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Actuator fault estimation and accommodation for switched systems with time delay: Discrete-time case
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Research Article
Actuator fault estimation and accommodation for switched systems with time delay: Discrete-time case$ Dongsheng Du a,b,n, Bin Jiang c,d a
College of Automation, Huaiyin Institute of Technology, 1 Meicheng Road, Huaian, 223003 Jiangsu, China College of Automation, Nanjing University of Science and Technology, Nanjing 210094, China c College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China d Jiangsu Key Laboratory of Internet of Things and Control Technologies (Nanjing University of Aeronautics and Astronautics), China b
art ic l e i nf o
a b s t r a c t
Article history: Received 5 June 2015 Received in revised form 4 November 2015 Accepted 5 February 2016
This paper investigates the problems of actuator fault estimation and accommodation for discrete-time switched systems with state delay. By using reduced-order observer method and switched Lyapunov function technique, a fault estimation algorithm is achieved for the discrete-time switched system with actuator fault and state delay. Then based on the obtained online fault estimation information, a switched dynamic output feedback controller is employed to compensate for the effect of faults by stabilizing the closed-loop systems. Finally, an example is proposed to illustrate the obtained results. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Discrete-time switched system Fault estimation Reduced-order observer Output feedback controller
1. Introduction In modern complex systems, dependability is as important as performances. Once a fault (sensor, actuator, or component failures) occurred in the system, the system's behavior may be drastically decreased, ranging from performance degradation to instability. Making a general survey of the related literature, fault tolerant control (FTC) technique is an efficient method in order to reach the system objectives, or if this turns to be impossible, to achievable new objectives to avoid catastrophic behaviors. Therefore, FTC have been the subjects of intensive investigations over the past two decades, and a number of achievements have been obtained, which can be found in several excellent papers, please refer to [1–9] and the references therein. As for FTC, which can be divided into two main approaches: passive and active. While passive fault tolerance considers systems faults as a special kind of uncertainties [10], active fault tolerance is based on fault detection and isolation (FDI) and accommodation technique [11]. Though FDI can give information whether there exist faults occurring [12], the magnitude of the fault can not be precisely provided. Therefore, the next step is to search an efficient approach to estimate the magnitude of the fault, which is called fault estimate or fault reconstruction [13]. Finally, using the ☆
This paper was recommended for publication by Dr. Steven Ding. Corresponding author at: College of Automation, Huaiyin Institute of Technology, 1 Meicheng Road, 223003 Jiangsu, China. E-mail addresses: [email protected] (D. Du), [email protected] (B. Jiang). n
estimated fault information, a fault-tolerant controller can be designed to compensate the effect of the fault. From the above discussion, it can be seen that fault estimation plays a very important role in active FTC. Therefore, the study of fault estimation has become a hot research topic owing to its importance in active FTC. During the past decade, various effective methods, such as sliding mode observer approach using equivalent output injection signal [14], adaptive technique [15], and learning method based on neural network [16] and so on, have been developed to realize fault estimation. Generally speaking, adaptive technique is usually used to obtain actuator fault information for continuous case [11]. Another commonly employed to realize fault estimation is based on descriptor observer technique, but it is usually utilized to estimate sensor fault [26]. Recently, reduced-order observer method is proposed to solve actuator fault estimation for discrete time systems, this motives us to investigate this issue for discretetime switched systems. Switched system is a class of hybrid systems, which consists of a finite number of subsystems and an associated switching signal governing the switching among them. Many real-world process and systems can be modeled as switched systems, including chemical processes, computer controlled systems, switched circuits, and so on. Therefore, it has been intensively investigated in the past decades [17– 23]. Compared with fruitful stability and stabilization results for switched systems, fault diagnosis and FTC achievements are relatively few. In [24,25], fault detection for switched system was separately investigated for continuous case and discrete case. Du et al. [26] investigated sensor fault estimation and accommodation approaches for continuous-
http://dx.doi.org/10.1016/j.isatra.2016.02.004 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
time switched systems, and [27,28] separately studied actuator fault estimation and accommodation approaches for continuous-time switched systems with constant time-delay case and time-varying delay case. However, to the best of our knowledge, the problems of actuator fault estimation and compensation for discrete-time switched systems with state delay are still under research, which motivate us to study this meaningful and challenging topic. Based on the above discussion, our objective of this paper is to analyze and develop a general framework of fault estimation and accommodation for discrete-time switched systems with actuator faults and state delay. The advantages of this paper are shown as the following aspects: firstly, the actuator fault estimation algorithm for discrete-time switched systems with state delay is resolved by using reduced-order observer approach; secondly, a switched dynamic output feedback controller is designed to compensate for the effect of actuator faults; thirdly, all the obtained efficient conditions are formulated in the form of linear matrix inequality technique, which can be easily solved; finally, a simulation example is provided to illustrate the effectiveness of the proposed results. The rest of this paper is organized as follows. Section 2 presents the system description. In Section 3, a switched reduced-order observer design, including an H 1 performance index is proposed to estimate the actuator fault. Furthermore, in Section 4, fault accommodation is given. An example is illustrated in Section 5 to show the effectiveness of the proposed approach, and the paper is concluded in Section 6.
2. Problem statements and preliminaries Consider the following discrete-time switched linear system with state delay: xðt þ 1Þ ¼
N X
δi ðtÞAi xðtÞ þ
i¼1
þ
N X
N X
δi ðtÞAdi xðt dÞ þ
i¼1
N X
δi ðtÞBi ðuðtÞ þ f ðtÞÞ
Since the matrix C is of fullh collum rank, there always exist a i ? matrix C ? A Rðn pÞn such that CC A Rnn is a nonsingular matrix.
3. Fault estimation design 3.1. State transformation We construct nonsingular matrix T ¼ CT ¼ 0pðn pÞ I p
δi ðtÞDi ωðtÞ
ð1Þ
ð2Þ
δi ðtÞC 2i xðtÞ
ð3Þ
where xðtÞ A Rn is the state vector, uðtÞ A Rm is the control input vector, yðtÞ A Rp is the measurable output vector, zðtÞ A Rl is the controlled output, f ðtÞ A Rm represents the additive actuator fault, ωðtÞ A Rr is the disturbance which is assumed to belong to l2 ½0; 1Þ, and the positive integer d denotes the known state delay. The vector δi ðtÞ is called switching signal, which specifies which subsystem will be activated at the discrete time t, where
δi ðtÞ : Z ¼ f0; 1; 2⋯g-f0; 1g;
N X
δi ðtÞ ¼ 1;
i¼1
N X
δi ðtÞ
"
# " # N X B1i D1i ðuðtÞ þ f ðtÞÞ þ δi ðtÞ ωðtÞ B2i D2i
ð6Þ
i¼1
" # x 1 ðtÞ yðtÞ ¼ 0pðn pÞ I p x 2 ðtÞ where " xðtÞ ¼
x 1 ðtÞ
" T 1 Bi ¼
"
# T 1 Ai T ¼
;
x 2 ðtÞ
B1i
#
" T 1 Di ¼
;
B2i
ð7Þ
# A12i ; A22i
A11i A21i
D1i
#
A11di A21di
# A12di ; A22di
CT ¼ 0pðn pÞ I p :
;
D2i
" T 1 Adi T ¼
Then, the system (6) and (7) can be rewritten as x 1 ðt þ 1Þ ¼
N X
δi ðtÞA11i x 1 ðtÞ þ
i¼1
8t AZ :
N X
δi ðtÞA12i yðtÞ
i¼1
N X
δi ðtÞA11di x 1 ðt dÞ þ
i¼1
ð4Þ
, which satisfies:
By using the condition in (5), the switched systems (1) and (2) can be transformed as follows: " #" " # # N X x 1 ðt þ 1Þ x 1 ðtÞ A11i A12i δi ðtÞ ¼ x 2 ðt þ 1Þ x 2 ðtÞ A21i A22i i¼1 # " #" N X A11di A12di x 1 ðt dÞ þ δi ðtÞ A21di A22di x 2 ðt dÞ
i¼1
xðt þ 1Þ ¼ f ðxðtÞÞ
i1
ð5Þ
þ
Lemma 1 (Vidyasagar [30]). The equilibrium 0 of
? C ðn pÞn C pn
xðtÞ ¼ TxðtÞ
þ
At an arbitrary continuous time t, δi ðtÞ is dependent on t or x(t), or both, or other switching rules. Similar to the switched signal adopted in [29], we assume that the sequence of subsystems in switching signal δi ðtÞ is a priori, that is its instantaneous value is available in real time. Matrices Ai, Adi, Bi, C, C2i, and Di are constant matrices with appropriate dimensions. For the purpose of this note, we give the following lemma and assumption:
h
Then define the following state transformation:
i¼1
i¼1
þ
Assumption 1. It is supposed that matrices Bi and C are of full rank, i.e., rankðBi Þ ¼ m and rankðCÞ ¼ p, and the pairs ðAi ; Bi Þ and ðAi ; CÞ are, respectively, controllable and observable.
þ
yðtÞ ¼ CxðtÞ zðtÞ ¼
(i) V is a positive definite function, decrescent, and radially unbounded; (ii) ΔVðt; xðtÞÞ ¼ Vðt þ 1; xðt þ1ÞÞ Vðt; xðtÞÞ is negative definite along the solutions of (4).
i¼1
i¼1
N X
is globally uniformly asymptotically stable if there is a function V : Z þ Rn -R such that
þ
N X
N X
δi ðtÞA12di yðt dÞ
i¼1
δi ðtÞB1i uðtÞ þ
i¼1
N X
δi ðtÞB1i f ðtÞ þ
i¼1
N X
δi ðtÞD1i ωðtÞ
i¼1
ð8Þ yðt þ1Þ ¼
N X
δi ðtÞA21i x 1 ðtÞ þ
i¼1
þ
N X
N X
δi ðtÞA21di x 1 ðt dÞ þ
i¼1
þ
N X i¼1
δi ðtÞA22i yðtÞ
i¼1 N X
δi ðtÞA22di yðt dÞ
i¼1
δi ðtÞB2i uðtÞ þ
N X i¼1
δi ðtÞB2i f ðtÞ þ
N X
δi ðtÞD2i ωðtÞ
ð9Þ
i¼1
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
By introducing the virtual input ηðtÞ and ρðtÞ:
ηðtÞ ¼
N X
δi ðtÞA12i yðtÞ þ
i¼1
N X
δi ðtÞA12di yðt dÞ þ
i¼1 N X
ρðtÞ ¼ yðt þ 1Þ
N X
δi ðtÞB1i uðtÞ
δi ðtÞA22i yðtÞ
i¼1
¼
δi ðtÞA22di yðt dÞ
i¼1
N X
δi ðtÞB2i uðtÞ
i¼1
þ
x 1 ðt þ1Þ ¼
δi ðtÞA11i x 1 ðtÞ þ
i¼1
þ
δi ðtÞB1i f ðtÞ þ
i¼1
ρðtÞ ¼
N X
N X
δi ðtÞA11di x 1 ðt dÞ þ ηðtÞ δi ðtÞD1i ωðtÞ
δi ðtÞA21i x 1 ðtÞ þ
N X
δi ðtÞA21di x 1 ðt dÞ þ
i¼1
N X
ð12Þ N X
þ
ð13Þ
For the dynamics (12) and (13), we construct the following reduced-order switched state observer:
δi ðtÞA11i x^ 1 ðtÞ þ
i¼1
i¼1
N X
þ
δi ðtÞB1i f^ ðtÞ
i¼1
ρ^ ðtÞ ¼
N X
δi ðtÞA11di x^ 1 ðt dÞ þ ηðtÞ
N X
δi ðtÞGi ðρ^ ðtÞ ρðtÞÞ
ð14Þ
i¼1
δi ðtÞA21i x^ 1 ðtÞ þ
i¼1
N X
δi ðtÞA21di x^ 1 ðt dÞ þ
i¼1
N X
δi ðtÞB2i f^ ðtÞ
i¼1
ð15Þ N X
f^ ðt þ 1Þ ¼ f^ ðtÞ
δi ðtÞF i ðρ^ ðtÞ ρðtÞÞ
ð16Þ
i¼1
where x^ ðtÞ A Rn p is the reduced-order observer state, ρ^ ðtÞ A Rp is the reduced-order observer output, f^ ðtÞ A Rm is the fault estimation of the fault f(t), Gi and Fi are reduced-order observer gain matrices to be designed. Let eðtÞ ¼ x^ 1 ðtÞ x 1 ðtÞ;
ef ðtÞ ¼ f^ ðtÞ f ðtÞ
ð17Þ
then the error dynamics are given as follows: eðt þ 1Þ ¼
N X
δi ðtÞðA11i Gi A21i ÞeðtÞ þ
i¼1
þ
N X
N X
δi ðtÞðA11di Gi A21di Þeðt dÞ
i¼1
δi ðtÞðB1i Gi B2i Þef ðtÞ þ
i¼1
N X
δi ðtÞðGi D2i D1i ÞωðtÞ
i¼1
ð18Þ ef ðt þ1Þ ¼ f^ ðt þ 1Þ f ðt þ 1Þ ¼ f^ ðtÞ
N X
δi ðtÞF i A21i eðtÞ
i¼1
N X
δi ðtÞF i B2i ef ðtÞ þ
i¼1
¼ f^ ðtÞ f ðtÞ
N X
N X
i¼1
δi ðtÞðA 1di G i A 2di Þeðt dÞ
i¼1
N X
δi ðtÞðG i D 1i D 2i ÞνðtÞ
ð20Þ
# eðtÞ eðtÞ ¼ ; ef ðtÞ
"
"
#
ωðtÞ νðtÞ ¼ ; Δf ðtÞ
A 2i ¼ ½A21i B2i ; " A11di A 1di ¼ 0mðn pÞ
# 0ðn pÞm ; 0mm
D 1i ¼ D2i 0pm ; " # 0ðn pÞm D1i D 2i ¼ ; 0md Im
A 1i ¼
A11i
B1i
0mðn pÞ
Im
# ;
A 2di ¼ A21di 0ðn pÞm ;
" Gi ¼
Gi
#
Fi
:
ð21Þ
The objective of this section is to determine the observer gains in order to guarantee asymptotic convergence of the estimated state in (14) to the state in (12). That is to say, the estimated error eðt þ 1Þ tends to zero when t-1. In the following, a reduced-order observer design method under an H 1 performance index is proposed to achieve robust fault estimation for discrete-time switched systems with state delay. Theorem 1. Given a constant γ 1 4 0, if there exists positive definite matrices Pi, Qi, nonsingular matrices Ω, and matrix Hi, for any i; j; l A f1; 2; …; Ng, such that 2 6 6 6 6 6 6 6 4
P j ðΩ þ Ω Þ T
ΩT A 1i Hi A 2i ΩT A 1di Hi A 2di Hi D 1i ΩT D 2i
nn
Pi þ Q i
0
0
nn
n
Ql
nn
n
n
0 γ 21 I r þ m
nn
n
n
n
0
3
7 Im 7 7 7 0 7o0 7 0 7 5 Im
ð22Þ T where I m ¼ 0ðn pÞm I m . Then the augmented switched system (20) is stable with H 1 performance index γ1, and the reduced-order fault estimation observer is given by T
ð23Þ
Hi
i¼1
Proof. Suppose that (22) holds, then it is easy to see from (22) that
δi ðtÞF i D2i ωðtÞ f ðt þ 1Þ
ðP j ΩÞT P j 1 ðP j ΩÞ Z 0
i¼1 N X
ð19Þ
i¼1
Gi ¼ Ω
δi ðtÞF i A21di eðt dÞ
N X
δi ðtÞðA 1i G i A 2i ÞeðtÞ þ
where "
3.2. Reduced-order observer design
x^ 1 ðt þ 1Þ ¼
δi ðtÞðIm F i B2i Þef ðtÞ Δf ðtÞ
i¼1
i¼1
N X
N X
N X
i¼1
N X
δi ðtÞF i D2i ωðtÞ
where Δf ðtÞ ¼ f ðt þ 1Þ f ðtÞ. Combining the dynamics in (18) and (19), one can get the following augmented system: eðt þ 1Þ ¼
δi ðtÞB2i f ðtÞ
δi ðtÞD2i ωðtÞ
δi ðtÞF i A21di eðt dÞ
i¼1
i¼1
i¼1
i¼1
þ
þ
i¼1
N X
N X
δi ðtÞF i A21i eðtÞ
i¼1
one can get N X
N X
N X
δi ðtÞF i D2i ωðtÞ ðf ðt þ 1Þ f ðtÞÞ
i¼1
i¼1
ð11Þ
N X
N X
δi ðtÞF i B2i ef ðtÞ þ
i¼1
ð10Þ
i¼1 N X
N X
3
δi ðtÞF i A21i eðtÞ
N X i¼1
ð24Þ
which implies that
δi ðtÞF i A21di eðt dÞ
P j ðΩ þ Ω Þ Z Ω P j 1 Ω T
T
ð25Þ
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Combining the condition in (23), (22) can be transformed into the following formation: 2 1 3 A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 0 Pj 6 7 6 nn P i þ Q i 0 0 Im 7 6 7 6 7 ð26Þ 6 nn n Q l 0 0 7o0 6 7 6 nn 7 2 n n γ I 0 4 5 1 r þm nn n n n Im
It is noted that 02
γ ν
i¼1
ð27Þ
i¼1
s ¼ t d
and define ΔV ðtÞ ¼ Vðt þ 1Þ VðtÞ. Then along the error dynamics (20), ! N X ΔVðtÞ ¼ e T ðt þ 1Þ δi ðt þ1ÞP i eðt þ 1Þ i¼1 t X
þ
e T ðsÞ N X
e ðtÞ
δi ðtÞP i eðtÞ
i¼1 N X
T
¼ e ðt þ 1Þ
T
e ðsÞ
δi ðt þ 1ÞP i eðt þ 1Þ e ðtÞ
N X
N X i¼1
s ¼ t d
!
T
i¼1
þ e T ðtÞ
t 1 X
N X i¼1
!
δi ðtÞQ i eðtÞ e T ðt dÞ
i¼1
N X
δi ðsÞQ i eðsÞ
3.3. Fault estimation algorithm design
!
δi ðtÞP i eðtÞ !
δi ðt dÞQ i eðt dÞ
i¼1
It follows that for any nonzero vector eðtÞ and the particular case δi ðtÞ ¼ 1, δk a i ðtÞ ¼ 0, δj ðt þ 1Þ ¼ 1, δk a j ðt þ 1Þ ¼ 0, and δl ðt dÞ ¼ 1, δk a l ðt dÞ ¼ 0, we have 02 3 ðA 1i G i A 2i ÞT T B6 ΔVðtÞ ¼ ϑ ðtÞ@4 ðA 1di G i A 2di ÞT 7 5 ðG i D 1i D 2i ÞT h i P j A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 2 31 0 0 Pi þ Q i 6 7C 0 Q l 0 5AϑðtÞ ð29Þ þ4 0
eTf ðtÞef ðtÞ γ 21 νT ðtÞνðtÞ
ð30Þ
i eTf ðtÞef ðtÞ γ 21 νT ðtÞνðtÞ þ ΔVðtÞ VðKÞ
r
t¼0
þ
N X
f ðtÞ
i¼1
δi ðtÞðA 1di G i A 2di Þ
T
I m I m e T ðtÞeðtÞ γ 21 νT ðtÞνðtÞ þ ΔVðtÞ
i¼1 N X
x~^ 1 ðt dÞ f^ ðt dÞ
δi ðtÞG i ρðtÞ þ I ηðtÞ
ð36Þ
h i In p . Substituting (10) and (11) into (36), one has where I ¼ 0mðn pÞ N ^ P x~ 1 ðt þ 1Þ x~^ 1 ðtÞ δ ðtÞðA G A Þ ¼ i 1i i 2i ^ ^ f ðt þ 1Þ
þ
N X
f ðtÞ
i¼1
δi ðtÞðA 1di G i A 2di Þ
þ
N X
δi ðtÞG i yðt þ 1Þ þ
þ
N X
x~^ 1 ðt dÞ f^ ðt dÞ
N X
δi ðtÞðIA12i G i A22i ÞyðtÞ
i¼1 N X
δi ðtÞðIA12di G i A22di Þyðt dÞ þ
i¼1
δi ðtÞðIB1i G i B2i ÞuðtÞ
i¼1
ð37Þ
t¼0 KX 1
f ðt þ 1Þ
i¼1
where K is an arbitrary positive integer. For any nonzero νðtÞ A l2 ½ 0; 1Þ and zero initial condition eðtÞ ¼ 0, one has KX 1 h
Combining (15) and (21), one has N P x~^ 1 ðt þ 1Þ x~^ 1 ðtÞ ¼ δ ðtÞðA G A Þ i 1i i 2i ^ ^
i¼1
i
t¼0
J 1K ¼
i¼1
i¼1
Then from the condition in (28), one can get that ΔVðtÞ o 0, which concludes from Lemma 1 that augmented switched system (20) is stable. Let KX 1 h
Based on the obtained results in the subsection 3.2, we construct the following augmented system: " # N A11i B1i x~^ ðtÞ P x~^ 1 ðt þ 1Þ 1 δ ðtÞ ¼ i f^ ðt þ 1Þ 0mðn pÞ I m f^ ðtÞ i¼1 " # N X 0ðn pÞm x~^ ðt dÞ A11di 1 δi ðtÞ 0 þ 0m f^ ðt dÞ mðn pÞ i¼1 " # " # N X In p Gi δi ðtÞ ηðtÞ ð35Þ ðρ^ ðtÞ ρðtÞÞ þ 0mðn pÞ Fi
þ
0
h iT where ϑðtÞ ¼ e T ðtÞ e T ðt dÞ νT ðtÞ .
J 1K ¼
ð34Þ
which implies, for any K, J 1K o0. Then, one has that for any nonzero νðtÞ A l2 ½0; 1Þ, J ef ðtÞ J 2 o γ 1 J νðtÞ J 2 . □
!
ð28Þ
0
By Schur complement formula, the inequality is equivalent to the following inequality: 2 1 3 Pj A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 6 7 T 6 7 P i þ Q i þI m I m 0 0 6 nn 7 ð33Þ 6 7o0 6 nn 7 n Ql 0 4 5 nn n n γ 21 I r þ m
eTf ðtÞef ðtÞ γ 21 νT ðtÞνðtÞ þ ΔVðtÞ o0
δi ðsÞQ i eðsÞ
!
γ 21 I r þ m
0
It follows from (33) and Schur complement formula that:
!
i¼1
s ¼ t dþ1 T
N X
3
νðtÞ þ ΔVðtÞ
0 Consider the following switched Lyapunov function: ! ! N t 1 N X X X δi ðtÞP i eðtÞ þ e T ðsÞ δi ðsÞQ i eðsÞ VðtÞ ¼ e T ðtÞ
ðA 1i G i A 2i ÞT
T B6 7 ¼ ϑ ðtÞ@4 ðA 1di G i A 2di ÞT 5 T ðG i D 1i D 2i Þ h i ; P j A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 2 31 T Pi þ Q i þ I m I m 0 0 6 7C 7CϑðtÞ þ6 ð32Þ 0 0 Q l 4 5A
T I m I m e T ðtÞeðtÞ 21 T ðtÞ
ð31Þ
Define the following invariable: N P ~^ ζ ðt þ1Þ ¼ x^1 ðt þ 1Þ δi ðtÞG i yðt þ1Þ f ðt þ 1Þ
ð38Þ
i¼1
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Then, it follows from (37) that N X
ζ ðt þ 1Þ ¼
δi ðtÞðA 1i G i A 2i Þζ ðtÞ þ
i¼1
þ
N X
N X
δi ðtÞðA 1di G i A 2di Þζ ðt dÞ
δi ðtÞðIA12i G i A22i ÞyðtÞ þ
N X
δi ðtÞðIA12di G i A22di Þyðt dÞ
δi ðtÞðIB1i G i B2i ÞuðtÞ
ð39Þ
Furthermore, it follows from that N P x~^ 1 ðtÞ ¼ ζ ðtÞ δi ðt 1ÞG i yðtÞ ^ f ðtÞ
ð40Þ
i¼1
PN where i ¼ 1 δi ðt 1Þ means the subsystem of time t 1 is activated.
0
;
C~ i ¼ ½C 2i 0;
On the basis of the obtained online fault estimation information, we design a fault tolerant controller to guarantee the stability in the presence of faults. Since the state x(t) is unavailable directly, we utilize the switched dynamical output feedback controller scheme to construct the fault tolerant controller. For the discretetime switched system (1)–(3), we consider the following dynamic output feedback controller: N X
^ þ 1Þ ¼ xðt
^ þ δi ðtÞAKi xðtÞ
i¼1
ð47Þ
4.1. H 1 performance analysis
N X
Lemma 2. Given a constant γ 2 4 0, the augmented switched systems (45) and (46) is stable with H 1 performance γ2, if there exist positive matrices P and Q , such that the following linear matrix inequality holds: 2 3 P A~ i P A~ di P B~ i P 6 7 6 nn P þ H T Q H þ C~ T C~ 0 0 7 6 7 i i ð48Þ 6 7 o0 6 nn 7 n Q 0 4 5 nn n n γ 22 I
ξðt þ 1Þ ¼
VðtÞ ¼ ξ ðtÞP ξðtÞ þ T
ð49Þ
ξT ðsÞHT Q H ξðsÞ
ð50Þ
s ¼ t d
t X
ΔVðtÞ ¼ ξT ðt þ1ÞP ξðt þ 1Þ þ ð42Þ
^ þ δi ðtÞAKi xðtÞ
N X
δi ðtÞBKi CxðtÞ
ð43Þ
i¼1
Then substituting u(t) into (1), one has N X
N X
δi ðtÞAi xðtÞ þ
i¼1
þ
δi ðtÞAdi xðt dÞ þ
i¼1
N X
N X
δi ðtÞBi ðf ðtÞ f^ ðtÞÞ þ
ξT ðsÞHT Q HξðsÞ ξT ðtÞP ξðtÞ
i¼1
N X
N X i¼1
δi ðtÞBi ef ðtÞ þ
i¼1
N X
^ δi ðtÞBi C Ki xðtÞ
δi ðtÞDi ωðtÞ
ð44Þ
i¼1
δi ðtÞA~ i ξðtÞ þ
i¼1
T
N X
T
T
ð51Þ
This with the relationship xðt dÞ ¼ H ξðt dÞ
J 2K ¼
ð52Þ
KX 1
zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ
ð53Þ
t¼0
δi ðtÞA~ di Hξðt dÞ þ
i¼1
N X
δi ðtÞB~ i μðtÞ
J 2K ¼
i¼1
KX 1
zT ðtÞξðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ VðKÞ
t¼0
ð45Þ r
δi ðtÞC~ i ξðtÞ
ð46Þ
i¼1
KX 1
zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ
ð54Þ
t¼0
It is noted that 02
where xðtÞ
T
ξ ðt dÞH Q H ξðt dÞ T
where K is an arbitrary positive integer. For any nonzero μðtÞ A ½0; 1Þ and zero initial condition ξðtÞ ¼ 0, one has
It follows that N X
¼ ξ ðt þ 1ÞP ξðt þ1Þ ξ ðtÞP ξðtÞ þ ξ ðtÞH T Q H ξðtÞ
i¼1 N X
ξT ðsÞHT Q HξðsÞ
s ¼ td
3 " #1 T h i A~ i T P þ HT Q H 0 ~ ~ ~ 4 5 @ Aξ~ ðtÞ ΔVðtÞ ¼ ξ ðtÞ P A i A di þ T 0 Q A~ di h iT T where ξ~ ðtÞ ¼ ξ ðtÞ xT ðt dÞ , which follows from (65) and Schur complete lemma that the system (49) is stable. Let
^ δi ðtÞBi C Ki xðtÞ
δi ðtÞDi ωðtÞ
δi ðtÞAdi xðt dÞ þ
t1 X
which implies 02
i¼1
δi ðtÞAi xðtÞ þ
N X
N X i¼1
i¼1
N X
t 1 X
ð41Þ
^ f^ ðtÞ δi ðtÞC Ki xðtÞ
N X
ξðt þ 1Þ ¼
δi ðtÞA~ di Hξðt dÞ
i¼1
Then along the error dynamics (49),
δi ðtÞBKi yðtÞ
i¼1
i¼1
¼
N X
Define Lyapunov function:
^ A Rn is the controller state vector, AKi, BKi and CKi are where xðtÞ controller gains to be determined later. Substituting (2) into (41), one has
xðt þ 1Þ ¼
δi ðtÞA~ i ξðtÞ þ
i¼1
i¼1
^ þ 1Þ ¼ xðt
N X
s ¼ t dþ1
N X
"
H ¼ ½I 0:
Proof. Firstly, we establish the stability of augmented switched system (45). When μðtÞ ¼ 0, (45) becomes
4. Fault accommodation design
zðtÞ ¼
i¼1
i¼1
uðtÞ ¼
Bi
i¼1
i¼1
þ
N X
Di B~ i ¼ 0
5
#
ξðtÞ ¼ ^ ; xðtÞ
"
#
ωðtÞ μðtÞ ¼ e ðtÞ ; f
" A~ i ¼
Ai
Bi C Ki
BKi C
AKi
# ;
Adi A~ di ¼ ; 0
zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ
3 T A~ B6 Ti 7 h i B6 7 ¼ z T ðtÞB6 A~ di 7P A~ i A~ di B~ i @4 5 T B~ i
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2 6 þ6 4
T P þ H T Q H þ C~ i C~ i
0
0
0
Q
0
0
γ
0
31
2 2I
Observe that
7C 7CzðtÞ 5A
Ξ ST X 1 S ¼ ST ðYX IÞ 1 ðY X 1 ÞðXY IÞ 1 S 4 0
Then we have P 4 0. Considering the notations in (47) with, and AKi, BKi, and CKi in (57), we can verify that the matrix inequality in (56) can be rewritten as
h iT T where zðtÞ ¼ ξ ðtÞ xT ðt dÞ μT ðtÞ . It follows from (65) and Schur complement formula that zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ o0
2
ð55Þ
which implies, for any K, J 2K o 0. Then, one has that for any nonzero νðtÞ A ½0; 1Þ, J zðtÞ J 2 o γ 2 J μðtÞ J 2 . □ Based on Lemma 1, a dynamic output feedback controller design method is developed in the following theorem. Theorem 2. Given a constant γ 2 4 0, then there exists a dynamic output feedback controller in the form of (41) and (42) such that the augmented switched system (45) and (46) is stable with H 1 performance γ2 if there exist positive definite matrices X, Y, matrices Ωi, Mi, Ni, such that the following linear matrix inequalities: 2 3 Λ11 Λ12 Λ13 0 0 6 nn Λ 0 Λ24 H 7 6 7 22 6 7 2 6 nn n γ2I 0 0 7 ð56Þ 6 7o0 6 nn n n I 0 7 4 5 nn
n
n
n
Q
1
i A f1; 2; …; Ng and some given positive matrix Q , where " # Ai Adi Ai Y þBi Ni I ; Λ12 ¼ ; Ωi XAi þ M i C XAdi X # Bi ; XBi 3 2 T T 3 2 3 I 0 Y Y C 2i 7 6 7 7 X 0 5; Λ24 ¼ 4 C T 5; H ¼ 6 4 I 5: 2i 0 Q 0 0
holds, for all Y Λ11 ¼ I " Di Λ13 ¼ XDi 2 Y 6 I Λ22 ¼ 4 0
In this case, a desired stable dynamic output feedback H 1 controller is given the parameters as follows: AKi ¼ S 1 ðΩi XAi Y M i CY XBi N i ÞW T ; T
C Ki ¼ Ni W
BKi ¼ S 1 M i ;
:
ð57Þ
where S and W are any nonsingular matrices satisfying SW T ¼ I XY ð58Þ Proof. Suppose that the inequalities (56) holds, which implies Y I o0 ð59Þ I X which, by the Schur complement formula, gives that X Y 1 4 0. Therefore, I XY is nonsingular, hence there exist nonsingular matrices S and W such that (58) holds. Now, we introduce the following nonsingular matrices: Y I I X ; Π2 ¼ ð60Þ Π1 ¼ T ; T 0 W 0 S Let 1
P ¼ Π2Π1 then X P¼ T S
S
ð61Þ
6 6 6 6 6 6 6 6 6 6 4
Π2P T
Π T2 A~ i Π 1
Π2
ð62Þ
0
0
0
Π T1 C~ i
nn
n
Q
nn
n
n
0 γ 22 I
0 0
nn
n
n
n
I
nn
n
n
n
n
T
T
3
0
Π 1 P Π 1 þ Π 1 C~ i
T
7 7 Π T1 H T 7 7 0 0
0 1 Q
7 7o0 7 7 7 7 5
ð65Þ T 2 ;
T 1 ; I; I; I; Ig
Pre- and post-multiplying this inequality by diagfΠ Π and its transpose, respectively, which, by the Schur complement, the result then follows from Lemma 1. □
Remark 1. For the sake of clarity, only a class of actuator fault case is considered in this paper. However, note that such additive actuator faults considered in this paper can be readily extended to general class of additive faults [31]. Meanwhile, the additive faults representation is more general than the multiplicative ones, which can be modeled as additive actuator ones [6]. Remark 2. The time delay is assumed to be constant in this paper. Moreover, by using the technique in [32], the results may be extended to time-varying case. Remark 3. For the existence of the terms Adi, switched Lyapunov function is not suitable in designing dynamic output feedback controller. A common Lyapunov function is employed to achieve the results. This may result in some conservatism.
5. An illustrate example Consider the discrete-time switched linear system (1)–(3) consisting of three subsystems described by 0:5 0:3 0:2 0:1 1 0:2 A1 ¼ ; A2 ¼ ; A3 ¼ ; 0:1 1 1 0:3 2 0:05 0:1 0:3 Ad1 ¼ ; 0 1 0:2 0:1 0:2 1 0:1 Ad2 ¼ ; Ad3 ¼ ; B1 ¼ ; 0 0:3 0 0:05 0:2 0:1 0:1 B2 ¼ ; B3 ¼ ; 0:2 0:2 0:1 0:1 0:1 D1 ¼ ; D2 ¼ ; D3 ¼ ; C 11 ¼ C 12 ¼ C 13 ¼ ½1 2; 0:2 0:2 0:2 C 21 ¼ C 22 ¼ C 23 ¼ ½2 1: The state transformation matrix can follows that the augmented matrices 0:65 0:1 0:25 A 11 ¼ ; A 12 ¼ 0 1 0
0 . It be chosen as T ¼ 10:5 0:5 in (21) can be constructed as 0:1 1:1 0:1 ; A 13 ¼ ; 1 0 1
A 21 ¼ ½0:15 0:5;
A 1d2 ¼
ð63Þ
Π T2 A~ di Π T2 B~ i
nn
where
Ξ ¼ W 1 YðX Y 1 ÞYW T 40
1
A 22 ¼ ½ 0:05 0:5;
Ξ
ð64Þ
A 1d3 ¼
0:25
0
0 0:3
0
0
0
0
A 23 ¼ ½ 0:1 0:5; ; ;
A 2d1 ¼ 0:15
0 ;
A 1d1 ¼
0:05
0
0
0
;
A 2d2 ¼ ½0:05 0;
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
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2.5 2 1.5 1 0.5 0 −0.5
0
10
20
30
40
50
60
70
80
90
100
90
100
Fig. 1. Fault f 1 ðtÞ (solid line) and its estimate f ^1 ðtÞ (dotted line).
3.5 3 2.5 2 1.5 1 0.5 0 −0.5
0
10
20
30
40
50
60
70
80
Fig. 2. Fault f 2 ðtÞ (solid line) and its estimate f ^2 ðtÞ (dotted line).
A 2d3 ¼ ½0:5 0; D 11 ¼ ½0:5 0;
0:1
0
0 0:1
1 0 : 1
D 22 ¼ D 23 ¼
D 12 ¼ ½0:5 0;
0
;
D 13 ¼ ½0:5 0;
D 21 ¼
0:1 0
0 ; 1
By solving the condition (22) in Theorem 2, one can get the optimal value γ 1 ¼ 1:82 with reduced-order observer parameters as follows: 0:4401 0:1357 0:0551 G1 ¼ ; G2 ¼ ; G3 ¼ : 0:0039 0:0007 0:0026 1 0 Then, the positive definite matrix Q is set up as 0 2 , by solving the conditions in Theorem 2, one obtains the minimum attenuation value γ 2 ¼ 2:16 with the dynamic output feedback controller parameters as follows: 0:2997 2:9905 0:1028 1:5334 ; AK2 ¼ ; AK1 ¼ 0:1125 2:2276 0:0709 1:6935 0:1191 0:1245 AK3 ¼ ; 0:1377 4:0849 0:8266 0:6498 0:4300 BK1 ¼ ; BK2 ¼ ; BK3 ¼ ; 0:4915 0:3593 0:9382 C K1 ¼ ½0:4399 11:6092; C K2 ¼ ½ 1:2777 0:5005; C K3 ¼ ½ 3:0098 5:7662: For simulation, an actuator constant fault signal f 1 ðtÞ is set up as ( 0; 0 r t r 20 f 1 ðtÞ ¼ 2; 20 o t r 100 If the external disturbance is set as a step signal, Fig. 1 depicts the fault f 1 ðtÞ (solid line) and its estimate f^ 1 ðtÞ (dotted line). An actuator time-varying fault signal f 2 ðtÞ is set up as ( 0; 0 r t r 20 f 2 ðtÞ ¼ sin ðtÞ þ 2; 20 o t r 100 Fig. 2depicts the fault f 2 ðtÞ (solid line) and its estimate f^ 2 ðtÞ (dotted line). It follows from Figs. 1 and 2 that the proposed reduced-order
0.3
0.2 x1(t) x2(t)
0.1
0
−0.1
−0.2
0
10
20
30
40
50
60
70
80
90
100
time step k Fig. 3. State x(t) with fault accommodation.
switched observer has good performance to estimate the actuator fault. Therefore, the simulation results exhibit well estimated operation. Fig. 3 shows the simulation result of state response with fault accommodation.
6. Conclusion By utilizing the reduced-order rank observer method and switched Lyapunov function technique, fault estimation and compensation for discrete-time switched system with state delay are studied in this work, and an efficient reduced-order rank observer design is obtained in the form of linear matrix inequality. Through a dynamic output feedback controller fault compensation, an fault-tolerant operation is realized. Finally, simulation results are presented to verify the effectiveness of the proposed method.
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
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Acknowledgments This work was partially supported by the Natural Science Foundation of Jiangsu Province (BK20140457), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Fund of Huaiyin Institute of Technology (HGC1309).
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[14] Edwards C, Spurgeon SK, Patton RJ. Sliding mode observers for fault detection and isolation. Automatica 2000;36(4):541–53. [15] Tan C, Tao G, Qi R. A discrete-time parameter estimation based adaptive actuator failure compensation control scheme. Int J Control 2013;86(2):276–89. [16] Polycarpou MM. Fault accommodation of a class of multivariable nonlinear dynamical systems using a learning approach. IEEE Trans Autom Control 2001;46(5):736–42. [17] Zhao J, Hill DJ. Passivity and stability of switched systems: a multiple storage function method. Syst Control Lett 2008;57(2):158–64. [18] Vesely V, Rosinova D. Robust MPC controller design for switched systems using multi-parameter dependent Lyapunov function. Int J Innov Comput Inf Control 2014;10(1):269–80. [19] Wu LG, Zheng WX. Weighted H1 model reduction for linear switched systems with time-varying delay. Automatica 2009;45(1):186–93. [20] Attia SB, Salhi S, Ksouri M. Static switched output feedback stabilization for linear discrete-time switched systems. Int J Innov Comput Inf Control 2012;8 (5 (A)):3203–13. [21] Zhang LX, Gao HJ. Asynchronously switched control of switched linear systems with average dwell time. Automatica 2010;46:953–8. [22] Zhang LX, Shi P, Boukas E, Wang C. Hinfty model reduction for uncertain switched linear discrete-time systems. Automatica 2008;44:2944–9. [23] Cheng D, Guo L, Lin Y, Wang Y. Stabilization of switched linear systems. IEEE Trans Autom Control 2005;50(5):661–6. [24] Wang D, Wang W, Shi P. Robust fault detection for switched linear systems with state delays. IEEE Trans Syst Man Cybern 2009;39(3):800–5. [25] Du D, Jiang B, Shi P. Fault detection for discrete-time switched systems with intermittent measurements. Int J Control 2012;85(1):78–87. [26] Du D, Jiang B, Shi P. Sensor fault estimation and compensation for time-delay switched systems. Int J Syst Sci 2012;43(4):629–40. [27] Du D, Jiang B, Shi P. Active fault-tolerant control for switched systems with time delay. Int J Adapt Control Signal Process 2011;25:466–80. [28] Du D, Jiang B, Shi P. Fault estimation and accommodation for switched systems with time-varying delay. Int J Control Autom Syst 2011;9(3):442–51. [29] Daafouz J, Riedinger P, Iung C. Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans Autom Control 2002;47(11):1883–7. [30] Vidyasagar M. Nonlinear systems analysis. Englewood Cliffs, NJ: Prentice-Hall International Editions; 1993. [31] Patton RJ, Chen J, Lopez-Toribio CJ. Fuzzy observers for nonlinear dynamic systems fault diagnosis. In: Proceedings of 37th IEEE conference on decision control, Tampa, FL, 1998. p. 84–9. [32] Du D, Jiang B, Zhou S. Delay-dependent robust stabilization of uncertain discrete-time switched systems with time-varying state delay. Int J Syst Sci 2008;39(3):305–13.
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Research Article
Actuator fault estimation and accommodation for switched systems with time delay: Discrete-time case$ Dongsheng Du a,b,n, Bin Jiang c,d a
College of Automation, Huaiyin Institute of Technology, 1 Meicheng Road, Huaian, 223003 Jiangsu, China College of Automation, Nanjing University of Science and Technology, Nanjing 210094, China c College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China d Jiangsu Key Laboratory of Internet of Things and Control Technologies (Nanjing University of Aeronautics and Astronautics), China b
art ic l e i nf o
a b s t r a c t
Article history: Received 5 June 2015 Received in revised form 4 November 2015 Accepted 5 February 2016
This paper investigates the problems of actuator fault estimation and accommodation for discrete-time switched systems with state delay. By using reduced-order observer method and switched Lyapunov function technique, a fault estimation algorithm is achieved for the discrete-time switched system with actuator fault and state delay. Then based on the obtained online fault estimation information, a switched dynamic output feedback controller is employed to compensate for the effect of faults by stabilizing the closed-loop systems. Finally, an example is proposed to illustrate the obtained results. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Discrete-time switched system Fault estimation Reduced-order observer Output feedback controller
1. Introduction In modern complex systems, dependability is as important as performances. Once a fault (sensor, actuator, or component failures) occurred in the system, the system's behavior may be drastically decreased, ranging from performance degradation to instability. Making a general survey of the related literature, fault tolerant control (FTC) technique is an efficient method in order to reach the system objectives, or if this turns to be impossible, to achievable new objectives to avoid catastrophic behaviors. Therefore, FTC have been the subjects of intensive investigations over the past two decades, and a number of achievements have been obtained, which can be found in several excellent papers, please refer to [1–9] and the references therein. As for FTC, which can be divided into two main approaches: passive and active. While passive fault tolerance considers systems faults as a special kind of uncertainties [10], active fault tolerance is based on fault detection and isolation (FDI) and accommodation technique [11]. Though FDI can give information whether there exist faults occurring [12], the magnitude of the fault can not be precisely provided. Therefore, the next step is to search an efficient approach to estimate the magnitude of the fault, which is called fault estimate or fault reconstruction [13]. Finally, using the ☆
This paper was recommended for publication by Dr. Steven Ding. Corresponding author at: College of Automation, Huaiyin Institute of Technology, 1 Meicheng Road, 223003 Jiangsu, China. E-mail addresses: [email protected] (D. Du), [email protected] (B. Jiang). n
estimated fault information, a fault-tolerant controller can be designed to compensate the effect of the fault. From the above discussion, it can be seen that fault estimation plays a very important role in active FTC. Therefore, the study of fault estimation has become a hot research topic owing to its importance in active FTC. During the past decade, various effective methods, such as sliding mode observer approach using equivalent output injection signal [14], adaptive technique [15], and learning method based on neural network [16] and so on, have been developed to realize fault estimation. Generally speaking, adaptive technique is usually used to obtain actuator fault information for continuous case [11]. Another commonly employed to realize fault estimation is based on descriptor observer technique, but it is usually utilized to estimate sensor fault [26]. Recently, reduced-order observer method is proposed to solve actuator fault estimation for discrete time systems, this motives us to investigate this issue for discretetime switched systems. Switched system is a class of hybrid systems, which consists of a finite number of subsystems and an associated switching signal governing the switching among them. Many real-world process and systems can be modeled as switched systems, including chemical processes, computer controlled systems, switched circuits, and so on. Therefore, it has been intensively investigated in the past decades [17– 23]. Compared with fruitful stability and stabilization results for switched systems, fault diagnosis and FTC achievements are relatively few. In [24,25], fault detection for switched system was separately investigated for continuous case and discrete case. Du et al. [26] investigated sensor fault estimation and accommodation approaches for continuous-
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Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
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time switched systems, and [27,28] separately studied actuator fault estimation and accommodation approaches for continuous-time switched systems with constant time-delay case and time-varying delay case. However, to the best of our knowledge, the problems of actuator fault estimation and compensation for discrete-time switched systems with state delay are still under research, which motivate us to study this meaningful and challenging topic. Based on the above discussion, our objective of this paper is to analyze and develop a general framework of fault estimation and accommodation for discrete-time switched systems with actuator faults and state delay. The advantages of this paper are shown as the following aspects: firstly, the actuator fault estimation algorithm for discrete-time switched systems with state delay is resolved by using reduced-order observer approach; secondly, a switched dynamic output feedback controller is designed to compensate for the effect of actuator faults; thirdly, all the obtained efficient conditions are formulated in the form of linear matrix inequality technique, which can be easily solved; finally, a simulation example is provided to illustrate the effectiveness of the proposed results. The rest of this paper is organized as follows. Section 2 presents the system description. In Section 3, a switched reduced-order observer design, including an H 1 performance index is proposed to estimate the actuator fault. Furthermore, in Section 4, fault accommodation is given. An example is illustrated in Section 5 to show the effectiveness of the proposed approach, and the paper is concluded in Section 6.
2. Problem statements and preliminaries Consider the following discrete-time switched linear system with state delay: xðt þ 1Þ ¼
N X
δi ðtÞAi xðtÞ þ
i¼1
þ
N X
N X
δi ðtÞAdi xðt dÞ þ
i¼1
N X
δi ðtÞBi ðuðtÞ þ f ðtÞÞ
Since the matrix C is of fullh collum rank, there always exist a i ? matrix C ? A Rðn pÞn such that CC A Rnn is a nonsingular matrix.
3. Fault estimation design 3.1. State transformation We construct nonsingular matrix T ¼ CT ¼ 0pðn pÞ I p
δi ðtÞDi ωðtÞ
ð1Þ
ð2Þ
δi ðtÞC 2i xðtÞ
ð3Þ
where xðtÞ A Rn is the state vector, uðtÞ A Rm is the control input vector, yðtÞ A Rp is the measurable output vector, zðtÞ A Rl is the controlled output, f ðtÞ A Rm represents the additive actuator fault, ωðtÞ A Rr is the disturbance which is assumed to belong to l2 ½0; 1Þ, and the positive integer d denotes the known state delay. The vector δi ðtÞ is called switching signal, which specifies which subsystem will be activated at the discrete time t, where
δi ðtÞ : Z ¼ f0; 1; 2⋯g-f0; 1g;
N X
δi ðtÞ ¼ 1;
i¼1
N X
δi ðtÞ
"
# " # N X B1i D1i ðuðtÞ þ f ðtÞÞ þ δi ðtÞ ωðtÞ B2i D2i
ð6Þ
i¼1
" # x 1 ðtÞ yðtÞ ¼ 0pðn pÞ I p x 2 ðtÞ where " xðtÞ ¼
x 1 ðtÞ
" T 1 Bi ¼
"
# T 1 Ai T ¼
;
x 2 ðtÞ
B1i
#
" T 1 Di ¼
;
B2i
ð7Þ
# A12i ; A22i
A11i A21i
D1i
#
A11di A21di
# A12di ; A22di
CT ¼ 0pðn pÞ I p :
;
D2i
" T 1 Adi T ¼
Then, the system (6) and (7) can be rewritten as x 1 ðt þ 1Þ ¼
N X
δi ðtÞA11i x 1 ðtÞ þ
i¼1
8t AZ :
N X
δi ðtÞA12i yðtÞ
i¼1
N X
δi ðtÞA11di x 1 ðt dÞ þ
i¼1
ð4Þ
, which satisfies:
By using the condition in (5), the switched systems (1) and (2) can be transformed as follows: " #" " # # N X x 1 ðt þ 1Þ x 1 ðtÞ A11i A12i δi ðtÞ ¼ x 2 ðt þ 1Þ x 2 ðtÞ A21i A22i i¼1 # " #" N X A11di A12di x 1 ðt dÞ þ δi ðtÞ A21di A22di x 2 ðt dÞ
i¼1
xðt þ 1Þ ¼ f ðxðtÞÞ
i1
ð5Þ
þ
Lemma 1 (Vidyasagar [30]). The equilibrium 0 of
? C ðn pÞn C pn
xðtÞ ¼ TxðtÞ
þ
At an arbitrary continuous time t, δi ðtÞ is dependent on t or x(t), or both, or other switching rules. Similar to the switched signal adopted in [29], we assume that the sequence of subsystems in switching signal δi ðtÞ is a priori, that is its instantaneous value is available in real time. Matrices Ai, Adi, Bi, C, C2i, and Di are constant matrices with appropriate dimensions. For the purpose of this note, we give the following lemma and assumption:
h
Then define the following state transformation:
i¼1
i¼1
þ
Assumption 1. It is supposed that matrices Bi and C are of full rank, i.e., rankðBi Þ ¼ m and rankðCÞ ¼ p, and the pairs ðAi ; Bi Þ and ðAi ; CÞ are, respectively, controllable and observable.
þ
yðtÞ ¼ CxðtÞ zðtÞ ¼
(i) V is a positive definite function, decrescent, and radially unbounded; (ii) ΔVðt; xðtÞÞ ¼ Vðt þ 1; xðt þ1ÞÞ Vðt; xðtÞÞ is negative definite along the solutions of (4).
i¼1
i¼1
N X
is globally uniformly asymptotically stable if there is a function V : Z þ Rn -R such that
þ
N X
N X
δi ðtÞA12di yðt dÞ
i¼1
δi ðtÞB1i uðtÞ þ
i¼1
N X
δi ðtÞB1i f ðtÞ þ
i¼1
N X
δi ðtÞD1i ωðtÞ
i¼1
ð8Þ yðt þ1Þ ¼
N X
δi ðtÞA21i x 1 ðtÞ þ
i¼1
þ
N X
N X
δi ðtÞA21di x 1 ðt dÞ þ
i¼1
þ
N X i¼1
δi ðtÞA22i yðtÞ
i¼1 N X
δi ðtÞA22di yðt dÞ
i¼1
δi ðtÞB2i uðtÞ þ
N X i¼1
δi ðtÞB2i f ðtÞ þ
N X
δi ðtÞD2i ωðtÞ
ð9Þ
i¼1
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
By introducing the virtual input ηðtÞ and ρðtÞ:
ηðtÞ ¼
N X
δi ðtÞA12i yðtÞ þ
i¼1
N X
δi ðtÞA12di yðt dÞ þ
i¼1 N X
ρðtÞ ¼ yðt þ 1Þ
N X
δi ðtÞB1i uðtÞ
δi ðtÞA22i yðtÞ
i¼1
¼
δi ðtÞA22di yðt dÞ
i¼1
N X
δi ðtÞB2i uðtÞ
i¼1
þ
x 1 ðt þ1Þ ¼
δi ðtÞA11i x 1 ðtÞ þ
i¼1
þ
δi ðtÞB1i f ðtÞ þ
i¼1
ρðtÞ ¼
N X
N X
δi ðtÞA11di x 1 ðt dÞ þ ηðtÞ δi ðtÞD1i ωðtÞ
δi ðtÞA21i x 1 ðtÞ þ
N X
δi ðtÞA21di x 1 ðt dÞ þ
i¼1
N X
ð12Þ N X
þ
ð13Þ
For the dynamics (12) and (13), we construct the following reduced-order switched state observer:
δi ðtÞA11i x^ 1 ðtÞ þ
i¼1
i¼1
N X
þ
δi ðtÞB1i f^ ðtÞ
i¼1
ρ^ ðtÞ ¼
N X
δi ðtÞA11di x^ 1 ðt dÞ þ ηðtÞ
N X
δi ðtÞGi ðρ^ ðtÞ ρðtÞÞ
ð14Þ
i¼1
δi ðtÞA21i x^ 1 ðtÞ þ
i¼1
N X
δi ðtÞA21di x^ 1 ðt dÞ þ
i¼1
N X
δi ðtÞB2i f^ ðtÞ
i¼1
ð15Þ N X
f^ ðt þ 1Þ ¼ f^ ðtÞ
δi ðtÞF i ðρ^ ðtÞ ρðtÞÞ
ð16Þ
i¼1
where x^ ðtÞ A Rn p is the reduced-order observer state, ρ^ ðtÞ A Rp is the reduced-order observer output, f^ ðtÞ A Rm is the fault estimation of the fault f(t), Gi and Fi are reduced-order observer gain matrices to be designed. Let eðtÞ ¼ x^ 1 ðtÞ x 1 ðtÞ;
ef ðtÞ ¼ f^ ðtÞ f ðtÞ
ð17Þ
then the error dynamics are given as follows: eðt þ 1Þ ¼
N X
δi ðtÞðA11i Gi A21i ÞeðtÞ þ
i¼1
þ
N X
N X
δi ðtÞðA11di Gi A21di Þeðt dÞ
i¼1
δi ðtÞðB1i Gi B2i Þef ðtÞ þ
i¼1
N X
δi ðtÞðGi D2i D1i ÞωðtÞ
i¼1
ð18Þ ef ðt þ1Þ ¼ f^ ðt þ 1Þ f ðt þ 1Þ ¼ f^ ðtÞ
N X
δi ðtÞF i A21i eðtÞ
i¼1
N X
δi ðtÞF i B2i ef ðtÞ þ
i¼1
¼ f^ ðtÞ f ðtÞ
N X
N X
i¼1
δi ðtÞðA 1di G i A 2di Þeðt dÞ
i¼1
N X
δi ðtÞðG i D 1i D 2i ÞνðtÞ
ð20Þ
# eðtÞ eðtÞ ¼ ; ef ðtÞ
"
"
#
ωðtÞ νðtÞ ¼ ; Δf ðtÞ
A 2i ¼ ½A21i B2i ; " A11di A 1di ¼ 0mðn pÞ
# 0ðn pÞm ; 0mm
D 1i ¼ D2i 0pm ; " # 0ðn pÞm D1i D 2i ¼ ; 0md Im
A 1i ¼
A11i
B1i
0mðn pÞ
Im
# ;
A 2di ¼ A21di 0ðn pÞm ;
" Gi ¼
Gi
#
Fi
:
ð21Þ
The objective of this section is to determine the observer gains in order to guarantee asymptotic convergence of the estimated state in (14) to the state in (12). That is to say, the estimated error eðt þ 1Þ tends to zero when t-1. In the following, a reduced-order observer design method under an H 1 performance index is proposed to achieve robust fault estimation for discrete-time switched systems with state delay. Theorem 1. Given a constant γ 1 4 0, if there exists positive definite matrices Pi, Qi, nonsingular matrices Ω, and matrix Hi, for any i; j; l A f1; 2; …; Ng, such that 2 6 6 6 6 6 6 6 4
P j ðΩ þ Ω Þ T
ΩT A 1i Hi A 2i ΩT A 1di Hi A 2di Hi D 1i ΩT D 2i
nn
Pi þ Q i
0
0
nn
n
Ql
nn
n
n
0 γ 21 I r þ m
nn
n
n
n
0
3
7 Im 7 7 7 0 7o0 7 0 7 5 Im
ð22Þ T where I m ¼ 0ðn pÞm I m . Then the augmented switched system (20) is stable with H 1 performance index γ1, and the reduced-order fault estimation observer is given by T
ð23Þ
Hi
i¼1
Proof. Suppose that (22) holds, then it is easy to see from (22) that
δi ðtÞF i D2i ωðtÞ f ðt þ 1Þ
ðP j ΩÞT P j 1 ðP j ΩÞ Z 0
i¼1 N X
ð19Þ
i¼1
Gi ¼ Ω
δi ðtÞF i A21di eðt dÞ
N X
δi ðtÞðA 1i G i A 2i ÞeðtÞ þ
where "
3.2. Reduced-order observer design
x^ 1 ðt þ 1Þ ¼
δi ðtÞðIm F i B2i Þef ðtÞ Δf ðtÞ
i¼1
i¼1
N X
N X
N X
i¼1
N X
δi ðtÞF i D2i ωðtÞ
where Δf ðtÞ ¼ f ðt þ 1Þ f ðtÞ. Combining the dynamics in (18) and (19), one can get the following augmented system: eðt þ 1Þ ¼
δi ðtÞB2i f ðtÞ
δi ðtÞD2i ωðtÞ
δi ðtÞF i A21di eðt dÞ
i¼1
i¼1
i¼1
i¼1
þ
þ
i¼1
N X
N X
δi ðtÞF i A21i eðtÞ
i¼1
one can get N X
N X
N X
δi ðtÞF i D2i ωðtÞ ðf ðt þ 1Þ f ðtÞÞ
i¼1
i¼1
ð11Þ
N X
N X
δi ðtÞF i B2i ef ðtÞ þ
i¼1
ð10Þ
i¼1 N X
N X
3
δi ðtÞF i A21i eðtÞ
N X i¼1
ð24Þ
which implies that
δi ðtÞF i A21di eðt dÞ
P j ðΩ þ Ω Þ Z Ω P j 1 Ω T
T
ð25Þ
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Combining the condition in (23), (22) can be transformed into the following formation: 2 1 3 A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 0 Pj 6 7 6 nn P i þ Q i 0 0 Im 7 6 7 6 7 ð26Þ 6 nn n Q l 0 0 7o0 6 7 6 nn 7 2 n n γ I 0 4 5 1 r þm nn n n n Im
It is noted that 02
γ ν
i¼1
ð27Þ
i¼1
s ¼ t d
and define ΔV ðtÞ ¼ Vðt þ 1Þ VðtÞ. Then along the error dynamics (20), ! N X ΔVðtÞ ¼ e T ðt þ 1Þ δi ðt þ1ÞP i eðt þ 1Þ i¼1 t X
þ
e T ðsÞ N X
e ðtÞ
δi ðtÞP i eðtÞ
i¼1 N X
T
¼ e ðt þ 1Þ
T
e ðsÞ
δi ðt þ 1ÞP i eðt þ 1Þ e ðtÞ
N X
N X i¼1
s ¼ t d
!
T
i¼1
þ e T ðtÞ
t 1 X
N X i¼1
!
δi ðtÞQ i eðtÞ e T ðt dÞ
i¼1
N X
δi ðsÞQ i eðsÞ
3.3. Fault estimation algorithm design
!
δi ðtÞP i eðtÞ !
δi ðt dÞQ i eðt dÞ
i¼1
It follows that for any nonzero vector eðtÞ and the particular case δi ðtÞ ¼ 1, δk a i ðtÞ ¼ 0, δj ðt þ 1Þ ¼ 1, δk a j ðt þ 1Þ ¼ 0, and δl ðt dÞ ¼ 1, δk a l ðt dÞ ¼ 0, we have 02 3 ðA 1i G i A 2i ÞT T B6 ΔVðtÞ ¼ ϑ ðtÞ@4 ðA 1di G i A 2di ÞT 7 5 ðG i D 1i D 2i ÞT h i P j A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 2 31 0 0 Pi þ Q i 6 7C 0 Q l 0 5AϑðtÞ ð29Þ þ4 0
eTf ðtÞef ðtÞ γ 21 νT ðtÞνðtÞ
ð30Þ
i eTf ðtÞef ðtÞ γ 21 νT ðtÞνðtÞ þ ΔVðtÞ VðKÞ
r
t¼0
þ
N X
f ðtÞ
i¼1
δi ðtÞðA 1di G i A 2di Þ
T
I m I m e T ðtÞeðtÞ γ 21 νT ðtÞνðtÞ þ ΔVðtÞ
i¼1 N X
x~^ 1 ðt dÞ f^ ðt dÞ
δi ðtÞG i ρðtÞ þ I ηðtÞ
ð36Þ
h i In p . Substituting (10) and (11) into (36), one has where I ¼ 0mðn pÞ N ^ P x~ 1 ðt þ 1Þ x~^ 1 ðtÞ δ ðtÞðA G A Þ ¼ i 1i i 2i ^ ^ f ðt þ 1Þ
þ
N X
f ðtÞ
i¼1
δi ðtÞðA 1di G i A 2di Þ
þ
N X
δi ðtÞG i yðt þ 1Þ þ
þ
N X
x~^ 1 ðt dÞ f^ ðt dÞ
N X
δi ðtÞðIA12i G i A22i ÞyðtÞ
i¼1 N X
δi ðtÞðIA12di G i A22di Þyðt dÞ þ
i¼1
δi ðtÞðIB1i G i B2i ÞuðtÞ
i¼1
ð37Þ
t¼0 KX 1
f ðt þ 1Þ
i¼1
where K is an arbitrary positive integer. For any nonzero νðtÞ A l2 ½ 0; 1Þ and zero initial condition eðtÞ ¼ 0, one has KX 1 h
Combining (15) and (21), one has N P x~^ 1 ðt þ 1Þ x~^ 1 ðtÞ ¼ δ ðtÞðA G A Þ i 1i i 2i ^ ^
i¼1
i
t¼0
J 1K ¼
i¼1
i¼1
Then from the condition in (28), one can get that ΔVðtÞ o 0, which concludes from Lemma 1 that augmented switched system (20) is stable. Let KX 1 h
Based on the obtained results in the subsection 3.2, we construct the following augmented system: " # N A11i B1i x~^ ðtÞ P x~^ 1 ðt þ 1Þ 1 δ ðtÞ ¼ i f^ ðt þ 1Þ 0mðn pÞ I m f^ ðtÞ i¼1 " # N X 0ðn pÞm x~^ ðt dÞ A11di 1 δi ðtÞ 0 þ 0m f^ ðt dÞ mðn pÞ i¼1 " # " # N X In p Gi δi ðtÞ ηðtÞ ð35Þ ðρ^ ðtÞ ρðtÞÞ þ 0mðn pÞ Fi
þ
0
h iT where ϑðtÞ ¼ e T ðtÞ e T ðt dÞ νT ðtÞ .
J 1K ¼
ð34Þ
which implies, for any K, J 1K o0. Then, one has that for any nonzero νðtÞ A l2 ½0; 1Þ, J ef ðtÞ J 2 o γ 1 J νðtÞ J 2 . □
!
ð28Þ
0
By Schur complement formula, the inequality is equivalent to the following inequality: 2 1 3 Pj A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 6 7 T 6 7 P i þ Q i þI m I m 0 0 6 nn 7 ð33Þ 6 7o0 6 nn 7 n Ql 0 4 5 nn n n γ 21 I r þ m
eTf ðtÞef ðtÞ γ 21 νT ðtÞνðtÞ þ ΔVðtÞ o0
δi ðsÞQ i eðsÞ
!
γ 21 I r þ m
0
It follows from (33) and Schur complement formula that:
!
i¼1
s ¼ t dþ1 T
N X
3
νðtÞ þ ΔVðtÞ
0 Consider the following switched Lyapunov function: ! ! N t 1 N X X X δi ðtÞP i eðtÞ þ e T ðsÞ δi ðsÞQ i eðsÞ VðtÞ ¼ e T ðtÞ
ðA 1i G i A 2i ÞT
T B6 7 ¼ ϑ ðtÞ@4 ðA 1di G i A 2di ÞT 5 T ðG i D 1i D 2i Þ h i ; P j A 1i G i A 2i A 1di G i A 2di G i D 1i D 2i 2 31 T Pi þ Q i þ I m I m 0 0 6 7C 7CϑðtÞ þ6 ð32Þ 0 0 Q l 4 5A
T I m I m e T ðtÞeðtÞ 21 T ðtÞ
ð31Þ
Define the following invariable: N P ~^ ζ ðt þ1Þ ¼ x^1 ðt þ 1Þ δi ðtÞG i yðt þ1Þ f ðt þ 1Þ
ð38Þ
i¼1
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
D. Du, B. Jiang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Then, it follows from (37) that N X
ζ ðt þ 1Þ ¼
δi ðtÞðA 1i G i A 2i Þζ ðtÞ þ
i¼1
þ
N X
N X
δi ðtÞðA 1di G i A 2di Þζ ðt dÞ
δi ðtÞðIA12i G i A22i ÞyðtÞ þ
N X
δi ðtÞðIA12di G i A22di Þyðt dÞ
δi ðtÞðIB1i G i B2i ÞuðtÞ
ð39Þ
Furthermore, it follows from that N P x~^ 1 ðtÞ ¼ ζ ðtÞ δi ðt 1ÞG i yðtÞ ^ f ðtÞ
ð40Þ
i¼1
PN where i ¼ 1 δi ðt 1Þ means the subsystem of time t 1 is activated.
0
;
C~ i ¼ ½C 2i 0;
On the basis of the obtained online fault estimation information, we design a fault tolerant controller to guarantee the stability in the presence of faults. Since the state x(t) is unavailable directly, we utilize the switched dynamical output feedback controller scheme to construct the fault tolerant controller. For the discretetime switched system (1)–(3), we consider the following dynamic output feedback controller: N X
^ þ 1Þ ¼ xðt
^ þ δi ðtÞAKi xðtÞ
i¼1
ð47Þ
4.1. H 1 performance analysis
N X
Lemma 2. Given a constant γ 2 4 0, the augmented switched systems (45) and (46) is stable with H 1 performance γ2, if there exist positive matrices P and Q , such that the following linear matrix inequality holds: 2 3 P A~ i P A~ di P B~ i P 6 7 6 nn P þ H T Q H þ C~ T C~ 0 0 7 6 7 i i ð48Þ 6 7 o0 6 nn 7 n Q 0 4 5 nn n n γ 22 I
ξðt þ 1Þ ¼
VðtÞ ¼ ξ ðtÞP ξðtÞ þ T
ð49Þ
ξT ðsÞHT Q H ξðsÞ
ð50Þ
s ¼ t d
t X
ΔVðtÞ ¼ ξT ðt þ1ÞP ξðt þ 1Þ þ ð42Þ
^ þ δi ðtÞAKi xðtÞ
N X
δi ðtÞBKi CxðtÞ
ð43Þ
i¼1
Then substituting u(t) into (1), one has N X
N X
δi ðtÞAi xðtÞ þ
i¼1
þ
δi ðtÞAdi xðt dÞ þ
i¼1
N X
N X
δi ðtÞBi ðf ðtÞ f^ ðtÞÞ þ
ξT ðsÞHT Q HξðsÞ ξT ðtÞP ξðtÞ
i¼1
N X
N X i¼1
δi ðtÞBi ef ðtÞ þ
i¼1
N X
^ δi ðtÞBi C Ki xðtÞ
δi ðtÞDi ωðtÞ
ð44Þ
i¼1
δi ðtÞA~ i ξðtÞ þ
i¼1
T
N X
T
T
ð51Þ
This with the relationship xðt dÞ ¼ H ξðt dÞ
J 2K ¼
ð52Þ
KX 1
zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ
ð53Þ
t¼0
δi ðtÞA~ di Hξðt dÞ þ
i¼1
N X
δi ðtÞB~ i μðtÞ
J 2K ¼
i¼1
KX 1
zT ðtÞξðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ VðKÞ
t¼0
ð45Þ r
δi ðtÞC~ i ξðtÞ
ð46Þ
i¼1
KX 1
zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ
ð54Þ
t¼0
It is noted that 02
where xðtÞ
T
ξ ðt dÞH Q H ξðt dÞ T
where K is an arbitrary positive integer. For any nonzero μðtÞ A ½0; 1Þ and zero initial condition ξðtÞ ¼ 0, one has
It follows that N X
¼ ξ ðt þ 1ÞP ξðt þ1Þ ξ ðtÞP ξðtÞ þ ξ ðtÞH T Q H ξðtÞ
i¼1 N X
ξT ðsÞHT Q HξðsÞ
s ¼ td
3 " #1 T h i A~ i T P þ HT Q H 0 ~ ~ ~ 4 5 @ Aξ~ ðtÞ ΔVðtÞ ¼ ξ ðtÞ P A i A di þ T 0 Q A~ di h iT T where ξ~ ðtÞ ¼ ξ ðtÞ xT ðt dÞ , which follows from (65) and Schur complete lemma that the system (49) is stable. Let
^ δi ðtÞBi C Ki xðtÞ
δi ðtÞDi ωðtÞ
δi ðtÞAdi xðt dÞ þ
t1 X
which implies 02
i¼1
δi ðtÞAi xðtÞ þ
N X
N X i¼1
i¼1
N X
t 1 X
ð41Þ
^ f^ ðtÞ δi ðtÞC Ki xðtÞ
N X
ξðt þ 1Þ ¼
δi ðtÞA~ di Hξðt dÞ
i¼1
Then along the error dynamics (49),
δi ðtÞBKi yðtÞ
i¼1
i¼1
¼
N X
Define Lyapunov function:
^ A Rn is the controller state vector, AKi, BKi and CKi are where xðtÞ controller gains to be determined later. Substituting (2) into (41), one has
xðt þ 1Þ ¼
δi ðtÞA~ i ξðtÞ þ
i¼1
i¼1
^ þ 1Þ ¼ xðt
N X
s ¼ t dþ1
N X
"
H ¼ ½I 0:
Proof. Firstly, we establish the stability of augmented switched system (45). When μðtÞ ¼ 0, (45) becomes
4. Fault accommodation design
zðtÞ ¼
i¼1
i¼1
uðtÞ ¼
Bi
i¼1
i¼1
þ
N X
Di B~ i ¼ 0
5
#
ξðtÞ ¼ ^ ; xðtÞ
"
#
ωðtÞ μðtÞ ¼ e ðtÞ ; f
" A~ i ¼
Ai
Bi C Ki
BKi C
AKi
# ;
Adi A~ di ¼ ; 0
zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ
3 T A~ B6 Ti 7 h i B6 7 ¼ z T ðtÞB6 A~ di 7P A~ i A~ di B~ i @4 5 T B~ i
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
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6
2 6 þ6 4
T P þ H T Q H þ C~ i C~ i
0
0
0
Q
0
0
γ
0
31
2 2I
Observe that
7C 7CzðtÞ 5A
Ξ ST X 1 S ¼ ST ðYX IÞ 1 ðY X 1 ÞðXY IÞ 1 S 4 0
Then we have P 4 0. Considering the notations in (47) with, and AKi, BKi, and CKi in (57), we can verify that the matrix inequality in (56) can be rewritten as
h iT T where zðtÞ ¼ ξ ðtÞ xT ðt dÞ μT ðtÞ . It follows from (65) and Schur complement formula that zT ðtÞzðtÞ γ 22 μT ðtÞμðtÞ þ ΔVðtÞ o0
2
ð55Þ
which implies, for any K, J 2K o 0. Then, one has that for any nonzero νðtÞ A ½0; 1Þ, J zðtÞ J 2 o γ 2 J μðtÞ J 2 . □ Based on Lemma 1, a dynamic output feedback controller design method is developed in the following theorem. Theorem 2. Given a constant γ 2 4 0, then there exists a dynamic output feedback controller in the form of (41) and (42) such that the augmented switched system (45) and (46) is stable with H 1 performance γ2 if there exist positive definite matrices X, Y, matrices Ωi, Mi, Ni, such that the following linear matrix inequalities: 2 3 Λ11 Λ12 Λ13 0 0 6 nn Λ 0 Λ24 H 7 6 7 22 6 7 2 6 nn n γ2I 0 0 7 ð56Þ 6 7o0 6 nn n n I 0 7 4 5 nn
n
n
n
Q
1
i A f1; 2; …; Ng and some given positive matrix Q , where " # Ai Adi Ai Y þBi Ni I ; Λ12 ¼ ; Ωi XAi þ M i C XAdi X # Bi ; XBi 3 2 T T 3 2 3 I 0 Y Y C 2i 7 6 7 7 X 0 5; Λ24 ¼ 4 C T 5; H ¼ 6 4 I 5: 2i 0 Q 0 0
holds, for all Y Λ11 ¼ I " Di Λ13 ¼ XDi 2 Y 6 I Λ22 ¼ 4 0
In this case, a desired stable dynamic output feedback H 1 controller is given the parameters as follows: AKi ¼ S 1 ðΩi XAi Y M i CY XBi N i ÞW T ; T
C Ki ¼ Ni W
BKi ¼ S 1 M i ;
:
ð57Þ
where S and W are any nonsingular matrices satisfying SW T ¼ I XY ð58Þ Proof. Suppose that the inequalities (56) holds, which implies Y I o0 ð59Þ I X which, by the Schur complement formula, gives that X Y 1 4 0. Therefore, I XY is nonsingular, hence there exist nonsingular matrices S and W such that (58) holds. Now, we introduce the following nonsingular matrices: Y I I X ; Π2 ¼ ð60Þ Π1 ¼ T ; T 0 W 0 S Let 1
P ¼ Π2Π1 then X P¼ T S
S
ð61Þ
6 6 6 6 6 6 6 6 6 6 4
Π2P T
Π T2 A~ i Π 1
Π2
ð62Þ
0
0
0
Π T1 C~ i
nn
n
Q
nn
n
n
0 γ 22 I
0 0
nn
n
n
n
I
nn
n
n
n
n
T
T
3
0
Π 1 P Π 1 þ Π 1 C~ i
T
7 7 Π T1 H T 7 7 0 0
0 1 Q
7 7o0 7 7 7 7 5
ð65Þ T 2 ;
T 1 ; I; I; I; Ig
Pre- and post-multiplying this inequality by diagfΠ Π and its transpose, respectively, which, by the Schur complement, the result then follows from Lemma 1. □
Remark 1. For the sake of clarity, only a class of actuator fault case is considered in this paper. However, note that such additive actuator faults considered in this paper can be readily extended to general class of additive faults [31]. Meanwhile, the additive faults representation is more general than the multiplicative ones, which can be modeled as additive actuator ones [6]. Remark 2. The time delay is assumed to be constant in this paper. Moreover, by using the technique in [32], the results may be extended to time-varying case. Remark 3. For the existence of the terms Adi, switched Lyapunov function is not suitable in designing dynamic output feedback controller. A common Lyapunov function is employed to achieve the results. This may result in some conservatism.
5. An illustrate example Consider the discrete-time switched linear system (1)–(3) consisting of three subsystems described by 0:5 0:3 0:2 0:1 1 0:2 A1 ¼ ; A2 ¼ ; A3 ¼ ; 0:1 1 1 0:3 2 0:05 0:1 0:3 Ad1 ¼ ; 0 1 0:2 0:1 0:2 1 0:1 Ad2 ¼ ; Ad3 ¼ ; B1 ¼ ; 0 0:3 0 0:05 0:2 0:1 0:1 B2 ¼ ; B3 ¼ ; 0:2 0:2 0:1 0:1 0:1 D1 ¼ ; D2 ¼ ; D3 ¼ ; C 11 ¼ C 12 ¼ C 13 ¼ ½1 2; 0:2 0:2 0:2 C 21 ¼ C 22 ¼ C 23 ¼ ½2 1: The state transformation matrix can follows that the augmented matrices 0:65 0:1 0:25 A 11 ¼ ; A 12 ¼ 0 1 0
0 . It be chosen as T ¼ 10:5 0:5 in (21) can be constructed as 0:1 1:1 0:1 ; A 13 ¼ ; 1 0 1
A 21 ¼ ½0:15 0:5;
A 1d2 ¼
ð63Þ
Π T2 A~ di Π T2 B~ i
nn
where
Ξ ¼ W 1 YðX Y 1 ÞYW T 40
1
A 22 ¼ ½ 0:05 0:5;
Ξ
ð64Þ
A 1d3 ¼
0:25
0
0 0:3
0
0
0
0
A 23 ¼ ½ 0:1 0:5; ; ;
A 2d1 ¼ 0:15
0 ;
A 1d1 ¼
0:05
0
0
0
;
A 2d2 ¼ ½0:05 0;
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
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2.5 2 1.5 1 0.5 0 −0.5
0
10
20
30
40
50
60
70
80
90
100
90
100
Fig. 1. Fault f 1 ðtÞ (solid line) and its estimate f ^1 ðtÞ (dotted line).
3.5 3 2.5 2 1.5 1 0.5 0 −0.5
0
10
20
30
40
50
60
70
80
Fig. 2. Fault f 2 ðtÞ (solid line) and its estimate f ^2 ðtÞ (dotted line).
A 2d3 ¼ ½0:5 0; D 11 ¼ ½0:5 0;
0:1
0
0 0:1
1 0 : 1
D 22 ¼ D 23 ¼
D 12 ¼ ½0:5 0;
0
;
D 13 ¼ ½0:5 0;
D 21 ¼
0:1 0
0 ; 1
By solving the condition (22) in Theorem 2, one can get the optimal value γ 1 ¼ 1:82 with reduced-order observer parameters as follows: 0:4401 0:1357 0:0551 G1 ¼ ; G2 ¼ ; G3 ¼ : 0:0039 0:0007 0:0026 1 0 Then, the positive definite matrix Q is set up as 0 2 , by solving the conditions in Theorem 2, one obtains the minimum attenuation value γ 2 ¼ 2:16 with the dynamic output feedback controller parameters as follows: 0:2997 2:9905 0:1028 1:5334 ; AK2 ¼ ; AK1 ¼ 0:1125 2:2276 0:0709 1:6935 0:1191 0:1245 AK3 ¼ ; 0:1377 4:0849 0:8266 0:6498 0:4300 BK1 ¼ ; BK2 ¼ ; BK3 ¼ ; 0:4915 0:3593 0:9382 C K1 ¼ ½0:4399 11:6092; C K2 ¼ ½ 1:2777 0:5005; C K3 ¼ ½ 3:0098 5:7662: For simulation, an actuator constant fault signal f 1 ðtÞ is set up as ( 0; 0 r t r 20 f 1 ðtÞ ¼ 2; 20 o t r 100 If the external disturbance is set as a step signal, Fig. 1 depicts the fault f 1 ðtÞ (solid line) and its estimate f^ 1 ðtÞ (dotted line). An actuator time-varying fault signal f 2 ðtÞ is set up as ( 0; 0 r t r 20 f 2 ðtÞ ¼ sin ðtÞ þ 2; 20 o t r 100 Fig. 2depicts the fault f 2 ðtÞ (solid line) and its estimate f^ 2 ðtÞ (dotted line). It follows from Figs. 1 and 2 that the proposed reduced-order
0.3
0.2 x1(t) x2(t)
0.1
0
−0.1
−0.2
0
10
20
30
40
50
60
70
80
90
100
time step k Fig. 3. State x(t) with fault accommodation.
switched observer has good performance to estimate the actuator fault. Therefore, the simulation results exhibit well estimated operation. Fig. 3 shows the simulation result of state response with fault accommodation.
6. Conclusion By utilizing the reduced-order rank observer method and switched Lyapunov function technique, fault estimation and compensation for discrete-time switched system with state delay are studied in this work, and an efficient reduced-order rank observer design is obtained in the form of linear matrix inequality. Through a dynamic output feedback controller fault compensation, an fault-tolerant operation is realized. Finally, simulation results are presented to verify the effectiveness of the proposed method.
Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i
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8
Acknowledgments This work was partially supported by the Natural Science Foundation of Jiangsu Province (BK20140457), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Fund of Huaiyin Institute of Technology (HGC1309).
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Please cite this article as: Du D, Jiang B. Actuator fault estimation and accommodation for switched systems with time delay: Discretetime case. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.004i