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Ind. Eng. Chem. Res. 2008, 47, 2636-2642
PROCESS DESIGN AND CONTROL Fault Diagnosis and Isolation of Multi-Input-Multi-Output Networked Control Systems Yingwei Zhang,*,†,§ Jie Sheng,‡ S. Joe Qin,§,| and Tim Hesketh‡ Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern UniVersity, Shenyang, Liaoning 110004, P. R. China, School of Electrical Engineering and Telecommunications, UniVersity of New South Wales, Sydney 2052, NSW, Australia, Department of Chemical Engineering, UniVersity of Texas at Austin, Austin, Texas 78712, and The Mork Family Department of Chemical Engineering and Materials Science, UniVersity of Southern California, 925 Bloom Walk, HED 211, Los Angeles, CA 90089-1211
In this article, we consider a type of multi-input-multi-output networked control systems (NCSs) in which the network-induced time delays for different inputs are different. For those unstable and/or poorly damped eigenvalues, we propose a reduced-order memoryless state observer design approach. The full-order state vector of the original systems using the reduced-order observer is reconstructed. We prove that the γ-stability of the state error dynamics can be guaranteed. Meanwhile, by computing the residual between the output of the practical systems and the output of the reduced-order observer, the fault-detection scheme of the NCSs is given. An illustrative example shows the effectiveness of the proposed method. 1. Introduction The study of networked control systems (NCSs) has recently received much attention. The uses of the NCSs have several advantages such as reconfigurability, low installation cost, and easy maintenance; it is also well suited for large geographically distributed systems.1-4 However, data networks operate in a discrete fashion, delivering information only at specific instants in time, which means that the controller cannot have access to the plant output at all times. In traditional system models, information from sensors is assumed to be instantaneously available. Thus, network-induced time delays, packet dropout, limited bandwidth, information loss due to encoding, and other characteristics of networks could influence the performance of the system designed without taking them into account. So, the characteristics of networks cannot be ignored. Compared with conventional control systems, these problems make the analysis and design of the NCSs more complex. There are some important achievements in the stability control of NCSs.5-13 The study on fault diagnosis of the NCSs also becomes necessary for practical control systems, particularly in safetycritical systems. The fault diagnosis theory for the NCSs is different from the ones for traditional control systems in many aspects.14-20 The designer of fault diagnosis should take all of these network-induced characteristics and limitation into account. There have been some results on the fault diagnosis of NCSs, which consider the NCSs as a discrete-time model with multiple inputs and multiple outputs and multiple distributed communication delays.14-16 The missing data problem is taken into account and the data packet dropout as a discretetime Markov chain with state spaces and the stationary transition * To whom correspondence should be addressed. Tel.: +1-512-4714417. Fax: +1-512-471-7060. E-mail:
[email protected]. † Northeastern University, Shenyang. ‡ University of New South Wales. § University of Texas at Austin. | University of Southern California.
Figure 1. Diagram of an NCS.
probability is modeled.17 The fault of the sensor failing to transfer the data is considered, and the controller is designed that can tolerate the sensor’s faults to guarantee system stability.18 The fault-detection strategy of the NCSs with short introduced delays is investigated.16 An H-infinity fault-detection filter design scheme is proposed for the NCSs modeled by discrete Markovian jump systems.19 A number of aspects and approaches to control the network for the NCSs are proposed. To analyze and derive the best strategies for fault tolerant NCSs, it is initially assumed that the network communication bandwidth is infinite.20 But in these articles, the introduced delays for different control inputs are the same and fault isolation is not considered. Compared with fault diagnosis, fault isolation is not an easy task. In this article, the fault diagnosis and faultisolation problems of the NCSs with different long introduced delays in different control inputs are considered, and the effects of the disturbance to the NCSs also are considered. Time delay τ is time variant, and the maximum of τ is known. Then the fault-detection scheme is proposed. The organization of the article is as follows. The model of the networked systems with different long introduced delays in different control inputs is given in Section 2, followed by a
10.1021/ie070779m CCC: $40.75 © 2008 American Chemical Society Published on Web 03/14/2008
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2637
system-decomposition scheme of the NCSs. The state observer with γ-stability for the NCSs is designed in Section 4. The fault diagnosis and isolation scheme of the systems by computing the residual between the output of the practical systems and the output of the observer is proposed in Section 5. The simulation example indicates that the proposed method is effective for the networked systems in Section 6. Conclusions are given at the end of the article. 2. Model of the Networked Systems. We consider the two kinds of network-induced time delays: sensor-controller delay τSC and controller-actuator delay τCA, as shown in Figure 1. τSC and τCA are time-variant and random variables. If the controller node is clock-driven, then at a sample instant the packet arrives at the controller/actuator and its sequence will not be determined, so that it is difficult to get the distinct state model of the NCSs. To overcome this difficulty, an even-driven mode for the controller node can be utilized. Let the total induced time delay τ ) τSC + τCA and the τ be known. When the controller is event-driven, the sequence and integrity of the data is guaranteed. Considering long delays and multi-inputs, we can design the networked systems as follows: x˘ (t) ) Ax(t) + B1u1(t - τ1) + ‚‚‚ + Bquq(t - τq) + Bdd(t) + Bf fa(t) y(t) ) Cx(t) + Dfs(t) (1)
here x(t) ∈ Rn is the state vector, y(t) ∈ Rr is the output vector, ui ∈ Rmi(i ) 1,‚‚‚,q) are the control vector of ith channel, and d(t) ∈ Rl is the process disturbance. fa(t) denotes actuator fault, fs(t) denotes sensor fault, and fa(t) and fs(t) are unknown. Matrices A, C, Bd, Bf, D, and Bi(i ) 1,2,‚‚‚,q) are all with appropriate dimensions. For the time delay, let τi ) liT - m j i, where li is a positive integer,0 e m j i < T, i ) 1, ‚‚‚, q. Discretizing systems (1) with period T, and considering the timedelay τi between plant and control input, an equivalent discretetime model for eq 1 is as follows:
as shown in Figure 2 by shadows. Obviously, space Ω contains those unstable and/or poorly damped eigenvalues of systems (eq 2). Once eigenvalues of matrix F are computed, Ω can be determined by choosing those satisfying |z| gγ for the specified γ. Upon the determination of Ω, the corresponding right eigenvector matrix V can be obtained:
V ) [V1,V2,...,Vr] ∈ Cn × r
(5)
[ziIn - F]Vi ) 0 (6)
(6)
∀zi ∈ Ω
(7)
where
where we have assumed that Ω has r(r < n) eigenvalues. Without loss of generality, providing that Ω consists of r distinct eigenvalues; from (4), we can easily derive that,
det[R(F)] ) det(zIn - F) ) Φ(z)det(zIr - Λr), ∀z ∈ Ω
x(k + 1) ) q
Fx(k) +
Figure 2. γ-Stability and the space Ω.
Gbiui(k - li + 1) + ∑ Gaiui(k - li) + i∑ )1
i)1
Pd(k) + Mfa(k) y(k) ) Cx(k) + Dfs(k) (2) where
F ) eAT,Gai )
(8)
q
∫mjT eAtBdt,Gbi ) ∫0mˆ eAtBdt,P ) ∫0T eA(T-t)Bddt,M ) ∫0T eA(T-t)Bfdt i
i
(3)
3. System Decomposition Based on γ-Stability. A discretetime system with the properties that all of its eigenvalues satisfy |z| < γ(∀ γ ∈ (0,1)) is said to be γ-stable. Referring to Figure 2, γ-stability means that all of the eigenvalues of the systems are within a disk with radius γ. In this section, a space Ω on the z plane related to γ-stability will first be introduced. After that, decomposition of systems (2) will be given according to z ∈ Ω and z ∉Ω. Given a critical value 0 < γ < 1, eigenvalues of the matrix F in systems (2) can be derived into two groups: |z| gγ and |z| < γ. The set of eigenvalues can be obtained by first computing the eigenvalues of matrix F. For a specified value γ, a space Ω can be defined as,
Ω ) {z:det(zIn - F) ) 0,|z| gγ} ,0 < γ < 1
(4)
where Φ(z) is an entire function on the plane |z| gγ, so ∀z ∈ Ω, Φ(z) * 0. If all of the r eigenvalues in space Ω are real numbers, Λr is a diagonal matrix
Λr ) diag(λ1, λ2,‚‚‚, λr)
(9)
If Ω has repeated eigenvalues, Λr will then have the Jordan canonical form. When Λr contains complex eigenvalues, then Λr can be changed to real matrix.21 Assume that when one complex conjugate pair in Λr is zh ) Rh ( jωh, a corresponding Jordan sub-block matrix Jh can be constructed as follows:
[
R w Jh ) -h w R h h h
]
(10)
Λr will be a real matrix in the block diagonal form
Λr ) block - diag(λ1, λ2,...,Jh,...λr)
(11)
and the real eigenvector matrix V becomes
V ) [V1,V2,..., x2 Re(Vh), x2 Im(Vh),...Vr]
(12)
where Vh is the complex right eigenvector associated with the complex eigenvalue pair zh. Remark 1. Assuming systems (eq 2) are observable for ∀z ∈ Ω, we can decompose the systems (eq 2) according to z ∈ Ω
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Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008
and z ∉ Ω. The observable part can be derived as -1
xo(k + 1) ) Λrxo(k) + V Pd(k) + V V
-1
-1
Using eqs 19 and 22, we get
q
∑ Gaiui(k - li) + i)1
q
∑ Gbiui(k - li + 1)
(z) ) (zIn - F)-1
(13)
The following state observer is with a γ-stability margin for systems (eq 2):
xr(k + 1) ) Λrxr(k) + L[y(k) - CVxr(k)]
(16)
∑ Eirui(k)
(17)
i)1
yˆ(k) ) Cxˆ(k)
(18)
where L should be chosen so that matrix Λr - LCV is γ-stable. Proof. Applying z-transform to (2), we have
x(z) ) (zI - F)
-1
q
∑ (Gaiz
i)1
-li
+ Gbiz
)ui (z) +
(zI - F)-1Pd(z) + (zI - F)-1Mfa(z) (19) y(z) ) Cx(z) + Dfs(z)
(20)
Applying z-transform to eqs 16-18, we get
xr(z) ) (zIr + LCV - Λr)-1Ly(z)
yˆ(z) ) CV(zIr + LCV - Λr)-1Ly(z) + C
(21)
q
∑ Eirui(z) i)1
(22)
q
∑ Eirui(z)
(23)
i)1
Define the error between the actual state x(k) and the observed state xˆ (k) as
(k) ) x(k) - xˆ (k)
q
Eirui(z) + LDfs(z)) +
∑ Eiru(z)
(25)
i)1
Equation 25 can be rewritten as follows after some manipulations,
(zIn - F)[In + V(zIr - Λr)-1LC](z) ) q
(Gaiz-l + Gbiz-l +1 - (zIn - F)(Eir + ∑ i)1 i
i
V(zIr - Λr)-1LCEir))ui(z) + Pd(z) + Mfa(z) (zIn - F)V(zIr - Λr)-1LDfs(z) (26) To ensure γ-stability of the state error, the following two conditions should be satisfied: (1) The dynamics of (k) must be γ-stable; this is determined by the zeros of det{ (zIn - F)[In + V(zIr - Λr)- 1LC]}; (2) The action of control input, that is, ui(z) on the right side of eq 26 should vanish as z f 1 for the steady-state tracking. For condition (1), define
ψ(z) ) det{ (zIn - F)[In + V(zIr - Λr)-1LC]}
(27)
A series of conversions are applied in the following:
ψ(z) ) det{(zIn - F)[In + V(zIr - Λr)-1LC]} ) Φ(z) det(zIr - Λr) det[Ir + (zIr - Λr)-1LCV] ) Φ(z) det[Ir - Λr + LCV] (28) Note that from eqs 26-28, the determinant identity
det(ST) ) det(S) det(T)
(29)
for any two n × n matrices S and T is used. From eq 28, we applied the results in eq 8 and the determinant identity
-li+1
xˆ (z) ) V(zIr + LCV - Λr)-1Ly(z) +
i
det(zIn - F) det[In + V(zIr - Λr)-1LC] )
q
xˆ(k) ) Vxr(k) +
∑
i)1
(14)
Eir ) [(In - F)(In + V(Ir - Λr)-1LC)]-1(Gai + Gbi) (15)
i
q
LC
where Gai, Gbi, P, C can be determined by eqs 2 and 3, and xo(k) is the state vector of the observable part. This shows that the observable part of the systems eq 2 is Λr, CV. 4. Design of State Observer with γ-Stability for the NCSs with Long Time Delays. For discrete-time NCSs with a long induced time delay described by (2), a state observer design approach will be proposed in this section. The full-order state vector is reconstructed by a reduced-order observer whose order is equal only to the number of unstable and/or poorly damped eigenvalues of the original systems. We will first present a theorem on the structure of the observer followed by a proof and then a summary on the design procedure. Theorem 1. Assume systems (2) are observable for ∀z ∈ Ω define
(Gaiz-l + Gbiz-l +1)ui(z) + (zIn ∑ i)1
F)-1Pd(z) + (zIn - F)-1Mfa(z) - V(zIr - Λr)-1(LC(z) +
i)1
y(k) ) CVxo(k)
q
(24)
det(In + HJ) ) det(Ir + JH)
(30)
by letting H ) V and J ) (zIr - Λr)-1LC. Using the determinant identity (eq 29) again for any two r × r matrices, ψ(z) defined in eq 27 is finally reduced to eq 30. Because ψ(z) is an entire function that does not vanish for ∀z ∈ Ω; it is then obvious that ψ(z) will be γ-stable if and only if the matrix
Λr - LCV
(31)
are γ-stable where 0 < γ < 1. It is immediately concluded that gain matrix L in the reduced-order observer (eq 22) should be properly chosen so that all eigenvalues of (Λr - LCV) are between 0 and γ, which guarantees the γ-stability of the error dynamics (eq 25). To find such a gain matrix L, it is required that the pair (Λr, CV) is observable; if systems (eq 2) are observable for ∀z ∈ Ω, as assumed by theorem 1, the condition on Λr, CV is always true (remark 1 in section 3). For condition 2, steady-state tracking is equivalent to making all of the terms preceding ui(z) in the error dynamics (eq 25) tend to zero when z f 1, that is,
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2639 q
(zI - F)x(z) )
∑ (Gaiz-l + Gbiz-l +1)ui(z) + i)1 i
i
Pd(z) + Mfa(z) (36)
Eir ) [(In - F)(In + V(Ir - Λr)-1LC)]-1(Gai + Gbi)
Substitute eq 35 into eq 36, and we have exactly same as given in eq 18 in Theorem 1. As a summary, the proposed reduced-order observer (eq 16) and the full-order state reconstruction (eqs 17-18) with Eir defined in eq 18 and L being properly chosen will ensure γ-stability of the error dynamics (eq 25) as well as the steadystate tracking. Remark 2. From the discussion on the choice of L, we can see the freedom in assigning the γ-stability margin of the observer (eqs 16-18). Because γ can be any positive value between 0 and 1, and the observer eigenvalues are chosen satisfying |z| < γ, the convergence speed of the observer can be arbitrarily determined by choosing difference values of γ. Remark 3. It is easy to see that steady-state tracking can always be achieved provided that (zIn - F)[In + V(zIr Λr)-1LC] is invertible. This is always possible by the assumption that F does not have eigenvalues equal to one and by the property det[In + V(zIr - Λr)-1LC] * 0, which can be guaranteed by the development from eqs 26 to 28. On the basis of the preceding proof, the design of the reduced-order observer with γ-stability can be summarized as follows: Step 1. Specify a value of γ and form the space Ω by computing the eigenvalues of eq 2. Step 2. Calculate Λr and V, respectively. Step 3. Choose gain matrix L so that (Λr - LCCr) is with γ-stability. Step 4. Calculate Eir by eq 15. The designed rth-order observer is thus obtained by eqs 16-18. 5. Generation of Residual and Fault Diagnosis and Isolation. (1) Generation of Residual. By computing the residual between the output of the practical systems and the output of the reduced-order observer, the fault-detection scheme of the NCSs is given in this section. From eq 2, we can get q
x(k - 1) ) Fx(k - 2) +
Gaiui(k - li - 2) + ∑ i)1
(zIn - F + PP′CFz-1)x(z) ) (I - PP′Cz-1)
x(z) ) T-1(In - PP′Cz-1)
q
(Gaiz-l -1 + Gbiz-l )ui(z) + ∑ i)1 i
i
CPz-ld(z) + CMfa(z)z-l + Dfs(z) (34) From eqs 33 and 34, we can calculate d(z) as follows:
d(z) ) P′y(z) - P′CFz-lx(z) - P′C
q
∑ (Gaiz-l + Gbiz-l +1)ui(z) + i)1 i
i
T-1PP′y(z) + T-1(M - PP′CMz-1)fa(z) - T-1PP′Dfs(z) (38) where T ) zIn - F + PP′CFz-1 is a nonsingular matrix. And we can get
y(z) ) CT-1(I - PP′Cz-1)
q
(Gaiz-l + Gbiz-l +1)ui(z) + ∑ i)1 i
i
CT-1PP′y(z) + CT-1(M - PP′CMz-1)fa(z) + (I - CT-1PP′)Dfs(z) (39) Let S ) I - CT-1PP′, then we have
S-1y(z) ) S-1CT-1(I - PP′Cz-1)
q
(Gaiz + Gbiz-l +1) ∑ i)1 i
ui(z) + S-1CT-1(M - PP′CMz-1)fa(z) + Dfs(z) (40) Combine eqs 23 and 40 and we have the following result
r(z) ) y(z) - yˆ (z) -1
) (I - CV(zIr + LCV - Λr L)y(z) - C
q
Eirui(z) ) ∑ i)1
q
((I - CV(zIr + LCV - Λr)-1L)S-1CT-1(I ∑ i)1 PP′Cz-1)(Gaiz-li + Gbiz-li+1) CEir)ui(z) + (I - CV(zIr + LCV - Λr)-1L)S-1CT-1(M PP′CMz-1)fa(z) + (I - CV(zIr + LCV - Λr)-1L)Dfs(z) (41)
The z-transform of eq 33 is as follows
y(z) ) CFz x(z) + C
i
From eq 37, we can easily obtain the following equation:
Gbiui(k - li - 1) + Pd(k - 2) + Mfa(k - 2) ∑ i)1
-1
(Gaiz-l + ∑ i)1
Gbiz-li+1)ui(z) + PP′y(z) + (M - PP′CMz-1)fa(z) PP′Dfs(z) (37)
q
y(k - 1) ) Cx(k - 1) + Dfs(k - 1) (33)
q
Let
Hfa(z) ) (I - CV(zIr + LCV - Λr)-1L)S-1CT-1(M PP′CMz-1) Hfs(z) ) (I - CV(zIr + LCV - Λr)-1L)D Then eq 41 becomes
q
(Gaiz-l -1 + ∑ i)1 i
Gbiz-li)ui(z) - P′CMz-lfa(z) - P′Dfs(z) (35) where P′ ) (CPz-1)-1. From eq 16, we can get
q
r(z) )
((I - CV(zIr + LCV - Λr)-1L)S-1CT -1(I ∑ i)1 PP′Cz-1)(Gaiz-li + Gbiz-li+1) - CEir)ui(z) + Hfa(z)fa(z) + Hfs(z)fs(z) (42)
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(2) Fault Diagnosis and Isolation. For the residue between the output of the practical systems and the output of the reducedorder observer, we can get from eq 42,
jr(k) ) Q(y(k) - yˆ(k))
(43)
where the coefficient Q increases the residual design freedom, improves the residual robust performance, and reduces its sensitivity to disturbance. Then
rˆ(z) ) Q(y(z) - yˆ (z)) q
)Q
((I - CV(zIr + LCV -Λr)-1L)S-11CT-1(I ∑ i)1 PP′Cz-1)(Gaiz-li + Gbiz-li+1) - CEir)ui(z) + QHfa(z)fa(z) + QHfs(z)fs(z) (44)
Equation 44 indicates that residue r is only relevant to ui, fa, and fs. The effect of ui can be removed when the faults occur, and it is required that
Figure 3. Simulation result when there is no fault.
q
Q
((I - CV(zIr + LCV - Λr)-1L)S-1CT-1(I ∑ i)1 PP′Cz-1)(Gaiz-li + Gbiz-li+1) - CEir) ) 0 (45)
Thus we can easily obtain
(I - CV(zIr + LCV - Λr)-1L)S-1CT-1(I PP′Cz-1)(Gaiz-li + Gbiz -li+1) - CEir ) 0 (46) Equation 46 is necessary for fault diagnosis. Let Hfa (z) ) 0, and the sensor fault fs is isolated. Similarly, let Hfs (z) ) 0, and the actuator fault fa (z) is isolated. 6. Numerical Example. Let us now consider the following systems:
x˘ (t) )
[
]
[]
[]
-1 -2 0 1 x(t) + u (t) + u (t) + 0 - 0.5 1 1 2 2 0 0 d(t) + f (t) 1 1 a y(t) ) [1 0 ] x(t) + fs(t) (47)
[]
[]
Set T ) 1s, τ1 ) 1.2s, τ2 ) 2.6s, so l1 ) 2, m j 1 ) 0.8s; l2 ) 3, m j 2 ) 0.4s. Discretizing eq 47 by the systems in eq 2 and we can get
[
]
[
]
0.3679 0.1353 0.0333 x(k) + u (k - 2) + 1.0000 0.6065 0.1276 1 0.3991 0.1480 u (k - 1) + u (k - 3) + 0.6594 1 0.4552 2 1.2757 0.4323 u (k - 2) + d(k) + 1.3042 2 0.7869 0.4323 f (k) 0.7869 a y(k) ) [1 0 ] x(k) + fs(k) (48)
x(k + 1) )
[
[
]
]
[
[
]
[
]
Now let us design the observer:
[
]
[
]
-0.2015 from eq 6. -0.7534 1 (3) The observability matrix of systems (3) is C ) , and 0 0.3679 0.1353 0.3679 , so CF ) and F ) 1.0000 0.6065 0.1353 1 0 (C CF) ) , and the rank of the system is 2. 0.3679 0.1353 (2) Get Λr ) 0.8740, derive V )
[
]
[
[] [ ]
]
This indicates the systems are observable, so the pair Λr, CV is observable. From theorem 1, L can be found so that (Λr -LCCr) is γ-stable. Set the eigenvalue of Λr - LCCr to be 0.5, and substitute it into det(zI - Λr + LCV) ) 0. Accordingly, L is obtained as L ) -1.8561. (4) From eq 24, we obtain E11 and E21.
]
0.3679 0.1353 , 1.0000 0.6065 which are 0.1005 and 0.8740. Set γ ) 0.7, so Ω ) {0.8740}. (1) Compute the eigenvalues of F )
Figure 4. Magnifying the figure of residual r(k).
E11 )
[
]
[
0.6146 1.7734 , E21 ) 1.3770 2.6734
]
Therefore, the reduced-order state observer is as follows:
xr(k + 1) ) 0.5xr(k) - 1.8561y(k) xˆ (k) ) -0.2015 0.6146 x (k) + u (k) + -0.7534 r 1.3770 1 1.7734 u (k) yˆ (k) ) [1 0] x(k) (49) 2.6734 2
[
]
[
[ ]
]
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2641
Figure 8. Magnifying the figure of residual r(k). Figure 5. Simulation result of the case with the actuator fault at 7s.
When a sensor fault occurs at 7s, the system’s actual output, observer output, and residual are shown in Figures 7 and 8. From it we observe that the residual increases rapidly at 7s, with no delays, which indicates the sensor fault occurs. 7. Conclusions
Figure 6. Magnifying the figure of residual r(k).
In this article, the discussions are based on the assumptions that the delay-time is more than one sampling period and the introduced-delays in different control inputs are different. A reduced-order memoryless state observer for the NCSs has been constructed. Considering long delays and multi-input-multioutput, we deal with them by using z-transform in this article. Then, we derive the equation of the residue and isolate the fault of the networked systems. At last, a numerical example has been given to illustrate our result. The approach proposed in this article is mainly focusing on network-introduced delay and the fault diagnosis and faultisolation problems of the NCSs with different long introduced delays in different control inputs. Many other features of the NCSs have not been concerned. There are some further research topics. Similar to linear systems, the effect produced by networkintroduced delay, packet dropout, asynchronous clock, and other features of the network should be discussed for nonlinear NCSs, and new fault diagnosis theory for nonlinear NCSs should also be developed. NCSs with packet dropout can be modeled as a switched system and then develop the related theory. The question of how to deal with the combined impacts of all these features is a big challenge in future research. Literature Cited
Figure 7. Simulation result of the case with the sensor fault at 7s.
We set coefficient Q ) 1 and used MATLAB for simulation. When there is no fault in the systems, the simulation results are shown in Figures 3 and 4 for the case where u1(k) is a unit input and u2(k) ) 5. The initial conditions for the systems and observer are taken to be x(k) ) 0, xr(k) ) 0. It is clear from Figure 3 that the reduced-order state observer tracks the true states of the system well, and from the magnifier Figure 4 we observe that the residual drifts in a region around 0. When an actuator fault occurs at 7s, the system’s actual output, observer output, and residual are shown in Figures 5 and 6. From it we observe that the residual increases rapidly after 8.8s, and delays 1.8s, which indicates that the actuator fault occurs.
(1) Zhang, L.; Shi, Y.; Chen, T.; Huang, B. A New Method for Stabilization of Networked Control Systems with Random Delays. IEEE Trans. Autom. Control 2005, 50, 1177-1181. (2) Beldiman, O.; Walsh, G. C.; Bushnell, L. Predictors for Networked Control Systems, Proceedings of ACC, 2000; 2347-2351. (3) Madievski, A. G.; Anderon, B. D. O.; Gevers, M. Optimum Realization of Sampled-Data Control for FWL Sensitivity Minimization. Automatica 1995, 31, 367-379. (4) Shoarinejad, K.; Speyer, J. L. A Two-Station Decentralized LQG Problem with Noisy Communications, Proceedings of the 38th CDC, 1999; 4929-4932. (5) Montestruque, A. L.; Antsaklis, M. P. A. Stability of Model-Based Networked Control Systems with Time-Varying Transmission Times. IEEE Trans. Autom. Control 2004, 49, 1562-1572. (6) Goodwin, C. G.; Haimovich, H.; Quevedo, E. D.; Welsh, S. J. A Moving Horizon Approach to Networked Control System Design. IEEE Trans. Autom. Control 2004, 49, 1427-1444. (7) Zhang, W.; Branicky, M. S.; Phillings, S. M. Stability of Networked Control Systems. IEEE Control Systems Magazine 2001, 21, 84-99.
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(8) Walsh, G. C.; Ye, H.; Bushnell, L. Stability Analysis of Networked Control Systems, Proc. of the American Control Conference, 2000; 28762880. (9) Zhivoglyadov, P. V.; Middleton, R. H. Networked Control Design for Linear Systems. Automatica 2003, 39, 743-750. (10) Hu, S.; Zhu, Q. Stochastic Optimal Control and Analysis of Stability of Networked Control Systems with Long Delay. Automatica 2003, 39, 1877-1884. (11) Zhang, Y.; Liu, J.; Hesketh, T. Output Feedback Stabilization of Networked Control Systems with Long Delay, 1st International Symposium on Systems and Control in Aerospace and Astronautics, 2006; 19-21. (12) Ma, C. L.; Fang, H. J. Research on Mean Square Exponential Stability of Networked Control Systems with Multi-Step Delay. Applied Mathematical Modeling 2006, 30, 941-950. (13) Young, S. S.; Vinh, H. N.; Young, S. Modified Kalman Filter for Networked Monitoring Systems Employing a Send-on-Delta Method. Automatica 2007, 43, 332-338. (14) Winstead, V.; Kolmanovsky, I. ObserVers for Fault Detection in Networked Systems with Random Delays. Proc. of the American Control Conference, 2004; 2457-2462. (15) Ye, H.; Ding, S. X. Fault Detection of Networked Control Systems with Networked-Induced Delay. Proc. of the 8th International Conference on Control, Automation, Robotics and Vision, 2004; 1366-1369.
(16) Zheng, Y.; Fang, H. J.; Wang, H.; Xie, L. B. Fault Detection Approach for Networked Control System Based on a Memoryless ReducedOrder Observer. ACTA Automatica Sinica 2003, 29, 84-89. (17) Zhang, P.; Ding, S. X.; Frank, P. M.; Sader, M. Fault Detection of Networked Control Systems with Missing Measurements, Proc. of the 5th Asian Control Conference, 2004; 1258-1263. (18) Fan, W. H.; Yang, S.; Cai, H.; Hu, W. L. MIMO Networked Control Systems Based on Fault-Tolerant Control. Proc. of the 5th World Conference on the Intelligent Control and Automation, 2004; 13661369. (19) Mao, Z.; Jiang, B.; Shi, P. H-Infinity Fault Detection Filter Design for Networked Control Systems Modelled by Discrete Markovian Jump Systems. IEE Proc. Control Theory and Application 2007, 1, 1336-1343. (20) Patton, R. J., Kambhampati, C., Casavola, A. A Generic Strategy for Fault-Tolerance in Control Systems Distributed over a Network. Eur. J. Control 2007, 13, 280-296. (21) Trinh, H.; Aldeen, M. D-Stability Analysis of Discrete-Delay Perturbed Systems. Int. J. Cont. 1995, 61, 493-505.
ReceiVed for reView June 5, 2007 ReVised manuscript receiVed November 5, 2007 Accepted January 28, 2008 IE070779M
