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or actuators
Author's Accepted Manuscript
Reliable H 1 control for uncertain nonlinear discrete-time Systems subject to multiple intermittent Faults in Sensors and/or Actuators Yuan Tao, Dong Shen, Youqing Wang, Yinzhong Ye
www.elsevier.com/locate/jfranklin
PII: DOI: Reference:
S0016-0032(15)00305-1 http://dx.doi.org/10.1016/j.jfranklin.2015.07.012 FI2402
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
10 January 2015 22 May 2015 13 July 2015
Cite this article as: Yuan Tao, Dong Shen, Youqing Wang, Yinzhong Ye, Reliable H 1 control for uncertain nonlinear discrete-time Systems subject to multiple intermittent Faults in Sensors and/or Actuators, Journal of the Franklin Institute, http://dx.doi.org/ 10.1016/j.jfranklin.2015.07.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Reliable
Control for Uncertain Nonlinear Discrete-Time
H∞
Systems Subject to Multiple Intermittent Faults in Sensors and/or Actuators Yuan Tao, Dong Shen, Youqing Wang*, Yinzhong Ye2 1. College of Information Science and Technology Beijing University of Chemical Technology, Beijing 100029, China 2. School of Electrical and Electronic Engineering, Shanghai Institute of Technology, Shanghai 201418, China *E-mail: [email protected] Abstract: This study focuses on passive fault-tolerant control for a class of uncertain nonlinear discrete-time systems subject to multiple intermittent faults. The considered intermittent faults are assumed to be additive ones in sensors and/or actuators. To achieve fault-tolerant control, a dynamic output-feedback controller is designed such that the closed-loop system remains stable and satisfies acceptable performance, even when there are parameter uncertainties, nonlinearities of specific type, and multiple additive intermittent sensor and/or actuator faults. The linear matrix inequality method is employed to obtain sufficient conditions for achieving fault tolerance and ensuring the prescribed H∞ performance index. Finally, the effectiveness of the proposed method is demonstrated by simulation examples. Keywords: fault-tolerant control; intermittent fault; sensor fault; actuator fault; robust H ∞ performance.
1
1. INTRODUCTION Fault-tolerant control (FTC) aims to ensure that a controlled system retains an acceptable performance even if some system components (including sensors and actuators) malfunction. With the growing scale and complexity of industrial processes, the advent of faults may endanger system security and cause serious economic loss, especially in safety-critical systems, such as chemical plants, aircrafts, spacecraft, and nuclear power plants. The engineering significance of FTC has been well recognized, and a number of practical systems have adopted the fault-tolerant strategy, thus creates ample scope for FTC studies within the system control community [1-4]. Usually, all reported FTC studies can be divided into two categories: active FTC (AFTC) [5-11] and passive FTC (PFTC) [12-18]. To achieve the purpose of AFTC, fault detection and diagnosis, which itself also draws considerable concern in control community [19, 20], is necessary to provide real-time fault information to reconfigure the controller. In contrast, the design of PFTC relies on a priori knowledge of failure information (e.g. location, amplitude). System faults are the focus of FTC design, and they are present in different forms within systems. In most cases, the term “fault” in the above-mentioned papers refers to a permanent fault (PF), which tends to remain faulty and may even worsen without intervention. However, many faults are not permanent, and a few types of such non-permanent faults are called intermittent faults (IFs). IFs exist widely in practical systems, including electronic equipment [21], spacecraft [22], mechanical devices [23], and communication systems [24], and they threaten system reliability and 2
security. The intermittent nature of a fault is reflected in the fact that it can recover without any corrective action. Therefore, as compared with the PF, the occurrence of the IF exhibits special properties such as randomness, intermittence, and repeatability [25, 26]. Owing to the existence of these special properties, traditional methods for addressing PFs cannot be applied to address IFs directly, regardless of the fault diagnosis technique or fault-tolerant scheme employed. From the viewpoint of FTC, the first hindrance is the lack of effective mathematical description. To solve the problem in the FTC framework, usually, a PF is described as an analytical model in deterministic form. However, it is difficult to depict an IF using a deterministic analytical form, because a suitable IF model must be able to switch frequently and randomly between fault occurrence and disappearance. In fact, the majority of IFs occur randomly, thus necessitating the use of stochastic variables to describe them [22, 27-29]. To the best of the authors’ knowledge, few studies on the FTC of multiple IFs have been reported. Motivated by the aforementioned state of the art, this study investigates PFTC problem for a class of uncertain nonlinear discrete-time systems subject to multiple IFs, where nonlinearities are assumed to satisfy the sector-bounded condition and parameter uncertainty is norm bounded. The multiple IFs, represented by a set of independent Bernoulli-distributed stochastic variables, are formulated as probabilistic events occurring on each sensor and/or actuator of the system. The aim is to design a dynamic output-feedback controller such that the closed-loop system maintains asymptotical stability and satisfies a given H∞ performance index. Using the linear 3
matrix inequality (LMI) approach, a sufficient condition for controller existence is obtained. Finally, two examples are provided to illustrate the effectiveness of the proposed scheme. The remainder of this paper is organized as follows. In Section 2, the mathematical model of the considered system is formulated, and some preliminaries are given. Section 3 presents the main results of PFTC of an uncertain nonlinear discrete-time system against multiple IFs, and scenarios of sensor faults, actuator faults, and both of them are investigated in sequence. In Section 4, two simulation example are provided to demonstrate the effectiveness of the proposed method. Finally, the conclusions of this study are summarized in Section 5. Notations: Throughout the paper, for real symmetric matrices, the notation X > Y indicates that the matrix X − Y is positive definite. R n represents n -dimensional Euclidean space, while R n×m is the set of all n × m real matrices. The superscript T denotes transpose. E {} represents the expectation of event . Prob{} represents the occurrence probability of event
. diag {M1 , M 2 , …, M n } denotes a block
diagonal matrix composed of square matrices M1 , M2 , …, Mn . l2 [ 0, ∞ ) is the space of square summable vectors. The symbol ∗ is used to represent the corresponding transposed block in the symmetry block matrix. I denotes an identity matrix with appropriate dimensions.
4
2. PROBLEM FORMULATION AND PRELIMINARIES Consider a class of uncertain nonlinear discrete-time systems subject to multiple intermittent faults: x ( k + 1) = ( A + ∆A ) x ( k ) + B ( u ( k ) + Ξ a ( k ) f a ( k ) ) + h ( x ( k ) ) , y ( k ) = C1 x ( k ) + Ξ s ( k ) f s ( k ) z ( k ) = C2 x ( k )
(1)
where x ( k ) ∈ R n is the system state, u ( k ) ∈ R m is the control input, y ( k ) ∈ R p is the measured output, and z ( k ) ∈ R r is the desired controlled output. A , B , C1 , and C2 are known system matrices of appropriate dimensions. The unknown matrix ∆A represents modeling uncertainty, and is assumed to satisfy ∆A = MFN , where M and N
are known real matrices with appropriate dimensions, and F follows the
constraint FF T ≤ I . In addition, h ( x ( k ) ) is a nonlinear function satisfying the following sector-bounded condition: T
h ( x ( k ) ) − T1 x ( k ) h ( x ( k ) ) − T2 x ( k ) ≤ 0 ∀x ( k ) ∈ Rn ,
(2)
where T1 , T2 ∈ R n×n are known real matrices. The considered faults are reflected in the terms Ξs ( k ) f s ( k ) and Ξ a ( k ) f a ( k ) , in which the vectors f s ( k ) ∈ R p and f a ( k ) ∈ R m represent multiple sensor and actuator faults, respectively. The stochastic variables π i ( k )( i = 1, , p ) and τ j ( k )( j = 1, , m ) in diagonal matrices Ξs ( k ) := diag {π1 ( k ) , , π p ( k )} and Ξ a ( k ) := diag {τ1 ( k ) , , τ m ( k )} account for the occurrence of IFs in the i th sensor signal and the jth actuator signal, obeying Bernoulli distribution and taking values of 1 or 0 as follows. 5
Pr ob {π i ( k ) = 1} = π i , Pr ob {π i ( k ) = 0} = 1 − π i
Pr ob {τ j ( k ) = 1} = τ j . Pr ob {τ j ( k ) = 0} = 1 − τ j
Here, π i ∈ [ 0, 1] ( i = 1, , p ) and τ j ∈ [ 0, 1] ( j = 1, , m ) are known constants. In the following, the expectations of Ξs and Ξa are denoted as Ξs := diag {π1 , , π p } and Ξ a := diag {τ1 , , τ m } , respectively.
Remark 1: IFs exist in many practical systems such as electronic equipment, spacecraft, mechanical devices, and communication systems. Many of these faults are in the nature of randomness. The majority of the random IFs may result from the system itself, such as net congestion in networked systems and electromagnetic interference in electronic systems, and a few of them are caused by the environment. Because it is difficult to describe these faults in a deterministic fashion, stochastic variables are employed widely to describe them [22, 27, 28, 30]. Remark 2: In theory, with the proposed IF description, the fault itself can be of any form, such as a square wave, triangular wave, or any other regular or irregular signal. However, in the framework of reliable H∞ control, it is assumed that the considered faults are square summable, i.e., f s ( k ) and f a ( k ) belong to l2 [ 0, ∞) . This study aims to design a class of dynamic full-order output feedback controllers of the following form: xc ( k + 1) = Ac xc ( k ) + Bc y ( k ) , u ( k ) = Cc xc ( k )
(3)
where xc ( k ) ∈ R n is the state vector of the controller, and Ac , Bc , and Cc are controller matrices with appropriate dimensions to be designed later. 6
Define x(k ) . xc ( k )
η (k ) =
(4)
Three closed-loop system situations, which correspond to the cases of only sensor faults, only actuator faults, and both these two kinds of faults, respectively, can be formulated by substituting (3) and (4) into the corresponding scenario descriptions revised from (1). A general form of the closed-loop system is summarized as follows: η ( k + 1) = Agη ( k ) + Bg Ξ g ( k ) f g ( k ) + Eh ( Dη ( k ) ) , z ( k ) = Cgη ( k )
(5)
where subscript “g” denotes the general form, and Ag , Bg , and Cg are matrices of the closed-loop system.
Ξa ( k ) fa ( k ) u (k )
z (k )
y (k )
Ξs ( k ) fs ( k ) Fig. 1. The structure of the closed-loop system. The study objective is to design dynamic output feedback controller (3) for system (1) with multiple sensor and/or actuator faults, such that the closed-loop system (5) (Fig. 1) meets the following H∞ performance index: 7
(1) Under the condition of Ξ g ( k ) ≡ 0 , the closed-loop system (5) is asymptotically stable. (2) Under the zero-initial condition, the controlled output satisfies ∞
∑ E{ z ( k ) k =0
2
∞
} < γ ∑ E{ Ξ ( k ) f ( k ) } 2
2
g
g
k =0
where γ > 0 is a prescribed scalar. Before presenting the main results, it is necessary to introduce some lemmas, which will be used frequently. Lemma 1 [31]: Given constant matrices S1 , S2 , and S3 , where S1 = S1T and S2 = S2T > 0 , S1 + S3T S2−1S3 < 0
if and only if S1 S3
S3T <0 −S2
or
− S2 ST 3
S3 < 0. S1
Lemma 2 [32]: Let M , N , and F be real matrices of appropriate dimensions with F satisfying FF T ≤ I . Then, for any scalar ε > 0 , one obtains T
MFN + ( MFN ) ≤ ε −1MM T + ε N T N .
Lemma 3: The sector-bounded condition (2) is equivalent to T
x ( k ) R1 T h ( x ( k ) ) R2
R2 x ( k ) ≤ 0. I h ( x ( k ) )
Here, R1 = (T1T T2 + T2T T1 ) 2 and R2 = − (T1T + T2T ) 2 . The proof of lemma 3 is omitted because it is straightforward from (2). Lemma 4 [33]: Let W0 ( x ) , W1 ( x ) , , Wl ( x ) be quadratic functions of x ∈ R n , i.e., Wi ( x ) = xT Φ i x, i = 0, 1, , l 8
with ΦTi = Φi . Then, the implication W1 ( x ) ≤ 0, …, Wl ( x ) ≤ 0 ⇒ W0 ( x ) < 0,
holds, if there exist scalars τ 1 ≥ 0, τ 2 ≥ 0, … , τ l ≥ 0 such that l
Φ 0 − ∑ τ i Φ i < 0. i =1
In the following, reliable H∞ control of an uncertain nonlinear discrete-time system subject to multiple IFs is investigated. Three fault scenarios, sensor fault, actuator fault, and both of them, are considered in sequence. 3. MAIN RESULTS Case A: multiple sensor faults Firstly, consider a class of systems that only involves multiple sensor IFs as follows: x ( k + 1) = ( A + ∆A ) x ( k ) + Bu ( k ) + h ( x ( k ) ) , y ( k ) = C1 x ( k ) + Ξ s ( k ) f s ( k ) z ( k ) = C2 x ( k )
(6)
where all parameters in (6) have the same definitions as those in Eq. (1). Substituting (3) and (4) into (6) yields the following closed-loop system: η ( k + 1) = Aη ( k ) + B Ξ s ( k ) f s ( k ) + Eh ( Dη ( k ) ) =Aη ( k ) + B1 f s ( k ) + Eh ( Dη ( k ) ) , z ( k ) = Cη ( k )
(7)
where A + ∆A A= Bc C1
BCc A = Ac Bc C1
BCc M + F [N Ac 0
0 0 I B = , B1 = , E = , C= [C2 0 Bc Bc Ξ s ( k )
, 0 ] = A + MFN
0] , D = [ I
0 ].
According to Section 2, the control objective is to design a dynamic output feedback controller of the form (3) such that 9
(1) Under the condition Ξ s ( k ) ≡ 0 , the closed-loop system (7) is asymptotically stable; and (2) under the zero-initial condition, the controlled output satisfies the following performance index: ∞
∑ E{ z ( k ) k =0
2
∞
} < γ ∑ E{ Ξ ( k ) f ( k ) } 2 s
2
s
(8)
s
k =0
where γ s > 0 is a prescribed scalar. Theorem 1. For a prescribed scalar γ s > 0 , if there exists matrix P = PT > 0 such that − P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗
holds, where B2 = 0
PA PE C 0 − P − τ DT R1 D −τ DT R2 ∗ −τ I ∗ ∗ T
(B Ξ ) c
s
PB2 0 0 <0 0 −γ s2 Ξ s
(9)
T
, then the closed-loop system (7) is asymptotically
stable, and H ∞ performance index (8) is satisfied. Proof: Define the Lyapunov function as V ( k ) = η T ( k ) Pη ( k ) .
(10)
Under the condition Ξ s = 0 , the difference of the Lyapunov function is ∆V ( k ) := V (η ( k + 1) ) − V (η ( k ) )
(
= Aη ( k ) + Eh ( Dη ( k ) )
T
) P ( Aη ( k ) + Eh ( Dη ( k ) ) ) − η
T
( k ) Pη ( k )
T
η ( k ) AT PA − P APE η ( k ) = . ∗ E T PE h ( Dη ( k ) ) h ( Dη ( k ) )
Applying Lemma 1 to (9) implies that AT PA − P + C T C ∗
AT PE τ DT R1 D τ DT R2 − < 0. E T PE ∗ τI 10
According to lemmas 3 and 4, it is straightforward that ∆V ( k ) < 0 . Hence, the closed-loop system (7) is asymptotically stable. To evaluate the H ∞ performance of the closed-loop system, the following index J is introduced under the zero-initial condition. ∞
{ = ∑ E{ z ( k ) ≤ ∑ E{ z ( k )
J = ∑ E z ( k ) − γ s2 Ξ s ( k ) f s ( k ) 2
2
k =0 ∞
}
2
− γ s2 Ξ s ( k ) f s ( k ) + ∆V ( k ) +E {V ( 0 )} − E {V ( ∞ )}
2
− γ s2 Ξ s ( k ) f s ( k )
k =0 ∞
} + ∆V ( k )}.
2
2
k =0
(11)
We, therefore, construct (12).
{
2
}
2
E z ( k ) − γ s2 Ξ s ( k ) f s ( k ) + ∆V ( k ) = E{ Aη ( k ) + Eh ( Dη ( k ) ) + B1 f s ( k )
(
T
) P ( Aη ( k ) + Eh ( Dη ( k ) ) + B f −η ( k ) Pη ( k ) + η ( k ) C Cη ( k ) − γ f ( k ) Ξ f ( k )} = E{ϕ ( k ) (φ ( k ) Pφ ( k ) + diag {− P + C C , 0, − γ Ξ }) ϕ ( k )} = E {ϕ ( k ) Ωϕ ( k )} , T
T
T
T
1 s
2 T s s
T
s
T
s
2 s
( k )) (12)
s
T
where ϕ ( k ) = η T ( k ) hT ( Dη ( k ) )
T
f sT ( k ) ,
φ ( k ) = A E B2 .
Applying lemma 1 to (9) implies that τ DT R1 D τ DT R2 τI Ω− ∗ 0 0
0 0 < 0 0
(13)
According to (13) and using lemmas 3 and 4 again, it can be inferred that ϕ T ( k ) Ωϕ ( k ) < 0 , which demonstrates that J < 0 . Therefore, inequality (8) holds,
which completes the proof.
□
Obviously, expression (9) is a bilinear matrix inequality (BMI) rather than a LMI. It cannot be solved directly. Hence, the method adopted in [34] is used to solve BMI (9) 11
and derive the controller parameters. Theorem 2. For given scalar γ s > 0 , symmetric matrix X > 0 , and scalar ε > 0 , if there exists a symmetric matrix Y > 0 , real matrices Aˆ ∈ R n×n , Bˆ ∈ R n× p , and Cˆ ∈ R m × n , and a positive scalar τ such that
− X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0
AX + BCˆ Aˆ
A YA + Bˆ Ξ −1C
∗ ∗
−I ∗
C2 X − X − τ XR1 X
∗ ∗
∗ ∗
∗ ∗ ∗
−I
0
Y
0 Bˆ
−Y
YM
C2 − I − τ XR1
0 −τ XR2
0 0
0 0
∗ ∗
−Y − τ R1 ∗
−τ R2 −τ I
0 0
0 0
∗ ∗
∗ ∗
∗ ∗
∗ ∗
−γ s2 Ξ s ∗
0 −ε I
∗
∗
∗
∗
∗
∗
s
I
1
M
0 0 T ε XN ε NT < 0 0 0 0 −ε I 0
(14)
holds, then closed-loop system (7) is asymptotically stable and H ∞ performance (8) can be satisfied. Furthermore, the controller matrices are designed as follows: Ac = S −1 Aˆ − Bˆ Ξ s−1C1 X − YBCˆ − YAX R −T −1 −1 , Bc = S Bˆ Ξ s −T ˆ Cc = CR
(
)
(15)
where R and S are matrices of appropriate dimensions satisfying RS T = I − XY .
(16)
Proof: Adopting a similar method as in [34], partition-symmetric matrices P and P −1 are given as Y P= T S
S −1 X , P = T R
R ,
(17)
where X and Y are symmetric matrices of dimensions n × n . Construct the following matrices X Π1 = T R
I I Y . , Π2 = T 0 0 S 12
(18)
Recalling the fact that PP −1 = I , (17) and (18) imply that (19)
PΠ 1 = Π 2 .
Then, some auxiliary matrices are defined as follows: Aˆ = SAc RT + SBc C1 X + YBc RT + YAX . Bˆ = SBc Ξ s ˆ T C = Cc R
(20)
Based on Eqs. (18)–(20), inequality (14) can be rewritten as −Π1T PΠ1 ∗ ∗ ∗ ∗ ∗ ∗
0 −I
Π1T PA Π1 C Π1
Π1T PE 0
Π1T PB2 0
Π1T PM 0
∗
Φ
−τΠ1T DT R2
0
0
∗ ∗
∗ ∗
−τ I ∗
0 −γ 2 Ξ s
0 0
∗ ∗
∗ ∗
∗ ∗
∗ ∗
−ε I ∗
εΠ1T N T 0 <0 0 0 −ε I 0 0
(21)
,
where Φ = −Π1T PΠ1 − τΠ1T DT R1 DΠ1 . Applying the congruence transformation diag {Π1−T , I , Π1−T , I , I , I , I } to (21) yields the following inequality (22). − P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
PA C
PE 0
− P − τ DT R1 D −τ DT R2
PB2 0
PM 0
0
0
∗ ∗
−τ I ∗
0 −γ 2 Ξ s
0 0
∗ ∗
∗ ∗
∗ ∗
−ε I ∗
0 0 ε N T 0 <0 0 0 −ε I
According to the lemma 1, (22) is equivalent to Σ + ε −1 HH T + ε G T G < 0.
where − P 0 ∗ −I Σ= ∗ ∗ ∗ ∗ ∗ ∗
PA
PE
C 0 − P − τ DT R1 D −τ DT R2 ∗ −τ I ∗
∗ 13
PB2 0 0 , 0 −γ 2 Ξ s
(22)
T H = M T P 0 0 0 0 ,
G = 0 0 N 0 0 . T
Note that the left hand of (9) can be constructed as Σ + HFG + ( HFG ) . According to Lemma 2, it follows that T
Σ + HFG + ( HFG ) ≤ Σ + ε −1 HH T + ε G T G < 0.
This implies that (9) is satisfied. Based on Theorem 1, closed-loop system (7) is asymptotically stable, and the prescribed H ∞ performance index is satisfied. This completes the proof.
□
Case B: multiple actuator faults Consider a class of uncertain nonlinear discrete-time systems subject to multiple actuator faults as follows: x ( k + 1) = ( A + ∆A ) x ( k ) + B ( u ( k ) + Ξ a ( k ) f a ( k ) ) + h ( x ( k ) ) , y ( k ) = C1 x ( k ) z ( k ) = C2 x ( k )
(23)
where all parameters in (23) have the same definitions as those in Equation (1). Sorting out (3) and (23), and using definition (4), a closed-loop system can be derived as η ( k + 1) = Aη ( k ) + B Ξ a ( k ) f a ( k ) + Eh ( Dη ( k ) ) =Aη ( k ) + B1 f a ( k ) + Eh ( Dη ( k ) ) , z ( k ) = Cη ( k )
(24)
where A + ∆A A= Bc C1
BCc A = Ac Bc C1
BCc M + F [N Ac 0
BΞ ( k ) B I B = , B1 = a , E = , C= [C2 0 0 0
14
, 0 ] = A + MFN
0] , D = [ I
0 ].
According to Section 2, the control objective is to design a dynamic output feedback controller of form (3) such that (1) Under the condition Ξ a ( k ) ≡ 0 , closed-loop system (24) is asymptotically stable; and (2) with the zero-initial condition, the controlled output satisfies the following performance index: ∞
∑ E{ z ( k )
2
k =0
∞
} < γ ∑ E{ Ξ ( k ) f ( k ) } 2 a
2
a
a
(25)
k =0
where γ a > 0 is a prescribed scalar. Theorem 3. For a prescribed scalar γ a > 0 , if there exists matrix P = PT > 0 such that −P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗
holds, where B2 = ( BΞ a )
T
PA
PE
0 − P − τ D R1 D −τ DT R2 ∗ −τ I ∗ ∗ C
T
PB2 0 0 <0 0 −γ a2 Ξa ,
(26)
T
0 , then closed-loop system (24) is asymptotically stable,
and H ∞ performance (25) is satisfied. Due to the proof of Theorem 3 is similar to that of Theorem 1, it is omitted. Theorem 4. For given scalar γ a > 0 , symmetric matrix X > 0 , and scalar ε > 0 , if there exist a symmetric matrix Y > 0 , real matrices Aˆ ∈ R n×n , Bˆ ∈ R n× p , and Cˆ ∈ R m× n , and a positive scalar τ such that
15
− X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
−I
0
−Y ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 −I ∗ ∗ ∗ ∗ ∗ ∗
AX + BCˆ Aˆ C2 X − X − τ XR1 X ∗ ∗ ∗ ∗ ∗
A
I
BΞ a
M
ˆ YA + BC 1 C2 − I − τ XR1 −Y − τ R1 ∗ ∗ ∗ ∗
Y 0 −τ XR2 −τ R2 −τ I ∗ ∗ ∗
YBΞ a 0 0 0 0 −γ a2 Ξ a ∗ ∗
YM 0 0 0 0 0 −ε I ∗
0 0 ε XN T ε NT < 0 0 0 0 −ε I 0
(27)
,
holds, then closed-loop system (24) is asymptotically stable and H ∞ performance (25) can be achieved. Furthermore, the controller parameters are designed as follows: ˆ X − YBCˆ − YAX R −T Ac = S −1 Aˆ − BC 1 −1 ˆ , Bc = S B −T ˆ Cc = CR
(
)
(28)
where R and S are matrices of appropriate dimensions satisfying (16). The proof of Theorem 4 is the same as that of Theorem 2 and is therefore omitted.
Case C: multiple sensor and actuator faults Consider a class of uncertain nonlinear discrete-time systems subject to multiple sensor and actuator faults simultaneously as described by (1). By substituting (3) and (4) into (1), the following closed-loop system is derived.
η ( k + 1) = Aη ( k ) + B1Ξ ( k ) f ( k ) + Eh ( Dη ( k ) ) z ( k ) = Cη ( k )
(29)
where
B B1 = 0
0 Ξ ( k ) 0 fa ( k ) , Ξ (k ) = a , f (k ) = Bc Ξ s ( k ) 0 0
0 . fs ( k )
Similarly, the control objective is to design a dynamic output feedback controller of form (3) such that (1) Under the condition Ξ ( k ) ≡ 0 , the closed-loop system (29) is asymptotically 16
stable; and (2) under the zero-initial condition, the controlled output satisfies the following performance index: ∞
∑ E{ z (k ) k =0
2
}<γ
∞
2 both
∑ E{ Ξ (k ) f (k ) k =0
2
}
(30)
where γ both > 0 is a prescribed scalar. Theorem 5. For a prescribed scalar γ both > 0 , if there exists matrix P = PT > 0 such that
− P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗
PA
PE 0
C T
− P − τ D R1D −τ DT R2 ∗ −τ I ∗
∗
PB2 0 0 <0 0 2 −γ both Ξ
(31)
holds, where B2 = B1Ξ , and Ξ = diag {Ξ a , Ξ s } , then the closed-loop system (29) is asymptotically stable, and H ∞ performance index (30) is satisfied. The proof is omitted for saving space. Theorem 6. For given scalar γ both > 0 , symmetric matrix X > 0 , and scalar ε > 0 , if there exist Y > 0 , real matrices Aˆ ∈ R n×n , Bˆ ∈ R n× p , and Cˆ ∈ R m×n , and a positive scalar τ such that
17
− X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 −I ∗
AX + BCˆ Aˆ
A YA + Bˆ Ξ −1C
C2 X − X − τ XR1 X
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
−I
0
−Y ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
I
BΞ a
C2 − I − τ XR1
Y 0 −τ XR2
YBΞ a 0 0
−Y − τ R1 ∗ ∗ ∗ ∗ ∗
−τ R2 −τ I ∗ ∗ ∗ ∗
0 0
1
s
2 −γ both Ξa ∗ ∗ ∗
0 Bˆ 0 0 0 0 0 2 −γ both Ξs ∗ ∗
M YM 0 0 0 0 0 0 −ε I ∗
0 0 ε XN T ε NT < 0 0 0 0 0 −ε I 0
(32) holds, then the closed-loop system (29) is asymptotically stable, and H ∞ performance index (30) is satisfied. Furthermore, the controller parameters are designed as Eq. (15). The proof is omitted for simplicity. Remark 3. This study is an extension of the up-to-date publication [29], which also studies the reliable H ∞ control problem for discrete-time system with IFs. In [29], there is an assumption that all sensor or actuator faults must occur simultaneously, which is not reasonable. By employing two sets of independent stochastic variables, this study proposes a more reasonable and general fault model, in which the sensor or actuator faults could occur separately. Remark 4. In practical systems, IFs occur on components, such as sensors and actuators. Usually, the dynamic performances of IFs are different, which depend on the IFs themselves and the circumstances. Under some situations, the IFs would deteriorate and even evolve into PFs while the IFs occur very frequently with the time going. In this study, it is assumed that the IFs are square summable such that the problem can be solved in the framework of reliable H ∞ control. In future study, the 18
dynamic properties of IFs will be paid more attention to, and more practical and precise mathematical descriptions (such as by means of Markovian system) of IFs will be proposed and investigated. 4. SIMULATION EXAMPLES In this section, two simulation examples are introduced to verify the effectiveness of the proposed control strategy. Example 1 Consider an uncertain nonlinear discrete-time system with the following system matrices:
−0.3 −0.2 −0.3 0.4 0.3 0.1 −0.2 0 A = 0.1 −1.1 0.1 , B = 0.1 0.5 , C1 = , 0.1 −0.1 0.4 0.3 0.2 −0.2 0.2 0.1 T
C2 = [ 0.3 0.5 0.7 ] , M = [ 0.03 0.03 0.03] , N = [ 0.1 0.1 0.1] . The nonlinear function is assumed as
0.02 x1 ( k ) sin 2 ( x1 ( k ) ) − 0.01( x1 ( k ) − x2 ( k ) + x3 ( k ) ) h ( x ( k )) = −0.01( x1 ( k ) − x3 ( k ) ) , −0.01( x2 ( k ) + x3 ( k ) ) which can be bounded by
0.01 0.01 −0.01 −0.01 0.01 −0.01 T1 = 0.01 0.02 0.04 , T2 = −0.03 −0.02 −0.02 . 0.02 −0.03 −0.04 −0.02 0.01 0.02 In the simulation, the probability of occurrence of a sensor and actuator fault is T assumed to be π1 = τ1 = 0.02 , π 2 = τ 2 = 0.03 , and f s ( k ) = f a ( k ) = [1 1] . Let F = sin ( k ) ,
and the prescribed H∞ performance level γ s = γ a = 0.9 . To solve matrix inequalities (14),
(27),
and
(32),
some
related 19
parameters
are
chosen
as
X = diag {0.6,0.6,0.6} , ε = 0.1 .
Solving matrix inequalities (14), (27), and (32), respectively, one can obtain the matrices for the dynamic output-feedback controller as below. For Case A −0.2694 Ac = 0.1070 0.5038 −0.1080 Cc = 0.4542
−0.2324 −0.0947 −0.1053 0.1471 0.1539 0.0326 , Bc = −0.1129 0.1135 , 6.7077 −3.9234 −1.2720 0.3233 −0.3451 0.3869 . −0.2142 −0.5639
For Case B 5.0548 −0.1876 0.1182 0.0945 8.4029 Ac = −0.2475 0.4720 −0.2399 , Bc = −12.9807 −4.6449 , 0.5622 −0.1825 0.8125 22.2821 10.4741 0.0336 −1.0417 0.4521 Cc = . 0.1735 0.9528 −1.1084 For Case C −0.2591 Ac = 0.0606 0.6565 −0.0875 Cc = 0.4977
−0.2487 −0.1107 −0.1636 0.1862 0.1644 0.0228 , Bc = −0.2254 0.2085 , 7.9145 −4.0537 −1.3604 0.3751 −0.3755 0.4015 . −0.1431 −0.5757
The simulation results of example 1 are shown in Figs. 2–8. The state trajectories of the uncontrolled system are given in Fig. 2, which demonstrates that the system is unstable. Figs. 3, 5 and 7 show the state trajectories of the controlled system for Cases A, B, and C, respectively. It is clear that the reliable controllers can maintain the closed-loop system in a stable stat and make the system converge rapidly to zero after the advent of multiple sensor and/or actuator faults. The designed controller signals of 20
Cases A, B, and C are shown in Figures 4, 6, and 8, respectively.
Fig. 2. State trajectories of uncontrolled system.
21
Fig. 3. State trajectories of the controlled system in Case A.
Fig. 4. Controller signal in Case A.
22
Fig. 5. State trajectories of the controlled system in Case B.
Fig. 6. Controller signal in Case B.
23
Fig. 7. State trajectories of the controlled system in Case C.
24
Fig. 8. Controller signal in Case C. Example 2
In this part, the proposed PFTC strategy is applied to an aircraft engine system. The original model is a linearized continuous-time system [35]. With a sampling period of 0.1 sec, one can obtain a discrete-time model, where its parameters are as following.
0.7225 1.0113 0.2315 0.3364 A= ,B = , 1.1317 0.6958 0.2678 0.4070 T
0 −0.1190 1 0 0.0056 C1 = . 0 1 −0.0021 0.0003 0.0858 It is assumed that when modeling the aircraft engine system, there exist norm bounded parameter uncertainties and sector-bounded nonlinear disturbance. Therefore, apart from the system parameters A , B , and C1 , other parameters are set as 25
follows:
C2 = [ 0.1 0.1] , T
M = [ 0.03 0.03] , N = [ 0.01 0.01] , 0.2 0.1 −0.1 0 , T2 = T1 = , 0 0.2 −0.1 0.1 ε = 0.1, X = diag {0.5, 0.5} , Ξ a = diag {0.02, 0.04} , Ξ s = diag {0.02, 0.04, 0.04, 0.04, 0.04} . In addition, the sector-bounded nonlinear function is of the form 1 0.3 ( x1k + x2 k ) h ( x ( k )) = + 0.1x1k + 0.1x2 k 2 1 + x12k + x22k
T
0.3 x1k + 0.3 x2 k .
In this simulation, only Case C is considered for simplicity, and the amplitude of the sensor and actuator in each channel is assumed to be 1. Let F = sin ( k ) , and the prescribed H ∞ performance level is taken as γ both = 9 . Solving LMI (32), one can obtain the matrices for the dynamic output-feedback controller as below.
0 −0.0273 0.2485 −0.2951 0.8454 0.0282 0 Ac = , Bc = , −0.1900 −1.8635 2.9979 3.0116 0 0.0018 0.1517 −15.8184 −3.1511 Cc = . 10.4989 0.5439 The simulation results of example 2 are shown in Figs. 9–11, which demonstrate the effectiveness of the proposed strategy.
26
Fig. 9. State trajectories of the uncontrolled aircraft engine system.
Fig. 10. State trajectories of the closed-loop aircraft engine system with intermittent faults.
27
Fig. 11. Controller signal of the closed-loop aircraft engine system with intermittent faults. 5. CONCLUSIONS
In this study, a PFTC strategy was investigated for a class of uncertain nonlinear discrete-time systems subject to multiple IFs, where the multiple IFs were depicted by a set of Bernoulli-distributed variables. In the framework of reliable H ∞ control, sensor IFs and/or actuator IFs are studied, and three dynamic output-feedback controllers were designed such that the closed-loop systems maintain asymptotical stability and satisfy a prescribed H∞ performance. The provided two simulation examples demonstrate the effectiveness of the proposed FTC method.
28
ACKNOWLEDGEMENT
This study was supported by the National Natural Science Foundation of China under Grant 61374099, the Program for New Century Excellent Talents in University under Grant NCET-13-0652, and Beijing Higher Education Young Elite Teacher Project under Grant YETP0505.
29
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33
Reliable H 1 control for uncertain nonlinear discrete-time Systems subject to multiple intermittent Faults in Sensors and/or Actuators Yuan Tao, Dong Shen, Youqing Wang, Yinzhong Ye
www.elsevier.com/locate/jfranklin
PII: DOI: Reference:
S0016-0032(15)00305-1 http://dx.doi.org/10.1016/j.jfranklin.2015.07.012 FI2402
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
10 January 2015 22 May 2015 13 July 2015
Cite this article as: Yuan Tao, Dong Shen, Youqing Wang, Yinzhong Ye, Reliable H 1 control for uncertain nonlinear discrete-time Systems subject to multiple intermittent Faults in Sensors and/or Actuators, Journal of the Franklin Institute, http://dx.doi.org/ 10.1016/j.jfranklin.2015.07.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Reliable
Control for Uncertain Nonlinear Discrete-Time
H∞
Systems Subject to Multiple Intermittent Faults in Sensors and/or Actuators Yuan Tao, Dong Shen, Youqing Wang*, Yinzhong Ye2 1. College of Information Science and Technology Beijing University of Chemical Technology, Beijing 100029, China 2. School of Electrical and Electronic Engineering, Shanghai Institute of Technology, Shanghai 201418, China *E-mail: [email protected] Abstract: This study focuses on passive fault-tolerant control for a class of uncertain nonlinear discrete-time systems subject to multiple intermittent faults. The considered intermittent faults are assumed to be additive ones in sensors and/or actuators. To achieve fault-tolerant control, a dynamic output-feedback controller is designed such that the closed-loop system remains stable and satisfies acceptable performance, even when there are parameter uncertainties, nonlinearities of specific type, and multiple additive intermittent sensor and/or actuator faults. The linear matrix inequality method is employed to obtain sufficient conditions for achieving fault tolerance and ensuring the prescribed H∞ performance index. Finally, the effectiveness of the proposed method is demonstrated by simulation examples. Keywords: fault-tolerant control; intermittent fault; sensor fault; actuator fault; robust H ∞ performance.
1
1. INTRODUCTION Fault-tolerant control (FTC) aims to ensure that a controlled system retains an acceptable performance even if some system components (including sensors and actuators) malfunction. With the growing scale and complexity of industrial processes, the advent of faults may endanger system security and cause serious economic loss, especially in safety-critical systems, such as chemical plants, aircrafts, spacecraft, and nuclear power plants. The engineering significance of FTC has been well recognized, and a number of practical systems have adopted the fault-tolerant strategy, thus creates ample scope for FTC studies within the system control community [1-4]. Usually, all reported FTC studies can be divided into two categories: active FTC (AFTC) [5-11] and passive FTC (PFTC) [12-18]. To achieve the purpose of AFTC, fault detection and diagnosis, which itself also draws considerable concern in control community [19, 20], is necessary to provide real-time fault information to reconfigure the controller. In contrast, the design of PFTC relies on a priori knowledge of failure information (e.g. location, amplitude). System faults are the focus of FTC design, and they are present in different forms within systems. In most cases, the term “fault” in the above-mentioned papers refers to a permanent fault (PF), which tends to remain faulty and may even worsen without intervention. However, many faults are not permanent, and a few types of such non-permanent faults are called intermittent faults (IFs). IFs exist widely in practical systems, including electronic equipment [21], spacecraft [22], mechanical devices [23], and communication systems [24], and they threaten system reliability and 2
security. The intermittent nature of a fault is reflected in the fact that it can recover without any corrective action. Therefore, as compared with the PF, the occurrence of the IF exhibits special properties such as randomness, intermittence, and repeatability [25, 26]. Owing to the existence of these special properties, traditional methods for addressing PFs cannot be applied to address IFs directly, regardless of the fault diagnosis technique or fault-tolerant scheme employed. From the viewpoint of FTC, the first hindrance is the lack of effective mathematical description. To solve the problem in the FTC framework, usually, a PF is described as an analytical model in deterministic form. However, it is difficult to depict an IF using a deterministic analytical form, because a suitable IF model must be able to switch frequently and randomly between fault occurrence and disappearance. In fact, the majority of IFs occur randomly, thus necessitating the use of stochastic variables to describe them [22, 27-29]. To the best of the authors’ knowledge, few studies on the FTC of multiple IFs have been reported. Motivated by the aforementioned state of the art, this study investigates PFTC problem for a class of uncertain nonlinear discrete-time systems subject to multiple IFs, where nonlinearities are assumed to satisfy the sector-bounded condition and parameter uncertainty is norm bounded. The multiple IFs, represented by a set of independent Bernoulli-distributed stochastic variables, are formulated as probabilistic events occurring on each sensor and/or actuator of the system. The aim is to design a dynamic output-feedback controller such that the closed-loop system maintains asymptotical stability and satisfies a given H∞ performance index. Using the linear 3
matrix inequality (LMI) approach, a sufficient condition for controller existence is obtained. Finally, two examples are provided to illustrate the effectiveness of the proposed scheme. The remainder of this paper is organized as follows. In Section 2, the mathematical model of the considered system is formulated, and some preliminaries are given. Section 3 presents the main results of PFTC of an uncertain nonlinear discrete-time system against multiple IFs, and scenarios of sensor faults, actuator faults, and both of them are investigated in sequence. In Section 4, two simulation example are provided to demonstrate the effectiveness of the proposed method. Finally, the conclusions of this study are summarized in Section 5. Notations: Throughout the paper, for real symmetric matrices, the notation X > Y indicates that the matrix X − Y is positive definite. R n represents n -dimensional Euclidean space, while R n×m is the set of all n × m real matrices. The superscript T denotes transpose. E {} represents the expectation of event . Prob{} represents the occurrence probability of event
. diag {M1 , M 2 , …, M n } denotes a block
diagonal matrix composed of square matrices M1 , M2 , …, Mn . l2 [ 0, ∞ ) is the space of square summable vectors. The symbol ∗ is used to represent the corresponding transposed block in the symmetry block matrix. I denotes an identity matrix with appropriate dimensions.
4
2. PROBLEM FORMULATION AND PRELIMINARIES Consider a class of uncertain nonlinear discrete-time systems subject to multiple intermittent faults: x ( k + 1) = ( A + ∆A ) x ( k ) + B ( u ( k ) + Ξ a ( k ) f a ( k ) ) + h ( x ( k ) ) , y ( k ) = C1 x ( k ) + Ξ s ( k ) f s ( k ) z ( k ) = C2 x ( k )
(1)
where x ( k ) ∈ R n is the system state, u ( k ) ∈ R m is the control input, y ( k ) ∈ R p is the measured output, and z ( k ) ∈ R r is the desired controlled output. A , B , C1 , and C2 are known system matrices of appropriate dimensions. The unknown matrix ∆A represents modeling uncertainty, and is assumed to satisfy ∆A = MFN , where M and N
are known real matrices with appropriate dimensions, and F follows the
constraint FF T ≤ I . In addition, h ( x ( k ) ) is a nonlinear function satisfying the following sector-bounded condition: T
h ( x ( k ) ) − T1 x ( k ) h ( x ( k ) ) − T2 x ( k ) ≤ 0 ∀x ( k ) ∈ Rn ,
(2)
where T1 , T2 ∈ R n×n are known real matrices. The considered faults are reflected in the terms Ξs ( k ) f s ( k ) and Ξ a ( k ) f a ( k ) , in which the vectors f s ( k ) ∈ R p and f a ( k ) ∈ R m represent multiple sensor and actuator faults, respectively. The stochastic variables π i ( k )( i = 1, , p ) and τ j ( k )( j = 1, , m ) in diagonal matrices Ξs ( k ) := diag {π1 ( k ) , , π p ( k )} and Ξ a ( k ) := diag {τ1 ( k ) , , τ m ( k )} account for the occurrence of IFs in the i th sensor signal and the jth actuator signal, obeying Bernoulli distribution and taking values of 1 or 0 as follows. 5
Pr ob {π i ( k ) = 1} = π i , Pr ob {π i ( k ) = 0} = 1 − π i
Pr ob {τ j ( k ) = 1} = τ j . Pr ob {τ j ( k ) = 0} = 1 − τ j
Here, π i ∈ [ 0, 1] ( i = 1, , p ) and τ j ∈ [ 0, 1] ( j = 1, , m ) are known constants. In the following, the expectations of Ξs and Ξa are denoted as Ξs := diag {π1 , , π p } and Ξ a := diag {τ1 , , τ m } , respectively.
Remark 1: IFs exist in many practical systems such as electronic equipment, spacecraft, mechanical devices, and communication systems. Many of these faults are in the nature of randomness. The majority of the random IFs may result from the system itself, such as net congestion in networked systems and electromagnetic interference in electronic systems, and a few of them are caused by the environment. Because it is difficult to describe these faults in a deterministic fashion, stochastic variables are employed widely to describe them [22, 27, 28, 30]. Remark 2: In theory, with the proposed IF description, the fault itself can be of any form, such as a square wave, triangular wave, or any other regular or irregular signal. However, in the framework of reliable H∞ control, it is assumed that the considered faults are square summable, i.e., f s ( k ) and f a ( k ) belong to l2 [ 0, ∞) . This study aims to design a class of dynamic full-order output feedback controllers of the following form: xc ( k + 1) = Ac xc ( k ) + Bc y ( k ) , u ( k ) = Cc xc ( k )
(3)
where xc ( k ) ∈ R n is the state vector of the controller, and Ac , Bc , and Cc are controller matrices with appropriate dimensions to be designed later. 6
Define x(k ) . xc ( k )
η (k ) =
(4)
Three closed-loop system situations, which correspond to the cases of only sensor faults, only actuator faults, and both these two kinds of faults, respectively, can be formulated by substituting (3) and (4) into the corresponding scenario descriptions revised from (1). A general form of the closed-loop system is summarized as follows: η ( k + 1) = Agη ( k ) + Bg Ξ g ( k ) f g ( k ) + Eh ( Dη ( k ) ) , z ( k ) = Cgη ( k )
(5)
where subscript “g” denotes the general form, and Ag , Bg , and Cg are matrices of the closed-loop system.
Ξa ( k ) fa ( k ) u (k )
z (k )
y (k )
Ξs ( k ) fs ( k ) Fig. 1. The structure of the closed-loop system. The study objective is to design dynamic output feedback controller (3) for system (1) with multiple sensor and/or actuator faults, such that the closed-loop system (5) (Fig. 1) meets the following H∞ performance index: 7
(1) Under the condition of Ξ g ( k ) ≡ 0 , the closed-loop system (5) is asymptotically stable. (2) Under the zero-initial condition, the controlled output satisfies ∞
∑ E{ z ( k ) k =0
2
∞
} < γ ∑ E{ Ξ ( k ) f ( k ) } 2
2
g
g
k =0
where γ > 0 is a prescribed scalar. Before presenting the main results, it is necessary to introduce some lemmas, which will be used frequently. Lemma 1 [31]: Given constant matrices S1 , S2 , and S3 , where S1 = S1T and S2 = S2T > 0 , S1 + S3T S2−1S3 < 0
if and only if S1 S3
S3T <0 −S2
or
− S2 ST 3
S3 < 0. S1
Lemma 2 [32]: Let M , N , and F be real matrices of appropriate dimensions with F satisfying FF T ≤ I . Then, for any scalar ε > 0 , one obtains T
MFN + ( MFN ) ≤ ε −1MM T + ε N T N .
Lemma 3: The sector-bounded condition (2) is equivalent to T
x ( k ) R1 T h ( x ( k ) ) R2
R2 x ( k ) ≤ 0. I h ( x ( k ) )
Here, R1 = (T1T T2 + T2T T1 ) 2 and R2 = − (T1T + T2T ) 2 . The proof of lemma 3 is omitted because it is straightforward from (2). Lemma 4 [33]: Let W0 ( x ) , W1 ( x ) , , Wl ( x ) be quadratic functions of x ∈ R n , i.e., Wi ( x ) = xT Φ i x, i = 0, 1, , l 8
with ΦTi = Φi . Then, the implication W1 ( x ) ≤ 0, …, Wl ( x ) ≤ 0 ⇒ W0 ( x ) < 0,
holds, if there exist scalars τ 1 ≥ 0, τ 2 ≥ 0, … , τ l ≥ 0 such that l
Φ 0 − ∑ τ i Φ i < 0. i =1
In the following, reliable H∞ control of an uncertain nonlinear discrete-time system subject to multiple IFs is investigated. Three fault scenarios, sensor fault, actuator fault, and both of them, are considered in sequence. 3. MAIN RESULTS Case A: multiple sensor faults Firstly, consider a class of systems that only involves multiple sensor IFs as follows: x ( k + 1) = ( A + ∆A ) x ( k ) + Bu ( k ) + h ( x ( k ) ) , y ( k ) = C1 x ( k ) + Ξ s ( k ) f s ( k ) z ( k ) = C2 x ( k )
(6)
where all parameters in (6) have the same definitions as those in Eq. (1). Substituting (3) and (4) into (6) yields the following closed-loop system: η ( k + 1) = Aη ( k ) + B Ξ s ( k ) f s ( k ) + Eh ( Dη ( k ) ) =Aη ( k ) + B1 f s ( k ) + Eh ( Dη ( k ) ) , z ( k ) = Cη ( k )
(7)
where A + ∆A A= Bc C1
BCc A = Ac Bc C1
BCc M + F [N Ac 0
0 0 I B = , B1 = , E = , C= [C2 0 Bc Bc Ξ s ( k )
, 0 ] = A + MFN
0] , D = [ I
0 ].
According to Section 2, the control objective is to design a dynamic output feedback controller of the form (3) such that 9
(1) Under the condition Ξ s ( k ) ≡ 0 , the closed-loop system (7) is asymptotically stable; and (2) under the zero-initial condition, the controlled output satisfies the following performance index: ∞
∑ E{ z ( k ) k =0
2
∞
} < γ ∑ E{ Ξ ( k ) f ( k ) } 2 s
2
s
(8)
s
k =0
where γ s > 0 is a prescribed scalar. Theorem 1. For a prescribed scalar γ s > 0 , if there exists matrix P = PT > 0 such that − P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗
holds, where B2 = 0
PA PE C 0 − P − τ DT R1 D −τ DT R2 ∗ −τ I ∗ ∗ T
(B Ξ ) c
s
PB2 0 0 <0 0 −γ s2 Ξ s
(9)
T
, then the closed-loop system (7) is asymptotically
stable, and H ∞ performance index (8) is satisfied. Proof: Define the Lyapunov function as V ( k ) = η T ( k ) Pη ( k ) .
(10)
Under the condition Ξ s = 0 , the difference of the Lyapunov function is ∆V ( k ) := V (η ( k + 1) ) − V (η ( k ) )
(
= Aη ( k ) + Eh ( Dη ( k ) )
T
) P ( Aη ( k ) + Eh ( Dη ( k ) ) ) − η
T
( k ) Pη ( k )
T
η ( k ) AT PA − P APE η ( k ) = . ∗ E T PE h ( Dη ( k ) ) h ( Dη ( k ) )
Applying Lemma 1 to (9) implies that AT PA − P + C T C ∗
AT PE τ DT R1 D τ DT R2 − < 0. E T PE ∗ τI 10
According to lemmas 3 and 4, it is straightforward that ∆V ( k ) < 0 . Hence, the closed-loop system (7) is asymptotically stable. To evaluate the H ∞ performance of the closed-loop system, the following index J is introduced under the zero-initial condition. ∞
{ = ∑ E{ z ( k ) ≤ ∑ E{ z ( k )
J = ∑ E z ( k ) − γ s2 Ξ s ( k ) f s ( k ) 2
2
k =0 ∞
}
2
− γ s2 Ξ s ( k ) f s ( k ) + ∆V ( k ) +E {V ( 0 )} − E {V ( ∞ )}
2
− γ s2 Ξ s ( k ) f s ( k )
k =0 ∞
} + ∆V ( k )}.
2
2
k =0
(11)
We, therefore, construct (12).
{
2
}
2
E z ( k ) − γ s2 Ξ s ( k ) f s ( k ) + ∆V ( k ) = E{ Aη ( k ) + Eh ( Dη ( k ) ) + B1 f s ( k )
(
T
) P ( Aη ( k ) + Eh ( Dη ( k ) ) + B f −η ( k ) Pη ( k ) + η ( k ) C Cη ( k ) − γ f ( k ) Ξ f ( k )} = E{ϕ ( k ) (φ ( k ) Pφ ( k ) + diag {− P + C C , 0, − γ Ξ }) ϕ ( k )} = E {ϕ ( k ) Ωϕ ( k )} , T
T
T
T
1 s
2 T s s
T
s
T
s
2 s
( k )) (12)
s
T
where ϕ ( k ) = η T ( k ) hT ( Dη ( k ) )
T
f sT ( k ) ,
φ ( k ) = A E B2 .
Applying lemma 1 to (9) implies that τ DT R1 D τ DT R2 τI Ω− ∗ 0 0
0 0 < 0 0
(13)
According to (13) and using lemmas 3 and 4 again, it can be inferred that ϕ T ( k ) Ωϕ ( k ) < 0 , which demonstrates that J < 0 . Therefore, inequality (8) holds,
which completes the proof.
□
Obviously, expression (9) is a bilinear matrix inequality (BMI) rather than a LMI. It cannot be solved directly. Hence, the method adopted in [34] is used to solve BMI (9) 11
and derive the controller parameters. Theorem 2. For given scalar γ s > 0 , symmetric matrix X > 0 , and scalar ε > 0 , if there exists a symmetric matrix Y > 0 , real matrices Aˆ ∈ R n×n , Bˆ ∈ R n× p , and Cˆ ∈ R m × n , and a positive scalar τ such that
− X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0
AX + BCˆ Aˆ
A YA + Bˆ Ξ −1C
∗ ∗
−I ∗
C2 X − X − τ XR1 X
∗ ∗
∗ ∗
∗ ∗ ∗
−I
0
Y
0 Bˆ
−Y
YM
C2 − I − τ XR1
0 −τ XR2
0 0
0 0
∗ ∗
−Y − τ R1 ∗
−τ R2 −τ I
0 0
0 0
∗ ∗
∗ ∗
∗ ∗
∗ ∗
−γ s2 Ξ s ∗
0 −ε I
∗
∗
∗
∗
∗
∗
s
I
1
M
0 0 T ε XN ε NT < 0 0 0 0 −ε I 0
(14)
holds, then closed-loop system (7) is asymptotically stable and H ∞ performance (8) can be satisfied. Furthermore, the controller matrices are designed as follows: Ac = S −1 Aˆ − Bˆ Ξ s−1C1 X − YBCˆ − YAX R −T −1 −1 , Bc = S Bˆ Ξ s −T ˆ Cc = CR
(
)
(15)
where R and S are matrices of appropriate dimensions satisfying RS T = I − XY .
(16)
Proof: Adopting a similar method as in [34], partition-symmetric matrices P and P −1 are given as Y P= T S
S −1 X , P = T R
R ,
(17)
where X and Y are symmetric matrices of dimensions n × n . Construct the following matrices X Π1 = T R
I I Y . , Π2 = T 0 0 S 12
(18)
Recalling the fact that PP −1 = I , (17) and (18) imply that (19)
PΠ 1 = Π 2 .
Then, some auxiliary matrices are defined as follows: Aˆ = SAc RT + SBc C1 X + YBc RT + YAX . Bˆ = SBc Ξ s ˆ T C = Cc R
(20)
Based on Eqs. (18)–(20), inequality (14) can be rewritten as −Π1T PΠ1 ∗ ∗ ∗ ∗ ∗ ∗
0 −I
Π1T PA Π1 C Π1
Π1T PE 0
Π1T PB2 0
Π1T PM 0
∗
Φ
−τΠ1T DT R2
0
0
∗ ∗
∗ ∗
−τ I ∗
0 −γ 2 Ξ s
0 0
∗ ∗
∗ ∗
∗ ∗
∗ ∗
−ε I ∗
εΠ1T N T 0 <0 0 0 −ε I 0 0
(21)
,
where Φ = −Π1T PΠ1 − τΠ1T DT R1 DΠ1 . Applying the congruence transformation diag {Π1−T , I , Π1−T , I , I , I , I } to (21) yields the following inequality (22). − P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
PA C
PE 0
− P − τ DT R1 D −τ DT R2
PB2 0
PM 0
0
0
∗ ∗
−τ I ∗
0 −γ 2 Ξ s
0 0
∗ ∗
∗ ∗
∗ ∗
−ε I ∗
0 0 ε N T 0 <0 0 0 −ε I
According to the lemma 1, (22) is equivalent to Σ + ε −1 HH T + ε G T G < 0.
where − P 0 ∗ −I Σ= ∗ ∗ ∗ ∗ ∗ ∗
PA
PE
C 0 − P − τ DT R1 D −τ DT R2 ∗ −τ I ∗
∗ 13
PB2 0 0 , 0 −γ 2 Ξ s
(22)
T H = M T P 0 0 0 0 ,
G = 0 0 N 0 0 . T
Note that the left hand of (9) can be constructed as Σ + HFG + ( HFG ) . According to Lemma 2, it follows that T
Σ + HFG + ( HFG ) ≤ Σ + ε −1 HH T + ε G T G < 0.
This implies that (9) is satisfied. Based on Theorem 1, closed-loop system (7) is asymptotically stable, and the prescribed H ∞ performance index is satisfied. This completes the proof.
□
Case B: multiple actuator faults Consider a class of uncertain nonlinear discrete-time systems subject to multiple actuator faults as follows: x ( k + 1) = ( A + ∆A ) x ( k ) + B ( u ( k ) + Ξ a ( k ) f a ( k ) ) + h ( x ( k ) ) , y ( k ) = C1 x ( k ) z ( k ) = C2 x ( k )
(23)
where all parameters in (23) have the same definitions as those in Equation (1). Sorting out (3) and (23), and using definition (4), a closed-loop system can be derived as η ( k + 1) = Aη ( k ) + B Ξ a ( k ) f a ( k ) + Eh ( Dη ( k ) ) =Aη ( k ) + B1 f a ( k ) + Eh ( Dη ( k ) ) , z ( k ) = Cη ( k )
(24)
where A + ∆A A= Bc C1
BCc A = Ac Bc C1
BCc M + F [N Ac 0
BΞ ( k ) B I B = , B1 = a , E = , C= [C2 0 0 0
14
, 0 ] = A + MFN
0] , D = [ I
0 ].
According to Section 2, the control objective is to design a dynamic output feedback controller of form (3) such that (1) Under the condition Ξ a ( k ) ≡ 0 , closed-loop system (24) is asymptotically stable; and (2) with the zero-initial condition, the controlled output satisfies the following performance index: ∞
∑ E{ z ( k )
2
k =0
∞
} < γ ∑ E{ Ξ ( k ) f ( k ) } 2 a
2
a
a
(25)
k =0
where γ a > 0 is a prescribed scalar. Theorem 3. For a prescribed scalar γ a > 0 , if there exists matrix P = PT > 0 such that −P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗
holds, where B2 = ( BΞ a )
T
PA
PE
0 − P − τ D R1 D −τ DT R2 ∗ −τ I ∗ ∗ C
T
PB2 0 0 <0 0 −γ a2 Ξa ,
(26)
T
0 , then closed-loop system (24) is asymptotically stable,
and H ∞ performance (25) is satisfied. Due to the proof of Theorem 3 is similar to that of Theorem 1, it is omitted. Theorem 4. For given scalar γ a > 0 , symmetric matrix X > 0 , and scalar ε > 0 , if there exist a symmetric matrix Y > 0 , real matrices Aˆ ∈ R n×n , Bˆ ∈ R n× p , and Cˆ ∈ R m× n , and a positive scalar τ such that
15
− X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
−I
0
−Y ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 −I ∗ ∗ ∗ ∗ ∗ ∗
AX + BCˆ Aˆ C2 X − X − τ XR1 X ∗ ∗ ∗ ∗ ∗
A
I
BΞ a
M
ˆ YA + BC 1 C2 − I − τ XR1 −Y − τ R1 ∗ ∗ ∗ ∗
Y 0 −τ XR2 −τ R2 −τ I ∗ ∗ ∗
YBΞ a 0 0 0 0 −γ a2 Ξ a ∗ ∗
YM 0 0 0 0 0 −ε I ∗
0 0 ε XN T ε NT < 0 0 0 0 −ε I 0
(27)
,
holds, then closed-loop system (24) is asymptotically stable and H ∞ performance (25) can be achieved. Furthermore, the controller parameters are designed as follows: ˆ X − YBCˆ − YAX R −T Ac = S −1 Aˆ − BC 1 −1 ˆ , Bc = S B −T ˆ Cc = CR
(
)
(28)
where R and S are matrices of appropriate dimensions satisfying (16). The proof of Theorem 4 is the same as that of Theorem 2 and is therefore omitted.
Case C: multiple sensor and actuator faults Consider a class of uncertain nonlinear discrete-time systems subject to multiple sensor and actuator faults simultaneously as described by (1). By substituting (3) and (4) into (1), the following closed-loop system is derived.
η ( k + 1) = Aη ( k ) + B1Ξ ( k ) f ( k ) + Eh ( Dη ( k ) ) z ( k ) = Cη ( k )
(29)
where
B B1 = 0
0 Ξ ( k ) 0 fa ( k ) , Ξ (k ) = a , f (k ) = Bc Ξ s ( k ) 0 0
0 . fs ( k )
Similarly, the control objective is to design a dynamic output feedback controller of form (3) such that (1) Under the condition Ξ ( k ) ≡ 0 , the closed-loop system (29) is asymptotically 16
stable; and (2) under the zero-initial condition, the controlled output satisfies the following performance index: ∞
∑ E{ z (k ) k =0
2
}<γ
∞
2 both
∑ E{ Ξ (k ) f (k ) k =0
2
}
(30)
where γ both > 0 is a prescribed scalar. Theorem 5. For a prescribed scalar γ both > 0 , if there exists matrix P = PT > 0 such that
− P 0 ∗ −I ∗ ∗ ∗ ∗ ∗ ∗
PA
PE 0
C T
− P − τ D R1D −τ DT R2 ∗ −τ I ∗
∗
PB2 0 0 <0 0 2 −γ both Ξ
(31)
holds, where B2 = B1Ξ , and Ξ = diag {Ξ a , Ξ s } , then the closed-loop system (29) is asymptotically stable, and H ∞ performance index (30) is satisfied. The proof is omitted for saving space. Theorem 6. For given scalar γ both > 0 , symmetric matrix X > 0 , and scalar ε > 0 , if there exist Y > 0 , real matrices Aˆ ∈ R n×n , Bˆ ∈ R n× p , and Cˆ ∈ R m×n , and a positive scalar τ such that
17
− X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 −I ∗
AX + BCˆ Aˆ
A YA + Bˆ Ξ −1C
C2 X − X − τ XR1 X
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
−I
0
−Y ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
I
BΞ a
C2 − I − τ XR1
Y 0 −τ XR2
YBΞ a 0 0
−Y − τ R1 ∗ ∗ ∗ ∗ ∗
−τ R2 −τ I ∗ ∗ ∗ ∗
0 0
1
s
2 −γ both Ξa ∗ ∗ ∗
0 Bˆ 0 0 0 0 0 2 −γ both Ξs ∗ ∗
M YM 0 0 0 0 0 0 −ε I ∗
0 0 ε XN T ε NT < 0 0 0 0 0 −ε I 0
(32) holds, then the closed-loop system (29) is asymptotically stable, and H ∞ performance index (30) is satisfied. Furthermore, the controller parameters are designed as Eq. (15). The proof is omitted for simplicity. Remark 3. This study is an extension of the up-to-date publication [29], which also studies the reliable H ∞ control problem for discrete-time system with IFs. In [29], there is an assumption that all sensor or actuator faults must occur simultaneously, which is not reasonable. By employing two sets of independent stochastic variables, this study proposes a more reasonable and general fault model, in which the sensor or actuator faults could occur separately. Remark 4. In practical systems, IFs occur on components, such as sensors and actuators. Usually, the dynamic performances of IFs are different, which depend on the IFs themselves and the circumstances. Under some situations, the IFs would deteriorate and even evolve into PFs while the IFs occur very frequently with the time going. In this study, it is assumed that the IFs are square summable such that the problem can be solved in the framework of reliable H ∞ control. In future study, the 18
dynamic properties of IFs will be paid more attention to, and more practical and precise mathematical descriptions (such as by means of Markovian system) of IFs will be proposed and investigated. 4. SIMULATION EXAMPLES In this section, two simulation examples are introduced to verify the effectiveness of the proposed control strategy. Example 1 Consider an uncertain nonlinear discrete-time system with the following system matrices:
−0.3 −0.2 −0.3 0.4 0.3 0.1 −0.2 0 A = 0.1 −1.1 0.1 , B = 0.1 0.5 , C1 = , 0.1 −0.1 0.4 0.3 0.2 −0.2 0.2 0.1 T
C2 = [ 0.3 0.5 0.7 ] , M = [ 0.03 0.03 0.03] , N = [ 0.1 0.1 0.1] . The nonlinear function is assumed as
0.02 x1 ( k ) sin 2 ( x1 ( k ) ) − 0.01( x1 ( k ) − x2 ( k ) + x3 ( k ) ) h ( x ( k )) = −0.01( x1 ( k ) − x3 ( k ) ) , −0.01( x2 ( k ) + x3 ( k ) ) which can be bounded by
0.01 0.01 −0.01 −0.01 0.01 −0.01 T1 = 0.01 0.02 0.04 , T2 = −0.03 −0.02 −0.02 . 0.02 −0.03 −0.04 −0.02 0.01 0.02 In the simulation, the probability of occurrence of a sensor and actuator fault is T assumed to be π1 = τ1 = 0.02 , π 2 = τ 2 = 0.03 , and f s ( k ) = f a ( k ) = [1 1] . Let F = sin ( k ) ,
and the prescribed H∞ performance level γ s = γ a = 0.9 . To solve matrix inequalities (14),
(27),
and
(32),
some
related 19
parameters
are
chosen
as
X = diag {0.6,0.6,0.6} , ε = 0.1 .
Solving matrix inequalities (14), (27), and (32), respectively, one can obtain the matrices for the dynamic output-feedback controller as below. For Case A −0.2694 Ac = 0.1070 0.5038 −0.1080 Cc = 0.4542
−0.2324 −0.0947 −0.1053 0.1471 0.1539 0.0326 , Bc = −0.1129 0.1135 , 6.7077 −3.9234 −1.2720 0.3233 −0.3451 0.3869 . −0.2142 −0.5639
For Case B 5.0548 −0.1876 0.1182 0.0945 8.4029 Ac = −0.2475 0.4720 −0.2399 , Bc = −12.9807 −4.6449 , 0.5622 −0.1825 0.8125 22.2821 10.4741 0.0336 −1.0417 0.4521 Cc = . 0.1735 0.9528 −1.1084 For Case C −0.2591 Ac = 0.0606 0.6565 −0.0875 Cc = 0.4977
−0.2487 −0.1107 −0.1636 0.1862 0.1644 0.0228 , Bc = −0.2254 0.2085 , 7.9145 −4.0537 −1.3604 0.3751 −0.3755 0.4015 . −0.1431 −0.5757
The simulation results of example 1 are shown in Figs. 2–8. The state trajectories of the uncontrolled system are given in Fig. 2, which demonstrates that the system is unstable. Figs. 3, 5 and 7 show the state trajectories of the controlled system for Cases A, B, and C, respectively. It is clear that the reliable controllers can maintain the closed-loop system in a stable stat and make the system converge rapidly to zero after the advent of multiple sensor and/or actuator faults. The designed controller signals of 20
Cases A, B, and C are shown in Figures 4, 6, and 8, respectively.
Fig. 2. State trajectories of uncontrolled system.
21
Fig. 3. State trajectories of the controlled system in Case A.
Fig. 4. Controller signal in Case A.
22
Fig. 5. State trajectories of the controlled system in Case B.
Fig. 6. Controller signal in Case B.
23
Fig. 7. State trajectories of the controlled system in Case C.
24
Fig. 8. Controller signal in Case C. Example 2
In this part, the proposed PFTC strategy is applied to an aircraft engine system. The original model is a linearized continuous-time system [35]. With a sampling period of 0.1 sec, one can obtain a discrete-time model, where its parameters are as following.
0.7225 1.0113 0.2315 0.3364 A= ,B = , 1.1317 0.6958 0.2678 0.4070 T
0 −0.1190 1 0 0.0056 C1 = . 0 1 −0.0021 0.0003 0.0858 It is assumed that when modeling the aircraft engine system, there exist norm bounded parameter uncertainties and sector-bounded nonlinear disturbance. Therefore, apart from the system parameters A , B , and C1 , other parameters are set as 25
follows:
C2 = [ 0.1 0.1] , T
M = [ 0.03 0.03] , N = [ 0.01 0.01] , 0.2 0.1 −0.1 0 , T2 = T1 = , 0 0.2 −0.1 0.1 ε = 0.1, X = diag {0.5, 0.5} , Ξ a = diag {0.02, 0.04} , Ξ s = diag {0.02, 0.04, 0.04, 0.04, 0.04} . In addition, the sector-bounded nonlinear function is of the form 1 0.3 ( x1k + x2 k ) h ( x ( k )) = + 0.1x1k + 0.1x2 k 2 1 + x12k + x22k
T
0.3 x1k + 0.3 x2 k .
In this simulation, only Case C is considered for simplicity, and the amplitude of the sensor and actuator in each channel is assumed to be 1. Let F = sin ( k ) , and the prescribed H ∞ performance level is taken as γ both = 9 . Solving LMI (32), one can obtain the matrices for the dynamic output-feedback controller as below.
0 −0.0273 0.2485 −0.2951 0.8454 0.0282 0 Ac = , Bc = , −0.1900 −1.8635 2.9979 3.0116 0 0.0018 0.1517 −15.8184 −3.1511 Cc = . 10.4989 0.5439 The simulation results of example 2 are shown in Figs. 9–11, which demonstrate the effectiveness of the proposed strategy.
26
Fig. 9. State trajectories of the uncontrolled aircraft engine system.
Fig. 10. State trajectories of the closed-loop aircraft engine system with intermittent faults.
27
Fig. 11. Controller signal of the closed-loop aircraft engine system with intermittent faults. 5. CONCLUSIONS
In this study, a PFTC strategy was investigated for a class of uncertain nonlinear discrete-time systems subject to multiple IFs, where the multiple IFs were depicted by a set of Bernoulli-distributed variables. In the framework of reliable H ∞ control, sensor IFs and/or actuator IFs are studied, and three dynamic output-feedback controllers were designed such that the closed-loop systems maintain asymptotical stability and satisfy a prescribed H∞ performance. The provided two simulation examples demonstrate the effectiveness of the proposed FTC method.
28
ACKNOWLEDGEMENT
This study was supported by the National Natural Science Foundation of China under Grant 61374099, the Program for New Century Excellent Talents in University under Grant NCET-13-0652, and Beijing Higher Education Young Elite Teacher Project under Grant YETP0505.
29
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