Optimal fault detection filter design for switched linear systems

Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Optimal fault detection filter design for switched linear systems Kamran Iftikhar ∗ , Abdul Qayyum Khan, Muhammad Abid Pakistan Institute of Engineering and Applied Sciences Islamabad, Pakistan

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Article history: Received 26 August 2013 Accepted 11 September 2014 Keywords: Switched systems Fault detection Fault isolation Average dwell time Multiple Lyapunov functions

abstract This report investigates the problem of fault detection filter design for switched linear systems. A mixed H− /H∞ fault detection filter is proposed and the solution is provided in the state space paradigm. The proposed solution has the advantage that not only faults can be detected but fault isolation is also possible. Linear matrix inequalities (LMIs) and Multiple Lyapunov Function (MLF) approach are used for the design of fault detection filter. It is assumed that the switching sequence is known and the fault detection filter is designed using the average dwell time constraints. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Safe and reliable operation of a plant/process is the core issue to be addressed in designing a control system. This can be achieved by employing sophisticated control strategies. Generally speaking every controlled process has sensors, actuators, controller hardware and the process itself. Ideally the operation of the plant should be in accordance with the desired specifications after addressing all the control and stability issues in the design of a controller. However, due to the malfunctioning of the sensors and actuators, deterioration of plant equipment or even sometimes in the controller hardware, faults are developed. These faults not only degrade the system performance but may sometimes have disastrous implications. In order to ensure safe, reliable and cost effective operation of the process, the role of process monitoring or diagnostic systems is inevitable. During the last three decades a great deal of attention has been given to model-based fault detection and isolation (FDI), see for instance [1–6] to list a few. It is evident that the process, in practice, is simultaneously influenced by unknown inputs (disturbances and noises) as well as by various faults which arise in sensors, actuators and/or process components. To this end a typical model based FDI scheme involves generation of residual signals (which carries the information of faults) and thresholds settings. The objectives of an FDI system are thus defined as (i) detection of faults as soon as possible (ii) avoidance of false alarms (which may be set due to unknown inputs). To this end, various optimization indices have been proposed in literature. These are H2 /H2 , H∞ /H∞ and H− /H∞ [2,4,7]. Among these indices H− /H∞ is of particular interest. H− -index measures the minimum influence of faults on the residual signal and H∞ -norm measures the maximum influence of unknown inputs on the residual signal. In the present work, a mixed H− /H∞ fault detection filter is proposed for switched linear systems. It is a very important class of hybrid systems [8,9]. A switched system consists of different modes that are either continuous or discrete and a

∗

Corresponding author. Tel.: +92 3465404025. E-mail addresses: [email protected] (K. Iftikhar), [email protected] (A.Q. Khan), [email protected] (M. Abid).

http://dx.doi.org/10.1016/j.nahs.2014.09.002 1751-570X/© 2014 Elsevier Ltd. All rights reserved.

K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

133

switching law for switching among the modes [10,11]. Stability and control of these systems have been greatly investigated in literature [10,12–15]. An important issue in switched systems is the switching. Stability is greatly affected by the switching signal. There are mainly two types of switching signals that are considered in literature: (1) Time dependent switching signal (2) State dependent switching signal [10,16]. An associated concept is the dwell time. Results on stability and stabilization of slow switched systems using dwell time and average dwell time constraints can be found in [10,17–21] and the references therein. To the best of our knowledge,very little attention has been given to fault detection of aforementioned class of systems. Some important contribution is made in the recent past in this area. See for instance [22–30]. In [23] fault detection problem for switched systems using the H− /H∞ performance index has been discussed with a constant threshold for detection purposes. In [24] fault detection problem for uncertain discrete switched systems has been considered and an adaptive threshold has been designed for efficient fault detection. In [25] the fault detection problem is solved for discrete switched linear systems with the assumption that the switching signal is unknown. In [26] fault detection problem for switched linear systems with state delays has been discussed using a switched Lyapunov function approach. The robust fault detection filter (RFDF) designed in [27] for slow switched systems uses H∞ optimization. The slow switched system considered consists of state delays and uncertainties in the system model. The robust fault detection filter is designed using Lyapunov–Krasovskii functional and average dwell time constraints for the slow switched system. In the present work fault detection scheme is designed using H− /H∞ optimization. The state delays and uncertainties in the system model have not been considered which will be incorporated in the future. In H− /H∞ optimization, the maximum influence of unknown inputs on the residual signal is represented by H∞ -norm and the minimum influence of faults is represented by H− -index. In order to ensure robustness, H∞ should be minimized and H− should be maximized simultaneously. The solution to the fault detection problem, thus, obtained using H− /H∞ performance index will ensure that the residual signal is maximally sensitive to faults and simultaneously highly robust against unknown inputs. The paper in hand, inspired by [31], presents a solution to the fault detection problem for switched linear systems using the H− /H∞ optimization. A state space solution is provided to the fault detection problem for slow switched systems. The state space solution to the H− /H∞ fault detection (FD) problem presented here has the advantage that it can also achieve fault isolation. The average dwell time constraints for the slow switched systems will be considered to design the optimal fault detection filter for linear switched systems. The rest of the paper is organized as follows: Preliminaries are given in Section 2. Section 3 states the fault detection problem for continuous switched systems and solution to the problem is then given in Section 4. Evaluation of residual and threshold computation is then given in Section 5. In Section 6 examples are provided showing the efficiency of the proposed method. The paper is then concluded in Section 7. 2. Notations and preliminaries The notations used in this paper are standard and are described here. The set of real and complex matrices with dimensions m × n is denoted by ℜm×n and Cm×n respectively. For a matrix A ∈ Cm×n the complex conjugate transpose is represented by A′ and the transpose is represented by AT . The upper singular value of a matrix A is represented by 

 ¯ AA′ ) and the lower singular value by σ (A) = λ(AA′ ) where λ and λ show the largest and smallest eigen λ( values of the matrix. A > 0 means that a matrix is positive definite and A < 0 means that it is negative definite. ℜ(s)m×n σ¯ (A) =

×n shows a set of real rational matrix functions of s and Lm ∞ shows a set of matrix functions that have entries bounded on the m×n ×n extended imaginary axis. H∞ ⊂ Lm are the matrix functions that are analytic in the closed right half plane. A stable real ∞ m×n . A transfer matrix function G(s) = C (sI − A)−1 B + D is shown as either rational matrix function of s is denoted by ℜH∞ s

G(s) = (A, B, C , D) or s

G(s) =



B D

A C



.

×n If we have G(s) ∈ ℜLm ∞ then the supremum of largest singular value known as the L∞ -norm is denoted by ∥G∥∞ = supw∈ℜ σ (G(jw)) and ∥G∥− = infw∈ℜ σ (G(jw)) is the infimum of the lower singular value. The pseudoinverse of a matrix A of order m × n is denoted by AĎ with order n × m, that is AĎ A = In where In is an identity matrix of order n.

Lemma 1 ([32]). A switched system x˙ (t ) = Ai x(t ) + Bi u(t ) y(t ) = Ci x(t ) + Di u(t ) where i ∈ {1, 2, . . . , N } is said to be globally uniformly asymptotically stable with average dwell time

τa > τa∗ =

ln µ

α

and satisfies the H∞ performance with index no greater than γ = max(γi ) as well if there exist Lyapunov functions Vi (x(t )) ∀i ∈ {1, 2, . . . , N } such that

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K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

• Vi (x(tk )) ≤ µVj (x(tk )) • V˙ i (x(t )) ≤ −α Vi (x(t )) − yT (t )y(t ) + γi2 uT (t )u(t ) ∀(i, j)i ∈ {1, 2, . . . , N } and i ̸= j. Lemma 2 ([33]). According to Schur’s complement following two statements are said to be equivalent 1.

φ11 T φ12



φ12 φ22



<0

−1 T 2. φ22 < 0, φ11 − φ12 φ22 φ12 < 0.

3. Problem formulation Consider a continuous time linear switched system with state space model given by



x˙ (t ) = Ai x(t ) + Bi u(t ) + Bfi f (t ) + Bdi d(t ) y(t ) = Ci x(t ) + Di u(t ) + Dfi f (t ) + Ddi d(t )

(1)

where i ∈ {1, 2, . . . , N } represents that which mode is active and N is the total number of modes in the system. The switching signal is assumed to be known here. Let us say that the switching signal is σ = i ∈ {1, 2, . . . , N }. Now depending upon σ there will be switching from one mode to another and at any time ith mode will be active. It should be noted here that the switching will take place depending on the average dwell time constraints. The switching signal σ can be a function of time or states of the system or some other logical function. x(t ) ∈ ℜn and u(t ) ∈ ℜnu are the system state vector and control input vector respectively. At any time depending on the switching signal σ , x(t ) will be representing the trajectory of the ith mode. y(t ) ∈ ℜny is the output vector. d(t ) ∈ ℜnd and f (t ) ∈ ℜnf are the disturbance and fault vectors. Ai ∈ ℜn×n , Bi ∈ ℜn×nu , Ci ∈ ℜny ×n and Di ∈ ℜny ×nu . Bfi ∈ ℜn×nf and Dfi ∈ ℜny ×nf are the fault coupling matrices and Bdi ∈ ℜn×nd and Ddi ∈ ℜny ×nd are the disturbance coupling matrices. In order to design a fault detection filter following assumptions are made.

• The pair (Ai , Ci ) is detectable ∀i ∈ {1, 2, . . . , N }. It is required so that observer can be designed for each mode of the system.

• Gfi (s) has no zeros on the imaginary axis ∀i ∈ {1, 2, . . . , N }. This assumption is needed so that fault at any frequency can have an effect on the residual signal. This assumption ensures that ∥Fi (s)Gfi (s)∥− > 0. The assumption on zeros of Gfi also implies that Dfi should be of full column rank for each mode of operation. • ny ≥ nf . (Number of outputs is greater than or equal to the number of faults.) • It is assumed that the switching sequence is known and faults in the switching signal are not considered in this paper. The model of switched observer for the switched system given in (1) is as follows:

 xˆ˙ (t ) = Ai xˆ (t ) + Bi u(t ) − Li (y(t ) − yˆ (t )) yˆ (t ) = Ci xˆ (t ) + Di u(t )  r (t ) = Hi (y(t ) − yˆ (t )).

(2)

Li ∈ ℜn×ny is the observer gain matrix and Hi ∈ ℜnf ×ny is the residual gain matrix for each mode i ∈ {1, 2, . . . , N }. Switching between different modes of the observer will take place according to the switching signal σ (which is assumed to be known). Each mode or subsystem of the given switched system is linear time invariant and therefore each isolated subsystem can s

be represented by a transfer function Gi (s) = (Ai , Bi , Ci , Di ) ∀i ∈ {1, 2, . . . , N }. Now with the error signal e(t ) = x(t ) − xˆ (t ) and residual signal r (t ) = Hi (y(t ) − yˆ (t )), a compact representation of the residual generator dynamics is obtained as



e˙ (t ) = (Ai + Li Ci )e(t ) + (Bdi + Li Ddi )d(t ) + (Bfi + Li Dfi )f (t ) r (t ) = Hi Ci e(t ) + Hi Ddi d(t ) + Hi Dfi f (t ).

There are now two main objectives that are required to be fulfilled for robust fault detection: 1. Minimization of effect of disturbances on the residual signal that is

∥rd (t )∥2 ≤ γi ∥d(t )∥2 ∀i ∈ {1, 2, .., N }. 2. Maximization of the effect of faults on the residual signal that is

∥rf (t )∥2 ≥ βi ∥f (t )∥2 ∀i ∈ {1, 2, .., N }.

(3)

K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

135

That is we require a fault sensitivity level greater than or equal to βi and at the same time it is required to minimize the disturbance attenuation level γi ∀i ∈ {1, 2, . . . , N }. Laplace transform of the residual signal R(s) during each mode is obtained as R(s) = Fi (s)(Gdi (s)d(s) + Gfi (s)f (s))

(4)

where the transfer function of the optimal post filter that is to be designed for each mode of the system is s

Fi (s) = (Ai + Li Ci , Li , Hi Ci , Hi ) and s

Gi (s) = (Ai , Bi , Ci , Di ) s

Gdi (s) = (Ai , Bdi , Ci , Ddi ) s

Gfi (s) = (Ai , Bfi , Ci , Dfi ). The H− /H∞ problem is solvable if a filter Fi (s) can be designed for each mode by finding Li and Hi such that γi is minimized and simultaneously rendering βi ≥ 1 under the dwell time or average dwell time constraints for slow switched systems. The solution to the given problem can now be obtained by setting ∥Fi (s)Gfi (s)∥− ≥ βi ≥ 1 ∀i ∈ {1, 2, . . . , N } and then finding γi under the average dwell time constraints. To achieve the desired objective, set Fi Gfi = βi I so that ∥Fi Gfi ∥− ≥ βi ∀i ∈ {1, 2, .., N }. The transfer function Fi (s)Gfi (s)is given as below: s

Fi (s)Gfi (s) = (Ai + Li Ci , Bfi + Li Dfi , Hi Ci , Hi Dfi ). Now to set Fi (s)Gfi (s) = βi I following two conditions need to be satisfied for each mode and Hi Dfi = βi Inf .

Bfi + Li Dfi = 0

So in this way Li is restricted so as to get Bfi + Li Dfi = 0. The problem then simply remains to find γi for each mode under the average dwell time constraints with the restriction that Fi (s)Gfi (s) = βi I. Writing the problem in a simpler form as follows: n ×ny

s

γi = inf{∥Fi (s)Gdi (s)∥∞ : ∥Fi (s)Gfi (s)∥− ≥ βi , Fi (s) = (Ai + Li Ci , Li , Hi Ci , Hi ) ∈ ℜH∞f

}.

It is important to mention here that even if the post filter Fi (s) has an unstable realization, a stable realization of the post nf ×ny

filter can still be obtained by some coprime factorization that is explained next. If there exists a filter Fi (s) ∈ ℜL∞ that achieves the required objective then a stable realization of the post filter can be found by a left coprime factorization of Fi (s) given by Fi (s) = (Mi (s))−1 F˜i (s) nf ×nf

where Mi (s) ∈ ℜH∞ reformulated as

nf ×ny

is inner which means that MiT (−s)Mi (s) = I and F˜i (s) ∈ ℜH∞ n ×ny

s

γi = inf{∥F˜i Gdi ∥∞ : ∥F˜i Gfi ∥− ≥ βi , F˜i (s) = (Ai + L˜ i Ci , L˜ i , H˜ i Ci , H˜ i ) ∈ ℜH∞f

nf ×ny

s

= inf{∥Fi Gdi ∥∞ : ∥Fi Gfi ∥− ≥ βi , Fi (s) = (Ai + Li Ci , Li , Hi Ci , Hi ) ∈ ℜL∞

. So the given problem can now be

}

},

(5)

γi will be obtained under the average dwell time constraints for the switched system. 4. State space solution to H− /H∞ problem Theorem 1. The Fault detection filter (3) for the switched system (1) with βi ≥ 1 and scalars µ ≥ 1, α > 0 and average dwell time

τa ≥ τa∗ =

ln µ

α

is said to be asymptotically stable and solves the H− /H∞ filter design problem if there exist some matrices Pi = PiT > 0, Zi and Si such that



Θ11 ∗ ∗

Θ12 Θ22 ∗

 Θ13 Θ23 < 0 Θ33 Ď

(6) Ď

⊥ T T Θ11 = α Pi + (Ai − Bfi Dfi Ci )T Pi + Pi (Ai − Bfi Dfi Ci ) + Zi D⊥ fi Ci + (Dfi Ci ) Zi Ď

Θ12 = Pi (Bdi − Bfi Dfi Ddi ) + Zi D⊥ fi Ddi Ď

T Θ13 = (βi Dfi Ci + βi Si D⊥ fi Ci )

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K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

Θ22 = −γi I Ď

T Θ23 = (βi Dfi Ddi + βi Si D⊥ fi Ddi ) Θ33 = −γi I Pi − µPj < 0

(7)

∀(i, j) ∈ {1, 2, . . . , N } and i ̸= j. Zi = Pi Ri , observer gain matrix Li and residual gain matrix Hi are given by Ď

Ď

Li = −Bfi Dfi + Ri D⊥ fi ,

Hi = βi (Dfi + Si D⊥ fi )

where Ri ∈ ℜn×(ny −nf ) and Si ∈ ℜnf ×(ny −nf ) . Ri and Si are additional variables that are introduced to provide an additional degree of freedom for designing Li and Hi . Also Ď



 

Dfi I Dfi = nf 0 D⊥ fi

,



Ď 

Dfi D⊥ fi

rank

= ny .

The optimal post-filter can be obtained by nf ×ny

s

Fi (s) = (Ai + Li Ci , Li , Hi Ci , Hi ) ∈ ℜH∞

.

The solution obtained in this way will have following characteristics 1. The given solution will ensure that the fault sensitivity level is greater than or equal to βi and at the same time the disturbance attenuation level is γi ∈ {1, 2, . . . , N } that is (a) ∥rd (t )∥2 ≤ γi ∥d(t )∥2 ∀i ∈ {1, 2, .., N } (b) ∥rf (t )∥2 ≥ βi ∥f (t )∥2 ∀i ∈ {1, 2, .., N } 2. The given solution will not only detect the presence of a fault in each mode but will also help in the isolation of fault if and only if we have Fi (s)Gfi (s) = βi Inf ∀i ∈ {1, 2, . . . , N }. 3. (Ai + Li Ci ) has no poles on the imaginary axis ∀i ∈ {1, 2, . . . , N }. 4. A non-singular solution for Pi ∀i ∈ {1, 2, . . . , N } will always exist. 4.1. Proof

Ď

As it has been assumed earlier that Dfi has full column rank ∀i ∈ {1, 2, . . . , N } so we can say that there exist matrices

ny −nf ×ny Dfi ∈ ℜnf ×ny and D⊥ such that the following condition is satisfied fi ∈ ℜ

Ď



 

Dfi I Dfi = nf 0 D⊥ fi

,



Ď 

Dfi D⊥ fi

rank

= ny .

As it is required that ∥Fi Gfi ∥− ≥ βi and we know that s

Fi (s)Gfi (s) = (Ai + Li Ci , Bfi + Li Dfi , Hi Ci , Hi Dfi ) so to achieve the desired objective we set Fi Gfi = βi I so that ∥Fi Gfi ∥− ≥ βi ∀i ∈ {1, 2, . . . , N }. This can easily be achieved if we set Bfi + Li Dfi = 0 and Hi Dfi = βi Inf . By this we can find the values for Li and Hi satisfying these constraints from Ď

Ď

Li = −Bfi Dfi + Ri D⊥ fi ,

Hi = βi (Dfi + Si D⊥ fi )

where Ri ∈ ℜn×(ny −nf ) and Si ∈ ℜnf ×(ny −nf ) are additional variables that are needed to be found out. Once Li and Hi have been chosen in a way described earlier, it is ensured that the fault sensitivity level for each mode will be greater than or equal to βi which will be some constant value greater than or equal to 1. Now with the given values for Li and Hi the problem now only remains to find out the H∞ -norm of Fi Gdi ∀i ∈ {1, 2, . . . , N }. The H∞ -norm of a slow switched system has to be found out under the switching constraints using dwell time or average dwell time. Average dwell time constraints are used in this paper. So the H∞ -norm of Fi Gdi is then obtained as follows using Lemma 1 V˙ i (e(t )) + rdT (t )rd (t ) < −α Vi (e(t )) + γi2 dT (t )d(t )

(8)

Vi (e(t )) − µVj (e(t )) < 0

(9)

∀i ∈ {1, 2, . . . , N } and α > 0 and µ ≥ 1 are given constants. γi is the H∞ -norm of Fi Gdi ∀i ∈ {1, 2, . . . , N }. Vi (e(t )) = eT (t )Pi e(t ) is the Lyapunov function ∀i ∈ {1, 2, . . . , N }. By putting the values of V˙ i (e(t )), Vi (e(t )) and rd (t ) in (8) we get e˙ T (t )Pi e(t ) + eT (t )Pi e˙ (t ) + (Hi Ci e(t ) + Hi Ddi d(t ))T (Hi Ci e(t ) + Hi Ddi d(t )) < −α eT (t )Pi e(t ) + γi2 dT (t )d(t ).

(10)

After putting the expressions for e˙ (t ) and rd (t ) in (10) we get eT (t )



dT (t )

  Π11 ∗

Π12 Π22

e(t ) <0 d(t )





(11)

K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

137

where

Π11 = α Pi + (Ai + Li Ci )T Pi + Pi (Ai + Li Ci ) + (Hi Ci )T (Hi Ci ) Π12 = Pi (Bdi + Li Ddi ) + (Hi Ci )T Hi Ddi Π22 = (Hi Ddi )T Hi Ddi − γi2 I . For (11) to hold we only require



Π11 ∗

 Π12 < 0. Π22

(12)

After applying Schur’s complement lemma on (12) and putting the expressions for Li and Hi we get



Φ11 ∗ ∗

Φ12 Φ22 ∗

 Φ13 Φ23 < 0 Φ33

(13)

Ď

Ď

T ⊥ Φ11 = α Pi + (Ai − Bfi Dfi Ci + Ri D⊥ fi Ci ) Pi + Pi (Ai − Bfi Dfi Ci + Ri Dfi Ci )

Ď

Φ12 = Pi (Bdi − Bfi Dfi Ddi + Ri D⊥ fi Ddi ) Ď

T Φ13 = βi (Dfi Ci + Si D⊥ fi Ci ) Φ22 = −γi I

Ď

T Φ23 = βi (Dfi Ddi + Si D⊥ fi Ddi ) Φ33 = −γi I .

Now for the FDF given in (3) we can find the values for γi for all the modes with average dwell time given by ln µ

τa ≥ τa∗ =

α

from the solution of following matrix inequalities (MIs)



Φ11 ∗ ∗

Φ12 Φ22 ∗

 Φ13 Φ23 < 0 Φ33

(14)

Ď

Ď

T ⊥ Φ11 = α Pi + (Ai − Bfi Dfi Ci + Ri D⊥ fi Ci ) Pi + Pi (Ai − Bfi Dfi Ci + Ri Dfi Ci )

Ď

Φ12 = Pi (Bdi − Bfi Dfi Ddi + Ri D⊥ fi Ddi ) Ď

T Φ13 = βi (Dfi Ci + Si D⊥ fi Ci ) Φ22 = −γi I

Ď

T Φ23 = βi (Dfi Ddi + Si D⊥ fi Ddi ) Φ33 = −γi I Pi − µPj < 0

(15)

∀i ∈ {1, 2, . . . , N } and α > 0 and µ ≥ 1 are given constants. In order to transform the MI (14) into an LMI following substitution is made Pi Ri = Zi . Now the proof is complete and it will ensure a fault sensitivity level of βi and the corresponding disturbance attenuation

γi ∀i ∈ {1, 2, . . . , N }. 4.2. Algorithm Case 1 (ny > nf )

1. Check if (Ai , Ci ) is detectable ∀i ∈ {1, 2, . . . , N }. Ď 2. Find Dfi and D⊥ fi such that



Ď

 

Dfi I Dfi = nf 0 D⊥ fi Ď

,

Ď 

 rank

Dfi D⊥ fi Ď

= ny .

⊥ 3. Set Li = −Bfi Dfi + Ri D⊥ fi and Hi = βi (Dfi + Si Dfi ). 4. Solve LMIs (6)–(7).

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K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

Case 2 (ny = nf ) 1. Check if (Ai , Ci ) is detectable ∀i ∈ {1, 2, . . . , N }. Ď

2. Find Dfi such that Ď

Dfi Dfi = Inf . Ď

Ď

3. Set Li = −Bfi Dfi and Hi = βi Dfi . 4. If Ai + Li Ci is stable then simply solve LMIs (6)–(7) and find γi ∀i ∈ {1, 2, . . . , N }. 5. (Ai + Li Ci , Hi Ci ) is unstable and detectable For this case we can easily find a left coprime factorization of Fi (s) by using the Theorem 13.34 given in [34] Mi (s)



F i ( s)





Ai + Li Ci + Li0 Hi Ci Hi Ci



Ai + L˜ i Ci H i Ci

= =

Li0 Inf



L˜ i Hi

Li0 Inf

Li + Li0 Hi Hi



.

Here L˜ i = Li + Li0 Hi and Li0 = −Xi CiT HiT and where Xi = XiT ∈ ℜn×n is the solution of the following Algebraic Riccati Equation (ARE)

(Ai + Li Ci )Xi + Xi (Ai + Li Ci )T − Xi CiT HiT Hi Ci Xi = 0.

(16)

Now we can say then that ∥Fi Gdi ∥∞ = ∥Mi−1 F˜i Gdi ∥∞ and also ∥Fi Gfi ∥− = ∥Mi−1 F˜i Gfi ∥− because Mi is inner ∀i ∈ {1, 2, . . . , N }. Now once (Ai + Li Ci ) has been stabilized, we have to find γi ∀i ∈ {1, 2, . . . , N } under the average dwell time constraints for the new observer gain matrix. 6. (Ai + Li Ci , Hi Ci ) is unstable but not detectable Now as Fi (s) is not detectable so to make it detectable we proceed as follows: Fi (s) =



Ai + Li Ci H i Ci



Ai11 + Li1 Ci1 0 0

=

Li Hi



Ai12 + Li1 Ci2 Ai22 + Li2 Ci2 Hi Ci2

Li1 Li2 Hi

 .

Here Ai11 + Li1 Ci1 is the part that is unstable and unobservable that makes Fi (s) undetectable. Ai22 + Li2 Ci2 is the part that is detectable. Now if we can prove that the pair (Ai11 , Ci1 ) is detectable then we can find some L˜ i1 so that we get a stable Ai11 + L˜ i1 Ci1 . For our assumption that the pair (Ai , Ci ) is detectable the following matrix should have full column rank Ai11 − sI Ai21 Ci1

Ai12





Ai22 − sI Ci2

for all s + s′ > 0. Now we see that I 0 0

0 I 0



0 Li2 I

Ai11 − sI Ai21 Ci1



Ai12



Ai22 − sI Ci2

 =

Ai11 − sI 0 Ci1

Ai22

Ai12 + Li2 C2 − sI Ci2



has full column rank for all s + s′ > 0 and hence it is proved that the pair (Ai11 , Ci1 ) is detectable. So we can find some L˜ i1 to get a stable (Ai11 + L˜ i1 Ci1 ). Now new observer gain L˜ i is obtained as L˜ i = L˜ Ti1



LTi2

T

.

So now



Ai11 + L˜ i1 Ci1 F˜s =  0 0 s

Ai12 + L˜ i1 Ci2 Ai22 + Li22 Ci2 Hi Ci2



L˜ i1 Li2  . Hi

With this new L˜ i1 , F˜i (s) is now detectable. F˜i (s) = Fi (s) after removal of the nonminimal part of the realizations of Fi (s) and F˜s (s). If the post filter is stable then find γi ∀i ∈ {1, 2, . . . , N } under the average dwell time constraints for the slow switched system and if it is still unstable then go to step (5).

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139

5. Residual evaluation and threshold computation Once the residual signal has been generated the next step is to evaluate the residual signal and compare it with some threshold value to detect the presence of fault in the system. Residual evaluation is necessary for efficient detection of faults because the residual signal may not be zero even if there are no faults in the system. There are different types of evaluation functions that can be used for evaluating the residual signal [2,24]. The evaluation function based on the RMS energy of the residual signal is used in this paper. So we have JRMS = ∥r (t )∥RMS =

t +T

  1

T

∥r (τ )∥2

 21

dτ .

t

T is the evaluation window. After evaluation of the residual signal we then need to compute the threshold value for efficient detection of the faults. Threshold setting is required to find out the maximum influence of unknown inputs (disturbances and noise) and model uncertainties on the residual signal when there are no faults present. Thresholds can either be fixed or adaptive [2,35,36]. In this paper norm based fixed threshold is used where we take the assumption that the L2 -norm of disturbance vector in all the modes is limited by a value say δd,2 . That is we have ∥di (t )∥2 < δd,2 ∀i ∈ {1, 2, . . . , N } where δd,2,i ≤ δd,2 . The threshold setting is then simply obtained by

γ

Jth,RMS,2,i = √ (δd,2 ) ∀i ∈ {1, 2, . . . , N } T where γ = max(γi ) is the worst-case performance index. Then after evaluating the residual and computing the threshold a decision can be made about the presence of fault in the system by

• JRMS < Jth,RMS,2 ⇒ No Fault • JRMS > Jth,RMS,2 ⇒ Alarm, Fault is detected. There can be some false alarms in case of fixed threshold setting. This is because the fixed threshold is not a function of the input signal. For switched systems there may be different unknown inputs with different upper limits in each mode. So instead of using δd,2 we can use δd,2,i for each mode. Moreover each mode has a different performance index γi . So rather than using the worst-case performance index γ , individual performance index γi can be used for each mode. As a result of which threshold setting will be different for each mode. The advantage in this case will be that a more efficient fault detection scheme is obtained because the threshold for each mode will be lowered as γi ≤ γ and δd,2,i ≤ δd,2 ∀i ∈ {1, 2, . . . , N }. So the threshold setting that is used in this paper is

γi

Jth,RMS,2,i = √ (δd,2,i ) ∀i ∈ {1, 2, . . . , N }. T 6. Simulation results Example 1. Consider a continuous time linear switched system with state space model represented by



x˙ (t ) = Ai x(t ) + Bi u(t ) + Bfi f (t ) + Bdi d(t ) y(t ) = Ci x(t ) + Di u(t ) + Dfi f (t ) + Ddi d(t ).

(17)

The switched system considered here has three modes that is N = 3. For each mode n = 3, nu = 2, ny = 3, nd = 2 and nf = 2. The state space matrices of the switched system considered here are as follows:

• Mode 1:    −4 2 1 1 0 −2 1 , 1 , A1 = −0.2 B1 = 0 −1 −0.5 −1 0.1 0.5     −0.2 0.1 0.2 0.01 −0.2 , −0.2 , Bd1 = 0.1 Dd1 = 0.01 0.1 0.03 −0.04 0.03 

1 0 0

0 1 0

 C1 =

Bf 1 = B1 ,

0 0 , 1



 D1 =

0.3 0 0

0 0.9 0



Df 1 = D1 .

• Mode 2:   −5 4 1 −1 0.5 , A 2 = −1 −2 −0.6 −1   −0.17 0.08 0.2 −0.2 , Bd2 = 0.1 0.01

1 0 0.5

 B2 =

0.1 0.1 −0.05

 Dd2 =

0 0.8 , 0.5



 C2 =

0.04 −0.1 , 0.09

1 0 0

0 1 0

0 0 , 1



D2 =



Bf 2 = B2 ,

0.7 0 0



Df 2 = D2 .

0 0.8 0



140

K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

Fig. 1. Residual signals with no faults.

Fig. 2. Actuator faults.

• Mode 3:       −4 2 2 0.8 0 1 0 0 0.6 −3 −2 , 1 , A3 = −1 B3 = 0 C3 = 0 1 0 , D3 = 0 −0.5 −1 −2 0.3 0.2 0 0 1 0     −0.2 0.03 0.12 0.1 0.2 −0.3 , Bd3 = 0.01 −0.1 , Dd3 = Bf 3 = B3 , Df 3 = D3 . 0.03 0.01 −0.025 0.027 

0 0.4 0



It can be seen from the state space model of the switched system that Bfi = Bi and Dfi = Di ∀i ∈ {1, 2, 3}. Hence there are two actuator faults that may be present in the given switched system. Setting a fault sensitivity level βi = 1 ∀i ∈ {1, 2, 3} with µ = 1.1 and α = 0.1, the disturbance attenuation levels for the switched system that were obtained after the application of Theorem 1 are γ1 = 0.6296, γ2 = 0.1925 and γ3 = 0.8004. The worst case performance index for the given switched system is therefore γ = 0.8004. δd,2,i = 3 ∀i ∈ {1, 2, 3}. Fig. 1 shows the residual signals when there are no faults present in the system. Actuator fault 1 was then introduced at time t = 2 s in Mode 1 whereas Actuator fault 2 was introduced in Mode 3 at time t = 12 s. Fig. 2 shows the two fault signals that were introduced in the given switched system. The residual signals for the system in the presence of faults are shown in Fig. 3. It can be seen in Fig. 3 that the residual signals are not zero even when there are no faults present in the system and so fault detection is not possible. Therefore the residual signals need to be evaluated and then a threshold level has to be set for detecting the faults. In Fig. 4 the evaluated

K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

141

Fig. 3. Residual signals in the presence of actuator faults.

Fig. 4. Evaluated residual signals with fixed threshold in the presence of actuator faults. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

residuals based on RMS energy of the residual signals have been plotted for T = 10. The threshold value for each mode is obtained by

γi

Jth,RMS,2,i = √ (δd,2,i ) ∀i ∈ {1, 2, 3}. T Now the threshold levels obtained for the given switched system are Jth,RMS,1 = 0.5973, Jth,RMS,2 = 0.1826, Jth,RMS,3 = 0.7593. The red line in Fig. 4 shows the threshold value for each mode. As individual mode performance index has been used to calculate the threshold so that is why the threshold is different for each mode. Now we can see from Fig. 4 that the evaluated residual signal crosses the threshold level only when a fault appears in the system making fault detection possible. The transfer function from the final residual to the fault inputs is βi Inf that is Fi (s)Gfi (s) = βi Inf ∀i ∈ {1, 2, 3}. Therefore Fault 1 affects the first residual and Fault 2 affects the second residual only that makes fault isolation possible. It can be seen in Fig. 4 that the two faults have been detected as well as isolated. The residual gain matrices for each mode are 3.33 0

 H1 =

0 1.11

10.27 , 2.84



 H2 =

1.43 0

0 1.25

0.19 , 1.65



1.67 0

 H3 =

0 2.5

0.37 . 24.19



We can see that there are two actuator faults present in the system and hence the residual gain matrices are of the order 2 × 3. In this way we will get only two residual signals.

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K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

Fig. 5. Evaluated residual signals with fixed threshold in the presence of actuator fault 1 (FDF is designed using the method in [23]).

Example 2. Consider a continuous time linear switched system with state space model represented by



x˙ (t ) = Ai x(t ) + Bi u(t ) + Bfi f (t ) + Bdi d(t ) y(t ) = Ci x(t ) + Di u(t ) + Dfi f (t ) + Ddi d(t ).

(18)

The switched system considered here has two modes that is N = 2. For each mode n = 3, nu = 2, ny = 3, nd = 2 and nf = 2. The state space matrices of the switched system considered here are as follows:

• Mode 1:   −3 1 −2 1 0.2 , A1 = −1 −0.5 B1 = 0 −2 −1 −0.6 0    −0.1 0.03 0.02 0.1 , Bd1 = −0.2 Dd1 = −0.01 −0.01 0.1 −0.01 

0 1 , 0



1 0 0

 C1 =

 −0.1 0.02 , −0.2

0 1 0

0 0 , 1



Bf 1 = B1 ,

0.5 0.7 0

0.1 0.9 0

0.9 0.4 0

0.5 0.7 0

 D1 =



Df 1 = D1 .

• Mode 2:  A2 =

Bd2

−2 2

−1 −0.7 −2

−1  −0.01 0.1 = −0.2

3 −1 , −0.4



 −0.03 −0.16 , 0.1

1 0 0

 B2 =

0.11 0.2 0.1

 Dd2 =

0 1 , 0



1 0 0

 C2 =

0.3 −0.01 , −0.05

0 1 0

0 0 , 1



 D2 =





Bf 2 = B2 ,

Df 2 = D2 .

Fault detection filter is first designed for the given switched system using the approach given in [23]. Setting a fault sensitivity level βi = 1 ∀i ∈ {1, 2} with µ = 1.5 and α = 0.5, the disturbance attenuation levels for the switched system are γ1 = 0.3760, γ2 = 0.8409. There are two actuator faults that may be present in the system. Fig. 5 shows the evaluated residual signals generated using the fault detection filter design by the method given in [23]. Switching from Mode 1 to Mode 2 takes place when t = 5 s. Actuator fault 1 was introduced t = 2 s. It can be clearly seen in Fig. 5 that the fault affects the first two residual signals. In Fig. 6 Actuator fault 2 has been introduced in the system and again we can see that both the residual signals are affected by the fault. So decision about fault isolation cannot be made. Now the Fault detection filter is designed for the given switched system using Theorem 1. Again setting a fault sensitivity level βi = 1 ∀i ∈ {1, 2} with µ = 1.5 and α = 0.5, the disturbance attenuation levels for the switched system are γ1 = 0.4019, γ2 = 0.6310. Figs. 7 and 8 show the evaluated residual signals obtained from the fault detection filter that is designed using Theorem 1. Here again actuator fault 1 was introduced at t = 2 s and Actuator fault 2 at t = 7 s. It can be clearly seen from Figs. 7 and 8 that not only both the faults can be detected but isolated as well. Actuator fault 1 affects the first residual only and Actuator fault 2 affects the second residual only. This is again because Fi (s)Gfi (s) = βi Inf ∀i ∈ {1, 2}.

K. Iftikhar et al. / Nonlinear Analysis: Hybrid Systems 15 (2015) 132–144

Fig. 6. Evaluated residual signals with fixed threshold in the presence of actuator fault 2 (FDF is designed using the method in [23]).

Fig. 7. Evaluated residual signals with fixed threshold in the presence of actuator fault 1. (FDF is designed using Theorem 1.)

Fig. 8. Evaluated residual signals with fixed threshold in the presence of actuator fault 2 (FDF is designed using Theorem 1).

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7. Conclusion In this paper a solution is provided to the fault detection (FD) problem for switched linear systems using the H− /H∞ optimization. A state space solution is provided to the fault detection problem using average dwell time constraints for slow switched systems and the results are presented in the form of Linear Matrix Inequalities (LMIs). The proposed solution has the advantage that in addition to fault detection it also provides fault isolation. This has been illustrated in the design examples. Moreover, the FDF is so designed that the generated residuals are highly robust to unknown inputs and maximally sensitive to faults. Recently, some useful results are presented on robust fault detection filter design (RFDF) for slow switched systems with state delays and uncertainties using H∞ optimization [27]. Our future work includes extending the proposed framework for the design of optimal fault detection filter in this paper to switched delay systems with model uncertainties in the paradigm of H− /H∞ . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

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