Robust Fault Detection Based-Observer for Linear Switched Systems

Robust Fault Detection Based-Observer for Linear Switched Systems D.E.C. BELKHIAT, N. MESSAI and N. MANAMANNI CReSTIC, Université de Reims Champagne-Ardenne, Moulin de la Housse BP 1039, 51687 REIMS cedex 2 France e -mail : {djamel.belkhiat, nadhir.messai, noureddine.manamanni}@univ-reims.fr

Abstract: This paper deals with the design of a robust hybrid observer for switched linear systems with unknown inputs and modeling error. The proposed observer is synthesized for the task of robust fault detection and concerns the case when the active mode is unknown. The basic idea of our approach is to formulate the design of the robust fault detection observer as a H ∞ model-matching problem. Then, some results of H ∞ optimization are exploited and an LMI solution procedure is proposed to synthesize the desired robust observer. Keywords: Switched systems; Observer design; Fault detection; H ∞ optimization; Linear Matrix Inequality (LMI).

1. INTRODUCTION In the last decades, a lot of works dealt with the design of control laws that improve the performances and guarantee the stability for many classes of hybrid dynamical systems (HDS), see (DeCarlo et al., 2000) (Antsaklis, 2000) and reference therein. However, when a fault occurs, these control laws become often inefficient and Fault Detection and Isolation (FDI) techniques should be implemented to safeguard the specified performances and to avoid the eventual shutdown or damage. Several approaches have been investigated for faults detection and isolation of HDS. In (Klein, 2000), a local linearization has been proposed in order to apply continuous methods. This technique has also been proposed in (Frisk, 2000) before using the parity space approach. Similar results have been obtained by using finite memory observers (Kratz and Aubry, 2003). In addition, the parity space approach was extended to hybrid systems (Cocquempot et al., 2004) in order to identify the current mode, to estimate the switching instants and to detect the faults that may occur. On the other hand, some results obtained for identification has been adapted in order to provide parameter estimation and fault detection for HDS (Messai et al., 2006). Finally, some attempts have considered non linear observer to estimate the state, the current mode and an unknown parameter for a class of switching systems (Saadaoui et al., 2006). However, to the best of our knowledge, there are only few works dealing with robust FDI for HDS. In fact, one of the key issues related to a FDI system is concerned with its robustness. This latter involves two aspects: The first one, concerns the robustness to modelling errors and disturbances and the second one is related with the sensitivity to the faults that should be detected, see (Zhang et

al., 2003). Thus, a suitable trade-off between robustness and sensitivity is essential to guarantee a good performance of a FDI system. In this work, we propose an approach to design a robust hybrid observer for switched linear systems with jumps. The observer herein proposed is synthesized for the task of robust fault detection without the knowledge of the active mode. For this task, we consider the robust fault detection problem as a standard H ∞ model matching one and we look to obtain the optimal trade-off between the robustness to the unknown inputs and the sensitivity to faults; see (Guo, et al., 2008). The paper is organized as follow: section 2 presents the considered class of HDS. Section 3, details the design of the proposed robust fault detection based-observer for linear switched systems. Finally, some numerical results illustrating the efficient of the approach are given in section 4. 2. SWITCHED LINEAR SYSTEM The switched linear systems considered in this paper are described as: x = Aq( t ) x + Bu + Bdq ( t ) d + B f q ( t ) f

(1)

y = Cx + Ddq ( t ) d + D f q ( t ) f

(2)

where x ∈ℜn is the state vector, u ∈ ℜm is the input vector, y ∈ℜ p is the measurement (output) vector, d ∈ ℜm is the unknown input vector (including disturbance, uninterested fault, noise or modeled uncertainty), f ∈ℜl is fault to be detected and isolated and q(t ) is an index function q : [0 ∞) → I N = {1,..., N } deciding which linear vectors fields is activate. Aq ( t ) , B, C , B f q ( t ) , Bd q ( t ) , D f q ( t ) , Ddq ( t ) are known

matrices with appropriate dimensions. Without loss of generality, we assume that d , f are L2 -norm bounded. Note that we consider that the change of the index function value occurs at defined switch sets Si , j , which are described by linear hyper planes according to: (3)

where I s is a set of tuple indicating which mode changes might occur in the switched system. Throughout this work, one supposes that there are only a finite number of mode changes in finite time and all couples ( Aq( t ) , C ) are observable. 3. ROBUST FAULT DETECTION BASED-OBSERVER DESIGN

xˆ = Aqˆ ( t ) xˆ + Bu + K qˆ ( t ) ( y − yˆ )

(4) (5)

r = V ( y − yˆ )

(6)

where xˆ ∈ℜ and yˆ ∈ℜ represent respectively the state and the output estimation vectors, r is the so-called residual signal. K qˆ and V are respectively the observer gain matrix1

Let e = x − xˆ be the estimation error. Thus, when the system state evolves in the mode q and the observer in the mode qˆ , both the dynamics of the observation error and the residual signal can be expressed as: e = ( Aqˆ − K qˆ C ) e + ( Aq − Aqˆ ) x +

) (

(7)

)

− K qˆ Ddq d + B f q − K qˆ D f q f

(

r = V ( y − yˆ ) = V Cx + Ddq d + D f q f − Cxˆ

)

(8)

One can remark that the dynamics of the residual signal depends not only on f and d but also on the state x . Thus, the problem of designing a robust fault detection observer (RFDO), which is one of the main objectives of this work, can be described as designing matrices K qˆ , V such that − K qˆ C ) are asymptotically stable. Nevertheless, there is

no guarantee that the estimation error will converge using only this condition because of the jumps at the switching instant. Hence, the estimated state will be also updated at the instant when the observer mode changes see e.g. (S. Pettersson, et al., 2002) and (S. Pettersson, 2005).

1

(10)

and G f ( s) are transfer functions

defined such that :

(

)

rx ( s ) = V C ( sI − Aqˆ ) ΔAq ,qˆ x ( s )  

−1

(11)

ex

( r ( s ) = V ( C ( sI − A )

) ) f (s)

−1

qˆ

B fq + D f q

(12) (13)

with: Aqˆ = Aqˆ − K qˆ C , Bdq = Bdq − K qˆ Ddq , B f q = B f q − K qˆ D f q , ΔAq ,qˆ = Aq − Aqˆ

p

and the residual weighting matrix.

qˆ

where Gx ( s ) , Gd ( s )

−1

y = Cxˆ

(A

r ( s ) = VGx ( s ) x ( s ) + VGd ( s ) d ( s ) + VG f ( s ) f ( s )

f

n

(9)

rd ( s ) = V C ( sI − Aqˆ ) Bdq + Ddq d ( s )

Let the observer for the system (1)-(2) be defined as:

dq

The dynamics of the residual generator can be formulated as: r ( s ) = rx ( s ) + rd ( s ) + rf ( s )

Si , j = { x ∈ℜn | si , j x = 0}, (i , j ) ∈ I s

(B

Moreover, the generated residual r should be as sensitive as possible to fault f and as robust as possible to the unknown input d .

The index time (t) will be omitted in the next when there is no ambiguity.

Hence, in order to ensure the robustness of the residuals, one can use the performance indexes H ∞ and H − . In fact, the performance H ∞ is used to attenuate the effect of the disturbances as well as the unknown signals: H ∞ = VGd

∞

= sup σ (VGd ( jω ) ) ω

(14)

While H − index is used as measurement of the worst case fault sensitivity of the residual generator on fault: H − = inf σ (VG f ( jω ) ) ω

(15)

where σ (. ) and σ (. ) denote the maximum and the minimum singular value of the matrices VGd and VG f , respectively. The objective now is to find the observer gain matrices K qˆ and the post filter matrix V , such that the error system is bounded and the conditions (16) –(17) hold for given scalars β qˆ > 0 and γ qˆ > 0 ,: H ∞ ≤ γ qˆ

(16)

H − ≥ β qˆ

(17)

3.1 Observer synthesis

Since the dynamics of the estimation error is a sum of three additive components; each term of (9) will be considered separately. Regarding ex in (11), one can exploit the results of (S. Pettersson, 2005). Thus, let us introduce the multiple

Lyapunov function, one for each observed mode corresponding to qˆ . Vi (ex ) = eTx Pe i ∈ IN i x,

i

(18)

where Pi ∈ℜn×n are symmetric positive definite matrices. The time derivative of (18) for the observed mode i , when the system state evolves according to mode j , becomes

(

)

T Vi (ex ) = exT [ Ai − K i C ] Pi + Pi [ Ai − K i C ] ex

(19)

+ e Pi ΔAj ,i x + x ΔA Pe T x

T

T j ,i i x

Finally the observer gains can be determined according to the following theorem: Theorem 1 (Pettersson, 2005) : If there exists a solution to (ε ≥ 0, α > 0, μi , j ≥ 0, vi , j ≥ 0) 1. α I ≤ Pi ≤ ξ I

2. Γi , j

⎡Γ11 = ⎢ i, j ⎣*

i ∈ IN

3. Pj = Pi + g C + C gi , j T

For more details about the proof of theorem 1 and the gain matrix determination see (S. Pettersson, 2005) and references therein. 4 Observer’s Robustness In this section, we formulate the RFDO design problem as a model-matching problem and solve it via a LMI formulation. Lemma 1. Consider the switched linear system (1) and (2) with the observer (4)-(6), system (12) is asymptotically stable and satisfies (16) if there exists singular matrices Pi > 0 H i and V such that the following LMI holds: ⎡ Θ11 ⎢ * ⎢ ⎢⎣ *

Θ12 Θ22 *

Θ13 ⎤ Θ23 ⎥ ≤ 0 ⎥ Θ33 ⎥⎦

(20)

(i, j ) ∈ I s

ed = Ai ed + Bd j d and rd = VCed + VDd j d .

Then : H ∞ ≤ γ i ⇔

Γ11 i , j = ( Ai − Ki C ) Pi + Pi ( Ai − Ki C ) + I + vi , j I

(CR )

)

−1 i

C xˆ + R

(CR )

−1 † i

y,

then if for T0 > 0 , sup x ( t ) ≤ xmax one has :

t

t

t

0

0

0

J = ∫ ( rdT rd ) dτ − γ i2 ∫ ( d T d )dτ = ∫ ( rdT rd − γ i2 d T d ) dτ t ⎛ dV ( ed ) ⎞ J = ∫ ⎜ rdT rd − γ i2 d T d + ⎟dτ − V ( ed ) dτ ⎠ 0⎝

t >T0

t

lim sup ex ( t ) ≤ t →∞

v 1+ v

ξ εx α max

(.) is the pseudo-inverse of , v is the largest vi , j , ( i , j ) ∈ I s and Ri ∈ℜn×n is a symmetric positive definite

where

0

function. Hence :

and ∀xˆ ∈ Si , j the state of the hybrid observer is updated

(

t

r dτ ≤ γ i2 ∫ ( d T d )dτ .

T d d

be a Lyapunov candidate Let V ( ed ) = edT Pe i d ≥ 0, Pi > 0

(*) indicates a transpose quantity

according to xˆ = I − R

t

∫r 0

22 2 Γ12 i , j = Pi ( A j − Ai ) , Γ i , j = μi , j Q j − ε vi , j I

−1 † i

; Θ 23 = DdTj V T , Θ33 = − I

Proof. According to (7) and (8), the following relations yield:

T

−1 i

(22)

Θ13 = C TV T ; Θ22 = −γ i2 I

where

+

(21)

with : Θ11 = AiT Pi + Pi Ai − H i C − C T H iT , Θ12 = Pi Bd j − H i Dd j ;

⎤ Γ12 i, j ⎥ ≤ 0 (i , j ) ∈ I s Γi22,j ⎥⎦

T i, j

⎡ λi2 I p× p * ⎤ ⎢ ⎥ ≥ 0, i ∈ I N I n×n ⎦⎥ ⎣⎢ H i

†

matrix. Note that, the LMI formulation of the observer synthesis problem is obtained via the introduction of new unknown variables H i ∈ℜn× p according to the following change of variable H i = PK i i . This leads to the following equivalent condition:

J = ∫ ⎡⎣edT 0

⎡ ⎡C T V T ⎤ ⎤ ⎡e ⎤ d T ⎤⎦ ⎢ ⎢ T T ⎥ ⎡⎣VC VDd j ⎤⎦ + Ed ⎥ ⎢ d ⎥ dτ − V ( ed ) ⎢⎣ ⎣⎢ Dd j V ⎦⎥ ⎥ ⎣d ⎦   

⎦ Rd

⎡ AiT Pi + Pi Ai where Ed = ⎢ * ⎢⎣

Pi Bd j ⎤ ⎥ −γ i2 I ⎥⎦

We can get J ≤ 0 , if Rd ≤ 0 . Then, using the Schur complement for Rd leads to: ⎡ AiT Pi + Pi Ai ⎢ Rd = ⎢ * ⎢ * ⎢⎣

Pi Bd j −γ i2 I *

CTV T ⎤ ⎥ DdTj V T ⎥ ≤ 0 ⎥ − I ⎥⎦

Finally, since Ai = Ai − Ki C and Bd j = Bd j − K i Dd j , the upper inequality can be written in the form of (22). „

Theorem 2. Consider the switched linear system (1) and (2) with observer (4)-(6), system (13) is asymptotically stable and satisfies (17) if there exists singular matrices Pi > 0 H i , and V such that the following LMI holds:

Lemma 2. Consider the switched linear system (1) and (2) with the observer (4)-(6), system (13) is asymptotically stable and satisfies (17) if there exists singular matrices Pi > 0 and H i , V such that the following LMI holds:

⎡Ter1 + 2ϕ1 (V , Vcn ) H i D f j − Pi B f j CTV T ⎤ ⎢ ⎥ ⎢ * β i2 I + 2ϕ 2 (V , V fn ) DTf j V T ⎥ ≤ 0 ⎢ ⎥ ⎢ * * −I ⎥ ⎣ ⎦

⎡ ϒ11 ⎢ * ⎣

With:

ϒ12 ⎤ ≤0 ϒ 22 ⎥⎦

(23)

Vcn = V n −1C , V fn = V n −1 Ff j , n = 1,2,...

with:

Ter1 = AiT Pi + Pi Ai − H i C − C T H iT

ϒ11 = AiT Pi + Pi Ai − H i C − C T H iT − C TV TVC ,

ϕ1 (V ,Vcn ) = (Vcn ) Vcn − (Vcn ) VC − C TV TVcn , T

ϒ12 = C V VD f j + H i D f j − Pi B f j , ϒ 22 = β I − D V VD f j T

2 i

T

T fj

T

t

∫r

t

r dτ ≥ βi2 ∫ f T f dτ . 0

By considering the candidate Lyapunov V ( e f ) = eTf Pe i f ≥ 0, Pi > 0 one obtains: t

t

t

0

0

0

function

⎛ dV ( e f J = ∫ ⎜ rfT rf − γ i2 f T f − ⎜ dτ 0⎝ t

J = ∫ ⎡⎣ eTf 0

) ⎞⎟dτ + V ⎟ ⎠

T

j

j

j

are

weighting matrices. Proof. According to Lemma 2

with: Ter1 = AiT Pi + Pi Ai − H i C − C T H iT , Ter 2 = H i D f j − Pi B f j ,

J = ∫ ( rfT rf ) dτ − βi2 ∫ ( f T f )dτ = ∫ ( rfT rf − βi2 f T f ) dτ t

ϕ2 (V , V fn ) = (V fn ) V fn − (V fn ) VD f − DTf V T V fn and Ff

⎡Ξ11 Ξ12 ⎤ ⎢ * Ξ ⎥≤0 ⎣ 22 ⎦

T f f

0

T

T

Proof. According to (7) and (8), the following relations yield: e f = Ai e f + B f j f and rf = VCe f + VD f j f .

Then H − ≤ βi ⇔

(24)

Ξ11 = Ter1− C TV TVC , Ξ12 = C T V T VD f j + Ter 2 Ξ 22 = β i2 I − DTf j V T VD f j + 2 DTf j V T VD f j − 2 DTf j V T VD f j

(e ) f

⎡ ⎡C T V T ⎤ f T ⎤⎦ ⎢ ⎢ T T ⎥ ⎡⎣VC VD f j ⎤⎦ − E f ⎢⎣ ⎣⎢ D f j V ⎦⎥

Then, ⎤ ⎡e f ⎥⎢ ⎥⎦ ⎣ f

⎤ ⎥ dτ + V ( e f ⎦

)

⎛ ⎡Ter1− 2C T V T VC ⎜⎢ ⎜⎢ * ⎝⎣

⎤ ⎥+Λf β I − 2 D V VD f j ⎥⎦ Ter 2 T fj

2 i

T

⎞ ⎟≤0 ⎟ ⎠

(25)

with: ⎡ A Pi + Pi Ai where E f = ⎢ * ⎣⎢ T i

Pi B f j ⎤ ⎥ β i2 I ⎦⎥

⎡C T V T VC − AiT Pi − Pi Ai Hence, if ⎢ * ⎢⎣

⎡C T V T VC Λf = ⎢ * ⎢⎣ C T V T VD f j − Pi B f j ⎤ ⎥ ≥ 0, DTf j V T VD f j − β i2 I ⎥⎦

Using Schur complement. The inequality (25) can be expressed in LMI form as:

we can get J ≥ 0 . Since

Ai = Ai − Ki C

and

C T V T VD f j ⎤ ⎥ DTf j V T VD f j ⎥⎦

Bd j = Bd j − K i Dd j , the latter

inequality can be written in the form of (23). „

Now, on the basis of Lemma 2, we can formulate the following theorem.

⎡Ter1 − 2C T V T VC ⎢ * ⎢ ⎢ * ⎣⎢

H i D f j − Pi B f j

β i2 I − 2 DTf V T VD f j

*

j

CTV T ⎤ ⎥ DTf j V T ⎥ ≤ 0 ⎥ − I ⎦⎥

with the following change of variables, n T T n T T ϕ1 (V ,Vc ) = −C V VC and ϕ1 (V , Vc ) = − D f j V VD f j , „

Theorem 3. Consider the switched linear system (1) and (2) with observer (4)-(6). The dynamics (7) and (8) are bounded and satisfy (16) and (17) if there exist matrices Pi > 0 , H i and V such that LMIs (20), (21), (22) and (24) hold. Corollary. In order to design the robust Fault Detection Based-Observer, one should find H i and V which satisfies

the performance index inf ω

γi . This task can be achieved βi

using the following iterative procedure 1- Begin a loop with proper values of γ i and βi . 2- Solve LMIs (20), (21) and (22) to find feasible solutions Pi , H i and V 3- Set V n −1 = V , Vcn = V n −1C , V fn = V n −1 Ff j and substitute the founded Pi into LMIs, (20), (21), (22) and (24), to find a feasible solution set of variables H i , V 4- Decrease γ i and increase βi and repeat the steps 2 and 3 until a feasible solution cannot be found. 4.1 Fault detection One of the widely adopted approaches to detect faults, is to choose a threshold J th > 0 and, based on this threshold, uses the following logical relationship for fault detection: r

2,T

> J th

(26)

r

2,T

≤ J th

(27)

where r

2,T

⎡0 ⎤ ⎡ −2 10 ⎤ , A2 = ⎢ B=⎢ ⎥, ⎥ ⎣ −2 − 1 ⎦ ⎣0 ⎦ ⎡1 0 ⎤ ⎡0 ⎤ C = [1 − 2.4] , Bd 2 = ⎢ ⎥ , B f2 = ⎢0 ⎥ , Dd2 = [1 0] , 0 1 ⎣ ⎦ ⎣ ⎦ D f2 = 0 . 2,

Mode

if

S2 = 0 :

where Si = ai x1 + bi x2 , i ∈ {1, 2} are the switching laws with a1 = 0.25 , b1 = 1 , a2 = 1 and b2 = 1 .

Note that the first subsystem is unstable while the second one is stable. However, the global state switched system is stable as shown in fig 1. According to the Corollary, we design the fault detection observer based on a repeated application of the theorem 3. This leads to the following solutions: γ 1 = γ 2 = 1.9667 , β1 = 3.4831 and λ1 = λ2 = 5 . ⎡3.5616 ⎤ ⎡ −0.0434 ⎤ K1 = ⎢ , K2 = ⎢ ⎥ ⎥ and V = 0.3330 . ⎣ −0.8557 ⎦ ⎣0.0545 ⎦

with ξ = 4.9 , α = 1 and ε = 6.78 . To show the efficiency of the proposed approach, an unknown input d is assumed to be band-limited white noise with power 0.2 (sampling time 0.01s). The fault signal f is simulated as a pulse of 0.05 amplitude occurred in the mode one from 5.2s to 5.4s (and zero otherwise). The generated residual r (t ) is illustrated in Fig 2, which shows that the residual converges to zero at time 2s. Then, it departs significantly from zero when the default occurs.

is determined by 2.5

1 2

r

2,T

⎡ t2 ⎤ T = ⎢ ∫ r ( t ) r ( t ) dt ⎥ = rd ( t ) + rf ( t ) ⎣⎢ t1 ⎦⎥

2,T

2

, T = t2 − t1

1.5 1

and rd ( t ) and rf ( t ) are defined as follows: rd ( t ) = r ( t ) | f = 0 , x2

rf ( t ) = r ( t ) |d = 0 .

0

Hence, one can choose J th = J th , d = sup d ∈L2 rd ( t )

-0.5

2,T

as the

threshold.

-2 -2.5

Let us consider the following state switched system with two discrete modes: 1,

if

C = [1 − 2.4] , D f1 = 14.72 .

-1 -1.5

4. SIMULATION AND RESULTS

Mode

0.5

S1 = 0 :

⎡1 0 ⎤ Bd1 = ⎢ ⎥ ⎣1 0 ⎦

2 ⎤ ⎡1.5 A1 = ⎢ ⎥, ⎣ −2 − 0.5⎦

,

⎡0 ⎤ B f1 = ⎢ ⎥ , ⎣0 ⎦

⎡0 ⎤ B=⎢ ⎥, ⎣0 ⎦

Dd1 = [1 0] ,

-2

-1.5

-1

-0.5

0 x1

0.5

1

1.5

2

2.5

Fig. 1. Phase plane

In addition, Fig. 3 illustrates the evolution of the residual evaluation function r 2,T and shows that one detects rapidly the simulated fault. Finally, to raise the efficiency of the approach, one depicts on Fig.4 the evolution of the residual obtained using the observer proposed in (S. Pettersson, 2005). The analysis of

Figs 2 and 4, shows that the noise effect has been attenuated in our case. 2.5 2

ACKNOWLEDGMENT

1.5

The authors would like to thank the Champagne Ardenne Region within the CPER MOSYP and the GIS3SGS within the project COSMOS for their support.

1 r(t)

unknown input and/or modelling error. The approach herein proposed is based on the use of some new results of H ∞ theory to provide robust fault detection. An illustrative example shows clearly the efficiency of the designed observer.

0.5 0

REFERENCES

-0.5 -1 -1.5

0

1

2

3

4 time t(sec)

5

6

7

8

Fig. 2. Generated residual value 80 70 60

residual eval

50 40 30 20 10 0

0

1

2

3

4 time t(sec)

5

6

7

8

Fig. 3. Evolution of residual evaluation function r

2,T

2.5 2 1.5

r(t)

1 0.5 0 -0.5 -1 -1.5

0

1

2

3

4 time t(sec)

5

6

7

8

Fig. 4. Generated residual value using only the observer 5. CONCLUSIONS In this paper, a robust hybrid observer is designed for the class of switched linear systems. The objective is to use this observer for fault detection when the system is subject to

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