- Email: [email protected]
Fault-tolerant Control Design for a Class of Linear Systems Under the Constrained Communications*
8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS) August 29-31, 2012. Mexico City, Mexico
Fault-tolerant Control Design for a Class of Linear Systems Under the Constrained Communications ? Zehui Mao ∗,∗∗ , Bin Jiang ∗ , Maoyin Chen ∗∗ , Huajun Gong ∗ , Ningyun Lu ∗ ∗ College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China. Emails: [email protected] (Z. Mao), [email protected] (B. Jiang), [email protected] (H. Gong), [email protected] (N. Lu). ∗∗ Department of Automation, Tsinghua University, Beijing, 100084, P. R. China. Email: [email protected].
Abstract: In this paper, the fault-tolerant controller design and the communication data rate are integrally investigated for a class of linear systems with the constrained communication channels. To guarantee the feasibility of proposed fault-tolerant control method, the necessary conditions for the faulty plant and the minimum data rate of the communication are analyzed. Taken into account the communication errors, the adaptive techniques based fault-tolerant controller is constructed to make the faulty system stable. Finally, an example of three-tank system is used to show the efficiency of the proposed method. Keywords: Fault-tolerant control, data rate, constrained communication, discrete-time systems. 1. INTRODUCTION With the developments of the communication, computer and network, the modern control systems adopt the communication channels instead of the traditional wires to connect the elements and modules in the systems. In distributed control systems, modules are loosely connected through some networks. For networked control systems, the networks are introduced into the systems to transfer the information between the sensors, controller and actuators. The used networks can be Internet, LAN, CAN and so on, which are considered as the communication channels. Faults exist in these systems are similar to other control systems. These faults may result in unsatisfactory performance or instability. So, it is clear that the faults must be promptly detected and appropriately tolerated. Fault detection and isolation (FDI) and fault-tolerant control (FTC) problems are generated, which have attracted persistent attention over the past decades Blanke et al. (2003); Tang et al. (2007); Jiang et al. (2006). In the communication based control systems, the introduced communication channels induce some new, such as transferred delays, data drop, quantity and time synchronization. These problems have been attracted many attentions and a lot of results can be obtained, such as Fang et al. (2007); He et al. (2008); Mao et al. (2009). In He et al. (2009), the robust fault detection problem ? Partially supported by the National Natural Science Foundation of China (61104020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20113218120010, 20113218110011), the Fundamental Research Funds for the Central Universities (NZ2012009) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education.
978-3-902823-09-0/12/$20.00 © 2012 IFAC
904
for a class of networked control systems with the random measurement delays and the stochastic data missing is investigated. In Tian et al. (2010), the reliable control design is considered for networked control systems against probabilistic actuator fault with different failure rates, measurements distortion, random network-induced delay and packet dropout. Among the obtained results, we can find that the FDI or FTC schemes are built on the given net environments, in which the characterizes of the network are known or satisfied some statistics probability. For example, the transfer delays in literature He et al. (2009) are modelled as a Markov chain. The relationships between these network conditions and FD/FTC scheme are not discussed, which make the existences of fault diagnosis and fault-tolerant controller unsure. For a communication based control system, the FTC is set at the terminal side of the communication channel, which leads to the limited information for FTC. So, it is clear that if the received information do not contain the necessary information, we could not construct any FTC scheme. Our interest is in presenting the necessary conditions for the faulty plant and communication. Further, we give the fault-tolerant controller design method. Recently, some researchers dedicate to study the communication conditions and the controller design. For examples, in Tatikonda et al. (2004), the channel encoder and channel decoder is designed along with the controller to achieve different control objectives. The upper and lower bounds on the channel rate required to achieve these different control objectives are provided. You et al. (2011) gives the results on data rate and packet dropout rate for stabilizability of linear systems over a lossy digital channel. These results show the relationship of the communication and controller design. 10.3182/20120829-3-MX-2028.00154
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
To the best of the authors’ knowledge to date, however, the problem of the relationships between the communication data and the fault-tolerant controller design has not been fully investigated yet. This motivates us to study this interesting and challenging problem. In this paper, we will consider the fault-tolerant controller design for a discrete-time linear systems with limited communication data rate. Based on the information theory Cover et al. (2006), the minimum data rate with related communication errors are obtained. Further, using the adaptive techniques, the fault-tolerant controller is designed to recover the faulty system. The rest of the paper is organized as follows: Section 2 describes the system model and presents the problem. The necessary conditions about the faulty plant and the minimum data rate are presented in Section 3. FTC scheme is proposed in Sections 4. Section 5 includes a simulation example, followed by some concluding remarks in Section 6. 2. SYSTEM DESCRIPTION AND PRELIMINARY yk
Plant x(k+1)=Ax(k)+Bu(k)
fault tolerant controller
Fig. 1.
y- k
Encoder
Decoder
C h a n n e l
The block of the networked control system with FTC
Let the discrete-time linear system without fault be represented by: x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)
(1) (2)
nx
where x ∈ R denotes the state vector with the initial value x0 , u ∈ Rnu is the input vector, y ∈ Rny is the measurable output vector. A, B and C are real constant matrices of appropriate dimensions. (A, B) is a controllable pair . It is assumed that system (1) is stable. The initial state x0 is a random vector on the open set Ω ⊆ Rnx with the probability density p(x0 ) and finite differential entropy h(x0 ). Suppose that the measurable output and the fault tolerant scheme are connected by a digital communication channel with limited data rate. Fig. 1 shows the configuration of the communication based fault-control system. Consider the system elements are subject to faults that could alter the system structure. Then, the system statespace representation with the faults can be defined by ˆ + Bu( ˆ ˆ k) ˆ k) x(kˆ + 1) = Ax( ˆ = Cx(k) ˆ y(k)
(3)
x(kˆ = 0) = x(k = τ ) ∈ R(τ )
(5)
(4)
where kˆ = k − τ and R(τ ) is the time set of fault occurrence. The above faulty system form is brought from 905
Dominguez-Garcia et al. (2009), which can be represent a lot of faulty systems with the component and actuator faults. According to Chen et al. (1999), we have that the component fault is expressed as a change in the system parameter A. For the gain lost fault of actuator, we always ˆ as BF , where F is a diagonal matrix with express B elements belonging to (0, 1]. Thus, the system (3)-(5) can be used to describe some faulty systems. For example, the RL circuit with a resistor failure can be represent in the form (3)-(5), see Dominguez-Garcia et al. (2009). For the healthy system (1)-(2), we assume that A is stable matrix, which implies that kλi (A)k < 1, for i = 1, . . . , n. Here, we focus on the conditions for fault-tolerant controller design. So it is assumed that the plant fault has been detected and diagnosed. Until now, there have been many literatures and publications about the fault detection and identification methods, such as Zhang et al. (2007), Zhong et al. (2007), Jiang et al. (2005), Mao et al. (2010) and so on. As the plant faults have been diagnosed, the parameters of the faulty plant are obtained. ˆ are known. For Thus, the parameter matrices Aˆ and B active FTC, the fault information, such as occurrence time, amplitude, position, should be obtained to reconstruct the controller. For passive FTC, perhaps the exact parameters could not be known, but the bounds about the parameters are available. In this paper, we want to design a faulttolerant controller which can make the faulty system stable even if the fault occurs. Thus, the available and full fault information is assumed here. Further, the faulty system is defined as a system that is not stable, i.e., kλi (A)k > 1, ∀1 ≤ i ≤ n. For the digital communication channel, it is assumed to be noiseless. The channel input and output alphabets are the same and denoted by S. The alphabet size is |S| = 2R , where R is called the data rate of the channel. At time k ∈ N , the symbol transmitted can depend on all past and present measurements and past symbols, i.e. s(k) = g(y(k), s(k − 1), k), where g(·) is the encode function. Then, on the other hand, output can depend on all past and present channel symbols y¯(k) = f (s(k), s(k − 1), k), where f (·) is the decode function. For fault-tolerant control scheme, only the obtained information y¯(k) can be used to reconstruct the controller. Now, we formulate the design problem as follows: (1) find the minimum data rate R, then the rate of the FTC channel must be larger than R; (2) under the data rate, design the fault-tolerant controller. 3. CONDITION ANALYSIS FOR FAULT-TOLERANT CONTROLLER DESIGN In this section, we will study the minimum data rate of the communication channel used in the FTC scheme. It is well-known that the transferred information determines the data rate. And the fault tolerant controller design method decides the transferred information. It is obvious that the measured vectors are used to construct the FTC scheme. Thus, for faulty system (3)-(5), the output ˆ must contain the fault information, which can be y(k) ˆ C) . But the easily achieved by the observable pair (A, full state observable condition could bring the redundant
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
information. To observe how the fault influences the state directly, it is convenient to transform Aˆ into the Jordan canonical form, which can refer the literature Horn et al. (1985) for details. ˆ not Let λ1 , . . . , λv , v ≤ n be the distinct eigenvalues of A, counting conjugates, and let the algebraic multiplicity of each λi be κi . The Jordan canonical form J of Aˆ has the block diagonal structure J , diag(J1 , . . . , Jv ) ∈ Rnx ×nx , where Ji ∈ Rni ×ni has either exactly one eigenvalue λi or a pair of complex conjugate eigenvalues λi , λ∗i , each ½ κi if λi ∈ R with multiplicity κi , and where ni , . 2κi otherwise Then, there exists a similarity matrix T ∈ Rnx ×nx such ˆ Defining the transformed state x that T −1 JT = A. ¯(k) , nx T x(k) ∈ R , the system equations (3)-(4) can be written as ˆ + T Bu( ˆ ˆ k) x ¯(kˆ + 1) = J x ¯(k) ˆ = CT −1 x ˆ y(k) ¯(k)
(6) (7)
By partitioning the transformed state vector, the dynamical equation above can be rewritten more explicitly as ˆ + (T Bu( ˆ i ˆ k)) x ¯i (kˆ + 1) = Ji x ¯i (k) −1 ˆ = [(CT )1 · · · (CT −1 )i · · · (CT −1 )v ] y(k) ˆ ···x ˆ ···x ˆ T ×[¯ x1 (k) ¯i (k) ¯v (k)]
(8) (9)
ˆ are the states of corresponding subsyswhere x ¯i (k) ˆ i denotes the control vector that feeds ˆ tems, (T Bu(k)) ˆ , [(CT −1 )1 · · · into the ith subsystem. CT −1 x ¯(k) −1 −1 −1 (CT )i · · · (CT )v ], (CT )i are the corresponding matrix blocks of output matrix. Without loss of generality, we assume that Ji is the only unstable Jordan block. ˆ the state x ˆ must be Then, from the output y(k), ¯i (k) observable. For a Jordan block, there exist four forms, which are determined by the eigenvalues λi . If λi is a signal eigenvalue, the observable condition should be described by that (Ji , (CT −1 )i ) is observable. If λi is the real multieigenvalue with κi multiplicity, then the subsystem (6) can be rewritten as 1 ˆ x ¯i (k) λi 1 x ¯1i (kˆ + 1) .. .. . . λi . . . = . .. .. . . . 1 . κi ˆ κi ˆ λ i κi ×κi x ¯i (k) x ¯i (k + 1) ˆ ˆ +(T Bu(k))i (10)
ˆ can be Here, we can observer that if the state x ¯κi i (k) obtained from output y¯(k), the fault information can be contained in the output. Thus, the condition is become as that the elements in the last row of matrix (CT −1 )i are not all zero. For a pair of complex conjugate eigenvalues λi , αi + βi j ·and λ∗i ,¸ αi − βi j, the matrix Ji can be αi −βi represented as . From now on, this condition is βi αi equivalent to that of the signal eigenvalue case. So the FTC condition is also that (Ji , (CT −1 )i ) is observable. For the last case that the multiplicity of the complex conjugate 906
eigenvalues λi and λ∗i is κi , the Jordan block Ji can be expressed as: αi −βi 1 0 βi αi 0 1 .. αi −βi . .. . βi αi .. . 1 0 .. . 0 1 αi −βi βi αi κi ×κi According to the above form, the condition for the faulttolerant controller design should be that the elements in the last row of matrix (CT −1 )i are not all zero. Conclusively, the fault-tolerant controller design condition for a plant is that the faulty subsystems of the plant ˆ C) can satisfy can be observed. As the faulty system (A, the previous condition, then the fault information can be transferred to the fault-tolerant control scheme. Next, we will analysis how many data rate the channel needs to transfer these information. It is well-known that there must exist the transferred errors called distortion (see Tatikonda et al. (2004)). Then, we define the distortion measures d(y, y¯) borrowed from Tatikonda et al. (2004) as: d(y, y¯) = ky − y¯k22
(11)
Then, we should find the data rate to make E[d(y, y¯)] ≤ D, where D is a given constant. Theorem 1. : Assume the initial value x(0) has density p(x0 ) with finite differential entropy h(x0 ). Consider the ˆ > 1 and the healthy plant faulty plant (3)-(4) with kλi (A)k (1)-(2) with kλi (A)k < 1, for i = 1, . . . , n. Then R, which represents the rate under the distortion measure d(y, y¯), must make the following condition hold ˆ R ≥ log|A|
(12)
Proof. In this proof process, some defines and methods are based on the information theoretics, which can referred the literatures Cover et al. (2006) and Tatikonda et al. (2004). According to the healthy system (1)-(2), it has that
τ
x(τ ) = A x(0) +
τ −1 X
Aτ −1−l Bu(l)
(13)
ˆ ˆ Aˆk−1−j Bu(j)
(14)
l=0
Further, we can obtain that
ˆ = Aˆkˆ x(τ ) + x(k)
ˆ k−1 X j=0
According to Eq.(13) and (14), it has that
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
ˆ y¯(k)) ˆ I(y(k); ˆ − h(y(k)|¯ ˆ y (k)) ˆ = h(y(k)) ˆ − h(y(k) ˆ − y¯(k)|¯ ˆ y (k)) ˆ = h(y(k)) a)
ˆ − h(y(k) ˆ − y¯(k)) ˆ ≥ h(y(k)) b)
ˆ − h(N (0, D)) ≥ h(y(k)) h i c) ˆ + h(y(τ )) − 1 log(2πe)D = log|C| + klog|A| 2 where a) follows because conditioning reduces entropy. Point b) follows from the fact that the normal distribution maximizes the entropy. Point c) follows because the inputs are not random variables. Further, for any given channel, define C cap , lim inf k→∞ 1/kCkcap (see Tatikonda et al. (2004)). We can present our necessary conditions data rate ˆ y¯(k)). ˆ This completes the proof. R ≥ k1 I(y(k); 2 In Theorem 1, it is assume that all eigenvalues of Aˆ are unstable. If we introduce Aˆ with some stable eigenvalues, ˆ we get R ≥ Σλ(A) ˆ max{0, log|λ(A)|}, which is as that in Tatikonda et al. (2004). 4. FAULT-TOLERANT CONTROLLER DESIGN In this section, we force on the design of fault-tolerant controller to stabilize the faulty plant through the communication. As the health system is stable, the input could be zero. Then, it is not need to transfer the information to the controller by the communication. When a fault occurs, the system is unstable. The controller must be construed to recover the faulty system. It is assumed that the communication channel satisfy the minimum data rate, and there is a fault detection and diagnosis scheme that can provide the effective fault information. Considering that the parameters of the faulty systems are known and (A, B) is controllable, we will design the fault-tolerant controller to stabilize the faulty system. Since only the information transferred though the communication channel can be obtained, the fault-tolerant conˆ to construct, which is represented troller should use y¯(k) as: ˆ = Kf y¯(k) ˆ + θ( ˆ k) ˆ uf (k)
(15)
ˆ , Kf [y(k) ˆ − where θˆ ∈ Rnu ×1 is the estimation of θ(k) ˆ y¯(k)], which is updated from the adaptive laws ˆ kˆ + 1) = αθ( ˆ k) ˆ + Γ¯ ˆ θ( y (k)
¯T 0 + C T P1T C T Γ 0 ˆ T P αI 0 0 B −P 0 0 0 <0 ∗ −¯ ε2 I 0 0 ∗ ∗ −P 0 ∗ ∗ ∗ −¯ ε1 I (17) ˆ + CT P T B ˆ + CT Γ ¯ T and ε¯1 + 1 − where (1.2) = AˆT P B 1 2α < 0. Moreover, if (17) is true, the feedback gain and the adaptive weighting matrix can be chosen as Kf = ˆ −1 P −1 P¯ , Γ = α−1 Γ, ¯ where B ˆ −1 is the left inverse of B ˆ matrix B. −P (1.2) AT P ∗ −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
(16)
T
where Γ = Γ > 0 and α > 0 are the parameters designed later. Theorem 2. : Consider the faulty plant (3)-(4) with the communication channel satisfying the minimum data rate in Theorem 1. The fault-tolerant controller (15) with the adaptive law (16) can make the the faulty plant (3)-(4) stable, if there exist scalars ε¯1 , ε¯2 > 0, α, and matrices P¯ , ¯ > 0 such that the following LMI holds. P > 0 and Γ 907
ˆ = θ( ˆ k) ˆ − θ(k). ˆ According to (16), it Proof. Define eθ (k) has that eθ (kˆ + 1) ˆ + ΓCx(k) ˆ + αθ(k) ˆ − θ(kˆ + 1) = αeθ (k) ˆ − y(k)] ˆ −Γ[¯ y (k) ˆ + ΓCx(k) ˆ +Υ , αeθ (k)
(18) From (3), (15) and (16), the dynamic equation of faulty system can be derived x(kˆ + 1) ˆ + BK ˆ +B ˆ k) ˆ ˆ k) ˆ f y¯(k) ˆ θ( = Ax( ˆ + BK ˆ − BK ˆ + BK ˆ +B ˆ k) ˆ ˆ k) ˆ f Cx(k) ˆ f y(k) ˆ f y¯(k) ˆ θ( = Ax( ˆ + Be ˆ ˆ f C)x(k) ˆ θ (k) = (Aˆ + BK (19) Consider the Lyapunov function candidate as ˆ = xT (k)P ˆ x(k) ˆ + eT (k)e ˆ θ (k) ˆ V (k) θ
(20)
then ∆V (kˆ + 1) ˆ = E{V (kˆ + 1)} − V (k) ˆ Aˆ + BK ˆ ˆ f C)T P (Aˆ + BK ˆ f C)x(k) = xT (k)( T ˆ T T ˆ + 2x (k)( ˆ Aˆ + BK ˆ ˆ f C) P Be ˆ θ (k) −x (k)P x(k) ˆ B ˆ + α2 eT (k)e ˆ θ (k) ˆ ˆ T P Be ˆ θ (k) +eTθ (k) θ T ˆ T ˆ + 2αe (k)Υ ˆ +2αe (k)ΓCx(k) θ
θ
ˆ T ΓT ΓCx(k) ˆ + 2xT (k)C ˆ T ΓT Υ +xT (k)C ˆ θ (k) ˆ +ΥT Υ − eT (k)e
(21) From the well-known inequality Mahmoud (2000), 2a b ≤ 1 T T ² a a + ²b b for any scalar ² > 0 and real vectors a and b, we have, ˆ ≤ ε1 α2 eT (k)e ˆ θ (k) ˆ + 1 ΥT Υ 2αeTθ (k)Υ θ ε1 T ˆ T T T ˆ T T ˆ + 1 ΥT Υ 2x (k)C Γ Υ ≤ ε2 x (k)C Γ ΓCx(k) ε2 θ
T
ˆ = [xT (k), ˆ eT (k)] ˆ T . Then, it follows from (21) Denote ξ(k) θ ∆V (kˆ + 1) · ¸ ˆ Ξ11 Ξ12 ξ(k) ˆ +γ ≤ ξ T (k) ∗ Ξ22 ˆ ˆ +γ , ξ T (k)Πξ( k)
(22)
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
"
where ˆ f C)T P (Aˆ + BK ˆ f C) Ξ11 = (Aˆ + BK +(1 + ε2 )C T ΓT ΓC − P, ˆ f C)T P B ˆ + αC T ΓT , Ξ12 = (Aˆ + BK ˆ T P B. ˆ Ξ22 = (ε1 α2 + α2 − 1)I + B γ = ( ε11 + ε12 + 1)ΥT Υ. θ(k) is norm bounded and α and Γ ˆ − θ(kˆ + 1) − Γ[¯ ˆ − y(k)] ˆ are constants, thus Υ = αθ(k) y (k) is also bounded, which implies γ > 0 is bounded. If Π < 0, we can obtain that ˆ 2 − c2 keθ (k)k ˆ 2+γ ∆V (kˆ + 1) ≤ −c1 kx(k)k
(23)
where c1 > 0 and c2 > 0. On the other hand, from (20), one has ˆ ≤ λmax (P ) k x(k) ˆ k2 + k eθ (k) ˆ k2 V (k)
(24)
Substituting (24) into (23) yields ˆ ≤ −αV + γ ∆V (k)
(25)
min(c1 ,c2 ) max[λmax (P ),1] .
where α = The data rate proposed in Theorem 1 guarantees the boundness of the transferred ˆ is not large error. If we assume that the norm of error θ(k) ˆ than θ0 , i.e., kθ(k)k ≤ θ0 , then it can define the set
# 0.9991 0.0000 0.0009 A = 0.0000 0.9982 0.0010 , 0.0009 0.0010 0.9981 " # · ¸ 6.4932 0.0000 10 0 B = 0.0000 6.4901 , C = . 01 0 0.0029 0.0033 When there is a fault occurrence at t = 0.5s, the matrix A is changed as " # 1.5991 0.0000 0.0009 Aˆ = 0.0000 1.1982 0.0010 . 0.0009 0.0010 0.9981 This change in matrix A could be considered as the pipe jam faults. Combining with B and C, we can verity that the faulty parts are observable and controllable. For this discrete plant, the sampling time is 0.1s. From Theorem 1, we can calculate the data rate R = 2, which determines the precision of quantization. The quantization, encode and decode are used from Li et al. (2011). Thus, with Theorem 2, the parameters for FTC are given as · ¸ −0.2463 −0.0000 Kf = , −0.0000 −0.1846 · ¸ −0.0563 −0.0032 α = 0.0065, Γ = . −0.2040 −0.0086 Fig. 2 shows the time response of the faulty system under the fault-tolerant scheme. It can be seen that the faulty system can also be stabilized, after some regulating process.
S1 = ¯ ½ ¾ ¯ ˆ θ( ˆ k)) ˆ ¯ λmin (P ) k y(k) ˆ k2 + 1 k θ( ˆ k) ˆ k2 > θ2 + γ (y(k), 0 ¯ 2kCk 2 α
0.1
y1 y2
0.08
0.06
ˆ θ( ˆ k)) ˆ ∈ S1 , using (20), it has that For (y(k),
0.04
y
ˆ V (k) ˆ k2 + k eθ (k) ˆ k2 ≥ λmin (P ) k x(k) λmin (P ) ˆ k2 + k eθ (k) ˆ k2 k y(k) ≥ 2kCk λmin (P ) ˆ k2 +[ 1 k θ( ˆ k) ˆ k2 − k θ0 k2 ] ≥ k y(k) 2kCk 2 γ ≥ α From the above inequality and (25), it can be seen that ˆ <0 ∆V (k)
0.02
0
−0.02
−0.04
−0.06
0
0.5
1
1.5
2
2.5 t/s
3
3.5
4
4.5
5
Fig. 2. Time response of the faulty system under FTC 6. CONCLUSIONS
ˆ θ( ˆ k)) ˆ ∈ S1 for (y(k),
As a result, the state (19) is bounded. Using Schur complement and post- and pre-multiplying with diag{I, I, I, θ, I, I}, ¯ < 0 can be converted to the LMI presented in Theorem Σ ¯ = θΓ, ε¯1 = α2 (1 + ε1 )−1 and ε¯2 = (1 + 2, with setting Γ −1 ε2 ) . Furthermore, from the definition of ε¯1 , we have that ε¯1 + 1 − 2θ < 0. This completes the proof. 2 5. AN ILLUSTRATIVE EXAMPLE In this section, the three-tank model is used to verity the effectiveness of the proposed method, which is borrowed from Zhou et al. (2011). The states are the hights of three tanks, while the outputs are the water currents of two pumps. The state-space model is 908
This paper studies the fault-tolerant controller design for a class of linear systems controlled under the constrained communication channels, which provides the necessary conditions that the communication channel and faulty plant must satisfy. Further, using the adaptive techniques, fault-tolerant controller is proposed to recover the faulty system, which is linear and discrete-time. Future research will address more general systems, in which the plants could be nonlinear and the communication channels could have noises. REFERENCES M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and fault-tolerant control. Springer Verlag, Berlin, Heidelberg, 2003.
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
J. Chen, and R. J. Patton. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Boston, 1999. T. Cover and J. Thomas. Elements of information theory. Wiley- Interscience, 2006. A. D. Dominguez-Garcia, J. G. Kassakian, and J. E. Schindall. A generalized fault coverage model for linear timeinvariant systems, IEEE Transactions on Reliability, 58: 553–567, 2009. H. J. Fang, H. Ye, and M. Y. Zhong. Fault diagnosis of networked control dystems. Annual Reviews in Control, 31:55–68, 2007. R. A. Horn, and C. R. Johnson. Matrix anakysis. Cambridge: Cambridge University Press, 1985. X. He, Z. Wang, and D. H. Zhou, Network-based fault detection for discrete-time state-delay systems: a new measurement model. Int. J. of Adaptive Control and Signal Processing, 22:510–528, 2008. X. He, Z. D. Wang, and D. H. Zhou. Robust fault detection for networked systems with communication delay and data missing. Automatica, 45:2634–2639, 2009. B. Jiang, and F. N. Chowdhuery. Fault estimation and accommodation for linear MIMO discrete-time systems. IEEE Trans. on Control System technology, 13:, 493– 499, 2005. B. Jiang, M. Staroswiecki, and V. Cocquempot. Fault accommodation for a class of nonlinear dynamic systems. IEEE Trans. on Automatic Control, 51:1578–1583, 2006. T. Li, M. Y. Fu, L. H. Xie, and J. F. Zhang. Distributed consensus with limited communication data rate, IEEE Trans. on Automatic Control, 56:279–292, 2011. M. S. Mahmoud. Robust control and filtering for time delay systems. New York: Marcel Dekker, Inc., 2000. Z. H. Mao, B. Jiang, and P. Shi. Protocol and fault detection design for nonlinear networked control systems. IEEE Transactions on Circuits and Systems II, 56:255– 259, 2009. Z. H. Mao, B. Jiang, and P. Shi. Fault-tolerant control for a class of nonlinear sampled-data systems via a Euler approximate observer, Automatica, 46:1852–1859, 2010. G. Nair, and R. Evans. Exponential stabilisability of finitedimensional linear systems with limited data rates. Automatica, 39:585–593, 2003. X. D. Tang, G. Tao, and S. M. Joshi. Adaptive actuator failure compensation for nonlinear MIMO systems with an aircraft control application. Automatica, 43:1869– 1883, 2007. S. Tatikonda, and S. Mitter. Control under communication constraints. IEEE Trans. on Automatic Control, 49: 1056–1068, 2004. E. G. Tian, D. Yue, and C. Peng. Reliable control for networked control systems with probabilistic actuator fault and random delays, Journal of the Franklin Institute, 347:1907–1926, 2010. K. Y. You, and L. H. Xie. Minimum data rate for mean square stabilizability of linear systems with Markovian packet losses. IEEE Trans. on Automatic Control, 56: 772–785, 2011. P. Zhang, S. X. Ding, G. Z. Wang and D. H. Zhou, An FDI approach for sampled-data systems. Proceedings of the American Control Conference, 2702–2707, 2001. M. Zhong, H. Ye, S. X. Ding, and G. Wang. Observerbased fast rate fault detection for a class of multirate
909
sampled-data systems. IEEE Trans. on Automatic Control, 52:520–525, 2007. D. H. Zhou, X. He, Z. Wang, G. P. Liu, and Y. D. Ji. Leakage fault diagnosis for an internetbased three-tank system: An experimental study, IEEE Trans. on Control Systems Technology, DOI: 10.1109/TCST.2011.2154383.
Fault-tolerant Control Design for a Class of Linear Systems Under the Constrained Communications ? Zehui Mao ∗,∗∗ , Bin Jiang ∗ , Maoyin Chen ∗∗ , Huajun Gong ∗ , Ningyun Lu ∗ ∗ College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China. Emails: [email protected] (Z. Mao), [email protected] (B. Jiang), [email protected] (H. Gong), [email protected] (N. Lu). ∗∗ Department of Automation, Tsinghua University, Beijing, 100084, P. R. China. Email: [email protected].
Abstract: In this paper, the fault-tolerant controller design and the communication data rate are integrally investigated for a class of linear systems with the constrained communication channels. To guarantee the feasibility of proposed fault-tolerant control method, the necessary conditions for the faulty plant and the minimum data rate of the communication are analyzed. Taken into account the communication errors, the adaptive techniques based fault-tolerant controller is constructed to make the faulty system stable. Finally, an example of three-tank system is used to show the efficiency of the proposed method. Keywords: Fault-tolerant control, data rate, constrained communication, discrete-time systems. 1. INTRODUCTION With the developments of the communication, computer and network, the modern control systems adopt the communication channels instead of the traditional wires to connect the elements and modules in the systems. In distributed control systems, modules are loosely connected through some networks. For networked control systems, the networks are introduced into the systems to transfer the information between the sensors, controller and actuators. The used networks can be Internet, LAN, CAN and so on, which are considered as the communication channels. Faults exist in these systems are similar to other control systems. These faults may result in unsatisfactory performance or instability. So, it is clear that the faults must be promptly detected and appropriately tolerated. Fault detection and isolation (FDI) and fault-tolerant control (FTC) problems are generated, which have attracted persistent attention over the past decades Blanke et al. (2003); Tang et al. (2007); Jiang et al. (2006). In the communication based control systems, the introduced communication channels induce some new, such as transferred delays, data drop, quantity and time synchronization. These problems have been attracted many attentions and a lot of results can be obtained, such as Fang et al. (2007); He et al. (2008); Mao et al. (2009). In He et al. (2009), the robust fault detection problem ? Partially supported by the National Natural Science Foundation of China (61104020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20113218120010, 20113218110011), the Fundamental Research Funds for the Central Universities (NZ2012009) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education.
978-3-902823-09-0/12/$20.00 © 2012 IFAC
904
for a class of networked control systems with the random measurement delays and the stochastic data missing is investigated. In Tian et al. (2010), the reliable control design is considered for networked control systems against probabilistic actuator fault with different failure rates, measurements distortion, random network-induced delay and packet dropout. Among the obtained results, we can find that the FDI or FTC schemes are built on the given net environments, in which the characterizes of the network are known or satisfied some statistics probability. For example, the transfer delays in literature He et al. (2009) are modelled as a Markov chain. The relationships between these network conditions and FD/FTC scheme are not discussed, which make the existences of fault diagnosis and fault-tolerant controller unsure. For a communication based control system, the FTC is set at the terminal side of the communication channel, which leads to the limited information for FTC. So, it is clear that if the received information do not contain the necessary information, we could not construct any FTC scheme. Our interest is in presenting the necessary conditions for the faulty plant and communication. Further, we give the fault-tolerant controller design method. Recently, some researchers dedicate to study the communication conditions and the controller design. For examples, in Tatikonda et al. (2004), the channel encoder and channel decoder is designed along with the controller to achieve different control objectives. The upper and lower bounds on the channel rate required to achieve these different control objectives are provided. You et al. (2011) gives the results on data rate and packet dropout rate for stabilizability of linear systems over a lossy digital channel. These results show the relationship of the communication and controller design. 10.3182/20120829-3-MX-2028.00154
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
To the best of the authors’ knowledge to date, however, the problem of the relationships between the communication data and the fault-tolerant controller design has not been fully investigated yet. This motivates us to study this interesting and challenging problem. In this paper, we will consider the fault-tolerant controller design for a discrete-time linear systems with limited communication data rate. Based on the information theory Cover et al. (2006), the minimum data rate with related communication errors are obtained. Further, using the adaptive techniques, the fault-tolerant controller is designed to recover the faulty system. The rest of the paper is organized as follows: Section 2 describes the system model and presents the problem. The necessary conditions about the faulty plant and the minimum data rate are presented in Section 3. FTC scheme is proposed in Sections 4. Section 5 includes a simulation example, followed by some concluding remarks in Section 6. 2. SYSTEM DESCRIPTION AND PRELIMINARY yk
Plant x(k+1)=Ax(k)+Bu(k)
fault tolerant controller
Fig. 1.
y- k
Encoder
Decoder
C h a n n e l
The block of the networked control system with FTC
Let the discrete-time linear system without fault be represented by: x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)
(1) (2)
nx
where x ∈ R denotes the state vector with the initial value x0 , u ∈ Rnu is the input vector, y ∈ Rny is the measurable output vector. A, B and C are real constant matrices of appropriate dimensions. (A, B) is a controllable pair . It is assumed that system (1) is stable. The initial state x0 is a random vector on the open set Ω ⊆ Rnx with the probability density p(x0 ) and finite differential entropy h(x0 ). Suppose that the measurable output and the fault tolerant scheme are connected by a digital communication channel with limited data rate. Fig. 1 shows the configuration of the communication based fault-control system. Consider the system elements are subject to faults that could alter the system structure. Then, the system statespace representation with the faults can be defined by ˆ + Bu( ˆ ˆ k) ˆ k) x(kˆ + 1) = Ax( ˆ = Cx(k) ˆ y(k)
(3)
x(kˆ = 0) = x(k = τ ) ∈ R(τ )
(5)
(4)
where kˆ = k − τ and R(τ ) is the time set of fault occurrence. The above faulty system form is brought from 905
Dominguez-Garcia et al. (2009), which can be represent a lot of faulty systems with the component and actuator faults. According to Chen et al. (1999), we have that the component fault is expressed as a change in the system parameter A. For the gain lost fault of actuator, we always ˆ as BF , where F is a diagonal matrix with express B elements belonging to (0, 1]. Thus, the system (3)-(5) can be used to describe some faulty systems. For example, the RL circuit with a resistor failure can be represent in the form (3)-(5), see Dominguez-Garcia et al. (2009). For the healthy system (1)-(2), we assume that A is stable matrix, which implies that kλi (A)k < 1, for i = 1, . . . , n. Here, we focus on the conditions for fault-tolerant controller design. So it is assumed that the plant fault has been detected and diagnosed. Until now, there have been many literatures and publications about the fault detection and identification methods, such as Zhang et al. (2007), Zhong et al. (2007), Jiang et al. (2005), Mao et al. (2010) and so on. As the plant faults have been diagnosed, the parameters of the faulty plant are obtained. ˆ are known. For Thus, the parameter matrices Aˆ and B active FTC, the fault information, such as occurrence time, amplitude, position, should be obtained to reconstruct the controller. For passive FTC, perhaps the exact parameters could not be known, but the bounds about the parameters are available. In this paper, we want to design a faulttolerant controller which can make the faulty system stable even if the fault occurs. Thus, the available and full fault information is assumed here. Further, the faulty system is defined as a system that is not stable, i.e., kλi (A)k > 1, ∀1 ≤ i ≤ n. For the digital communication channel, it is assumed to be noiseless. The channel input and output alphabets are the same and denoted by S. The alphabet size is |S| = 2R , where R is called the data rate of the channel. At time k ∈ N , the symbol transmitted can depend on all past and present measurements and past symbols, i.e. s(k) = g(y(k), s(k − 1), k), where g(·) is the encode function. Then, on the other hand, output can depend on all past and present channel symbols y¯(k) = f (s(k), s(k − 1), k), where f (·) is the decode function. For fault-tolerant control scheme, only the obtained information y¯(k) can be used to reconstruct the controller. Now, we formulate the design problem as follows: (1) find the minimum data rate R, then the rate of the FTC channel must be larger than R; (2) under the data rate, design the fault-tolerant controller. 3. CONDITION ANALYSIS FOR FAULT-TOLERANT CONTROLLER DESIGN In this section, we will study the minimum data rate of the communication channel used in the FTC scheme. It is well-known that the transferred information determines the data rate. And the fault tolerant controller design method decides the transferred information. It is obvious that the measured vectors are used to construct the FTC scheme. Thus, for faulty system (3)-(5), the output ˆ must contain the fault information, which can be y(k) ˆ C) . But the easily achieved by the observable pair (A, full state observable condition could bring the redundant
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
information. To observe how the fault influences the state directly, it is convenient to transform Aˆ into the Jordan canonical form, which can refer the literature Horn et al. (1985) for details. ˆ not Let λ1 , . . . , λv , v ≤ n be the distinct eigenvalues of A, counting conjugates, and let the algebraic multiplicity of each λi be κi . The Jordan canonical form J of Aˆ has the block diagonal structure J , diag(J1 , . . . , Jv ) ∈ Rnx ×nx , where Ji ∈ Rni ×ni has either exactly one eigenvalue λi or a pair of complex conjugate eigenvalues λi , λ∗i , each ½ κi if λi ∈ R with multiplicity κi , and where ni , . 2κi otherwise Then, there exists a similarity matrix T ∈ Rnx ×nx such ˆ Defining the transformed state x that T −1 JT = A. ¯(k) , nx T x(k) ∈ R , the system equations (3)-(4) can be written as ˆ + T Bu( ˆ ˆ k) x ¯(kˆ + 1) = J x ¯(k) ˆ = CT −1 x ˆ y(k) ¯(k)
(6) (7)
By partitioning the transformed state vector, the dynamical equation above can be rewritten more explicitly as ˆ + (T Bu( ˆ i ˆ k)) x ¯i (kˆ + 1) = Ji x ¯i (k) −1 ˆ = [(CT )1 · · · (CT −1 )i · · · (CT −1 )v ] y(k) ˆ ···x ˆ ···x ˆ T ×[¯ x1 (k) ¯i (k) ¯v (k)]
(8) (9)
ˆ are the states of corresponding subsyswhere x ¯i (k) ˆ i denotes the control vector that feeds ˆ tems, (T Bu(k)) ˆ , [(CT −1 )1 · · · into the ith subsystem. CT −1 x ¯(k) −1 −1 −1 (CT )i · · · (CT )v ], (CT )i are the corresponding matrix blocks of output matrix. Without loss of generality, we assume that Ji is the only unstable Jordan block. ˆ the state x ˆ must be Then, from the output y(k), ¯i (k) observable. For a Jordan block, there exist four forms, which are determined by the eigenvalues λi . If λi is a signal eigenvalue, the observable condition should be described by that (Ji , (CT −1 )i ) is observable. If λi is the real multieigenvalue with κi multiplicity, then the subsystem (6) can be rewritten as 1 ˆ x ¯i (k) λi 1 x ¯1i (kˆ + 1) .. .. . . λi . . . = . .. .. . . . 1 . κi ˆ κi ˆ λ i κi ×κi x ¯i (k) x ¯i (k + 1) ˆ ˆ +(T Bu(k))i (10)
ˆ can be Here, we can observer that if the state x ¯κi i (k) obtained from output y¯(k), the fault information can be contained in the output. Thus, the condition is become as that the elements in the last row of matrix (CT −1 )i are not all zero. For a pair of complex conjugate eigenvalues λi , αi + βi j ·and λ∗i ,¸ αi − βi j, the matrix Ji can be αi −βi represented as . From now on, this condition is βi αi equivalent to that of the signal eigenvalue case. So the FTC condition is also that (Ji , (CT −1 )i ) is observable. For the last case that the multiplicity of the complex conjugate 906
eigenvalues λi and λ∗i is κi , the Jordan block Ji can be expressed as: αi −βi 1 0 βi αi 0 1 .. αi −βi . .. . βi αi .. . 1 0 .. . 0 1 αi −βi βi αi κi ×κi According to the above form, the condition for the faulttolerant controller design should be that the elements in the last row of matrix (CT −1 )i are not all zero. Conclusively, the fault-tolerant controller design condition for a plant is that the faulty subsystems of the plant ˆ C) can satisfy can be observed. As the faulty system (A, the previous condition, then the fault information can be transferred to the fault-tolerant control scheme. Next, we will analysis how many data rate the channel needs to transfer these information. It is well-known that there must exist the transferred errors called distortion (see Tatikonda et al. (2004)). Then, we define the distortion measures d(y, y¯) borrowed from Tatikonda et al. (2004) as: d(y, y¯) = ky − y¯k22
(11)
Then, we should find the data rate to make E[d(y, y¯)] ≤ D, where D is a given constant. Theorem 1. : Assume the initial value x(0) has density p(x0 ) with finite differential entropy h(x0 ). Consider the ˆ > 1 and the healthy plant faulty plant (3)-(4) with kλi (A)k (1)-(2) with kλi (A)k < 1, for i = 1, . . . , n. Then R, which represents the rate under the distortion measure d(y, y¯), must make the following condition hold ˆ R ≥ log|A|
(12)
Proof. In this proof process, some defines and methods are based on the information theoretics, which can referred the literatures Cover et al. (2006) and Tatikonda et al. (2004). According to the healthy system (1)-(2), it has that
τ
x(τ ) = A x(0) +
τ −1 X
Aτ −1−l Bu(l)
(13)
ˆ ˆ Aˆk−1−j Bu(j)
(14)
l=0
Further, we can obtain that
ˆ = Aˆkˆ x(τ ) + x(k)
ˆ k−1 X j=0
According to Eq.(13) and (14), it has that
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
ˆ y¯(k)) ˆ I(y(k); ˆ − h(y(k)|¯ ˆ y (k)) ˆ = h(y(k)) ˆ − h(y(k) ˆ − y¯(k)|¯ ˆ y (k)) ˆ = h(y(k)) a)
ˆ − h(y(k) ˆ − y¯(k)) ˆ ≥ h(y(k)) b)
ˆ − h(N (0, D)) ≥ h(y(k)) h i c) ˆ + h(y(τ )) − 1 log(2πe)D = log|C| + klog|A| 2 where a) follows because conditioning reduces entropy. Point b) follows from the fact that the normal distribution maximizes the entropy. Point c) follows because the inputs are not random variables. Further, for any given channel, define C cap , lim inf k→∞ 1/kCkcap (see Tatikonda et al. (2004)). We can present our necessary conditions data rate ˆ y¯(k)). ˆ This completes the proof. R ≥ k1 I(y(k); 2 In Theorem 1, it is assume that all eigenvalues of Aˆ are unstable. If we introduce Aˆ with some stable eigenvalues, ˆ we get R ≥ Σλ(A) ˆ max{0, log|λ(A)|}, which is as that in Tatikonda et al. (2004). 4. FAULT-TOLERANT CONTROLLER DESIGN In this section, we force on the design of fault-tolerant controller to stabilize the faulty plant through the communication. As the health system is stable, the input could be zero. Then, it is not need to transfer the information to the controller by the communication. When a fault occurs, the system is unstable. The controller must be construed to recover the faulty system. It is assumed that the communication channel satisfy the minimum data rate, and there is a fault detection and diagnosis scheme that can provide the effective fault information. Considering that the parameters of the faulty systems are known and (A, B) is controllable, we will design the fault-tolerant controller to stabilize the faulty system. Since only the information transferred though the communication channel can be obtained, the fault-tolerant conˆ to construct, which is represented troller should use y¯(k) as: ˆ = Kf y¯(k) ˆ + θ( ˆ k) ˆ uf (k)
(15)
ˆ , Kf [y(k) ˆ − where θˆ ∈ Rnu ×1 is the estimation of θ(k) ˆ y¯(k)], which is updated from the adaptive laws ˆ kˆ + 1) = αθ( ˆ k) ˆ + Γ¯ ˆ θ( y (k)
¯T 0 + C T P1T C T Γ 0 ˆ T P αI 0 0 B −P 0 0 0 <0 ∗ −¯ ε2 I 0 0 ∗ ∗ −P 0 ∗ ∗ ∗ −¯ ε1 I (17) ˆ + CT P T B ˆ + CT Γ ¯ T and ε¯1 + 1 − where (1.2) = AˆT P B 1 2α < 0. Moreover, if (17) is true, the feedback gain and the adaptive weighting matrix can be chosen as Kf = ˆ −1 P −1 P¯ , Γ = α−1 Γ, ¯ where B ˆ −1 is the left inverse of B ˆ matrix B. −P (1.2) AT P ∗ −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
(16)
T
where Γ = Γ > 0 and α > 0 are the parameters designed later. Theorem 2. : Consider the faulty plant (3)-(4) with the communication channel satisfying the minimum data rate in Theorem 1. The fault-tolerant controller (15) with the adaptive law (16) can make the the faulty plant (3)-(4) stable, if there exist scalars ε¯1 , ε¯2 > 0, α, and matrices P¯ , ¯ > 0 such that the following LMI holds. P > 0 and Γ 907
ˆ = θ( ˆ k) ˆ − θ(k). ˆ According to (16), it Proof. Define eθ (k) has that eθ (kˆ + 1) ˆ + ΓCx(k) ˆ + αθ(k) ˆ − θ(kˆ + 1) = αeθ (k) ˆ − y(k)] ˆ −Γ[¯ y (k) ˆ + ΓCx(k) ˆ +Υ , αeθ (k)
(18) From (3), (15) and (16), the dynamic equation of faulty system can be derived x(kˆ + 1) ˆ + BK ˆ +B ˆ k) ˆ ˆ k) ˆ f y¯(k) ˆ θ( = Ax( ˆ + BK ˆ − BK ˆ + BK ˆ +B ˆ k) ˆ ˆ k) ˆ f Cx(k) ˆ f y(k) ˆ f y¯(k) ˆ θ( = Ax( ˆ + Be ˆ ˆ f C)x(k) ˆ θ (k) = (Aˆ + BK (19) Consider the Lyapunov function candidate as ˆ = xT (k)P ˆ x(k) ˆ + eT (k)e ˆ θ (k) ˆ V (k) θ
(20)
then ∆V (kˆ + 1) ˆ = E{V (kˆ + 1)} − V (k) ˆ Aˆ + BK ˆ ˆ f C)T P (Aˆ + BK ˆ f C)x(k) = xT (k)( T ˆ T T ˆ + 2x (k)( ˆ Aˆ + BK ˆ ˆ f C) P Be ˆ θ (k) −x (k)P x(k) ˆ B ˆ + α2 eT (k)e ˆ θ (k) ˆ ˆ T P Be ˆ θ (k) +eTθ (k) θ T ˆ T ˆ + 2αe (k)Υ ˆ +2αe (k)ΓCx(k) θ
θ
ˆ T ΓT ΓCx(k) ˆ + 2xT (k)C ˆ T ΓT Υ +xT (k)C ˆ θ (k) ˆ +ΥT Υ − eT (k)e
(21) From the well-known inequality Mahmoud (2000), 2a b ≤ 1 T T ² a a + ²b b for any scalar ² > 0 and real vectors a and b, we have, ˆ ≤ ε1 α2 eT (k)e ˆ θ (k) ˆ + 1 ΥT Υ 2αeTθ (k)Υ θ ε1 T ˆ T T T ˆ T T ˆ + 1 ΥT Υ 2x (k)C Γ Υ ≤ ε2 x (k)C Γ ΓCx(k) ε2 θ
T
ˆ = [xT (k), ˆ eT (k)] ˆ T . Then, it follows from (21) Denote ξ(k) θ ∆V (kˆ + 1) · ¸ ˆ Ξ11 Ξ12 ξ(k) ˆ +γ ≤ ξ T (k) ∗ Ξ22 ˆ ˆ +γ , ξ T (k)Πξ( k)
(22)
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
"
where ˆ f C)T P (Aˆ + BK ˆ f C) Ξ11 = (Aˆ + BK +(1 + ε2 )C T ΓT ΓC − P, ˆ f C)T P B ˆ + αC T ΓT , Ξ12 = (Aˆ + BK ˆ T P B. ˆ Ξ22 = (ε1 α2 + α2 − 1)I + B γ = ( ε11 + ε12 + 1)ΥT Υ. θ(k) is norm bounded and α and Γ ˆ − θ(kˆ + 1) − Γ[¯ ˆ − y(k)] ˆ are constants, thus Υ = αθ(k) y (k) is also bounded, which implies γ > 0 is bounded. If Π < 0, we can obtain that ˆ 2 − c2 keθ (k)k ˆ 2+γ ∆V (kˆ + 1) ≤ −c1 kx(k)k
(23)
where c1 > 0 and c2 > 0. On the other hand, from (20), one has ˆ ≤ λmax (P ) k x(k) ˆ k2 + k eθ (k) ˆ k2 V (k)
(24)
Substituting (24) into (23) yields ˆ ≤ −αV + γ ∆V (k)
(25)
min(c1 ,c2 ) max[λmax (P ),1] .
where α = The data rate proposed in Theorem 1 guarantees the boundness of the transferred ˆ is not large error. If we assume that the norm of error θ(k) ˆ than θ0 , i.e., kθ(k)k ≤ θ0 , then it can define the set
# 0.9991 0.0000 0.0009 A = 0.0000 0.9982 0.0010 , 0.0009 0.0010 0.9981 " # · ¸ 6.4932 0.0000 10 0 B = 0.0000 6.4901 , C = . 01 0 0.0029 0.0033 When there is a fault occurrence at t = 0.5s, the matrix A is changed as " # 1.5991 0.0000 0.0009 Aˆ = 0.0000 1.1982 0.0010 . 0.0009 0.0010 0.9981 This change in matrix A could be considered as the pipe jam faults. Combining with B and C, we can verity that the faulty parts are observable and controllable. For this discrete plant, the sampling time is 0.1s. From Theorem 1, we can calculate the data rate R = 2, which determines the precision of quantization. The quantization, encode and decode are used from Li et al. (2011). Thus, with Theorem 2, the parameters for FTC are given as · ¸ −0.2463 −0.0000 Kf = , −0.0000 −0.1846 · ¸ −0.0563 −0.0032 α = 0.0065, Γ = . −0.2040 −0.0086 Fig. 2 shows the time response of the faulty system under the fault-tolerant scheme. It can be seen that the faulty system can also be stabilized, after some regulating process.
S1 = ¯ ½ ¾ ¯ ˆ θ( ˆ k)) ˆ ¯ λmin (P ) k y(k) ˆ k2 + 1 k θ( ˆ k) ˆ k2 > θ2 + γ (y(k), 0 ¯ 2kCk 2 α
0.1
y1 y2
0.08
0.06
ˆ θ( ˆ k)) ˆ ∈ S1 , using (20), it has that For (y(k),
0.04
y
ˆ V (k) ˆ k2 + k eθ (k) ˆ k2 ≥ λmin (P ) k x(k) λmin (P ) ˆ k2 + k eθ (k) ˆ k2 k y(k) ≥ 2kCk λmin (P ) ˆ k2 +[ 1 k θ( ˆ k) ˆ k2 − k θ0 k2 ] ≥ k y(k) 2kCk 2 γ ≥ α From the above inequality and (25), it can be seen that ˆ <0 ∆V (k)
0.02
0
−0.02
−0.04
−0.06
0
0.5
1
1.5
2
2.5 t/s
3
3.5
4
4.5
5
Fig. 2. Time response of the faulty system under FTC 6. CONCLUSIONS
ˆ θ( ˆ k)) ˆ ∈ S1 for (y(k),
As a result, the state (19) is bounded. Using Schur complement and post- and pre-multiplying with diag{I, I, I, θ, I, I}, ¯ < 0 can be converted to the LMI presented in Theorem Σ ¯ = θΓ, ε¯1 = α2 (1 + ε1 )−1 and ε¯2 = (1 + 2, with setting Γ −1 ε2 ) . Furthermore, from the definition of ε¯1 , we have that ε¯1 + 1 − 2θ < 0. This completes the proof. 2 5. AN ILLUSTRATIVE EXAMPLE In this section, the three-tank model is used to verity the effectiveness of the proposed method, which is borrowed from Zhou et al. (2011). The states are the hights of three tanks, while the outputs are the water currents of two pumps. The state-space model is 908
This paper studies the fault-tolerant controller design for a class of linear systems controlled under the constrained communication channels, which provides the necessary conditions that the communication channel and faulty plant must satisfy. Further, using the adaptive techniques, fault-tolerant controller is proposed to recover the faulty system, which is linear and discrete-time. Future research will address more general systems, in which the plants could be nonlinear and the communication channels could have noises. REFERENCES M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and fault-tolerant control. Springer Verlag, Berlin, Heidelberg, 2003.
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
J. Chen, and R. J. Patton. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Boston, 1999. T. Cover and J. Thomas. Elements of information theory. Wiley- Interscience, 2006. A. D. Dominguez-Garcia, J. G. Kassakian, and J. E. Schindall. A generalized fault coverage model for linear timeinvariant systems, IEEE Transactions on Reliability, 58: 553–567, 2009. H. J. Fang, H. Ye, and M. Y. Zhong. Fault diagnosis of networked control dystems. Annual Reviews in Control, 31:55–68, 2007. R. A. Horn, and C. R. Johnson. Matrix anakysis. Cambridge: Cambridge University Press, 1985. X. He, Z. Wang, and D. H. Zhou, Network-based fault detection for discrete-time state-delay systems: a new measurement model. Int. J. of Adaptive Control and Signal Processing, 22:510–528, 2008. X. He, Z. D. Wang, and D. H. Zhou. Robust fault detection for networked systems with communication delay and data missing. Automatica, 45:2634–2639, 2009. B. Jiang, and F. N. Chowdhuery. Fault estimation and accommodation for linear MIMO discrete-time systems. IEEE Trans. on Control System technology, 13:, 493– 499, 2005. B. Jiang, M. Staroswiecki, and V. Cocquempot. Fault accommodation for a class of nonlinear dynamic systems. IEEE Trans. on Automatic Control, 51:1578–1583, 2006. T. Li, M. Y. Fu, L. H. Xie, and J. F. Zhang. Distributed consensus with limited communication data rate, IEEE Trans. on Automatic Control, 56:279–292, 2011. M. S. Mahmoud. Robust control and filtering for time delay systems. New York: Marcel Dekker, Inc., 2000. Z. H. Mao, B. Jiang, and P. Shi. Protocol and fault detection design for nonlinear networked control systems. IEEE Transactions on Circuits and Systems II, 56:255– 259, 2009. Z. H. Mao, B. Jiang, and P. Shi. Fault-tolerant control for a class of nonlinear sampled-data systems via a Euler approximate observer, Automatica, 46:1852–1859, 2010. G. Nair, and R. Evans. Exponential stabilisability of finitedimensional linear systems with limited data rates. Automatica, 39:585–593, 2003. X. D. Tang, G. Tao, and S. M. Joshi. Adaptive actuator failure compensation for nonlinear MIMO systems with an aircraft control application. Automatica, 43:1869– 1883, 2007. S. Tatikonda, and S. Mitter. Control under communication constraints. IEEE Trans. on Automatic Control, 49: 1056–1068, 2004. E. G. Tian, D. Yue, and C. Peng. Reliable control for networked control systems with probabilistic actuator fault and random delays, Journal of the Franklin Institute, 347:1907–1926, 2010. K. Y. You, and L. H. Xie. Minimum data rate for mean square stabilizability of linear systems with Markovian packet losses. IEEE Trans. on Automatic Control, 56: 772–785, 2011. P. Zhang, S. X. Ding, G. Z. Wang and D. H. Zhou, An FDI approach for sampled-data systems. Proceedings of the American Control Conference, 2702–2707, 2001. M. Zhong, H. Ye, S. X. Ding, and G. Wang. Observerbased fast rate fault detection for a class of multirate
909
sampled-data systems. IEEE Trans. on Automatic Control, 52:520–525, 2007. D. H. Zhou, X. He, Z. Wang, G. P. Liu, and Y. D. Ji. Leakage fault diagnosis for an internetbased three-tank system: An experimental study, IEEE Trans. on Control Systems Technology, DOI: 10.1109/TCST.2011.2154383.