Reliable stabilization of switched system with average dwell-time approach

Author’s Accepted Manuscript

Reliable stabilization of switched system with average dwell-time approach Ying Jin, Jun Fu, Youmin Zhang, Yuanwei Jing

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S0016-0032(12)00307-9 http://dx.doi.org/10.1016/j.jfranklin.2012.12.008 FI1627

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Journal of the Franklin Institute

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28 July 2012 8 November 2012 3 December 2012

Cite this article as: Ying Jin, Jun Fu, Youmin Zhang and Yuanwei Jing, Reliable stabilization of switched system with average dwell-time approach, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2012.12.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Reliable Stabilization of Switched Systems with Average Dwell-time Approach1 Ying Jin, Jun Fu, Youmin Zhang, and Yuanwei Jing

Abstract—This paper presents an average dwell-time approach for reliable stabilization of a class of switched nonlinear systems. Due to the nature of average dwell-time techniques and the representation of actuator faults, this paper has the following features compared with the existing methods in the literature: 1) the proposed method is independent on arbitrary switching polices provided that switching is on-the-average slow enough; 2) the proposed controller exponentially stabilizes this class of nonlinear systems with actuator faults and its nominal (i.e., without actuator faults) systems without necessarily changing any structures and/or parameters of the proposed controllers; 3) the proposed method treats all actuators in a unified way without necessarily classifying all actuators into faulty actuators and healthy ones. A numerical example is given to show the effectiveness of the proposed method. I. INTRODUCTION Switched systems are used to model many physical or man-made systems which have switching nature [1]. Theoretically, it consists of a limited number of subsystems, one of which is activated for a specific time interval under a switching law to achieve certain target [2]. As high performance requirements of control systems for handling nonlinearities, uncertainties and operating condition variations and also 1

Ying Jin is with the College of Information Science and Engineering, Northeastern University, Shenyang, 110819, China,

and also with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canada ([email protected]). Jun Fu is with the Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA ([email protected]). Youmin Zhang is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canada ([email protected]) Yuanwei Jing is with the College of Information Science and Engineering, Northeastern University, Shenyang, 110819, China ([email protected]). -1-

fault-induced dynamic changes, much attention has been paid on switched systems and available results on switched systems have been applied widely and practically [1-6]. However, most of the existing results are presented on theoretical research of switched linear systems and their applications [1, 2, 4, 6, 29, 30]. Since the nature of hybrid dynamic systems is inherently nonlinear, studies of switched nonlinear systems have become one of the key topics in the control community [21, 24, 25]. Undoubtedly, stability is the first requirement for a system to work properly, thus stability of switched systems is the first and important task in research interests on switched systems. Due to the complexity of switched nonlinear systems, the available results on stability of switched nonlinear systems are limited, see [3, 7-13] and the references therein as examples. Thus, switched nonlinear systems with special structures are explored for stability-related problems [4, 8-11]. Namely, Reference [8] considered a class of cascaded nonlinear systems by switching gains of controllers of the linear subsystems to achieve invariant control and semi-globally asymptotic stability. The result in [8] was extended to a multiple-input case in [9]. Reference [4] studied quadratic stability of a class of switched nonlinear systems without control input. Reference [10] considered adaptive neural control for a class of switched nonlinear systems in strict lower triangular form. Work in [11] concerned global stabilization for a class of switched nonlinear feedforward systems. As shown by these results, stability of switched nonlinear systems with special structures is still an important research area. On the other hand, maintenances or repairs in the highly automated industrial systems cannot be always achieved immediately, for preserving safety and reliability of the systems during operation, the possibility of occurrence and presence of uncertain faults must be taken into account during the system analysis and control design stages to avoid life-threatening losses and heavy economic costs caused by faults [7, 12-14, 19], which makes fault-tolerant control (FTC) attract more and more attention [12, 15-18]. Reference [15] considered decentralized fault-tolerant control for a class of interconnected nonlinear systems consisting of finite subsystems to achieve desired tracking objectives and guarantee stability of the closed-loop systems. However, [15] only considered bounded fault functions which satisfy matching conditions and -2-

bounded interconnected uncertain structures, but the authors did not explicitly consider faults of the actuators which transmit control signal into the plant. [16-18] designed fault-tolerant controllers for the nonlinear systems with actuator faults to guarantee robust reliable stability of the systems. Their common feature is that actuators are decomposed into two parts, one of which is susceptible to faults, the other of which is robust to faults, to compensate for actuator faults effectively. But in order to implement the controller designs, the two-part decomposition has to be known in advance. It is in general difficult to obtain an ideal decomposition practically due to the randomness and uncertainty of faults. From the above analysis on switched systems and fault-tolerant control, fault-tolerant control of switched nonlinear systems is one of promising research interests due to the fact that many practical systems can be cast into hybrid dynamic systems. Compared with available results on switched systems and traditional fault-tolerant control, there are very few results on fault-tolerant control of switched nonlinear system [7, 20, 21]. Reference [7] gave a sufficient condition on robust fault-tolerant control of a class of nonlinear switched systems by decomposing actuators into two parts, in the same way as [16], i.e., one part is robust to actuator faults, and the other is susceptible to actuator faults. Thus, the method in [7] has the above-mentioned disadvantages on fault modeling and characterization. In addition, structural uncertainties of input matrices were not considered for this class of nonlinear switched systems in [7]. In [20], by combining safe-parking method and reconfiguration-based approach the authors proposed two switching strategies to realize fault-tolerant controls of a class of switched nonlinear systems against actuator faults. But when actuator faults occur, the two methods both need to determine reparation time for faulty actuators, which is not easy to acquire in many real-world scenarios. Reference [21] considered an observer-based fault-tolerant control of a class of switched nonlinear systems with external disturbances. From a system structure point of view, the nonlinear item in this paper is connected to the system in a parallel way, which can be directly compensated for by control signals. However, it is worth noting that fault-tolerant control for nonlinear cascaded systems is much more complicated, where the nonlinear term, as a nonlinear subsystem, is cascaded to the other subsystem. -3-

In about last decades, average dwell-time switching techniques have been becoming one of popular methods to stabilize switched systems, due that it is more general and flexible than dwell-time techniques [22, 23]. It means that the number of switches in the finite interval is bounded and the average time between successive switchings is greater than or equal to a constant [1, 22]. Thus, there are several important results on applications of average dwell-time techniques [1, 23-28]. However, to the best of the authors’ knowledge, there are few results on applying average dwell-time techniques to fault-tolerant control of switched systems [1, 23, 24, 27, 28]. These references only considered fault-tolerant control of switched linear systems. Reference [25] applied average dwell-time method to fault-tolerant control of the switched nonlinear system where the nonlinear item is connected to the system in a parallel way, which can be directly compensated for by control signals. [26] proposes an adaptive logic-based switching fault-tolerant control method for a class of nonlinear uncertain systems against actuator faults, and also uses a similar way as [25] to deal with the unmodelled dynamics. This paper considers the problem of robust fault-tolerant control of a class of switched cascade nonlinear systems with structural uncertainties existing in both system matrices and input matrices, and proposes a fault-tolerant control method for this class of switched systems by using average dwell-time techniques. The main features and contributions of this paper are highlighted as follows: (1) The proposed control design works on both the switched nonlinear systems with actuator faults and its nominal systems (i.e., without actuator faults) without necessarily changing any structures and/or parameters of the proposed controllers. Therefore, it is classified as a reliable control, or passive fault-tolerant control strategy [12]; (2) The proposed method unlike [16-18], in a unified way for easy and practical applications, treats all actuators without necessarily classifying all actuators into faulty actuators and robust ones; (3) The proposed method is independent of arbitrary switching signals provided that the average switching time is greater than some constant time. The layout of the paper is as follows. Section II presents the problem statement. The details about -4-

designing the controllers of the nonlinear switched systems and its stability analysis are presented in Section III. A numerical example is given in Section IV. Section V concludes the paper. II. PROBLEM STATEMENT Consider a class of uncertain switched nonlinear systems described by: z x

where x R r and z R n

r

gi ( z , x), ( Ai

Ai ) x ( Bi

are system states, ui

(1)

Bi )ui ,

R qi is control input, i(t ) : 0,

M

1,2, , m is a

switching signal, g i z, x is a known nonlinear function, Ai and Bi are known constant matrices, and Ai and Bi are matrix functions representing structural uncertainties.

We now need following assumptions for system (1): Assumption 1: Assume that Ai , Bi is controllable and that all the states are available for feedback. Assumption 2: Assume that Ai and Bi are the structural uncertainties with bounded norms, i.e., Ai

i

and

Bi

(2)

i.

Assumption 3: Assume that g i z, x satisfies global Lipschitz condition, i.e., there exists constant Li

0 such that

gi z, x1

gi z, x2

Li x1

x2 ,

z , xi , x2 .

(3)

Assumption 4: There exist a smooth positive-definite function W z with W (0) 0, positive constants

k1 , k2 , constants

i

0,

0, and i M such that dW z gi z,0 dz dW z dz

k1 z

2

2

i

z ,

z ,

W ( z)

(4) 2

k2 z . -5-

Then, we design state feedback controllers as follows:

ui q where K i R i

r

Ki x

(5)

are constant matrices.

Given whether a fault occurs on each actuator or not, a matrix Lis is introduced to represent fault situation of the actuators of the i th subsystem as follows:

Lis where if l ij

1 actuator j , j 1, 2,

diag(l1i , l 2i , , l qi )

(6)

i , q , is normal and if l j =0 actuator j is faulty, and Lis

0.

Therefore, the closed-loop switched nonlinear systems involving uncertain structures and actuator faults are given as follows:

z

g i ( z , x),

x [( Ai

(7)

Bi ) Lis K i ]x.

Ai ) ( Bi

The control objective is to design feedback gain matrices K i i M such that switched nonlinear system (7) are globally exponentially stable for all uncertain matrices Ai , Bi and the actuator faults. III. CONTROLLER DESIGN OF THE SWITCHED SYSTEMS This section will present the main results on the robust fault-tolerant control of the nonlinear switched cascaded systems (1) and stability analysis of the closed-loop systems (7). Before presenting the main theorem, we need the following lemma. Lemma 1:

For

0 and symmetric positive matrix

x, y R r and constant

, the following

inequalities hold: xT y y T x

xT x

yT

1

y

xT x

yT y min ( )

Theorem 1: Suppose that Assumptions 1-4 are satisfied. If there exist

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(8)

0,

1 and symmetric

positive matrices H i ,U i and Qi , such that the following Riccati equations Qi

AiT Pi

Pi Ai

Pi

1

2

Hi

i

1

Bi

r

1

Ui min

2

Hi

min

Pi

Pj ,

i, j

i

P,

r

i

0 i

Ui

qi

qi

BiT Pi

(9.a)

M

(9.b)

M

have positive-definite solution Pi . Then, i. If average dwell-time * a

a

ln

,

[0,

0

(9.c)

)

the closed-loop systems (7) are globally exponentially stable under arbitrary switching rules with BiT P, i.e., ui K i x

controller gain K i

BiT Px are fault-tolerant feedback controllers which

stabilize switched systems (1) globally and exponentially. ii. State estimation of systems (7) x (t )

where x

T

x z , b1

min[ k1 , min i M

b2 e b1

min

(

0

)t

2

( Pi )], b2

(9.d)

x ( 0) ,

max[ k 2 , max i M

max

( Pi )], and Ki

BiT Pi .

Proof: Consider the function below as a piecewise Lyapunov function candidate: V ( x, z) Vi

xT Pi x W z ,

(10)

where Pi are positive definite solution matrices, and W (z ) is a smooth positive definite function with W ( 0)

0 . Along the trajectory of systems (7), the time derivate of V x, z is

V

xT ATi Pi Pi Ai

AiT Pi Pi Ai 2 Pi Bi Lis BT Pi

Pi BLis BiT Pi Pi Bi Lis BiT Pi x

-7-

dW z gi z, x dz

(11)

0 and symmetric positive definite matrices H i and U i ,

According to Lemma 1, for the constant also Assumption 2, one has xT

AiT Pi

xT

Pi Ai x

AiT H i

xT min

Hi

min

Hi

xT

xT

1

1

Ai

1

Lis

Pi Bi

( Lis )( Lis )

xT

xT 2 Pi Bi

Lis BiT Pi x 1

1

Ui

min

PH i Pi x

Lis U i ( Lis ) BiT Pi

Pi Bi

min

Ui

(12)

Pi H i Pi x,

r

2 Pi Bi Lis BiT Pi x

xT

1

AiT Ai 2

xT

Pi H i Pi x

Lis BiT Pi

Pi Bi BiT Pi

1

Pi BiU i BiT Pi x 1

(13)

Pi BiU i BiT Pi x

Pi BiU i BiT Pi x

and the following xT

Pi Bi Lis BiT Pi

Pi Bi Lis BiT Pi x

x T Pi Bi ( Lis ) BiT Pi xT

1

Pi Bi BiT Pi

xT

i

xT

i

2

Pi Pi

2

Pi Pi

1 1

Pi Bi ( Lis ) BiT Pi x Pi Bi ( Lis )( Lis ) BiT Pi x

(14) ( Lis )( Lis ) Pi Bi BiT Pi x Pi Bi BiT Pi x.

Please note that the last steps of (13) and (14) follow because of ( Lis )( Lis ) 1. Using (10)-(14) and (9), one has Vi

x T AiT Pi

Pi Ai

Pi

1

Hi

2 i

Lr

Bi

1

1

Ui min

2 min

Hi

i

r

x

dW z g i z, x dz

and thus,

-8-

Ui

qi

qi

BiT Pi

Vi

xT AiT Pi Pi Ai

V

0 i

1

P

2 min

Hi

xT Qi x

i

2

Hi

i

Ui

dW z gi z,0 dz dW z gi z, x dz

P x

r

1

Lr Bi

0 i

dW z g i z ,0 dz

1 qi

min U i dW z gi z, x dz

gi ( z,0)

BiT Pi

qi

gi ( z,0)

W z

0

W z

0

According to the global Lipschitz condition in Assumption 3, and (3) and (4), then one has V

Vi

0

min

min (Qi ) x

choosing

i

0k2

2 2 Li

4

min (Qi )

0,

2

(Qi ) x

z

i

2

Li z x

0

2

Li z 2 min (Qi )

0k2

i

k2 z 2 2 Li

4

min (Qi )

2

z

2

one has V

(15)

V

0 i

using (9.b) and (10), one has Vi (t )

To an arbitrarily given t 0 , let 0 t0

V j (t ),

i, j

t Ni (t0 ,t ) be the switching constants during (0, t ) .

t2  tk

t1

(16)

M

From (15) and (16), we can know that 0 (t

V (t ) V (tk )e

tk )

0 (t

V (tk 1 )e N i( t , 0 )

 0t

e

if

a

satisfies (9.c), i.e. for any t

N i ( t , 0 ) ln

0 (t

tk )

tk 1 )

0t

V (0)

V (0)

0

t

N i (t ,0) then N i (t ,0) ln

e

V (tk )e

* a

* a

,

ln

t hold, thus V (t )

e

(

0

)t

V (0) -9-

(17)

According to Assumption 4, there exist two positive constants b1 and b2 so that b1 ( x

2

x (t )

2

2

2

z ) V (t ) b2 ( x

2

(18)

z )

From (17) and (18), we have

1 V (t ) b1

b2 e b1

(

)t

0

x ( 0)

2

Hence,

x (t )

b2 e b1

(

0

)t

x (0)

2

The proof has completed. Remark 1: The controllers designed in Theorem 1 are suitable to both the actuators with faults ( Lis and those without faults ( Lis

I qi )

I qi ), with an advantage that no changes are needed to make on structures

and/or parameters of the controllers to guarantee globally asymptotical stability and satisfy other desired performances of the systems (1). Remark 2: Pre- and post-multiplying both sides of matrix inequalities (9) by Complement Lemma, and letting

P 1 , applying Schur

be a sufficiently small constant, inequality (9) can be transformed

into the following LMIs:

AiT

I

0 r

*

Ai

i

min

*

2

i

[

i

(H i ) *

where

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] 1 Ir

0 Qi 1

0

i

,

M

1 i

2

Hi

i

I r Bi

1

Ui min

(U i )

I qi

1

I qi B i

T

Remark 3: Please note that the proposed method in this paper can be easily extended to a wider class of nonlinear switched systems which has the following form:

However, it has to be in this cascade structure. Stabilization of a class of non-cascade nonlinear systems is out of the scope of this paper and is one of our future works. IV. SIMULATION STUDIES In this section, we use a numerical example to validate the proposed method. Consider the following uncertain nonlinear switched systems: z x

where i

g i ( z , x), ( Ai

Ai ) x ( Bi

Bi )u i ,

1, 2 and the matrices and parameters are listed below:

A1

4 0 0.4

A2

2 0 0.4

2 4 1

0 0 , 4

2 2.5 1

1 0 0.5 0.5 , 0.5 1

B1

0 0 , B2 2

H1

0.1 0.002 0 0.002 0.1 0 , 0 0 0.1

H2

0.2 0.002 0 0.002 0.2 0 , 0 0 0.2 - 11 -

1 0 0.6 0.5 , 0.6 0.9

U1

U2

1 0 , 0 1

2 0 , 0 2

(19)

Q1

Q2

z3

g1 ( z , x) and

1

0.7244 0.4067 .1153

0.4067 0.6398 0.3002

0.1153 0.3002 , L1s 0.6398

0 0 , 0 1

0.8244 0.4067 0.1153

0.4067 0.6398 0.3002

0.1153 0.3002 , L2s 0.7363

1 0 , 0 0

z3

x1 sin( z ), g 2 ( z, x)

x1 cos( z ),

2

1.5 ,

2

0.1 ,

2

0.2 ,

1.2 ,

1

1

0.2 ,

0.1.

Solving Riccati Eq. (9.a) and (9.b) with the parameters above gives a positive definite matrix solution as shown below:

P1

0.8662 0.0209 0.0209 0.8227 0.0696 0.2560

Then, according to ui

Ki x

BiT Pi x (i

be x(0) [ 3,3, 2]T and z(0) 2 . Also let * a

a

0.0696 0.2560 0.7889

P2

,

2.1576 0.3710 0.3710 0.2467 0.1492 0.3710

0.1492 0.3710 1.9508

.

1,2) , the controller can be designed. Let initial conditions

0.06 ,

1.1 and

0

0.1 . We have

* a

ln

1.5 . Take

2 , and choose the following switching rule:

As shown in Fig. 1, the state feedback control law guarantees that systems (15) are still asymptotically stable when the second actuator of the first subsystem and the first actuator of the second subsystem have faults, as indicated by L1s and L2s . Fig. 1 is the time history of the states, where the stars represent switching points. Fig. 2 and Fig. 3 represent switching sequences of controller gains and switching signal, respectively. From the figures, the effectiveness of the proposed control method is shown by this numerical example.

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Fig. 1 System state responses

Fig. 2 Switching sequences of gain matrices

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Fig. 3 Switching signal V. CONCLUSION This paper has introduced a fault-tolerant control method for a class of uncertain switched nonlinear systems by using average dwell-time techniques. A sufficient condition is carried out on globally exponential stabilization of the switched nonlinear systems against actuator faults under arbitrarily switching signals provided that switching is on-the-average slow enough. This method can also be easily applied to robust fault-tolerant control problems of uncertain switched systems with sensor faults, or with both sensor and actuator faults. The simulation results on a numerical example have shown the effectiveness of the proposed method. The proposed method can be applied to stabilization of the longitudinal dynamics of the F-18 aircraft in [31] and the rollover dynamics of steerable vehicles in [32].

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