A1 , B1 , C1 ), a model of actuator fault (A2 , B2 , C2 ) and a model of sensor faults (A3 , B3 , C3 ), with the various matrices defined by: » – 0.45 0 Ai = , i = 1...3 0 0.4 – » ˆ ˜T 1 0 , B1 = 0.1815 1.7902 , C1 = 0 1 » – ˆ ˜T 1 0 B2 = 1.1815 1.7902 , C2 = , 0 1 – » ˆ ˜T 1.5 0 . B3 = 0.1815 1.7902 , C3 = 0 1.5 For each operating model, we obtain τ1 = 0.2025, τ2 = 0.2025, τ3 = 0.3025. » – 0.5337 −0.0340 K1 = . −0.0840 0.4249 – » 0.5337 −0.0340 . K2 = −0.0840 0.4249 – » 1.3406 0.8184 . K3 = −1.1535 −0.6226

Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100

200

300

400

500 µ1

600

700

800

900

Figure: Activation probability of model 1

1000

Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100

200

300

400

500 µ2

600

700

800

900

Figure: Activation probability of model 2

1000

Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100

200

300

400

500 µ3

600

700

800

900

Figure: Activation probability of model 3

1000

◮

A structure of a Sliding Window observer (SWO) is proposed to handle the FDI issue.

◮

he SWO is based on a combination of the Finite Memory Observer (FMO) and the Luenberger Observer.

◮

Stability conditions of the proposed SWO is addressed under LMI formulation using a quadratic Lyapunov function.

◮

Guarantee the convergence of the SWO for a given interval of state estimation error.

◮

Weighting parameter τ is obtained from optimization procedure.

◮

Observer is applied within the framework of Markovian switching systems.

◮

Comparison of the obtained results with those of GPB approach.

Plan

Goals & Means Goals : ◮

Fault detection.

◮

State estimation.

◮

Linear systems.

◮

Subjected to noises.

Faults to be detected are a priori indexed and modelled. Means : ◮

◮

Markovian switching system.

◮

Model associated to a particular operating mode.

◮

Random commutations between operating modes.

◮

Linear matrix inequalities & quadratic Lyapunov function.

Finite memory observer Consider the system (

xk+1 = Axk + Buk + Gwk yk = Cxk + vk

On the time horizon [k, k + m − 1], where the observation window is of size m y k = Lm xk + Bm u k + Gm w k + v k with the following definitions h iT T T z k = zkT zk+1 . . . zk+m−1 , z ∈ {y , u, w , v } h iT Lm = C T (CA)T . . . (CAm )T 2

6 6 6 6 6 6 6 Bm = 6 6 6 6 6 6 4 2

6 6 6 6 6 6 6 Gm = 6 6 6 6 6

0

0

... .

CB

0 .

. .

... .

. . .

. .

. .

CAB

CB

. . . m−2 CA B

. . . m−3 CA B

...

CB

0

0

...

...

.

. CG

0

CAG

CG

.

.

. .

.

. .

. .

. . . . .

.

. .

.

. .

03 7 7 7 07 7 .7 7 .7 .7 7 7 7 7 05 0 03 7 7 7 07 7 7 .7 .7 .7 7 7 7

State estimate

by minimizing the criterion Jk = (Lm xˆk + Bm u ¯k − y¯k )T (Lm xˆk + Bm u ¯k − y¯k ) subject to xk . We obtain xˆk = αy k − βu k where ´−1 T ` Lm , α = LTm Lm

´−1 T ` Lm Bm β = LTm Lm

ek = xˆk − xk = αv k + γw k

`

´−1 LTm Lm

LTm Gm .

with γ = Express the state vector

xk = αy k − βu k − αv k − γw k

Sliding window observer structure Expression of the system state xk+1 =τ (αy k+1 − βu k+1 − αv k+1 − γw k+1 ) +(1 − τ )(Axk + Buk + wk ) where τ is a weighting scalar the following observer structure is proposed: 8 > < xˆk+1 = τ (αy k+1 − βu k+1 ) + (1 − τ )(Aˆ xk + Buk − K (ˆ yk − yk )) > : yˆk = C xˆk where K is the observer gain to be computed.

Stability conditions Estimation error ek = xˆk − xk concatenation of vectors noise g k = [vk , v k+1 , wk , w k+1 ]T is the vector of noise and F = [(1 − τ )K , τ α, (τ − 1)G , τ γ] Quadratic Lyapunov function Vk (ek ) = ekT Pek and its variation: T ∆Vk (ek ) = ek+1 Pek+1 − ekT Pek

Stability conditions

Proposition Suppose that there exists a gain matrix K , two positive definite matrices P > 0 and Q > 0, and a given positive scalar τ , satisfying the following matrix inequality: Q − P + (1 − τ )2 (A − KC )T P (A − KC ) + (1 − τ )2 (A − KC )T P 2 (A − KC ) < 0 then the observer (??) has a bounded estimation error, i.e. there exists a positive constant r 2 = k F k2 k g k k2 λmax (P + I )/λmin (Q) such that ∆Vk (ek ) < 0 for kek k > r , where I denotes the identity matrix.

Linearization To solve the nonlinear matrix inequality (??), the following sufficient LMI condition in the variables P > 0, Q > 0 and X is proposed: 3 2 (1 − τ )−2 (P − Q) (PA − XC )T (PA − XC )T 5>0 4 PA − XC P 0 PA − XC 0 I

The gain observer is given by K = P −1 X .

Development of the method Markovian process governed by an a priori known Markov transition matrix Π given by: 3 2 p11 · · · p1r 6 .. 7 .. Π = 4 ... . . 5 pr1 · · · prr where pij is the mode transition probability from the model Mi to the model Mj ; we note µkj the probability that the j th model is active at time k. Consider the j th model described by: xk+1 = Aj xk + Bj uk + Gj wk Mj : yk = Cj xk + vk The state estimation of this model is carried out using the SWO described in section ?? to give the following equations: 8 j xˆ = τj (αj y k−m+1 − βj u k−m+1 ) > > < k−m+1 +(1 − τj )(Aj xˆk−m + Bj uk−m − Kj (ˆ yk−m − yk−m )) > > : j yˆk−m = Cj xˆk−m

The state estimate at the final time k of the observation window is obtained by integrating the system (??): j xˆkj = Am−1 xˆk−m+1 + Tj,m u k−m+1 j h

i

The state estimate xˆk of the switching system is then computed as a weighted sum of the states of the “local” models: xˆk =

r X

xˆkj µjk

j=1

the following recurrence on the probability that the system operates according to the model j at the moment k can be established: P Lj (k) ri=1 pij µjk−1 j µk = Pr (1) Pr j l =1 Ll (k) i =1 pil µk−1

Fault models An actuator fault can be modelled by ”modifying” an appropriate column of the control input matrix B. Thus, a fault on the i th actuator is described by writing the following equation: xk+1 = Axk + (B + ∆Bi ) uk + wk On a same way, a sensor fault is described by: yk = (C + ∆Ci ) xk + vk

Simulation example we consider a model of normal operating (A1 , B1 , C1 ), a model of actuator fault (A2 , B2 , C2 ) and a model of sensor faults (A3 , B3 , C3 ), with the various matrices defined by: » – 0.45 0 Ai = , i = 1...3 0 0.4 – » ˆ ˜T 1 0 , B1 = 0.1815 1.7902 , C1 = 0 1 » – ˆ ˜T 1 0 B2 = 1.1815 1.7902 , C2 = , 0 1 – » ˆ ˜T 1.5 0 . B3 = 0.1815 1.7902 , C3 = 0 1.5 For each operating model, we obtain τ1 = 0.2025, τ2 = 0.2025, τ3 = 0.3025. » – 0.5337 −0.0340 K1 = . −0.0840 0.4249 – » 0.5337 −0.0340 . K2 = −0.0840 0.4249 – » 1.3406 0.8184 . K3 = −1.1535 −0.6226

Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100

200

300

400

500 µ1

600

700

800

900

Figure: Activation probability of model 1

1000

Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100

200

300

400

500 µ2

600

700

800

900

Figure: Activation probability of model 2

1000

Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100

200

300

400

500 µ3

600

700

800

900

Figure: Activation probability of model 3

1000

◮

A structure of a Sliding Window observer (SWO) is proposed to handle the FDI issue.

◮

he SWO is based on a combination of the Finite Memory Observer (FMO) and the Luenberger Observer.

◮

Stability conditions of the proposed SWO is addressed under LMI formulation using a quadratic Lyapunov function.

◮

Guarantee the convergence of the SWO for a given interval of state estimation error.

◮

Weighting parameter τ is obtained from optimization procedure.

◮

Observer is applied within the framework of Markovian switching systems.

◮

Comparison of the obtained results with those of GPB approach.