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Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method
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Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method Ying Jin a,n, Youmin Zhang b, Yuanwei Jing c a
The State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China The Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada H3G 1M8 c College of Information Science and Engineering, Northeastern University, Shenyang 110819, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 1 July 2015 Received in revised form 24 January 2016 Accepted 25 March 2016 Communicated by Lixian Zhang
This paper studies a robust fault tolerant control of a class of nonlinear switched systems with strong structural uncertainties and actuator faults. This paper presents a new robust fault tolerant state feedback method, by using the average dwell time technique to stabilize the switched nonlinear systems exponentially under an arbitrary switching law. Comparing with the existing results, the method of this paper has three features: (1) this method is applicable to the switched nonlinear systems with strong structural uncertainties and faulty actuators and also the nominal case of the considered systems, there is no need to change any structures and/or parameters of the controller. (2) The switching law is arbitrary provided that the average switching is slow enough in the average sense. (3) This method treats all actuators in the same way, no need to separate faulty actuators from the healthy ones. The simulation result verifies the effectiveness of this method. & 2016 Published by Elsevier B.V.
Keywords: Fault-tolerant control Switched systems Average dwell-time
1. Introduction Uncertainties are unavoidable in real applications. If the uncertainties are ignored during designing controllers then that may cause the closed-loop systems unstable. Uncertainties can be roughly classified into two categories: parameter uncertainties and structural uncertainties. For the parameter uncertainties, the way to compensate is to design adaptive estimator. For the structural uncertainties, designing robust controller is to stabilize the system with bounded uncertainties [29]. Switched systems are an important class of hybrid systems [27]. When uncertainties exist in switched systems, the methods which address the nominal switched systems degrade the system performance or even destabilize the systems. Therefore, robust control of switched systems has received more and more attention [30,11,17,24,4,12,9,26,27]. But most of the existing results are designed for switched linear systems and their applications [30,29,24]. Since the nonlinearities are an inherent nature of hybrid dynamic systems, robust control of switched nonlinear systems has been one of important research areas in control community. Undoubtedly, stability is an important and key precondition for systems to work normally, thus the stability of switched systems n
Correstponding author. E-mail address: [email protected] (Y. Jin).
has become a center task on control of switched nonlinear systems. Due to the structural complexity of the switched nonlinear systems, the existing results on stability of switched nonlinear systems are not fruitful, see Ref. [11,19,15,2,7,13,28,23]. The research mainly focuses on switched nonlinear systems with special structures [11,7,13,28,23,5,31]. Therefore, the stability of switched nonlinear systems with special structures is still a core research area since many industrial process systems can be modeled as switched nonlinear systems with special structures [22]. On the other hand, integrated automation of industrial processes has reached a new stage, the maintenances and repairs of faults cannot be achieved immediately, which affect system safety and feasibility. Taking the possibilities of occurrences of uncertainties and faults into account during systems analysis and controller designs to avoid huge economic costs and even life threaten safety caused by faults is very important [24,3,16,25,8,26]. With the fast development of modern software and computing techniques, proposing method by designing algorithm to achieve fault tolerant control can save more economic costs and provide more design freedom than replacing hardware. Thus, it is important to apply fault tolerant control to industrial process control [18,21,10,6]. Ref. [18] considered a class of interconnected nonlinear systems consisting of a limited subsystems by designing decentralized fault tolerant controller to realize tracking objective and guarantee the stability of the closed-loop systems, but ignored the fault models of actuators during the input control signal transmitting to actuators [18]. The common feature in Refs.
http://dx.doi.org/10.1016/j.neucom.2016.03.047 0925-2312/& 2016 Published by Elsevier B.V.
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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[21,10,6] is that actuators need to be classified that which part of actuators are faulty and which ones are robust to faults in advance during designing fault-tolerant controllers. Generally, it is hard to obtain the information about the two-part separation of actuators mentioned above in advance during implementing controllers practically. As shown above the analysis of robust stabilization on switched systems and fault tolerant control, and many practical systems modeled as hybrid dynamic systems, fault-tolerant control of switched nonlinear systems is one of the important research areas [3,20]. The available results combing switched systems and traditional fault tolerant method are limited [24,1,20,14]. Ref. [24] adopted the proposed method from [21] to deal with actuator faults under given conditions with stability of a class of cascade switched nonlinear systems, but this paper did not consider strong uncertainties in input matrices. Du and Mhaskar [4] combined safe-parking and reconfiguration method to address a class of switched nonlinear systems which introduced two switching strategies to achieve actuator fault-tolerant control. Both strategies need to determine the reparation times once actuator fault occurs, it is usually difficult to acquire practically. Ref. [20] studied a class of switched nonlinear systems with external disturbance and designed an observer-based fault tolerant controller. From a system structure view, the nonlinear item is connected to system in a parallel way to get compensation directly by control signals. Nonlinear subsystem cascaded to another system improve the complexity of cascaded switched nonlinear systems with fault tolerant control. In the past two decades, average dwell time technique is one of popular methods to stabilize switched systems [8,1]. The so-called average dwell time means that the number of switchings is bounded in a finite time interval, and average dwell time between continuous switching intervals is not less than an positive constant [27,8]. Many important results of using average dwell time technique on applications are available [27,1,20,25,14,9,26], the existing results on applying average dwell time method to deal with fault tolerant control of switched systems are only a few. Refs. [27,1,20,9,26] consider fault tolerant control of switched linear system, Yang et al. [25] presented fault tolerant control of switched nonlinear systems by using average dwell-time technique, where the nonlinear subsystem is cascaded to another system in a parallel way to obtain compensation directly by control signals. Ma and Yang [14] proposed adaptive-based fault tolerant control of a class of uncertain switched nonlinear systems with actuator faults, it adopted the way in Yang et al. [25] to address unmodeled dynamic systems. This paper develops a robust fault tolerant controller by using average dwell time technique to stabilize a class of nonlinear switched systems with both system matrices and input matrices existing strong disturbance uncertainties. The features are as follows: 1. The strong disturbance uncertainties are element-wise absolute-value bounded. 2. The switching law is arbitrary if the switching is slow enough in the average sense. 3. The proposed method treats all actuators by an unified way without classifying all actuators into two parts: faulty actuators and the actuators robust to faults, which is different from the mentioned methods in Veillette [18], Wang et al. [21], Liang et al. [10], Han and Yu [6], and Ma and Yang [14]. The layout of this paper is as follows: Section 2 states the problem. The details of designing controller of the switched nonlinear systems and stability analysis are presented in Section 3. Numerical simulation is given in Section 4. Section 5 concludes this paper.
2. Problem statement Consider a class of uncertain switched nonlinear systems z_ ¼ g i ðz; xÞ; x_ ¼ ðAi þ ΔAi Þx þ ðBi þ ΔBi Þui
ð1Þ
nr
r
qi
where x A R and z A R are system states, ui A R is control input, iðtÞ : ½0; þ 1Þ-M ¼ f1; 2; …; mg is a switching signal, g i ðz; xÞ is a known nonlinear function, Ai,Bi is a known nonlinear function, ΔAi ; ΔBi are uncertain matrix functions representing structural uncertainties. Remark 1. System (1) represents a special class of switched nonlinear systems [19,20,9], which is formed by a set of nonlinear cascade systems and a switching law to determine which nonlinear cascade system is active in use. When the set contains only one subsystem, System (1) is reduced to a non-switched nonlinear cascade system [15,2]. We make the following assumptions for system (1): Assumption 1. Assume that ðAi ; Bi Þ are controllable, and that the states are available for feedback. Assumption 2. Assume there exist non-negative constant matrices EiA and EiB, so that strong structural uncertain matrices ΔAi ; ΔBi are satisfied, i.e. j ΔAi j rEAi ;
j ΔBi j rEBi :
ð2Þ
Assumption 3. g i ðz; xÞ satisfies globally Lipschitz condition, i.e., there exists a constant Li 4 0 such that J g i ðz; x1 Þ g i ðz; x2 Þ J r Li J x1 x2 J ;
8 z; x1 ; x2
Assumption 4. There exist a smooth positive definition function W(z) with Wð0Þ ¼ 0, positive constant k1,k2, and constant β 4 0; γ 40; i A M, such that dWðzÞ g ðz; 0Þ r β i J z J 2 ; dz i dWðzÞ dz r γ J z J ;
ð3Þ ð4Þ
k1 J z J 2 r J WðzÞ J r k2 J z J 2 :
ð5Þ
Remark 2. The purpose of condition (4) is to assume, for stability proof, that the time derivative of W(z) corresponding to the z subsystem is satisfied linear growth condition with respect to J z J . Let the controller be in the form ui ¼ K i x
ð6Þ qi r
where K i A R is a constant matrix. Given whether a fault occurs on each actuator or not, propose a matrix Lis ðLis a0Þ representing the actuators fault situation of the ith subsystem of the switched nonlinear systems as follows: i i i Lis ¼ diag l1 ; l2 ; …; lq ð7Þ i
i
if lj ¼ 1, the jth actuator is normal; if lj ¼ 0, the jth actuator is i lj
faulty, where ðjA 1; 2; …; qÞ. Thus the model of nonlinear switched system with actuator faults and structural uncertainties is as follows: z_ ¼ g i ðz; xÞ; x_ ¼ ½ðAi þ ΔAi Þ þ ðBi þ ΔBi ÞLis K i x:
ð8Þ
The objective is to seek feedback a gain matrix K i ði A MÞ, such that the switched nonlinear system (8) with structural uncertainties and actuator faults existing in both system matrix and input matrix can be exponentially stabilized.
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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"
3. Fault tolerant control design
Lemma 1. For 8 x; y A Rr and constant ε 4 0 and symmetric positive definite matrix Π, the following inequalities hold xT y þ yT x r
x Πx
ε
þ εyT Π
1
yr
x Πx T
ε
þ
εy y : λmin ðΠ Þ T
P i r μP j ;
ε
λmin ðHi Þ
J EAi J 2 I r þ λ0 P i ¼ Q i ;
ði A MÞ
8 i; j A M
ð10Þ ð11Þ
have symmetric positive definite solution Pi. Then, (a) if average dwell-time
τa Z τna ¼
ln μ
λ
;
τa satisfies
λ A ð0; λ0 Þ
ð12Þ
Then the closed-loop system (8) with the controller gain K i ¼ BTi P i can be stabilized globally and exponentially under arbitrary switching law. (b) The norm estimation of state of system (8) is sffiffiffiffiffi b2 ðλ0 λÞt e 2 J xð0Þ J ð13Þ J xðtÞ J r b1
b1 ¼ min½k1 ; min λmin ðP i Þ; 8iAM b2 ¼ max k2 ; max λmax ðP i Þ ; K i ¼ BTi P i : 8iAM
Using (10)–(18), we have
1 V_ r xT ATi P i þ P i Ai þ P i H i þ ε J EBi J 2 I r ε 1 ε 1 I qi þ I qi BTi P i þ Bi U i þ ε ε λmin ðU i Þ ε dWðzÞ g ðz; xÞ J EA J 2 I x þ þ dz i λmin ðHi Þ i r Therefore,
1 V_ i þ λ0 V i r xT ATi P i þ P i Ai þP i H i þ ε J EBi J 2 I r ε 1 ε 1 þ Bi U i þ I þ I BT P ε λmin ðU i Þ qi ε qi i i ε dWðzÞ g ðz; 0Þ J EAi J 2 I r þ λ0 P i x þ þ dz i λmin ðHi Þ
Considering (3)–(5) and global Lipschitz condition in Assumption 3, we obtain
Proof. Consider the following piece wise candidate function: Vðx; zÞ ¼ V i ¼ xT P i x þ WðzÞ;
ð14Þ
where Pi is positive definite solution, and W(z) is a smooth positive definite function with Wð0Þ. Then along the trajectory of system (8), the time derivative of Vðx; zÞ is V_ ¼ xT ATi P i þ P i Ai þ ΔATi P i þ P i ΔAi 2P i Bi Lis BTi P i dWðzÞ g ðz; xÞ P i Bi Lis ΔBTi P i P i ΔBi Lis BTi P i x þ ð15Þ dz i According to inequality (9), and Assumption 2, we can obtain xT ΔATi P i þ P i ΔAi x 1 rxT εΔATi H i 1 ΔAi þ P i H i P i x ð16Þ
ε
ATi
1
xT ð 2P i Bi Lis BTi P i Þx ¼ xT ½2P i Bi ð Lis ÞBTi P i x 1 rxT εP i Bi ð Lis ÞU i 1 ð Lis ÞBTi P i þ P i Bi U i BTi P i x ε ε 1 i i T T P i Bi ð Ls Þð Ls ÞBi P i þ P i Bi U i BTi P i x rx ε λmin ðU i Þ
V_ i þ λ0 V i r λmin ðQ i Þ J x J 2 βi J z J 2 þ γ Li J z J J x J þ λ0 k2 J z J 2 ! 2 γ Li γ 2 L2i ¼ λmin ðQ i Þ J x J J z J β i λ 0 k2 JzJ2 2λmin ðQ i Þ 4λmin ðQ i Þ
Choosing βi λ0 k2 V_ i r λ0 V i
γ 2 L2i
4λmin ðQ i Þ
Z 0 gives ð19Þ
From (11) and (14), one can obtain V i ðtÞ r μV j ðtÞ;
8 i; jA M
ð20Þ
For any t 4 0, let 0 o t 1 o t 2 o ⋯ o t k ¼ t Ni ð0; on ð0; tÞ. Using (19) and (20) leads to
tÞ
be switching instants
VðtÞ r Vðt k Þe λ0 ðt tk Þ r μVðt k Þe λ0 ðt t k Þ
Δ ΔAi þ P i H i P i x rx ε λ ðH Þ min i ε 1 A rxT J Ei J 2 I r þ P i H i P i x ε λmin ðHi Þ
ε
ð18Þ
dWðzÞ ½g i ðz; xÞ g i ðz; 0Þ þ λ0 WðzÞ dz dWðzÞ dWðzÞ g ðz; 0Þ þ ½g i ðz; xÞ g i ðz; 0Þ ¼ xT Q i x þ dz i dz þ λ0 WðzÞ
x ¼ ðx zÞT ;
T
ε
þ
where
h i ¼ xT P i Bi ð Lis ÞΔBTi P i þ P i ΔBi ð Lis ÞBTi P i x 1 r xT εP i ΔBi ΔBTi P i þ P i Bi ð Lis Þð Lis ÞBTi P i x ε 1 B 2 T r x ε J Ei J P i P i þ J ð Lis Þð Lis Þ J P i Bi BTi P i x ε 1 ¼ xT ε J EBi J 2 P i P i þ P i Bi BTi P i x
ð9Þ
Theorem 1. Assume that Assumptions 1–4 are satisfied. If there exist ε 4 0; μ 4 1, λ0 4 0 and symmetric positive definite matrices Hi, Ui and Q i , such that Riccati equations 1 1 ε 1 I qi þ I qi BTi P i P i H i þ ε J EBi J 2 I r þ Bi U i þ ε ε ε λmin ðU i Þ þ ATi P i þ P i Ai þ
#
ε J ð Lis Þð Lis Þ J 1 rx P i Bi BTi P i þ P i Bi U i BTi P i x ε λmin ðU i Þ ε 1 P i Bi BTi P i þ P i Bi U i BTi P i x ¼ xT ε λmin ðU i Þ i T T x ð P i Bi Ls ΔBi P i P i ΔBi Lis BTi P i Þx T
Before presenting the main result, we give the following lemma.
T
3
r μVðt k 1 Þe λ0 ðtk t k 1 Þ e λ0 ðt tk Þ ¼ μVðt k 1 Þe λ0 ðt tk 1 Þ r ⋯ r μNi ðt; ¼e
λ0 t þ N i ðt; 0Þ ln μ
0Þ λ0 t
e
Vð0Þ
Vð0Þ
τa satisfies (12), for any t 4 0 t ln μ Ni ðt; 0Þ r n ; τna ¼ τa λ If
ð17Þ
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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4
3
State responses
2 x2
1
z 0 x3 −1
x1
−2 −3
0
5
10
15
20
25
20
25
t/s Fig. 1. States response.
0.8 0.6
Switching gain matrix
0.4 0.2 0
K1
K2
−0.2 −0.4 −0.6 −0.8 −1 −1.2
0
5
10
15 t/s
Fig. 2. Sequences of switching gain matrices.
Then Ni ðt; 0Þ ln μ r λt hold, therefore we have VðtÞ re ðλ0 λÞt Vð0Þ
4. Simulation results ð21Þ
According to (A4), there exist two positive constants b1 and b2, such that b1 ð J x J 2 þ J z J 2 Þ rVðtÞ r b2 ð J x J 2 þ J z J 2 Þ
ð22Þ
The proof is completed. □
2
2 0
2 2
2
Remark 3. Both sides of matrix inequalities (10) need to pre- and post-multiply Γ i ¼ P i 1 and apply to Schur complement lemma. Let σ be a sufficient small positive constant, inequalities (10) can be transformed into the following LMIs inequality equation: 2 3 Γ i ATi þ Ai Γ i þ Λi Γi Γi h i1 6 7 6 7 i n λ εðH J EAi J 2 σ Ir 0 7 o0; ði A MÞ 6 min iÞ 4 5 n n Q i 1
1 1 ε 1 I þ I BT Λi ¼ Hi þ ε J EBi J 2 Ir þBi U i þ ε ε λmin ðU i Þ qi ε qi i
ð23Þ
where i ¼ 1; 2, ε1 ¼ 1:2, ε2 ¼ 1:5,
0:4 1 2 0:02 sin ðtÞ 0 ΔA1 ¼ 6 4 0:12
sffiffiffiffiffi b2 ðλ0 λÞt J xðtÞ J r e 2 J xð0Þ J : b1
z_ ¼ g i ðz; xÞ; x_ ¼ ðAi þ ΔAi Þx þ ðBi þ ΔBi Þui
6 A1 ¼ 4
From (21) and (22), one has
where
Consider the following uncertain switched nonlinear system:
3 0 0 7 5; 2
2
0:1 0 0:02
3
3 0
2 1:5
0:4
1 2
6 A2 ¼ 4 3
0 7 0:1 5; 0
3 0 0 7 5; 3
2
1 6 B1 ¼ 4 0:5 0:5
0:04
0:06
0:14
0 0:06
ΔA2 ¼ 6 4 0
2
3 0 0:5 7 5; 1 3
0 7 0:16 cos ðtÞ 5; 0
3
2 3 1 0 0:02 0:1 0:04 0:08 6 7 6 7 6 7 0:5 5; ΔB1 ¼ 4 0:04 0:15 5; ΔB2 ¼ 4 0:02 0:02 5; B2 ¼ 4 0:6 0:6 0:9 0:12 0 0:06 0 2 3 2 3 2 1 1 1 0:1 0:002 0 0:2 0:002 6 7 6 7 6 A 0:1 0 5; H 2 ¼ 4 0:002 0:2 Ei ¼ 4 1 1 1 5; H 1 ¼ 4 0:002 0 0 0:1 0 0 1 1 1
3 0 7 0 5 0:2
2
3 0:7244 0:4067 0:1153 6 0:4067 0:6398 0:3002 7 Q1 ¼ 4 5; 0:1153 0:3002 0:6398 2 3 0:8244 0:4067 0:1153 6 0:4067 0:6398 0:3002 7 Q2 ¼ 4 5; 0:1153 0:3002 0:7363
U 1 ¼ ½1 0; 0 1; L2s
¼ ½0 0; 0 1;
U 2 ¼ ½2 0; 0 2;
μ ¼ 1:1;
j ΔAi j ¼
0:2EAi ;
L1s ¼ ½1 0; 0 0; j Δ Bi j ¼ Δ Bi ;
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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5
Switching signal
2
1 0
5
10
t/s
15
20
25
Fig. 3. Switching signal.
g 1 ðz; xÞ ¼ z þ x1 sin z; g 2 ðz; xÞ
λ ¼ 0:06;
¼ z þ x1 cos z;
References and
λ0 ¼ 0:1:
ln μ
We can easily obtain τa ¼ λ ¼ 1:5. By using Lemma 1, we can design state feedback controller to stabilize system (23) exponentially. By solving Riccati equations (11), the symmetric positive definite matrices are given as 2 3 0:9443 0:0409 0:0817 6 7 1:0162 0:2978 5; P 1 ¼ 4 0:0429 0:0817 0:2978 0:8669 2 3 1:1137 0:0132 0:1343 6 0:0132 1:6041 0:2744 7 P2 ¼ 4 5: 0:1343 0:2744 1:9508 n
Then the controller is ui ¼ K i x ¼ BTi P i x
ði ¼ 1; 2Þ:
Initial values are chosen as xð0Þ ¼ ½ 3 3 2T ;
zð0Þ ¼ 2:
The system can be stabilized exponentially under arbitrary switching law once the condition with the average dwell-time not less than 1.5 s is satisfied. For simplicity, assume switching law is dependent only on time. The designed controller can stabilize system (23) when actuator faults occur as in Ls1 and Ls2. Figs. 1–3 show states response of the closed-loop system, the switching gain sequence and the switching signals, respectively, and also verifies the effectiveness of the proposed control method. In the figures, stars represent switching points.
5. Conclusion This paper has presented a fault tolerant control method for a class of uncertain switched nonlinear system with strong structural uncertainties by using the average dwell time technique. The uncertain structural matrix functions are defined with element wise-absolute value bounded, which is different from traditional norm bounded [9]. This method also can be applied to robust fault tolerant control of uncertain switched system with sensor faults or the situation on both actuators and sensor faults [25].
Acknowledgment This paper was partially supported by NSFC under the grant 61503065. The authors would like to thank Professors Tianyou Chai and Jun Fu for their helpful discussion.
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[26] L. Zhang, N. Cui, M. Liu, Y. Zhao, Asynchronous filtering of discrete-time switched linear systems with average dwell time, IEEE Trans. Circuits Syst. I: Regul. Pap. 58 (5) (2011) 1109–1118. [27] L. Zhang, H. Gao, Asynchronously switched control of switched linear systems with average dwell time, Automatica 46 (5) (2010) 953–958. [28] Y. Zhang, J. Jiang, Fault tolerant control system design with explicit consideration of performance degradation, IEEE Trans. Aerosp. Electron. Syst. 39 (3) (2003) 838–848. [29] J. Zhao, G.M. Dimirovski, Quadratic stability of a class of switched nonlinear systems, IEEE Trans. Autom. Control 49 (4) (2004) 574–578 (0018-9286). [30] J. Zhao, D.J. Hill, Dissipativity theory for switched systems, IEEE Trans. Autom. Control. 53 (4) (2008) 941–953. [31] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Autom. Control. 57 (7) (2012) 1809–1815.
Ying Jin was a postdoctoral researcher at McGill University, Montreal, Quebec, Canada. She obtained her Ph. D. in 2013 in control theory and control engineering from Northeastern University, Shenyang, China. She was a visiting Ph.D. student at Concordia University of Canada during the time period starting from July 2011 to June 2013. She is currently a lecturer in the State Key Lab of Synthetical Automation for Process Industries (Northeastern University), China. Dr. Jin's research interests include fault-tolerant control of switched nonlinear systems and switching control of multi-body mechanical systems.
Youmin Zhang is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada, where he is an associate professor. He obtained his B.S., M.S. and Ph.D. all from Northwestern Polytechnical University, China. His research interests mainly lie in fault-tolerant control systems, reliability and fault-tolerant techniques for safetycritical engineering systems and avionics and flight control. He is senior member of AIAA and IEEE.
Yuanwei Jing received his B.S. degree in mathematics from Liaoning University, Liaoning, China, in 1981, and his M.S. and Ph.D. degrees in automatic control from Northeastern University, Shenyang, China, in 1984 and 1988, respectively. From 1998 to 1999, he was a senior visiting scholar with the Computer Science Telecommunication Program of University Missouri, Kansas City. He is currently with the School of Information Science and Engineering, Northeastern University. His current research interests include complex control systems and control problems in modern communication network systems.
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method Ying Jin a,n, Youmin Zhang b, Yuanwei Jing c a
The State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China The Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada H3G 1M8 c College of Information Science and Engineering, Northeastern University, Shenyang 110819, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 1 July 2015 Received in revised form 24 January 2016 Accepted 25 March 2016 Communicated by Lixian Zhang
This paper studies a robust fault tolerant control of a class of nonlinear switched systems with strong structural uncertainties and actuator faults. This paper presents a new robust fault tolerant state feedback method, by using the average dwell time technique to stabilize the switched nonlinear systems exponentially under an arbitrary switching law. Comparing with the existing results, the method of this paper has three features: (1) this method is applicable to the switched nonlinear systems with strong structural uncertainties and faulty actuators and also the nominal case of the considered systems, there is no need to change any structures and/or parameters of the controller. (2) The switching law is arbitrary provided that the average switching is slow enough in the average sense. (3) This method treats all actuators in the same way, no need to separate faulty actuators from the healthy ones. The simulation result verifies the effectiveness of this method. & 2016 Published by Elsevier B.V.
Keywords: Fault-tolerant control Switched systems Average dwell-time
1. Introduction Uncertainties are unavoidable in real applications. If the uncertainties are ignored during designing controllers then that may cause the closed-loop systems unstable. Uncertainties can be roughly classified into two categories: parameter uncertainties and structural uncertainties. For the parameter uncertainties, the way to compensate is to design adaptive estimator. For the structural uncertainties, designing robust controller is to stabilize the system with bounded uncertainties [29]. Switched systems are an important class of hybrid systems [27]. When uncertainties exist in switched systems, the methods which address the nominal switched systems degrade the system performance or even destabilize the systems. Therefore, robust control of switched systems has received more and more attention [30,11,17,24,4,12,9,26,27]. But most of the existing results are designed for switched linear systems and their applications [30,29,24]. Since the nonlinearities are an inherent nature of hybrid dynamic systems, robust control of switched nonlinear systems has been one of important research areas in control community. Undoubtedly, stability is an important and key precondition for systems to work normally, thus the stability of switched systems n
Correstponding author. E-mail address: [email protected] (Y. Jin).
has become a center task on control of switched nonlinear systems. Due to the structural complexity of the switched nonlinear systems, the existing results on stability of switched nonlinear systems are not fruitful, see Ref. [11,19,15,2,7,13,28,23]. The research mainly focuses on switched nonlinear systems with special structures [11,7,13,28,23,5,31]. Therefore, the stability of switched nonlinear systems with special structures is still a core research area since many industrial process systems can be modeled as switched nonlinear systems with special structures [22]. On the other hand, integrated automation of industrial processes has reached a new stage, the maintenances and repairs of faults cannot be achieved immediately, which affect system safety and feasibility. Taking the possibilities of occurrences of uncertainties and faults into account during systems analysis and controller designs to avoid huge economic costs and even life threaten safety caused by faults is very important [24,3,16,25,8,26]. With the fast development of modern software and computing techniques, proposing method by designing algorithm to achieve fault tolerant control can save more economic costs and provide more design freedom than replacing hardware. Thus, it is important to apply fault tolerant control to industrial process control [18,21,10,6]. Ref. [18] considered a class of interconnected nonlinear systems consisting of a limited subsystems by designing decentralized fault tolerant controller to realize tracking objective and guarantee the stability of the closed-loop systems, but ignored the fault models of actuators during the input control signal transmitting to actuators [18]. The common feature in Refs.
http://dx.doi.org/10.1016/j.neucom.2016.03.047 0925-2312/& 2016 Published by Elsevier B.V.
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
2
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[21,10,6] is that actuators need to be classified that which part of actuators are faulty and which ones are robust to faults in advance during designing fault-tolerant controllers. Generally, it is hard to obtain the information about the two-part separation of actuators mentioned above in advance during implementing controllers practically. As shown above the analysis of robust stabilization on switched systems and fault tolerant control, and many practical systems modeled as hybrid dynamic systems, fault-tolerant control of switched nonlinear systems is one of the important research areas [3,20]. The available results combing switched systems and traditional fault tolerant method are limited [24,1,20,14]. Ref. [24] adopted the proposed method from [21] to deal with actuator faults under given conditions with stability of a class of cascade switched nonlinear systems, but this paper did not consider strong uncertainties in input matrices. Du and Mhaskar [4] combined safe-parking and reconfiguration method to address a class of switched nonlinear systems which introduced two switching strategies to achieve actuator fault-tolerant control. Both strategies need to determine the reparation times once actuator fault occurs, it is usually difficult to acquire practically. Ref. [20] studied a class of switched nonlinear systems with external disturbance and designed an observer-based fault tolerant controller. From a system structure view, the nonlinear item is connected to system in a parallel way to get compensation directly by control signals. Nonlinear subsystem cascaded to another system improve the complexity of cascaded switched nonlinear systems with fault tolerant control. In the past two decades, average dwell time technique is one of popular methods to stabilize switched systems [8,1]. The so-called average dwell time means that the number of switchings is bounded in a finite time interval, and average dwell time between continuous switching intervals is not less than an positive constant [27,8]. Many important results of using average dwell time technique on applications are available [27,1,20,25,14,9,26], the existing results on applying average dwell time method to deal with fault tolerant control of switched systems are only a few. Refs. [27,1,20,9,26] consider fault tolerant control of switched linear system, Yang et al. [25] presented fault tolerant control of switched nonlinear systems by using average dwell-time technique, where the nonlinear subsystem is cascaded to another system in a parallel way to obtain compensation directly by control signals. Ma and Yang [14] proposed adaptive-based fault tolerant control of a class of uncertain switched nonlinear systems with actuator faults, it adopted the way in Yang et al. [25] to address unmodeled dynamic systems. This paper develops a robust fault tolerant controller by using average dwell time technique to stabilize a class of nonlinear switched systems with both system matrices and input matrices existing strong disturbance uncertainties. The features are as follows: 1. The strong disturbance uncertainties are element-wise absolute-value bounded. 2. The switching law is arbitrary if the switching is slow enough in the average sense. 3. The proposed method treats all actuators by an unified way without classifying all actuators into two parts: faulty actuators and the actuators robust to faults, which is different from the mentioned methods in Veillette [18], Wang et al. [21], Liang et al. [10], Han and Yu [6], and Ma and Yang [14]. The layout of this paper is as follows: Section 2 states the problem. The details of designing controller of the switched nonlinear systems and stability analysis are presented in Section 3. Numerical simulation is given in Section 4. Section 5 concludes this paper.
2. Problem statement Consider a class of uncertain switched nonlinear systems z_ ¼ g i ðz; xÞ; x_ ¼ ðAi þ ΔAi Þx þ ðBi þ ΔBi Þui
ð1Þ
nr
r
qi
where x A R and z A R are system states, ui A R is control input, iðtÞ : ½0; þ 1Þ-M ¼ f1; 2; …; mg is a switching signal, g i ðz; xÞ is a known nonlinear function, Ai,Bi is a known nonlinear function, ΔAi ; ΔBi are uncertain matrix functions representing structural uncertainties. Remark 1. System (1) represents a special class of switched nonlinear systems [19,20,9], which is formed by a set of nonlinear cascade systems and a switching law to determine which nonlinear cascade system is active in use. When the set contains only one subsystem, System (1) is reduced to a non-switched nonlinear cascade system [15,2]. We make the following assumptions for system (1): Assumption 1. Assume that ðAi ; Bi Þ are controllable, and that the states are available for feedback. Assumption 2. Assume there exist non-negative constant matrices EiA and EiB, so that strong structural uncertain matrices ΔAi ; ΔBi are satisfied, i.e. j ΔAi j rEAi ;
j ΔBi j rEBi :
ð2Þ
Assumption 3. g i ðz; xÞ satisfies globally Lipschitz condition, i.e., there exists a constant Li 4 0 such that J g i ðz; x1 Þ g i ðz; x2 Þ J r Li J x1 x2 J ;
8 z; x1 ; x2
Assumption 4. There exist a smooth positive definition function W(z) with Wð0Þ ¼ 0, positive constant k1,k2, and constant β 4 0; γ 40; i A M, such that dWðzÞ g ðz; 0Þ r β i J z J 2 ; dz i dWðzÞ dz r γ J z J ;
ð3Þ ð4Þ
k1 J z J 2 r J WðzÞ J r k2 J z J 2 :
ð5Þ
Remark 2. The purpose of condition (4) is to assume, for stability proof, that the time derivative of W(z) corresponding to the z subsystem is satisfied linear growth condition with respect to J z J . Let the controller be in the form ui ¼ K i x
ð6Þ qi r
where K i A R is a constant matrix. Given whether a fault occurs on each actuator or not, propose a matrix Lis ðLis a0Þ representing the actuators fault situation of the ith subsystem of the switched nonlinear systems as follows: i i i Lis ¼ diag l1 ; l2 ; …; lq ð7Þ i
i
if lj ¼ 1, the jth actuator is normal; if lj ¼ 0, the jth actuator is i lj
faulty, where ðjA 1; 2; …; qÞ. Thus the model of nonlinear switched system with actuator faults and structural uncertainties is as follows: z_ ¼ g i ðz; xÞ; x_ ¼ ½ðAi þ ΔAi Þ þ ðBi þ ΔBi ÞLis K i x:
ð8Þ
The objective is to seek feedback a gain matrix K i ði A MÞ, such that the switched nonlinear system (8) with structural uncertainties and actuator faults existing in both system matrix and input matrix can be exponentially stabilized.
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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"
3. Fault tolerant control design
Lemma 1. For 8 x; y A Rr and constant ε 4 0 and symmetric positive definite matrix Π, the following inequalities hold xT y þ yT x r
x Πx
ε
þ εyT Π
1
yr
x Πx T
ε
þ
εy y : λmin ðΠ Þ T
P i r μP j ;
ε
λmin ðHi Þ
J EAi J 2 I r þ λ0 P i ¼ Q i ;
ði A MÞ
8 i; j A M
ð10Þ ð11Þ
have symmetric positive definite solution Pi. Then, (a) if average dwell-time
τa Z τna ¼
ln μ
λ
;
τa satisfies
λ A ð0; λ0 Þ
ð12Þ
Then the closed-loop system (8) with the controller gain K i ¼ BTi P i can be stabilized globally and exponentially under arbitrary switching law. (b) The norm estimation of state of system (8) is sffiffiffiffiffi b2 ðλ0 λÞt e 2 J xð0Þ J ð13Þ J xðtÞ J r b1
b1 ¼ min½k1 ; min λmin ðP i Þ; 8iAM b2 ¼ max k2 ; max λmax ðP i Þ ; K i ¼ BTi P i : 8iAM
Using (10)–(18), we have
1 V_ r xT ATi P i þ P i Ai þ P i H i þ ε J EBi J 2 I r ε 1 ε 1 I qi þ I qi BTi P i þ Bi U i þ ε ε λmin ðU i Þ ε dWðzÞ g ðz; xÞ J EA J 2 I x þ þ dz i λmin ðHi Þ i r Therefore,
1 V_ i þ λ0 V i r xT ATi P i þ P i Ai þP i H i þ ε J EBi J 2 I r ε 1 ε 1 þ Bi U i þ I þ I BT P ε λmin ðU i Þ qi ε qi i i ε dWðzÞ g ðz; 0Þ J EAi J 2 I r þ λ0 P i x þ þ dz i λmin ðHi Þ
Considering (3)–(5) and global Lipschitz condition in Assumption 3, we obtain
Proof. Consider the following piece wise candidate function: Vðx; zÞ ¼ V i ¼ xT P i x þ WðzÞ;
ð14Þ
where Pi is positive definite solution, and W(z) is a smooth positive definite function with Wð0Þ. Then along the trajectory of system (8), the time derivative of Vðx; zÞ is V_ ¼ xT ATi P i þ P i Ai þ ΔATi P i þ P i ΔAi 2P i Bi Lis BTi P i dWðzÞ g ðz; xÞ P i Bi Lis ΔBTi P i P i ΔBi Lis BTi P i x þ ð15Þ dz i According to inequality (9), and Assumption 2, we can obtain xT ΔATi P i þ P i ΔAi x 1 rxT εΔATi H i 1 ΔAi þ P i H i P i x ð16Þ
ε
ATi
1
xT ð 2P i Bi Lis BTi P i Þx ¼ xT ½2P i Bi ð Lis ÞBTi P i x 1 rxT εP i Bi ð Lis ÞU i 1 ð Lis ÞBTi P i þ P i Bi U i BTi P i x ε ε 1 i i T T P i Bi ð Ls Þð Ls ÞBi P i þ P i Bi U i BTi P i x rx ε λmin ðU i Þ
V_ i þ λ0 V i r λmin ðQ i Þ J x J 2 βi J z J 2 þ γ Li J z J J x J þ λ0 k2 J z J 2 ! 2 γ Li γ 2 L2i ¼ λmin ðQ i Þ J x J J z J β i λ 0 k2 JzJ2 2λmin ðQ i Þ 4λmin ðQ i Þ
Choosing βi λ0 k2 V_ i r λ0 V i
γ 2 L2i
4λmin ðQ i Þ
Z 0 gives ð19Þ
From (11) and (14), one can obtain V i ðtÞ r μV j ðtÞ;
8 i; jA M
ð20Þ
For any t 4 0, let 0 o t 1 o t 2 o ⋯ o t k ¼ t Ni ð0; on ð0; tÞ. Using (19) and (20) leads to
tÞ
be switching instants
VðtÞ r Vðt k Þe λ0 ðt tk Þ r μVðt k Þe λ0 ðt t k Þ
Δ ΔAi þ P i H i P i x rx ε λ ðH Þ min i ε 1 A rxT J Ei J 2 I r þ P i H i P i x ε λmin ðHi Þ
ε
ð18Þ
dWðzÞ ½g i ðz; xÞ g i ðz; 0Þ þ λ0 WðzÞ dz dWðzÞ dWðzÞ g ðz; 0Þ þ ½g i ðz; xÞ g i ðz; 0Þ ¼ xT Q i x þ dz i dz þ λ0 WðzÞ
x ¼ ðx zÞT ;
T
ε
þ
where
h i ¼ xT P i Bi ð Lis ÞΔBTi P i þ P i ΔBi ð Lis ÞBTi P i x 1 r xT εP i ΔBi ΔBTi P i þ P i Bi ð Lis Þð Lis ÞBTi P i x ε 1 B 2 T r x ε J Ei J P i P i þ J ð Lis Þð Lis Þ J P i Bi BTi P i x ε 1 ¼ xT ε J EBi J 2 P i P i þ P i Bi BTi P i x
ð9Þ
Theorem 1. Assume that Assumptions 1–4 are satisfied. If there exist ε 4 0; μ 4 1, λ0 4 0 and symmetric positive definite matrices Hi, Ui and Q i , such that Riccati equations 1 1 ε 1 I qi þ I qi BTi P i P i H i þ ε J EBi J 2 I r þ Bi U i þ ε ε ε λmin ðU i Þ þ ATi P i þ P i Ai þ
#
ε J ð Lis Þð Lis Þ J 1 rx P i Bi BTi P i þ P i Bi U i BTi P i x ε λmin ðU i Þ ε 1 P i Bi BTi P i þ P i Bi U i BTi P i x ¼ xT ε λmin ðU i Þ i T T x ð P i Bi Ls ΔBi P i P i ΔBi Lis BTi P i Þx T
Before presenting the main result, we give the following lemma.
T
3
r μVðt k 1 Þe λ0 ðtk t k 1 Þ e λ0 ðt tk Þ ¼ μVðt k 1 Þe λ0 ðt tk 1 Þ r ⋯ r μNi ðt; ¼e
λ0 t þ N i ðt; 0Þ ln μ
0Þ λ0 t
e
Vð0Þ
Vð0Þ
τa satisfies (12), for any t 4 0 t ln μ Ni ðt; 0Þ r n ; τna ¼ τa λ If
ð17Þ
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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4
3
State responses
2 x2
1
z 0 x3 −1
x1
−2 −3
0
5
10
15
20
25
20
25
t/s Fig. 1. States response.
0.8 0.6
Switching gain matrix
0.4 0.2 0
K1
K2
−0.2 −0.4 −0.6 −0.8 −1 −1.2
0
5
10
15 t/s
Fig. 2. Sequences of switching gain matrices.
Then Ni ðt; 0Þ ln μ r λt hold, therefore we have VðtÞ re ðλ0 λÞt Vð0Þ
4. Simulation results ð21Þ
According to (A4), there exist two positive constants b1 and b2, such that b1 ð J x J 2 þ J z J 2 Þ rVðtÞ r b2 ð J x J 2 þ J z J 2 Þ
ð22Þ
The proof is completed. □
2
2 0
2 2
2
Remark 3. Both sides of matrix inequalities (10) need to pre- and post-multiply Γ i ¼ P i 1 and apply to Schur complement lemma. Let σ be a sufficient small positive constant, inequalities (10) can be transformed into the following LMIs inequality equation: 2 3 Γ i ATi þ Ai Γ i þ Λi Γi Γi h i1 6 7 6 7 i n λ εðH J EAi J 2 σ Ir 0 7 o0; ði A MÞ 6 min iÞ 4 5 n n Q i 1
1 1 ε 1 I þ I BT Λi ¼ Hi þ ε J EBi J 2 Ir þBi U i þ ε ε λmin ðU i Þ qi ε qi i
ð23Þ
where i ¼ 1; 2, ε1 ¼ 1:2, ε2 ¼ 1:5,
0:4 1 2 0:02 sin ðtÞ 0 ΔA1 ¼ 6 4 0:12
sffiffiffiffiffi b2 ðλ0 λÞt J xðtÞ J r e 2 J xð0Þ J : b1
z_ ¼ g i ðz; xÞ; x_ ¼ ðAi þ ΔAi Þx þ ðBi þ ΔBi Þui
6 A1 ¼ 4
From (21) and (22), one has
where
Consider the following uncertain switched nonlinear system:
3 0 0 7 5; 2
2
0:1 0 0:02
3
3 0
2 1:5
0:4
1 2
6 A2 ¼ 4 3
0 7 0:1 5; 0
3 0 0 7 5; 3
2
1 6 B1 ¼ 4 0:5 0:5
0:04
0:06
0:14
0 0:06
ΔA2 ¼ 6 4 0
2
3 0 0:5 7 5; 1 3
0 7 0:16 cos ðtÞ 5; 0
3
2 3 1 0 0:02 0:1 0:04 0:08 6 7 6 7 6 7 0:5 5; ΔB1 ¼ 4 0:04 0:15 5; ΔB2 ¼ 4 0:02 0:02 5; B2 ¼ 4 0:6 0:6 0:9 0:12 0 0:06 0 2 3 2 3 2 1 1 1 0:1 0:002 0 0:2 0:002 6 7 6 7 6 A 0:1 0 5; H 2 ¼ 4 0:002 0:2 Ei ¼ 4 1 1 1 5; H 1 ¼ 4 0:002 0 0 0:1 0 0 1 1 1
3 0 7 0 5 0:2
2
3 0:7244 0:4067 0:1153 6 0:4067 0:6398 0:3002 7 Q1 ¼ 4 5; 0:1153 0:3002 0:6398 2 3 0:8244 0:4067 0:1153 6 0:4067 0:6398 0:3002 7 Q2 ¼ 4 5; 0:1153 0:3002 0:7363
U 1 ¼ ½1 0; 0 1; L2s
¼ ½0 0; 0 1;
U 2 ¼ ½2 0; 0 2;
μ ¼ 1:1;
j ΔAi j ¼
0:2EAi ;
L1s ¼ ½1 0; 0 0; j Δ Bi j ¼ Δ Bi ;
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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5
Switching signal
2
1 0
5
10
t/s
15
20
25
Fig. 3. Switching signal.
g 1 ðz; xÞ ¼ z þ x1 sin z; g 2 ðz; xÞ
λ ¼ 0:06;
¼ z þ x1 cos z;
References and
λ0 ¼ 0:1:
ln μ
We can easily obtain τa ¼ λ ¼ 1:5. By using Lemma 1, we can design state feedback controller to stabilize system (23) exponentially. By solving Riccati equations (11), the symmetric positive definite matrices are given as 2 3 0:9443 0:0409 0:0817 6 7 1:0162 0:2978 5; P 1 ¼ 4 0:0429 0:0817 0:2978 0:8669 2 3 1:1137 0:0132 0:1343 6 0:0132 1:6041 0:2744 7 P2 ¼ 4 5: 0:1343 0:2744 1:9508 n
Then the controller is ui ¼ K i x ¼ BTi P i x
ði ¼ 1; 2Þ:
Initial values are chosen as xð0Þ ¼ ½ 3 3 2T ;
zð0Þ ¼ 2:
The system can be stabilized exponentially under arbitrary switching law once the condition with the average dwell-time not less than 1.5 s is satisfied. For simplicity, assume switching law is dependent only on time. The designed controller can stabilize system (23) when actuator faults occur as in Ls1 and Ls2. Figs. 1–3 show states response of the closed-loop system, the switching gain sequence and the switching signals, respectively, and also verifies the effectiveness of the proposed control method. In the figures, stars represent switching points.
5. Conclusion This paper has presented a fault tolerant control method for a class of uncertain switched nonlinear system with strong structural uncertainties by using the average dwell time technique. The uncertain structural matrix functions are defined with element wise-absolute value bounded, which is different from traditional norm bounded [9]. This method also can be applied to robust fault tolerant control of uncertain switched system with sensor faults or the situation on both actuators and sensor faults [25].
Acknowledgment This paper was partially supported by NSFC under the grant 61503065. The authors would like to thank Professors Tianyou Chai and Jun Fu for their helpful discussion.
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Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i
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Ying Jin was a postdoctoral researcher at McGill University, Montreal, Quebec, Canada. She obtained her Ph. D. in 2013 in control theory and control engineering from Northeastern University, Shenyang, China. She was a visiting Ph.D. student at Concordia University of Canada during the time period starting from July 2011 to June 2013. She is currently a lecturer in the State Key Lab of Synthetical Automation for Process Industries (Northeastern University), China. Dr. Jin's research interests include fault-tolerant control of switched nonlinear systems and switching control of multi-body mechanical systems.
Youmin Zhang is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada, where he is an associate professor. He obtained his B.S., M.S. and Ph.D. all from Northwestern Polytechnical University, China. His research interests mainly lie in fault-tolerant control systems, reliability and fault-tolerant techniques for safetycritical engineering systems and avionics and flight control. He is senior member of AIAA and IEEE.
Yuanwei Jing received his B.S. degree in mathematics from Liaoning University, Liaoning, China, in 1981, and his M.S. and Ph.D. degrees in automatic control from Northeastern University, Shenyang, China, in 1984 and 1988, respectively. From 1998 to 1999, he was a senior visiting scholar with the Computer Science Telecommunication Program of University Missouri, Kansas City. He is currently with the School of Information Science and Engineering, Northeastern University. His current research interests include complex control systems and control problems in modern communication network systems.
Please cite this article as: Y. Jin, et al., Fault-tolerant control of switched nonlinear systems with strong structural uncertainties: Average dwell-time method, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.03.047i