Fault Detection and Compensation for Linear ... - Semantic Scholar

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9, SEPTEMBER 2011

Fault Detection and Compensation for Linear Systems Over Networks With Random Delays and Clock Asynchronism Bo Liu and Yuanqing Xia

Abstract—This paper proposes a fault detection and compensation scheme based on likelihood ratios (LRs) for networked predictive control systems with random network-induced time delays and clock asynchronism. The compensator is applied to compensate for the network-induced time delays. The measured outputs are sent back to the local node with random delays and the observer updates based on the time schedule of a remote node clock to avoid the deficiency of asynchronism. Two schemes are proposed in this paper to update the LRs of fault. One of them is to set up a buffer in the local node to save the measured outputs out of sequence, and the observer processes the measured outputs one by one in their original sequence. The other scheme is to discard the measured outputs that are out of sequence; thus, the observer has to estimate the state and update the LRs with intermittent observations. The convergence analysis of a generalized LR test with intermittent observations is proposed as well. A numerical simulation is also given to validate the proposed method. Index Terms—Fault compensation, fault detection (FD), intermittent observations, likelihood ratio (LR), networked control systems (NCSs).

I. I NTRODUCTION

T

HE research of fault detection (FD) stems from its practical application to a variety of industries such as aerospace, energy systems, and process control to name a few. The main function of an FD scheme is to detect a fault when it happens, which may be acted on by sending alarm signals, taking protection measures, or reconfiguring a running control scheme [1]. The observer-based FD scheme is currently receiving much attention [2]. Generally speaking, an observer-based FD system consists of an observer-based residual generator and a residual evaluator. It is the state of the art that problems related to the observer-based FD system design are mainly addressed in the context of improving system robustness against unknown Manuscript received March 17, 2010; revised September 2, 2010, November 20, 2010, and December 8, 2010; accepted December 12, 2010. Date of publication January 6, 2011; date of current version August 12, 2011. The work of Y. Xia was supported in part by the National Natural Science Foundation of China under Grant 60974011, by the Program for New Century Excellent Talents in University of the People’s Republic of China under Grant NCET-08-0047, by the Ph.D. Programs Foundation of the Ministry of Education of China under Grant 20091101110023, by the Program for Changjiang Scholars and Innovative Research Team in University, and by the Beijing Municipal Natural Science Foundation under Grant 4102053. The authors are with the Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2010.2103533

disturbances and simultaneously enhancing system sensitivity to faults. A study on solving such problems builds the recent research which focuses on designing observer-based FD systems [3]–[5]. A networked control system (NCS) has many advantages over a traditional point-to-point control system, which includes low cost of installation, ease of maintenance, low cost, and great flexibility. For these reasons, the networked control architecture has been already used in many applications, particularly where weight and volume are of consideration, for example, in automobiles and aircraft [18]–[21]. However, there are many fundamental questions regarding the stable operation of an interconnected hard real-time system, such as signal coding and control information flow, peer-to-peer networking, effects of the network on the performance of the system, etc. Different control methods for NCSs have been reported (see, for example, [24]–[26]) [27], [28]. The network-induced time delay and data packet dropout cannot be ignored in FD and isolation (FDI) [13]. There are some important achievements in the FDI for NCSs (see, for example, [14]–[17]). The theory and practice of fault diagnosis and fault-tolerant control for NCSs are different from the ones for traditional control systems in many aspects. For example, it is apparent that networked-induced delay, packet dropout, and other characteristics of networks could influence the performance of a fault diagnosis system designed without taking them into account [30]. With some assumptions, the NCS was modeled as a simplified delay system, and then, many existing methods, such as state observer, filtering algorithm, etc., developed originally for ordinary timedelay systems could be used for or extended to fault diagnosis of NCSs. Furthermore, new algorithms dedicated to NCSs have been presented based on these models. In control systems, faults are defined as a kind of latent disordered dynamics, which may occur at any part of control systems. Usually, sensor and actuator faults as a sudden offset or drift can all be modeled as additive changes in state (for example, see [6] and [7]) or as unknown nonlinear functions (for example, see [8]–[10]). In addition, disturbances are traditionally modeled as additive state changes. In this paper, we model faults as abrupt changes in system states with unknown occurrence time and jump magnitude. Hence, the FD problem in this paper becomes the estimation of the changes. Maximum likelihood (ML) estimation is a powerful tool for estimating the unknown parameters by maximizing the likelihood of given measurements. However, it is more popular

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LIU AND XIA: FAULT DETECTION AND COMPENSATION FOR LINEAR SYSTEMS

to use the ratio of likelihoods under different hypotheses to estimate the unknown parameter [12]. As for estimating the occurrence time and magnitude of an unknown abrupt state change for linear systems, Willsky and Jones [11] proposed the generalized likelihood-ratio (GLR) test algorithm in the time domain. In their method, a Kalman filter is applied to estimate the states under the hypothesis that there is no state change; meanwhile, the innovations of the Kalman filter are refiltered under the hypothesis that a change occurs at the hypothetical time instant with unknown magnitude to obtain the estimate of magnitude, and then, the estimate of occurrence time is solved by maximizing the LR. This paper proposes a fault detection and compensation scheme based on LRs for networked predictive control systems with random time delays and clock asynchronism. A predictive control scheme based on a state observer is designed to compensate for the network time delay. Two schemes are proposed to update the fault estimation. One of them is conservative, and it needs a buffer to keep the data packet’s sequence. The other one is without a buffer, and the observer processes the packets in sequence and discards the others. Hence, the LRs of fault are computed, and if a fault is detected and identified, the estimate of the fault is sent to the controller to compensate for the fault. A numerical simulation is also given to validate the proposed method. II. N ETWORKED P REDICTIVE C ONTROL FOR S YSTEMS W ITH N ETWORK D ELAY This paper considers the case where the controller at the local node is far away from the plant and the manipulated variables and measured outputs of the plant are transmitted through networks. The plant studied in this paper is described by xt+1 = At xt + Bt ut + vt + σt−k ν yt = Ct xt + et

(1) (2)

where xt ∈ Rn , yt ∈ Rl , and ut ∈ Rm represent the state variables, measured outputs, and manipulated inputs, respectively. The additive fault ν ∈ Rn enters at time k as a step jump, where σt denotes the unit step function. vt and et are noises. Here, vt , et , and x0 are assumed to be independent Gaussian variables vt ∈ N (0, Qt )

et ∈ N (0, Rt )

x0 ∈ N (0, Π0 ).

(3)

Furthermore, they are assumed to be mutually independent. For the simplicity of the stability analysis, it is assumed that the reference input of the system is zero. Also, the following assumptions are made. Assumption 2.1: The pair (At , Bt ) is completely controllable, and the pair (At , Ct ) is completely observable. Assumption 2.2: The sum of the upper bounds of the network delays in the forward channel and the feedback channel is not greater than N1 , where N1 is a positive integer. Assumption 2.3: A buffer is set at the local node with a length of N2 , which is to gather the measured outputs transmitted from the sensors and where N2 is the upper bound of the network delay in the feedback channel. Note that if a generalized LR (GLR) with intermittent observation (proposed in Section V) is applied, this buffer is not needed.

Fig. 1.

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Both feedback and forward data packets are with the same time stamp.

Assumption 2.4: Clock asynchronism exists between the local clock and the remote clock. In the rest of this section, we present a predictive control scheme considering the clock asynchronism and random time delay. Liu et al. proposed a predictive control method to compensate for the network-induced time delays [19], [20], but the method is not suitable for the system considered in this paper since clock asynchronism exists. First, we explain why the predictive control method proposed by Liu et al. [19], [20] cannot be applied in this paper. The predictive controller gives predictive control signals based on the measurements and the controlled plant model. Assume that, at time t, the clock of the controller (at the local node) reads t . There is a constant shift, denoted by δ, that exists between t and t , i.e., t + δ = t, and the shift δ = kT , k = ±1, ±2, · · ·, is unknown and cannot be obtained easily since the time delay between the local and remote nodes is random. Two cases need to be considered: δ > 0 and δ < 0. If δ > 0, then the local node receives the measured outputs with time stamp t at time instant t + ∆b , but the clock at the local node reads t + ∆b − δ, where ∆b = kt T denotes the time delay in the feedback channel of this transmission. As for the case ∆b ≥ δ, the controller treats the delay as ∆b − δ, which is shorter than its real value. Thus, the controller gives the predictive control signals which are not based on t but based on t , i.e., ut +i|t , i = (kt − k)T, (kt − k + 1)T, . . .. This set of control signals does not guarantee that the controlled plant is stabilized. On the other hand, if ∆b < δ, then a serious logical error occurs since the controller receives the measured outputs before it is transmitted according to its clock. As for the other case that δ < 0, the analysis is very similar. In summary, the existence of clock asynchronism may incur some drawback and even serious error in predictive control systems. Due to the latent asynchronism between the local clock and the remote clock and the existence of random delay in the feedback channel, we present a scheme wherein the controller and observer generate the estimates and predictive manipulated variables based on the remote clock time, namely, all the time information subscripts in this paper denote the time shown by the remote clock, and thus, the asynchronism can be ignored. In Fig. 1, a data packet with time stamp t is transmitted to

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the local node at time instant t (remote node clock time) and it arrives at the local node at time instant t + ∆b (remote node clock time). Then, the controller at the local node gives a set of predictive manipulated variables which is put into a forward data packet with time stamp t and sends it to the remote node. The remote node receives the same packet at time instant t + ∆b + ∆f , where ∆f is the time delay in the forward channel. According to the assumptions, we have ∆b + ∆f ≤ N1 and the time delay compensator at the remote node chooses the newest u ˆt+∆b +∆f |k , where k is anyone available, as the input of the plant; thus, the time asynchronism is canceled. Based on the scheme and analysis aforementioned, the NCS considered in this paper is equal to a system in which the data packets are transmitted from the remote node to the local node without any time delay and the data packets are transmitted from the local node to the remote node with random time delay ∆b + ∆f . Since the network time delay is random, the data packet transmitted may be out of sequence. As aforementioned in Assumption 2.3, a buffer of length N2 is set at a local node and the measured outputs arriving at the local node are stored in the buffer. The observer chooses the measured observations from the buffer according to the sequence in which the measured outputs are sampled by sensors. For example, after the measured output yt is processed, the buffer deletes the same measured output, and the observer searches the next measured output yt+1 from the buffer. If yt+1 is not available in the buffer, then the observer holds on and searches the same one at the next step, but if yt+1 is available, then the observer uses it to update. This kind of application is to ensure that the observations are processed by their intrinsic order. However, an alternative method does not process those observations out of sequence. In this paper, the state observer is designed as x ˆt+1|t = At x ˆt|t−1 + Bt uot + Lt εt

(4)

where x ˆt+1|t ∈ Rn and uot ∈ Rm are the one-step-ahead state prediction and the input of the observer at time t, respectively. ˆt|t−1 is an innovation. The matrix Lt = At Mt ∈ εt = y t − C x Rn×l can be designed using observer design approaches. Note that, for all variables in (4) with subscripts like t + i|t or t, the time instant t denotes the time shown by the clock at the remote node. The estimator of the state is ˆt|t−1 + Mt εt x ˆt|t = x

(5)

Mt = Pt|t−1 CtT St−1 Pt+1|t = At [Pt|t−1 − Mt Ct Pt|t−1 ]AT t + Qt

ˆt|t−1 + Bt uot + Lt (yt − Ct x ˆt|t−1 ) x ˆt+1|t = At x x ˆt+2|t = At+1 x ˆt+1|t + Bt+1 u ˆt+1|t .. . x ˆt+N1 |t = At+N1 −1 x ˆt+N1 −1|t + Bt+N1 −1 u ˆt+N1 −1|t. (9) The gain of the feedback controller, i.e., Kt , can be designed based on the modern control theory in the case of no delay, for example, linear quadratic Gaussian, eigenstructure or pole assignment, H2 and H∞ in the presence of disturbance, etc. The predictive manipulated variables are calculated by u ˆt+i|t = Kt+i x ˆt+i|t ,

i = 0, 1, . . . , N1

(10)

and it follows from (9) that u ˆt+i|t = Kt+i

t+i−1

(Ah + Bh Kh )ˆ xt|t ,

i = 1, 2, . . . , N1 .

h=t

(11) At every step, the controller sends a set of predictive manipulated variables in a data packet: {ˆ ut+i|t |i = 0, 1, . . . , N1 }, while at the remote node, a compensator is designed to choose predictive manipulated variables to the plant as the actual manipulated input from its receiving buffer ˆt|t−i ut = u

(12)

where i = min{i|ˆ ut|t−i } is available. From the aforementioned, it is shown that, in the case of no network delay in the communication channel, the input to the plant actuator is the output of the controller. In the case of a delay iT , where T is the sampling period, the control input to the actuator is the ith-step-ahead control prediction received in the current sampling period. III. S TABILITY A NALYSIS OF C LOSED -L OOP S YSTEMS In the fault- and disturbance-free cases, namely, vt = 0 and ν = 0, the stability of the closed-loop system will be proven in this section. The predicted value is used as the control input to the actuator ˆt|t−i u t = Kt x

(13)

while the control input to the observer is

where Mt is the filter gain, and it can be obtained by St = Ct Pt|t−1 CtT + Rt

N1 are constructed as

(6) (7) (8)

where Pt+1|t is the covariance of the estimate before the measurement yt+1 is processed and (5)–(8) are obtained easily from the famous Kalman filter equations. Following the state observer described by (4), based on the output data up to t, the state predictions from time t + 1 to t +

ˆt|t = Kt x ˆt|t uot = u

(14)

where Kt ∈ Rm×n is the state feedback control matrix to be determined using modern control theory. Fig. 2 shows the structure of the proposed control method for networked predictive control systems. Remark 3.1: In some research works, the actual inputs ut of the plant are assumed to be sent back to the local node without time delay; thus, the actual inputs and outputs of the plant are available for the observer. However, this scheme cannot be easily applied in practice [22], so it is assumed in this paper that


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(11), and (12) is stable if there exists a positive definite matrix P ∈ R(2N1 +2)n×(2N1 +2)n such that T

A (i)P A(i) − P < 0

(18)

¯ is shown at the bottom of for i = 0, 1, 2, · · · , N1 , where A(i) the page, with i ∈ {1, 2, · · · , N1 }, A(i) ∈ R2(N1 +1)n×2(N1 +1)n Fig. 2. Structure of proposed control method for networked predictive control systems.

x ˆt+i|t =

(Ah + Bh Kh )ˆ xt+1|t .

× (Bt−i Kt−i Mt−i Ct−i + Lt−i Ct−i )

× (At−i + Bt−i Kt−i − Lt−i Ct−i − Bt−i Kt−i Mt−i Ct−i )

(15)

M3 (i) = Bt Kt Mt Ct + Lt Ct M4 (i) = At + Bt Kt − Lt Ct − Bt Kt Mt Ct ¯ and where A(0) ∈ R2(N1 +1)n×2(N1 +1)n is also shown at the top of the next page with

(Ah + Bh Kh )

h=t+1 t+i

× (Bt Kt Mt Ct + Lt Ct )xt +

M1 (0) = At + Bt Kt Mt Ct

(Ah + Bh Kh )

M2 (0) = Bt Kt − Bt Kt Mt Ct

h=t+1

× (At + Bt Kt − Lt Ct − Bt Kt Mt Ct )ˆ xt|t−1 u ˆt+i|t , = Kt+i

t+i

(16)

M3 (0) = Bt Kt Mt Ct + Lt Ct M4 (0) = At + Bt Kt − Lt Ct − Bt Kt Mt Ct .

(Ah + Bh Kh )

h=t+1

Proof: Since the control input to the actuator of the plant is ut = Kt x ˆt|t−i , then, based on (17), the closed-loop system can be written as

× (Bt Kt Mt Ct + Lt Ct )xt t+i + Kt+i (Ah + Bh Kh ) h=t+1

xt+1 = At xt + Bt ut

× (At + Bt Kt − Lt Ct − Bt Kt Mt Ct )ˆ xt|t−1 . (17)

= At xt + Bt u ˆt|t−i

Theorem 3.1: For the networked predictive control systems with random network-induced delays in both forward and backward channels, the closed-loop system described by (1),



(Ah + Bh Kh )

h=t−i+1

Based on (1), (2), and (4), it can be shown that t+i

t

M2 (i) = Bt Kt

h=t+1

x ˆt+i|t =

(Ah + Bh Kh )

h=t−i+1

the actual inputs ut of the plant cannot be sent back to the local node and are also not available for the observer. The estimate of the plant input u ˆt|t is in use for the observer. Xia et al. proposed an analysis on several schemes of NCS observer design [23] and compared two cases mentioned previously. Thus, it follows from (4) that t+i

t

M1 (i) = Bt Kt

A I .. .

0 ··· 0 ··· .. . ··· 0 ··· 0 ··· .. . ···

     0   0   ..  .   0 0 ··· A(i) =   M3 (i) 0 · · ·   0 0 ···  . ..  .  . . ···   0 0 ···   0 0 ···  . ..  .. . ··· 0 0 ···

··· ···

M1 (i) 0 .. .

0 0 .. .

I 0 .. .

0 I .. .

0 0 0 .. .

0 0 0 .. .

0 0 .. .

0 0 .. .

··· ··· ··· .. .

0

0

···

··· ··· ··· ··· ··· ··· ···

= At xt + M1 (i)xt−i + M2 (i)ˆ xt−i|t−i−1 .

0 0 .. .

0 0 .. .

0 0 .. .

0 0 .. .

I 0 0 .. . 0 0 ··· 0

0 0 ··· 0 0

··· ··· .. .

M2 (i) 0 .. .

··· ··· .. .

0 0 .. .

··· 0 0 ··· 0 M4 (i) · · · 0 I ··· .. .. . . ··· 0 0 ··· 0 0 ··· .. .. . . ··· 0 0 ···

0 0 ··· 0 0

0 0 0 .. .

··· 0 0 0 .. .

I 0 .. .

0 I .. .

0

0

0 0 .. .

 0 0    0  0     0  0  0 ..   .  0  0 ..  .

I

0

··· ··· .. .

0 0 .. .

··· ··· .. .

0 0 .. .

··· ··· ···

0 0 0 .. .

··· ··· ··· ··· ···

(19)

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

M1 (0)  I  .  ..   0   0   ..  .   0 A(0) =   M3 (0)   0  .  .  .   0   0  .  .. 0

0 ··· 0 ··· .. . ··· 0 ··· 0 ··· .. . ···

0 0 .. .

0 0 .. .

I 0 .. .

0 I .. .

··· ··· ··· ··· ···

0 0 .. .

··· ··· ··· ···

0 ··· 0 ··· 0 ··· .. . ··· 0 ··· 0 ··· .. . ···

0 0 0 .. .

0 0 0 .. .

0 0 .. .

0 0 .. .

··· ··· ··· .. .

0

0

0

···

···

0 0 .. .

I 0 0 .. . 0 0 ··· 0

Since the control input to the observer is uot = ut|t = K x ˆt|t , then, based on (4), the state observer has the following form: x ˆt+1|t = At x ˆt|t−1 + Bt ut|t + Lt (Ct xt − Ct x ˆt|t−1 ) xt|t−1 + Lt Ct xt = (At − Lt Ct )ˆ ˆt|t−1 + Mt (yt − Ct x + Bt Kt x ˆt|t−1 ) xt|t−1 + M3 (i)xt . (20) = M4 (i)ˆ Let

T T T xT (t) = xT t xt−1 · · · xt−i xt−(i+1) · · · · · · xT ˆT ˆT t−N x t|t−1 x t−1|t−2 · · · ··· x ˆT ˆT t−i|t−i−1 · · · x t−N |t−N −1

0 0 .. .

I 0 .. .

0 I .. .

0

0

0

0

··· ···

··· ···

0 0 .. . I

0

0 0 .. .

··· ··· .. .

0 0 .. .

··· ··· ···

0 0 0 .. .

··· ··· ··· ··· ···

From (18), it follows that Vt+1 − Vt < 0 for δ(k). Therefore, system (23) is stable for all switching sequences δ(k). IV. FD AND I DENTIFICATION BASED ON LRs The LR test is a multiple hypothesis test, where the different fault hypotheses are compared with the no-fault hypothesis pairwise. In the LR test, the fault magnitude is assumed to be known. The hypotheses under consideration are

= 2 log

(21)

It follows that the closed-loop system is a switched system which is composed of N1 + 1 discrete-time subsystems, i.e., (22)

where i = 0, 1, · · · , N1 . The switched system can be described as (23)

where δ(k) : {0, 1, · · ·} → {0, 1, 2, · · · , N1 } and δ(k) is the switching signal. Let Vt = xT t P xt , and then T Vt+1 − Vt = xT t+1 P xt+1 − xt P xt

T T = xT t Aδ(k) P Aδ(k) − P xt .

0 0 .. .

··· 0 0

 0 0    0  0     0  0  0 ..   .  0  0 ..  .

··· ··· .. .

The log LR for the hypotheses is given as the test statistic p y N |H1 (k, ν) (25) lN (k, ν) := 2 log p(y N |H0 )

When i = 0, based on (16), the augmented system can be expressed as

xt+1 = Aδ(k) xt

··· 0 0 0 .. .

0 0 .. .

0 0

H1 (k, ν) : a fault of magnitude ν at time k.

xt+1 = A(i)xt .

xt+1 = A(i)xt

0 0 0 .. .

0 0 .. .

H0 : no fault

and then, the augmented system can be expressed as

x ¯t+1 = A(0)xt .

0 M2 (0) · · · 0 0 ··· .. .. . ··· . 0 0 ··· 0 0 ··· .. .. . ··· . 0 0 ··· 0 M4 (0) · · · 0 I ··· .. .. . . ···

(24)

p(y N |k, ν) p(y N |k = N )

(26)

where the factor 2 is for notational convenience and y N denotes the set of measurements y1 , y2 , . . . , yN . In this paper, we use the convention that H1 (N, ν) = H0 ; hence, again, k = N means no fault. The double maximization over k and ν called GLR proposed by Willsky and Jones [11] is given to estimate the fault p(y N |k, ν) ν p(y N |k = N ) p y N |k, νˆ(k) ˆ k = arg max 2 log k p(y N |k = N )

νˆ(k) = arg max 2 log

(27) (28)

where νˆ(k) is the ML estimate of ν, given a fault at time k. The ˆ νˆ(k)) ˆ > h, fault candidate kˆ in the GLR test is accepted if lN (k, where the threshold h is a parameter to be tuned. The key point of the GLR test is that the innovation from the state observer can be expressed as a linear regression in ν εt (k) = ϕT t (k)ν + εt

(29)


where εt (k) is the actual innovation from the state observer if ν and k are known and εt is the virtual innovation of system (1) without additive fault. Before the fault occurs, the actual innovation is equal to the virtual innovation, i.e., εt (k) = εt , and the innovations get a bias εt ∈ N (0, St ), which means ϕT t (k) = 0, t = 0, 1, . . . , k. However, given a fault ν at time k, the innovations get a bias εt (k) ∈ N ϕT t (k)ν, St and the regressors ϕt (k) can be computed using

t−1 T ϕt (k) = Ct Ah − At−1 µt (k)

(30)

i=k

µt+1 (k) = At µt (k) + Mt+1 ϕT t (k)

(31)

initialized by zeros at time t = k [12]. Here, ϕt ∈ Rn and µt ∈ Rn×n . At the time t = N , the test statistic is given by T −1 (k)RN (k)fN (k) lN (k, νˆ(k)) = fN

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a fault after it occurs. Computing νˆt (t − W ) recursively needs data from step t − W to t, so that the computing time of every step is fixed. From (29), we can know that the innovations from the state observer are the linear combinations of fault ν, so that the recursive least squares algorithm can be used to estimate the fault ν for each time k. If the windowed LR estimate νˆt (t − W ) is greater than the threshold, the FD ˆ kˆ = (ˆ νˆ(k), νt (t − W ), t − W ) is sent to the predictive state observer to compensate for the fault. From the time t, the predictive state observer (4) is modified to ˆ x ˆt+1|t = At x ˆt|t−1 + Bt uot + Lt εt + νˆ(k)

and the state predictive equation and predictive manipulated variable equation are modified as follows:

(32)

ˆ ˆt|t−1 + Bt uot + Lt εt + νˆ(k) x ˆt+1|t = At x ˆ x ˆt+2|t = At+1 x ˆt+1|t + Bt+1 u ˆt+1|t + νˆ(k)

where the linear regression quantities are Rt (k) =

t

ϕi (k)Si−1 ϕT i (k)

.. .

(33)

i=1

ft (k) =

t

(36)

ˆt+N1 −1|t x ˆt+N1 |t = At+N1 −1 x ϕi (k)Si−1 εi

ˆ + Bt+N1 −1 u ˆt+N1 −1|t + νˆ(k)

(34)

(37)

i=1

and the predictive manipulated variables are as follows:

for each k, 1 ≤ k ≤ t. A change candidate is given by

u ˆt+i|t = Kt+i x ˆt+i|t ,

kˆ = arg max lN (k, νˆ(k)) . It is accepted if lN (k, νˆ(k)) is greater than some threshold h (otherwise, kˆ = N ) and the corresponding estimate of the change magnitude is given by ˆ = R−1 (k)f ˆ N (k). ˆ νˆN (k) N

From the aforementioned, the recursive computing equations of predictive states are given as follows: x ˆt+i|t =

t+i−1

(Ah + Bh Kh )ˆ xt|t

h=t

The formulation (32) is offline. Since the test statistic involves a matrix inversion of RN , a more efficient online method is as follows:

i = 0, 1, . . . , N1 .

+

i−1 m=1

t+i−1

ˆ + νˆ(k). ˆ (Ah + Bh Kh ) νˆ(k)

(38)

h=t+m

νt (k) lt (k, νˆ(k)) = ftT (k)ˆ

V. GLR W ITH I NTERMITTENT O BSERVATIONS

where t is used as the time index instead of N . νˆt (k) can be updated recursively, eliminating the matrix inversion of Rt (k). However, in real applications, the memories required to restore the computed data is square increasing and computation time increases step by step; hence, setting a suitable data structure becomes important. We recommend a tradeoff windowed online GLR method if the FDI system is with limited computation capacity

The previous method is easy to be applied, but with some drawbacks, for example, the computation burden is heavy at some time instants when many packets need to be processed simultaneously. An alternative scheme for the local node is to process the observations without a buffer. In this case, when an observation packet arrives at the local node, the observer checks the time stamp of this packet. If this time stamp is greater than every time stamp of the packet which has already arrived at the local node, the observer processes it; if the time stamp is not the newest at the present time instant, the observer discards it and treats the observation as zero. First, the state observer is designed as

νt (t − W ) lt (t − W, νˆ(t − W )) = ftT (t − W )ˆ

(35)

where W is the length of the data window. A better performance of the estimate can be obtained if a larger W is chosen; however, a larger W means that much time is needed for detecting

ˆt|t−1 + Bt uot + γt Lt εt x ˆt+1|t = At x

(39)

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where γt = {0, 1} denotes whether the observation with time stamp t is processed or is discarded, which is with probability distribution pγt (1) = λ. Note that a similar description of γt is presented in many works to denote whether the data are transmitted successfully. The meanings of these two descriptions of γt are different from each other but the same in mathematical analysis. Now, we give the convergence analysis of GLR with intermittent observations. However, the work in this section is limited to linear time-invariant (LTI) systems. First, by minimizing the following loss function: Vt (ν) =

t

γk εk −

T ϕT kν

Sk−1

εk −

ϕT kν

and from the Kalman filter equations, we have xt−1|t−1 + Mt CAt x0 x ˆt|t = (A − Mt CA)ˆ + Mt C

t

At−i vi + Mt C

i=1

i=1

xt−1|t−1 (0) + Mt CAt x0 x ˆt|t (0) = (A − Mt CA)ˆ t

(40)

At−i vi + Mt C

i=1

ϕ Rtϕ = Rt−1 + γt ϕt St−1 ϕT t .

Now, we define a function

gλ,t (X) = X − λXϕt ϕT t Xϕt + St

−1

ϕT t X.

¯ = AM ¯ AT −AM ¯ C T (C M ¯ C T +R)−1 C M ¯ AT +Q. M

(44)

(45)

¯ CA)| < 1, then the limit of regresIf |λ(A)| < 1 and |λ(A − M sors ϕt (k), t = k, k + 1, · · ·, ∀k, which is described as follows: t−k − Aµt (k) (46) ϕT t (k) = C A Mt+1 ϕT t (k)

−1 ¯ lim ϕT t (k) = (I − M C)(I − A) .

t→∞

xt+1 = At+1 x0 +

i=1

At+1−i vi +

i=1

At−i ν.

Thus, we define that the error between these two filters is x ˜t = x ˆt|t (0) − x ˆt|t = (A − Mt CA)˜ xt−1 + (I − Mt C)

t

At−i ν. (50)

i=1

From the concept of ϕT t (k) and (50), we have lim ϕT ˜t . t (0)ν = lim x

t→∞

t→∞

¯ = Since |λ(A)| < 1, the steady-state Kalman filter gain M lim t → ∞Mt exists and is found by solving

¯ CA)| < 1, the first term of the right side of Because |λ(A − M (50) will approach to zero when t → ∞. For the second term of the right side of (50), we need to analyze the existence of limit of the following: lim

t→∞

t

At−i .

i=1

Without loss of generality, we assume that all eigenvalues of A are distinct. Then, we can factorize A = T −1 ΛT , where Λ is a diagonal matrix with the diagonal elements being the eigenvalues. Then t t t−i −1 t−i A =T Λ lim T lim t→∞

t→∞

i=1

(47)

i=1

= T −1 (I + Λ + Λ2 + · · · + Λt + · · ·)T = T −1 (I − Λ)−1 T −1 = T −1 (I − Λ)T

(48)

Proof: Assuming that k = 0 without loss of generality, we have the following from (1): t+1

t i=1

where Mt is the Kalman filter gain, exists, and

t+1

At−i ν

¯ = AM ¯ AT − AM ¯ C T (C M ¯ C T + R)−1 C M ¯ AT + Q. M

From (44), we have gλ,t (X) ≤ X ∀t, X. Now, we state some properties of function (44) as the following lemmas. Lemma 5.1: For the LTI system described in (1), where At = A, Bt = B, Ct = C, Qt = Q, and Rt = R ∀t, let λ(A) ¯ denote the solution denote the eigenvalues of matrix A and M of the following:

µt+1 (k) = Aµt (k) +

+ Mt et + (I − Mt C)

(42)

Let Pt denote the covariance matrix of νˆ, according to [12], and we have Pt = (Rtϕ )−1 . The famous matrix inversion lemma applied to (42) gives −1 T ϕt Pt−1 . (43) Pt = Pt−1 − γt Pt−1 ϕt ϕT t Pt−1 ϕt + St

t i=1

k=1

where Rtϕ = tk=1 ϕk Sk−1 ϕT k . This is the recursive update for the estimate; we also can obtain the update of Rtϕ

At−i ν + Mt et .

The alternative Kalman filter which is assumed to have the information of the fault gives

+ Mt C

we have the least square estimate with intermittent observations −1 (41) ˆt−1 νˆt = νˆt−1 + γt (Rtϕ ) ϕt St−1 εt − ϕT t ν

t

= (I − A)−1 . Then, the limit of the regressor exists −1 ¯ lim ϕT t (k) = (I − M C)(I − A) .

At+1−i ν

t→∞

(49)


Lemma 5.2 [29]: Consider the function −1 T gλ,t (X) = X − λXϕt ϕT ϕt X. t Xϕt + St

(51)

Assume X, St ∈ S = {S ∈ Rn×n |S ≥ 0} and 0 < λ ≤ 1. Then, the following are true. 1) If X ≤ Y , then gλ,t (X) ≤ gλ,t (Y ). 2) If X ≥ gλ,t (X), then X > 0. 3) Let Xt+1 = gλ,t (Xt ). If X1 ≥ X0 , then Xt+1 ≥ Xt , and if X1 ≤ X0 , then Xt+1 ≤ Xt . Now, we formally state the convergence analysis of GLR with intermittent observations as the following theorem. Theorem 5.1: For the LTI system described in (1), where At = A, Bt = B, Ct = C, Qt = Q, and Rt = R ∀t, let λ(A) denote the eigenvalues of matrix A. If |λ(A)| < 1, then the GLR estimation covariance Pt with intermittent observation is bounded, and for any P0 , there exists a P¯ = 0 independent of P0 such that lim Pt = P¯ .

t→∞

4403

Fig. 3. Test statistics with fault compensation of numerical simulation in this paper.

(52)

Proof: From (43), we have P0 ≥ P1 ≥ · · · ≥ Pt ≥ · · · . Thus, for any t, 0 ≤ Pt ≤ P0 , which means for any initial condition P0 > 0, Pt is a monotonically decreasing sequence with a lower bound zero. According to the Dedekind Theorem, the sequence converges at a finite matrix. Letting P¯ denote the limit of Pt , we have t t P¯ = lim gλ,t (P0 ) = lim gλ,t (Pt ). t→∞

t→∞

(53)

Substituting (53) to (52) gives ¯ −1 ϕ¯T P¯ = 0 P¯ ϕ( ¯ ϕ¯T P¯ ϕ¯ + S) where ϕ¯ and S¯ denote the limits of ϕt (k) and St , respectively, for any k, when t approaches infinity. According to Lemma 1, ϕ¯ exists and does not always equal to zero. Sinopoli et al. proved that if A is stable, the state covariance Pt|t is convergent when t approaches infinity [29], namely, S¯ exists and S¯ = CA( lim Pt|t )AT C T + C T QC + R > 0. t→∞

This implies that the term

In this section, in order to validate the proposed method, a servo motor control system [19] that consists of a dc motor, load plate, and speed and angle sensors is considered. The model of the motor control plant at a sampling period of 0.04 s is identified to be 0.05409z −2 + 0.115z −3 + 0.001z −4 . 1 − 1.12z −1 − 0.213z −2 + 0.335z −3

0.1150

0.0001].

The state feedback matrix K is designed to be

VI. N UMERICAL S IMULATION

G(z −1 ) =

The system can also be written as the state-space form with the following system matrices:     1.12 0.213 −0.335 1 A= 1 0 0  B = 0 0 1 0 0 C = [0.0541

¯ −1 ϕ¯T ϕ( ¯ ϕ¯T P¯ ϕ¯ + S) does not always equal to zero, which proves that P¯ = 0.

Fig. 4. Test statistics without fault compensation of numerical simulation in this paper.

(54)

K = [0.027

0.575

0.0001]

which ensures that the close-loop system without time delay is stable. In this simulation, N1 = 4 and W = 5, the fault occurs at time k = 20, and its magnitude is ν = [8 0 0]T . The threshold h = 2000, and as Fig. 3 shows, the fault ˆ k) ˆ = (7.98, 26). Note that if the threshold detected is (ˆ ν (k), is designed to be smaller, kˆ is closer to its true value k, but the estimate of fault departs its true value, so a suitable threshold should be designed considering the tradeoff between the precisions of the estimated k and ν. Fig. 4 shows the test

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Fig. 5. States and their estimates with fault compensation of this simulation.

statistics without compensation of this simulation; the log LRs do not converge at zero after the fault is detected without compensation. Fig. 5 shows the states and their estimates with fault compensation in this simulation; the estimates depart from their true values when the fault occurs at k = 20, and after the fault is detected and compensated, the estimates of states converge at their true values. The simulation shows a remarkable phenomenon that the states of the controlled plant converge at another equilibrium points. The shift between the original points and the new equilibrium point of each state depends on the magnitude of the fault and the impact lasting time. VII. D ISCUSSION Network-induced time delay is the key issue in the investigation of NCSs. Generally speaking, there are two delays: sensor to controller delay (or delay in the backward channel) and controller to actuator delay (or delay in the forward channel). These two delays can be combined when the controller is linear and time invariant. It is shown in [31] that the networkinduced delay will prevent a traditional observer-based FD system from satisfying the essential requirement of a qualified residual generator, i.e., with the delay, the residual signal of a traditional residual generator will not be decoupled from the control input any longer. To solve this problem, some methods are proposed. If the network-induced time delay of an NCS is unknown and smaller than the sampling period, then the NCS with disturbance and latent fault can be described as a system without any time delay and the effect of time delay is modeled as an unknown input item [32]. The fault diagnosis system should be robust to this term, and some schemes are developed recently. For example, a traditional parity-relation-based lowpass postfiltering residual generator for NCSs with networkinduced delays is proposed in [33] to reduce the influence of the unknown input item. As for the case when the time delay is greater than one sampling period, the traditional robust methods do not have good performances since the influence

of the unknown input item caused by the time delay is great. Huang and Nguang proposed an observer-based FD scheme for NCSs with network-induced time delay and data packet dropout [34], wherein the lengths of delay are not limited to be smaller than the sampling period but are bounded. They also treated the time delays as time-varying input delays and developed a new disturbance attenuation notation according to the H∞ performance. There are two kinds of nuisances needed to be attenuated in the FDI systems; one is disturbance or unmodeled dynamics, and the other is unknown inputs caused by time delay. In this paper, the proposed predictive method faces only one nuisance and the influence of network-induced time delay is totally compensated theoretically. As aforementioned, predictive control is a good choice to design the FDI scheme without considering the attenuation of the influence of the network-induced time delay, but it needs the observer and controller to work precisely according to time. Many works have not paid attention to this practical problem, and their methods of predictive control have nice performance only in theory. The estimation of the clock asynchronism is not easily applied in practice when the network-induced time delay is unknown and random; the proposed scheme of using the time stamps in the measurement data packets is suitable and easy to be carried out. Furthermore, it converts an NCS with time delays both in the backward and forward channels to an equivalent system with a time delay in the forward channel if the controlled plant is linear. It is easy to analyze the stability of the NCS and to design the controller. On the other hand, this scheme involves heavier computation burden since the controller has to extend the predictive horizon to make it longer than the sum of the maximum of time delays in two channels. Two schemes for updating the LRs for the latent fault are also proposed in this paper. One is to set up a buffer at the local node to gather measurement data packets to keep them processed in sequence, i.e., an observation is processed after all other observations are processed, which are sampled by sensors earlier than the observation. This method is conservative and easily gives an analysis of stability. The other scheme is to process the intermittent observations that are in sequence and discard the observations that are out of sequence. Based on this scheme, the NCS with random network-induced time delay is modeled as an NCS with data packet dropout. Xiao et al. proposed a peak covariance stability analysis of a time-varying Kalman filter with possible packet losses in transmitting measurement outputs to the filter via a packet-based network [35]. It is shown that if the observability index of the discrete-time LTI system under investigation is one, the Kalman filter is peak covariance stable under no additional condition. In this paper, the updating of the LRs of the latent fault with intermittent observations is converted to the least squares problem with intermittent observations. The result shows that a GLR test with intermittent observations is convergent if the state transfer matrix and the covariance matrices of noise satisfy a proposed condition. However, limitations still exist. For example, the latent fault must be a step-type function and the convergence analysis of the GLR test with intermittent observations is only for LTI systems.


VIII. C ONCLUSION A fault detection and compensation scheme based on LRs for networked predictive control systems with random networkinduced time delays and clock asynchronism has been proposed in this paper. The NCS is stabilized by designing a state feedback gain and the sets of predictive manipulated variables. The measured outputs are sent back to the local node. The LRs of fault are computed step by step. If a fault is detected and identified, the fault estimate will be sent to the controller to compensate for the fault. Two methods are proposed for the local node to update the estimates. One of them is to process the observations according to the sequence in which the observation packets are sent from the remote node. The other method is to process the observations if they are the newest and discard the other observations. However, the convergence analysis of the second method is limited to LTI systems. ACKNOWLEDGMENT The authors would like to thank Prof. Z. Deng for her very helpful suggestions which have improved the presentation of this paper. The work of Prof. Z. Deng was supported by the National Natural Science Foundation of China under Grant 60904086. R EFERENCES [1] C. P. Tan, F. Crusca, and M. Aldeen, “Extended results on robust state estimation and fault detection,” Automatica, vol. 44, no. 8, pp. 2027–2033, Aug. 2008. [2] P. Zhang and S. X. Ding, “An integrated tradeoff design of observer based fault detection systems,” Automatica, vol. 44, no. 7, pp. 1886–1894, Jul. 2008. [3] B. Akin, U. Orguner, S. Choi, and A. Toliyat, “A simple real-time fault signature monitoring tool for motor drive embedded fault diagnosis systems,” IEEE Trans. Ind. Electron., vol. 58, no. 5, pp. 1990–2001, 2011. [4] Y. Wang, W. Wang, and D. Wang, “LMI approach to design fault detection filter for discrete-time switched systems with state delays,” Int. J. Innov. Comput., Inf. Control, vol. 6, no. 1, pp. 387–398, Jan. 2010. [5] I. M. Jaimoukha, Z. Li, and V. Papakos, “A matrix factorization solution to the H_/H∞ fault detection problem,” Automatica, vol. 42, no. 11, pp. 1907–1912, Nov. 2006. [6] A. P. Deshpande, S. C. Patwardhan, and S. S. Narasimhan, “Intelligent state estimation for fault tolerant nonlinear predictive control,” J. Process Control, vol. 19, no. 2, pp. 187–204, Feb. 2009. [7] C. W. Chan, H. Song, and H. Y. Zhang, “Application of fully decoupled parity equation in fault detection and identification of DC motors,” IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1277–1284, Aug. 2006. [8] G. Tortora, B. Kouvaritakis, and D. W. Clarke, “Simultaneous optimization of tracking performance and accommodation of sensor faults,” Int. J. Control, vol. 75, no. 3, pp. 163–176, 2002. [9] Y. Zhang and S. J. Qin, “Adaptive actuator fault compensation for linear systems with matching and unmatching uncertainties,” J. Process Control, vol. 19, no. 6, pp. 985–990, Jun. 2009. [10] T. J. Kim, W. C. Lee, and D. S. Hyun, “Detection method for opencircuit fault in neutral-point-clamped inverter systems,” IEEE Trans. Ind. Electron., vol. 56, no. 7, pp. 2754–2763, Jul. 2009. [11] A. S. Willsky and H. L. Jones, “A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems,” IEEE Trans. Autom. Control, vol. AC-21, no. 1, pp. 108–112, Feb. 1976. [12] F. Gustafsson, Adaptive Filtering and Change Detection. Chichester, U.K.: Wiley, 2000. [13] Y. Zhang, Y. Chen, J. Sheng, and T. Hesketh, “Fault detection and diagnosis of networked control system,” Int. J. Syst. Sci., vol. 39, no. 10, pp. 1017–1024, Oct. 2008. [14] Y. Wang, S. X. Ding, H. Ye, L. Wei, P. Zhang, and G. Wang, “Fault detection of networked control systems with packet based periodic communication,” Int. J. Adapt. Control Signal Process., vol. 23, no. 8, pp. 682–698, Aug. 2009.

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Bo Liu was born in the Ningxia Hui Autonomous Region, China, in 1982. He received the B.S. degree in automation from the North China University of Technology, Beijing, China, in 2005 and the M.S. degree in automatic control from Beijing Institute of Technology, Beijing, in 2008, where he is currently working toward the Ph.D. degree in control science and engineering. His research interests include networked control systems, active disturbance rejection control, process control, and fault diagnosis.

4406


Yuanqing Xia was born in Anhui Province, China, in 1971. He received the B.S. degree from the Department of Mathematics, Chuzhou University, Chuzhou, China, in 1991, the M.S. degree in fundamental mathematics from Anhui University, Hefei, China, in 1998, and the Ph.D. degree in control theory and control engineering from Beijing University of Aeronautics and Astronautics, Beijing, China, in 2001. From 1991 to 1995, he was a Teacher with Tongcheng Middle School, Tongcheng, China. During January 2002–November 2003, he was a Postdoctoral Research Associate with the Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, where he worked on navigation, guidance, and control. From November 2003 to February 2004, he was with the National University of Singapore, Singapore, Singapore, as a Research Fellow, where he worked on variable structure control. From February 2004 to February 2006, he was with the University of Glamorgan, Pontypridd, U.K., as a Research Fellow, where he worked on networked control systems (NCSs). From February 2007 to June 2008, he was a Guest Professor with Innsbruck Medical University, Innsbruck, Austria, where he worked on biomedical signal processing. Since July 2004, he has been with the Department of Automatic Control, Beijing Institute of Technology, Beijing, first as an Associate Professor and then, since 2008, as a Professor. His current research interests are in the fields of NCSs, robust control, sliding-mode control, active disturbance rejection control, and biomedical signal processing.