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Fuzzy adaptive state-feedback fault-tolerant control for switched stochastic nonlinear systems with faults
Author’s Accepted Manuscript Fuzzy Adaptive State-feedback Fault-Tolerant Control for Switched Stochastic Nonlinear Systems with Faults Zhiyao Ma, Shaocheng Tong, Yongming Li www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)02053-6 http://dx.doi.org/10.1016/j.neucom.2015.12.080 NEUCOM16604
To appear in: Neurocomputing Received date: 18 November 2015 Revised date: 20 December 2015 Accepted date: 22 December 2015 Cite this article as: Zhiyao Ma, Shaocheng Tong and Yongming Li, Fuzzy Adaptive State-feedback Fault-Tolerant Control for Switched Stochastic Nonlinear Systems with Faults, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.12.080 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.
Fuzzy Adaptive State-feedback Fault-Tolerant Control for Switched Stochastic Nonlinear Systems with Faults Zhiyao Ma*, Shaocheng Tong, Yongming Li Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, Liaoning, 121001, P. R. China Abstract: In this paper, the fault-tolerant control (FTC) problem is investigated for a class of uncertain switched stochastic nonlinear systems. The controlled system contains unknown nonlinear functions and actuator faults, which are modeled as both lock-in-place and loss of effectiveness. With the help of fuzzy logic systems to approximate the unknown nonlinear functions and by utilizing the common Lyapunov function (CLF) method, a fuzzy adaptive state-feedback fault-tolerant tracking control approach is developed. It is shown that all the variables of the closed-loop system are semi-globally uniformly bounded (SGUUB) in probability, the tracking error converges to an arbitrary small neighborhood of the origin. A simulation is given to demonstrate the effectiveness of the proposed approach. Keywords: Nonlinear switched stochastic systems; common Lyapunov function; fault-tolerant control; backstepping technique 1. Introduction In the past decades, adaptive control design for stochastic nonlinear systems has attracted more and more attention and many important control methods have been developed, see [1-6] and references therein. Among them, [1, 2] proposed the backstepping control design method for stochastic single-input-single-output (SISO) nonlinear systems by utilizing Quartic Lyapunov function. In [3, 4], according to changing supply function and small-gain theorem, the adaptive fuzzy backstepping design schemes were developed for nonlinear stochastic SISO systems in the presence of unmodeled dynamics. Then, [5, 6] proposed the adaptive fuzzy output-feedback decentralized control methods for stochastic nonlinear large-scale systems. However, the main
This work was supported by the National Natural Science Foundation of China (Nos. 61374113, 61573175). †Corresponding author. Tel.: +86-416-4199101; Fax: +86-416-4199415. E-mail: [email protected].
1
limitation in [1-6] is that these approaches are only suitable for those stochastic nonlinear systems, in which the nonlinear uncertainties must be linearly parameterized. To cope with the above limitation, many novel adaptive control methods have been developed, see [7-14]. [7-10] proposed some adaptive state-feedback control methods for the nonlinear stochastic systems that the states can be directly used, and [11-14] proposed several adaptive output-feedback control schemes for the nonlinear stochastic systems in the presence of the unmeasured states. On the other hand, switched systems have gained considerable interest in the past decades and many physical and engineering systems can be constructed as nonlinear switched systems [15-17]. As we know that the stability of the switched system can be guaranteed if a CLF exists under arbitrary switching (see [18]). In recently years, some backstepping control schemes have been investigated [19-21] for the nonlinear switched systems on the basis of the CLF stability theory, but the nonlinear functions of considered systems are both known. Furthermore, due to the existence of stochastic disturbances in practical systems, some researchers begin to put more time and energy into the investigations of switched stochastic systems. Many remarkable consequences of stability analysis (see [22-24]) and controller design (see [25-27]) have been acquired. Among them, [24] proposed an adaptive control scheme for a class of stochastic nonlinear systems in the presence of irreducible homogenous Markovian switching. Based on the common Lyapunov function method, [27] proposed the state feedback and output feedback control approaches for a class of stochastic nonlinear systems with measured and unmeasured states, respectively. However, the above control schemes assume that all the components of the considered systems are in good operating conditions, and they did not consider the problem of faults. In general, there always exist some faults [28] in the practical control systems, like actuator and sensor faults, these faults often reduce the control characteristics and influence the stability of the considered system, and more serious result is the catastrophic accidents. Thus, many FTC design methods have been proposed to accommodate the actuator faults involved in the nonlinear systems [29-34]. The works in [29, 30] investigated the adaptive fuzzy FTC schemes for a class of unknown SISO nonlinear strict-feedback systems in the presence of actuator failures. Based on their results, a fuzzy adaptive FTC design method was proposed in [31] for a class of nonlinear multi-input-multi-output (MIMO) systems. To handle the unmeasured states problems, [32] proposed a fuzzy adaptive output-feedback tracking FTC method for large-scale nonlinear systems. 2
The fuzzy adaptive output feedback tracking FTC approaches were proposed in [33, 34] for the stochastic nonlinear MIMO and large-scale systems. However, the aforementioned methods are only applied to the nonlinear non-switched systems. Inspired by the above observations, in this paper, a fuzzy adaptive state-feedback fault-tolerant tracking control scheme is developed for a class of switched stochastic nonlinear systems, under arbitrary switching. It is shown that the proposed control approach can guarantee that all the variables of the closed-loop switched stochastic system are SGUUB in probability and the tracking error regulates to a small neighborhood of the origin regardless of actuator faults. Compared with previous results, this paper owns two contributions: (1) This paper proposed a fuzzy adaptive FTC design scheme for a class of switched stochastic nonlinear systems. The proposed adaptive control method has solved the actuator faults problem. Note that the previous References [19-27] assumed that all the components in the considered systems are in good operating conditions. Thus, they can not be used for dealing with the switched stochastic nonlinear systems with actuator faults. (2) The considered nonlinear systems of this paper contain stochastic disturbance, actuator faults and switched signals. Consequently, based on the common Lyapunov function stability theory and FTC design method, the proposed adaptive controller can not only have the robustness to stochastic disturbance and accommodate the actuator faults, but also guarantee the stability of the whole controlled systems, thus the existing results in [33-38] can not be applied to control the switched stochastic nonlinear systems considered in this paper. 2. Problem Formulations and Preliminaries 2.1 System descriptions and assumptions Consider a class of uncertain switched stochastic nonlinear systems:
dxi ( f (t ), i ( xi ) xi 1 )dt gT (t ), i ( xi )dw 1 i n 1 T T n2 dxn ( f (t ), n ( xn ) qn ( xn )u )dt g (t ), n ( xn )dw y x1
(1)
where xi [ x1 ,, xi ] R , i 1,, n are the state vectors, u [u1 , u2 ,, um ] R T
i
T
def
m
is
the input vector of the system, y R is the system output. (t ) : [0, ) {1,2,, N } 3
is the switching signal, which takes its values in the compact set . Moreover, (t ) p ,
p 1,, N means that p -th subsystem is active. qn ( xn ) [qn1 ( xn ),, qnm ( xn )]T Rm ,
f p ,i , qnj and g p ,i for i 1,, n , j 1,, m are unknown nonlinear smooth functions. w R is an independent r -dimension standard Wiener process. The actuator faults involved in this paper are both lock-in-place and loss of effectiveness are defined by [29-34] as follows. Lock-in-place fault model:
u j (t ) u j , t t j , j { j1 , j2 ,, jh } {1,2,, m}
(2)
Loss of effectiveness fault model:
ui (t ) i vi (t ) t ti , i { j1 , j2 ,, jh } {1,2,, m}, i [ i ,1] , 0 i 1
(3)
where u j is the unknown constant where the j -th actuator stuck at, vi (t ) is the applied control signal which experiences loss of effectiveness fault, t j and t i are the time instants at which the j -th lock-in-place and i -th loss of effectiveness fault occurs respectively. effective proportion of the actuator after losing some effectiveness, In particular,
i is still
i is the lower bound of i .
i is 1 means that the actuator u (t ) is in the failure-free case.
Taking the actuator faults (2) and (3) into account, the input vector u (t ) can be expressed as
u(t ) v(t ) s(u v(t ))
(4)
where u [u1, , um ] , v(t ) [v1 (t ),, vm (t )] , and T
T
diag{1 ,2 ,,m } s diag{s1 , s2 ,, sm }
(5)
1, if the j th actuator fails as (2) i.e. u j u j , sj 0, otherwise. In this paper, our control objective is to design a fuzzy adaptive state-feedback fault-tolerant tracking controller
u0
with parameter adaptive laws for the systems (1) with actuator faults (2) 4
and (3) to ensure all the variables of the closed-loop system are bounded in probability and the output y (t ) follows the ideal reference signal yr under arbitrary switching. In order to accomplish the control purpose, the nominal plant is proposed in the following form:
dxi ( f (t ), i ( xi ) xi 1 )dt gT (t ), i ( xi )dw 1 i n 1 T n2 dxn ( f (t ), n ( xn ) qn ( xn )u0 )dt g (t ), n ( xn )dw y x1
(6)
where qn ( xn ) is an unknown smooth nonlinear function. u0 is the designed controller by the following backstepping proof procedures. Starting from the control target, it is required that at least one actuator does not stuck at, of course, the remaining actuators can lose effectiveness or be normal, Supposed that all but one actuator have
been
stuck
and
the
values
are
equal
to
zero,
i.e.,
u j (t ) u j (t ) 0 ,
j 1,2,, i 1, i 1,, m , the considered system can still match the nominal system (6), i.e.,
qni ( xn )ui qn ( xn )u0 This will require such constants
(7)
1*j such that 1*j qnj ( xn ) qn ( xn ) , j 1,, m . The values
of 1 j are unknown, but it is necessary to design parameters adaptive laws for the sign of *
each 1 j . *
To facilitate the control design, the following assumptions are given. Assumption 1 [29]: The system (1) is constructed that for any up to m 1 actuators stuck at some places and the remaining actuators may lose effectiveness, the controlled system can still accomplish the control purpose. Assumption 2[30]: There exist constants 0 qn 0 qn1 such that qn 0 qn ( xn ) qn1 , qn ( xn ) is given in (6). Assumption 3[30]: There exist constants qnd 0 such that qn ( xn ) qnd , xn i R . n
5
Assumption 4[30]: The known values sign[1 j ] , j {1,2,, m} take the place of the sign of the *
constant
1*j .
2.2 Fuzzy logic systems In this paper, we employ fuzzy logic systems in [35] to approximate the unknown nonlinear functions. The common form of fuzzy logic systems is described as y( x)
T
( x) , where
R N is the estimate parameter vector, and (x) is the vector of the fuzzy basis function. Lemma 1[35,40-42]: Let F (x) be a real smooth function defined on a compact set R N , and for a positive constant , there exists a FLS y( x)
T
( x) such as
sup F ( x) T ( x)
(8)
x
Define the optimal parameter vector as follows:
arg min{sup F ( x) T ( x) } R N
(9)
x
Define the fuzzy minimum approximation error
as:
F ( x) T ( x)
(10)
Lemma 2[2]: (Young’s inequality) For any vectors x, y R , the following inequality must exist: n
xT y
op p 1 x q y p qo
q
(11)
where o 0 , p 1 , q 1 and ( p 1)(q 1) 1. 2.3 Stochastic System and Stability Definition Consider the nonlinear stochastic system
dX (t ) F ( X (t ))dt G( X (t ))d(t ) where X R
n
is the state, F : R R and G : R R n
n
n
which satisfied F (0) 0 and G(0) 0 , and Wiener process.
6
nr
(12) are locally Lipschitz functions,
is an r -dimensional independent standard
Definition 1[36]: For any given V ( X ) C , combined with (12), the differential operator is 2
defined in the following form:
V 1 T 2V V ( X ) F ( X ) Tr G ( X ) 2 G( X ) X 2 X
(13)
Lemma 3[36]: It is assumed that there exists a C function V : R R , two positive constants n
2
A1 and A2 , two functions 1, 2 K , one has
1 ( X ) V ( X ) 2 ( X ) [V ( X )] A1V ( X ) A2 For all X R
n
and t t0 . The system (12) for each X 0 R
(14)
n
has a unique strong solution
and the following inequality is satisfied
E[V ( X )] V ( X 0 ) E A1t
A2 , t t0 . A1
(15)
In addition, if inequality (15) satisfies, the signals of the system (12) are SGUUB in mean square. 3. Adaptive Fuzzy Control Design and Stability Analysis In this section, by utilizing the backstepping design approach [37, 38], and combing the CLF stability theory, an adaptive baskstepping state-feedback FTC method will be developed in detail, and the stability of the controlled system will be proved. With the solution to the nominal plant, the special controller structure is constructed in [30]:
v(t ) 1 1*u0 1 2* where
(16)
1* R m and 2* R m are the special controller parameter vectors, which are meet some
matching conditions for the control objective. Suppose that the lock-in-place fault (2) only occur at time instant t k , k 1,, l , and t0 t1 tl , t0 0 , tl 1 . Up to t k , there are h actuators stuck at some unknown places, that is u j (t ) u j , j j1 , jh , 0 h m 1 , meanwhile, the others may lose effectiveness, i.e., u j (t ) j v j , j j1 ,, jh , as long as 7
j [ j ,1] . From (4) and (16), one has
qnT u qnT ( I s) 1 1*u0 qnT ( I s) 1 2* qnT su qnT ( I s) 1*u0 qnT ( I s) 2* qnT su
(17)
If the switched stochastic nonlinear system (1) can match the nominal plant (6) when all but one actuator take place faults, the following equations can be obtained:
q ( I s) T n
* 1
j j1 j h
q
* 1 j nj
qnT ( I s ) 2* qnT su
j j1 j h
* 2j
2* j q 1*j n
j j1 j h
1*j q qn 1*j n
j j1 j h
j
u
j j1 j h
j
j j1 j h
(18)
j
uj
1*j
qn
(19)
0 The controller parameters
1*j and 2* j ( j {1,2,, m} ) of the matching conditions are
required as follows
j j1 j h
j j1 j h
The choice of the controller parameters
1*j 1 1*j
2* j 1*j
j j1 j h
(20)
u
1*j
0
(21)
1*j and 2* j , j j1,, jh has no relationship with the
, m} , their values can be chosen as considered system, when j { j | j j1 ,, jh } {1,2，
1*j 0 , 2* j 0 . Note that the parameters 1 , *
2* and j are all unknown, therefore the special controller
structure (16) cannot be applied. Let
1 1* 1* and 1 2* 2* , then (16) can be written as v(t ) 1u0 2
where 1 and 2 are the estimate values of
1* and 2* , respectively.
8
(22)
In this paper, the n -step adaptive backstepping [46, 47] state-feedback FTC controller design is based on the following change of coordinates:
z1 y yr
where
(23)
zi xi i 1 , i 2,, n 1
(24)
zn xn n 1
(25)
i 1 are the virtue control functions, which will be obtained in the following design
procedures. Step 1 : From (1), by using z1 y yr , one has
dz1 ( f p,1 ( x1 ) x2 )dt g Tp,1 ( x1 )dw y r dt
(26)
Similar to [39], consider the Lyapunov function candidate as:
V1
1 4 1 ~2 1 ~2 z1 1 1 4 2 1 21
~
(27)
~
where 1 0 and 1 0 are design parameters. 1 1 1 , and 1 1 1 are the *
*
parameters estimation errors, respectively. And 1 and 1 being the estimations of 1 and *
1* , respectively. By Itˆ formula and (26), one has
V1 z13 ( z2 1 f p ,1 ( x1 ) y r )
3 2 T 1 ~ 1 ~ z1 g p ,1 ( x1 ) g p ,1 ( x1 ) 1 1 2 1 1 1 1
(28)
where z2 x2 1 is used. From Lemma 2, the following inequality holds: 4 3 2 T 3 3 z1 g p ,1 ( x1 ) g p ,1 ( x1 ) b1 2 z14 g p ,1 ( x1 ) b12 2 4 4
where b1 0 is a design parameter. By (29), (28) becomes
9
(29)
4 3 3 1 ~ 1 ~ V1 z13 ( z2 1 f p ,1 ( x1 ) y r b1 2 z1 g p ,1 ( x1 ) ) b12 1 1 4 4 1 1 1 1
where
(30)
f p ,1 ( x1 ) and g p ,1 ( x1 ) cannot be straightway applied to construct virtual control
function, then fuzzy logic systems
pT,1 p,1 ( x1 ) and pT,1 p,1 ( x1 , yr ) are used to approximate
4 3 2 b1 z1 g p ,1 ( x1 ) , respectively, i.e., 4 4 3 f p,1 ( x1 ) p*T,1 p,1 ( x1 ) p,1 ( x1 ) , b12 z1 g p ,1 ( x1 ) p*T,1 p ,1 ( x1 , yr ) p ,1 ( x1 , yr ) 4
f p ,1 ( x1 ) and
(31)
By substituting (31) into (30), it produces
V1 z13 ( z2 1 p*T,1 p ,1 ( x1 ) p*T,1 p ,1 ( x1 , yr ) 3 1 ~ 1 ~ d p ,1 () y r ) b12 1 1 4 1 1 1 1
(32)
where d p ,1 () p ,1 ( x1 ) p ,1 ( x1 , yr ) . d p ,1 () is bounded, i.e., an unknown positive constant
d1* will be found such that max d p ,1 () d1* . p
By using equation (11) and the fact that p ,1 p ,1 1 , p ,1 p ,1 1 , we have T
z13 p*T,1 p ,1 ( x1 )
z16 * T z6 1 p ,1 p ,1 1 1* 4 4
z13 p*T,1 p ,1 ( x1 , yr ) where 1 max p ,1 *
*
p
2
T
z16 * T z6 1 p ,1 p ,1 1 1* 4 4
and 1 max *
p
(33)
(34)
2
p*,1 , is a positive designed constant.
By Lemma 2, we can obtain the following inequality: 4
3 z4 d *4 z ( z2 d p ,1 ) 13 z14 2 4 1 4 2 4 1 4 1 3 1
(35)
Substituting inequalities (33)-(35) into (32) produces 4
z13 z3 3 1 1 1 1 13 z1 y r ) 4 4 2 4 *4 z2 d1 3 2 1 ~ 1 z16 1 ~ 1 z16 2 b1 1 (1 ) 1 (1 ) 4 14 4 14 4 1 4 1 4
V1 z13 (
Design the virtue control signal 1 , the adaptive laws of the parameters 1 and 1 as: 10
(36)
4
3 3 z13 z13 1 c1 z1 1 z1 1 1 y r 2 4 4
(37)
1 z1 1 1 1 4
(38)
6 1 z1 1 1 1 4
(39)
6
where c1 0 , 1 0 and 1 0 are design constants. By equations (37)-(39), (36) becomes in the following form:
V1 c1 z14
z24 ~ ~ H1 1 11 1 11 4 4 1 1 1
(40)
d1*4 3 where H1 2 b12 . 4 4 1 4 Step i (2 i n 1) : By zi xi i 1 , it produces
i 1 g p , j ( x j )]T dw j 1 x j
i 1
dzi ( f p ,i ( xi ) xi 1 i 1 )dt [ g p ,i ( xi )
(41)
where
i 1
i 1 i 1 i 1 ( j 1) i 1 i 1 ( f ( x ) x ) y j p, j j j 1 ( j) r j 1 x j j 0 yr j 1 j i 1
1 i 1 2 i 1 T i 1 g p , m ( xm ) g p , k ( xk ) j 2 m, k 1 xmxk j 1 j i 1
(42)
Consider the Lyapunov function candidate as:
Vi Vi 1 where
1 4 1 ~2 1 ~2 zi i i 4 2 i 2i
~
(43)
~
i and i are positive design parameters. i *i i and i *i i are
parameters estimation errors, respectively. By Itˆ formula and (41), it follows
11
i 1 f p, j ( x j ) j 1 x j
i 1
Vi Vi 1 zi3[ zi 1 i f p ,i ( xi )
1 i 1 2 i 1 T 1 ~ 1 ~ g p , m ( xm ) g p , k ( xk ) Pi ] i i 2 m, k 1 xmxk i i i i
i 1 i 1 3 2 zi [ g p ,i ( xi ) i 1 g p , j ( x j )]T [ g p ,i ( xi ) i 1 g p , j ( x j )] 2 j 1 x j j 1 x j
(44)
where i 1 i 1 i 1 i 1 i 1 x j 1 (i j )1 yr( j 1) i 1 j j j 1 x j j 0 yr j 1 j j 1 j
i 1
Pi
(45)
Firstly, by applying Lemma 2 to the last term in (44), one has i 1 i 1 3 2 zi [ g p ,i ( xi ) i 1 g p , j ( x j )]T [ g p ,i ( xi ) i 1 g p , j ( x j )] 2 j 1 x j j 1 x j
3 3 bi2 bi 2 zi4 g p ,i ( xi ) i 1 g p , j ( x j ) 4 4 j 1 x j i 1
4
(46)
where bi 0 is a constant. By substituting (46) into (44), one has i 1 j ~ 1 ~ j j j j j H i 1 zi3 ( zi 4 i 1 j 1 j j 1 j
i 1
i 1
Vi c j z 4j j 1
(47)
1 ~ 1 ~ 3 zi 1 i M i Ni Pi ) i i i bi2 i i i 4 where
i 1 f p, j ( x j ) j 1 x j
i 1
M i f p ,i ( xi )
(48)
4
i 1 3 1 i 1 2 i 1 T Ni bi 2 zi g p ,i ( xi ) i 1 g p , j ( x j ) g p , m ( xm ) g p , k ( xk ) 4 2 m, k 1 xmxk j 1 x j
(49)
and i 1
i 1 3 2 2 ( i 1 ) b j , i 2,, n 1 4 j 1 4 j j 1 4
H i 1 Then,
fuzzy
logic
d *j 4
systems
(50)
pT,i p,i ( xi , yr(i 1) , i 1, i 1 ) , yr(i1) [ yr , y r ,, yr(i1) ]T , 12
i1 [1 ,, i1 ]T , i 1 [1,, i 1 ]T and pT,i p ,i ( xi ) are used to approximate N i and M i , respectively, i.e.,
Ni p*T,i p,i () p,i () , M i p*T,i p,i ( xi ) p ,i ( xi )
(51)
Substituting (51) into (47), one has i 1 j ~ 1 ~ j j j j j H i 1 zi3[ zi 4 i 1 j 1 j j 1 j
i 1
i 1
Vi c j z 4j j 1
(52)
1 ~ 1 ~ 3 zi 1 i p ,i ( xi ) p ,i () d p ,i () Pi ] i i i bi2 i i i 4 *T p ,i
*T p ,i
where d p ,i () p ,i ( xi ) p ,i () . Similar to step 1 , an unknown positive constant d i will be *
found such that max d p ,i () di* . p
Similar to inequality (33), (34), we have
zi6 * T zi6 * z p ,i ( xi ) i p ,i p ,i i 4 4
(53)
zi6 * T z6 i p ,i p ,i i *i 4 4
(54)
3 *T i p ,i
zi3 p*T,i p ,i ()
4
3 z4 d *4 z ( zi 1 d p ,i ) i3 zi4 i 14 i 4 2 4 i 4 i 3 i
where i max p ,i *
*
p
2
, i max *
p
(55)
2
p*,i .
Substituting inequalities (53)-(55) into (52) produces i 1 j ~ ~ j j j j j H i 1 j 1 j j 1 j
i 1
i 1
Vi c j z 4j j 1
4
z3 z3 3 z [ zi i Pi i i i i i3 zi ] 4 i 1 4 4 2 3 i
1
(56)
1 ~ i zi6 1 ~ i zi6 3 2 zi41 di*4 i ( i ) i ( ) bi 2 i i 4 i 4 4 4 i4 4 i4
Design the virtue control function i , the adaptive laws of parameters i and i as:
i ci zi
4
zi3 z3 3 1 i i i i3 zi 4 zi Pi 4 4 2 4 i 1 13
(57)
i z 6 i i i i 4
(58)
i i zi6 i i 4 where ci 0 ,
(59)
i 0 are design constants.
By substituting (57)-(59) into (56), one has i j ~ z4 ~ j j j j j i 14 H i 4 i j 1 j j 1 j
i
i
Vi c j z 4j j 1
(60)
where i
d *j 4
i 3 2 2 i bj 4 j 1 4 j j 1 4
Hi
(61)
Step n : In this step, the actual control input u0 appears. According to (1) and zn xn n 1 , one has
i 1 g p , j ( x j )]T dw x j 1 j
n 1
dzn ( f p , n ( xn ) qnT u n 1 )dt [ g p , n ( xn )
(62)
where
n 1
n 1 n 1 n 1 ( j 1) n 1 n 1 ( f ( x ) x ) y j p, j j j 1 ( j) r j 1 x j j 0 yr j 1 j n 1
1 n 1 2 n 1 T n 1 g p , m ( xm ) g p , k ( xk ) j 2 m, k 1 xmxk j 1 j n 1
(63)
Note that by (4) and (22), u (t ) can be showed as
u ( I s)1u0 ( I s) 2 su
(64)
Substituting (64) into (62), one has
dzn [
q
j j1 j h
u
j nj 1 j 0
q
j j1 j h
2 j
j nj
[ g p , n ( xn ) i 1 g p , j ( x j )]T dw j 1 x j n 1
From (17)-(21), (65) can be expressed as
14
q
j j1 j h
u j f p , n ( xn ) n 1 ]dt
nj
(65)
dzn qn ( xn )[u0 qn1 ( xn )( f p , n ( xn ) n 1 ) n 1 j~1 j j~2 j i 1 u ] dt [ g ( x ) g p , j ( x j )]T dw 0 p,n n * * j j1 j h 1 j j j1 j h 1 j j 1 x j
(66)
* * where ~1 1 1 and ~2 2 2 are the parameter estimate errors.
Take the common Lyapunov function candidate:
Vn Vn 1 where
1 1 ~2 1 ~2 zn4 n n 4qn ( xn ) 2 n 2n
j j1 j h
j~12j j~22j * 2j 1j 2 1*j j j j 2 1 j 1
(67)
h
n , n , 1 j and 2 j are positive design parameters.
From the Itˆ formula and (47), we can easily obtain the following form: n 1 j ~ ~ j j j j j H n 1 j 1 j j 1 j
n 1
n 1
Vn c j z 4j j 1
zn3[
1 4 n 1
z n u0
j j1 j h
j~1 j u 1*j 0
j j1 j h
j~2 j M n Nn ] 1*j
1 ~ 1 ~ 3 i n n bn2 i i n 4 ~ j 1 j1 j j~2 j 2 j qn ( xn ) zn4 1 j 2j 2 4qn ( xn ) 1*j 1*j j j1 j h j j1 j h
(68)
where
M n qn1 ( xn )( f p, n ( xn ) n 1 ) n1 3 N n bn2 zn g p ,n ( xn ) n1 g p , j ( x j ) 4 j 1 x j
Fuzzy logic systems
(69) 4
(70)
pT,n p,n ( xn , yr( n) , n1 , n1 ) and pT, n p, n ( xn , yr( n 1) , n 1, n 1 ) are
used to approximate M n and N n , i.e.,
M n p*T, n p, n () p, n ()
(71)
N n p*T, n p, n () p, n ()
(72)
Substituting (71) and (72) into (68), and using mathematical induction, we have
15
n 1 j ~ ~ j j j j j H n 1 j 1 j j 1 j
n 1
n 1
Vn c j z 4j j 1
zn3[
1 4 n 1
zn u0 p*T, n p , n () p*T, n p , n () d p , n ()] (73)
1 ~ 1 ~ qnd zn4 3 2 nn n n bn n n 4qn20 4 ~1 j (
j j1 j h
j1 j 1 j 1*j
j zn3 j 2 j u0 ) ~2 j ( 2j * 1 j 1*j j j j
j j1 j h
1
h
j j1 j h
j zn3 ) 1*j
where d p , n () p , n () p , n () . We find an unknown positive constant d n such that *
max d p , n () d n* . p
Similar to above step, the following inequalities can be easily obtained:
zn6 * T z6 n p , n p , n n *n 4 4
(74)
zn6 * T zn6 * z p , n () n p , n p , n n 4 4
(75)
zn3 p*T, n p , n ()
3 *T n p,n
4
zn3d p , n 2
where n max p , n , n max *
*
*
p
p
3 3 4 d n*4 n zn 4 2 4n
(76)
2
p*, n .
By the inequalities (74)-(76), (73) becomes: n 1 j ~ ~ j j j j j H n 1 j 1 j j 1 j
n 1
n 1
Vn c j z 4j j 1
z [ 3 n
1 4 n 1
4
z3 z3 3 zn u0 n n n n n3 zn ] 4 4 2 (77)
1 ~ n zn6 1 ~ n zn6 3 2 d n*4 qnd zn4 n ( ) ( ) bn 4 n n n n 4 n 4 4 4 n 4qn20 ~1 j (
j j1 j h
j1 j 1*j
1 j
j j1 j h
j zn3 j 2 j j zn3 ~ u0 ) 2 j ( 2j ) * 1*j 1*j j j j j j j 1 j 1
h
where H n 1 is given in inequality (47) with i n . By selecting real control input and the adaptive laws of parameters as
16
1
h
u0 cn zn lz n
1 4 n 1
4
zn
zn3 z3 3 n n n n3 zn 4 4 2
(78)
n zn n n n 4
(79)
n n zn n n 4
(80)
6
6
1 j sign[1*j ]
(81)
2 j 2 j 2j
(82)
zn3
2 j sign[1*j ] where cn ,
1 j 1 j 1 j
u0 zn3
1 j
2j
n , n , 1 j and 2 j are design positive constants with l qnd /( 4qn20 ) .
Substituting (78)-(82) into (77), one has
j ~2 n j ~2 j j j 1 2 j j 1 2 j
n
n
Vn c j z 4j j 1
j j1 j h
j 1 j~12j 2 1*j
j j1 j h
j 2 j~22j 2 1*j
(83)
Hn
where
Hn
n 3 2 n j *2 n j *2 2 n b j j j 4 j 1 4 j j 1 4 j 1 2 j j 1 2 j
d *j 4
n
j j1 j h
j 1 j 1*j2 2 1*j
j j1 j h
j 2 j 2*2j 2 1*j
and the facts n n j ~ j ~2 j *j2 j j j j 1 j j 1 2 j j 1 2 j n
n n j ~ j ~2 j *j2 j j j 2 2 j 1 j 1 j 1 j j j n
j j1 j h
j 1 j~1 j1 j j1 j~12j 1*j 2 1*j j j j 1
h
17
j j1 j h
j 1 j1*2j 2 1*j
(84)
j j1 j h
j 2 j~2 j 2 j j 2 j~22j 1*j 2 1*j j j j 1
h
j j1 j h
j 2 j 2*2j 2 1*j
are taken into account. For any given positive parameter
, we should choose appropriate parameters such that
1 j 1 j , 2 j 2 j , j 1,, m . Then (83) can be expressed as n 1
z 4j
z 4j
1 ~2 n 1 ~2 j j 4qn 0 j 1 2 j j 1 4 j 1 2 j j~12j j~22j 2j) 1j * * j j1 j h 2 1 j j j1 j h 2 1 j
V (
n
(85)
V where
min{4c1,,4cn 1,4cn qn0 ,1,,n ,1,, n , } and H n . t
Multiplying V by e
and by Itˆ formula, one has
d (etV ) et (V V )dt et M (t )dw
(86)
where
M (t )
n i 1 V T V g1 ( x1 ) ( gi ( xi ) i 1 g j ( x j ))T z1 i 2 zi j 1 x j
From (85) and (86), one has
d (etV ) et dt et M (t )dw
(87)
Integrating (87) over [0, T ] , we get
V (T ) etV (0)
T eT et 1dw(t ) 0
(88)
Taking expectation on (88), it follows that
E[V (t )] V (0)e t
(89)
By Lemma 3 and utilizing the similar conclusions in [8-14], then all the signals of the considered system are bounded in probability.
18
The above control design and analysis are concluded in the following theorem. Theorem 1: For uncertain switched stochastic nonlinear system (1) with actuator faults, under Assumptions 1-4, the controller (78) with the virtue control signals (37), (57) and parameter laws (38), (39), (58), (59), (79), (80), (81) and (82) guarantee that all the signals in the closed-loop system is SGUUB in probability. Moreover, the tracking error can be made to converge to a small neighborhood of zero by selecting the design parameters appropriately. Remark 1: In this paper, we not only consider stochastic nonlinear systems but also consider the switched stochastic nonlinear systems with actuator faults, it is more complexity and difficult than common stochastic systems. We adopt common Lyapunov function stability theory and fault-tolerant control technique to solve the switched problem and actuator faults, respectively. Therefore, it is more challenge than previous investigations in the literatures [10-12]. And also, the literature [43-45] have obtained great achievements on the Takagi-Sugeno model systems, however, they can not applied in the system which considered in this paper. The parameters design guideline of the proposed control scheme is summarized as follows. 1) choose the design parameters
i , i , 1 j , 2 j , ci , i , i , l , 1 j and 2 j
(i 1,, n) , j ( j1,, jh ) in (27), (37-39), (43), (57-59), (67) and (78-82) to satisfied that
i 0 , i 0 , 1 j 0 , 2 j 0 , ci 0 , i 0 , i 0 , l 0 , 1 j 0 and
2 j 0 , respectively. 2) Choose the design parameters i ,
i , 1 j and 2 j (i 1,, n) , j ( j1,, jh ) in
(39), (58), (59) and (79)-(82), which are positive constants and are used for the According to [32]–[34], the appropriate choices of i , parameters
(38),
-modification.
i , 1 j and 2 j can
prevent the
i i 1 j and 2 j to drift.
4. Simulation Study In this section, the effectiveness and the control performances of the design method are 19
showed by the numerical example. Consider the second-order switched stochastic nonlinear systems with actuator faults:
dx1 ( f1,1 ( x1 ) x2 )dt g1,1 ( x1 )dw dx2 ( f1, 2 ( x2 ) q21( x2 )u1 q22 ( x2 )u2 )dt g 2,1 ( x2 )dw y x 1
(90)
dx1 ( f 2,1 ( x1 ) x2 )dt g1, 2 ( x1 )dw dx2 ( f 2, 2 ( x2 ) q21( x2 )u1 q22 ( x2 )u2 )dt g 2, 2 ( x2 )dw y x 1
(91)
where q21( x2 ) 2 cos( x1x2 ) , q22 ( x2 ) 4 sin( x1 ) , f1,1 ( x1 ) 0.6 x1 , f1, 2 ( x2 ) x1x2 , 2
f 2,1 ( x1 ) 0.9 sin( x1 ) , f 2, 2 ( x2 ) 1.1sin( x1x2 ) , g1,1 ( x1 ) x12 cos x12 , g2,1 ( x2 ) x12 sin x2 , g1, 2 ( x1 ) x12 sin x1 /(1 x12 ) , g2, 2 ( x2 ) 5x2e x1 / 2 /(1 x12 ) . The given reference signal is yr (t ) sin(0.2t ) . Parameters in controller and adaptive laws are chosen as c1 20 , c2 30 , l 10 , 1 1 ,
2 1 , 1 , 1 0.1 , 2 0.2 , 1 0.3 , 2 0.5 , 11 0.001 , 12 0.001 ,
21 0.001 , 22 0.001 , 11 0.01 , 12 0.01 , 21 0.01 , 22 0.01 , 1 0.01 , 2 0.01 , r1 0.05 , r2 0.05 . For simulation purpose, the actuator faults involved in this paper are defined as u1 (t ) 1 when t 5 , and u2 (t ) 0.6v2 for t 10 . The initial conditions are chosen as x1 (0) 0.01 , x2 (0) 0.2 , 11(0) 0.3 , 12 (0) 0.6 and the others initial values are chosen as zeros. The simulation results are shown in Figs. 1-5, where Fig. 1 exhibits the trajectories of the output and tracking signal; Fig. 2 expresses the trajectory of x2 ; Fig. 3 shows the trajectories of actuators u1 and u 2 ; Fig. 4 exhibits the trajectory of the actual input u0 ; Fig. 5 exhibits the trajectory of switched signal.
20
Fig. 1. The trajectories of x1 (red) and yr (black).
Fig. 2. The trajectory of x2 .
21
Fig. 3. The trajectories of u1 (black) and u 2 (red).
Fig. 4. The trajectory of u0 .
Fig. 5. The trajectory of the switched signal. 5. Conclusions In this paper, a fuzzy adaptive state-feedback fault-tolerant tracking control method has been developed for a class of uncertain switched stochastic nonlinear systems in the presence of actuator faults. Based on the baskstepping technique, nonlinear FTC design approach and common Lyapunov function method, a fuzzy adaptive fault-tolerant tracking control scheme has been
22
developed. The proposed control scheme can guarantee that all signals of the closed-loop system are SGUUB in mean square and the tracking error converges to an arbitrarily small neighborhood of the origin. Our further research works will concentrate on the switched stochastic nonlinear MIMO and large-scale systems based on this paper.
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uncertain systems, Control Theory and Applications, vol. 9, no. 2, pp. 244-250, 2011. [32] S. Tong, B. Huo, and Y. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Transactions on Fuzzy Systems, vol. 22, no. 1, pp. 1-15, 2014. [33] S. Sui, S. C. Tong and Y. M. Li, Fuzzy adaptive fault-tolerant tracking control of MIMO stochastic pure-feedback nonlinear systems with actuator failures, Journal of the Franklin Institute, vol. 351, no. 6, pp. 3424-3444, 2014. [34] S. C. Tong, S. Sui and Y. M. Li, Adaptive fuzzy decentralized tracking fault-tolerant control for stochastic nonlinear large-scale systems with unmodeled dynamics, Information Sciences, vol. 289, pp. 225-240, 2014. [35] L. X. Wang, Adaptive fuzzy systems and control: Design and stability analysis, Prentice-Hall, Englewood Cliffs, NJ, 1994. [36] H. B. Ji and H. S. Xi, Adaptive output-feedback tracking of stochastic nonlinear systems, IEEE Transaction Automatic Control, vol. 51, no. 2, pp. 355-360, 2006. [37] Q. Zhou, H. Y. Li and P. Shi, Decentralized adaptive fuzzy tracking control for robot finger dynamics, IEEE Transactions on Fuzzy Systems, vol. 23, no. 3, pp. 501-510, 2015. [38] T. Wang, H. Gao and J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control, IEEE Transactions on Neural Networks and Learning Systems, 10.1109/TNNLS.2015.2411671, 2015. [39] H. Y. Li, P. Shi, D. Y. Yao and L. G. Wu, Observer-based adaptive sliding mode control of nonlinear Markovian jump systems, Automatica, 10.1016/j.automatica.2015.11.007. [40] Y. P. Pan and M. J. Er, Enhanced adaptive fuzzy control with optimal approximation error convergence, IEEE Transactions on Fuzzy Systems, vol. 21, no. 6, pp. 1123-1132, 2013. [41] Y. P. Pan, Y. Zhou, T. R. Sun and M. J. Er, Composite adaptive fuzzy H tracking control of uncertain nonlinear systems, Neurocomputing, vol. 99, pp. 15-24, 2013. [42] Y. P. Pan, M. J. Er, D. P. Huang, and T. R. Sun, Practical adaptive fuzzy H tracking control of uncertain nonlinear systems, International Journal of Fuzzy Systems, vol. 14, no. 4, pp. 463-473, 2012. [43] H. Y. Li, Y. Shen, Y. N. Pan and H. K. Lam, Model reduction for interval type-2 26
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Zhiyao Ma, received the B.S. degree in Information and Computing Science from Liaoning University of Technology, Jinzhou, China, in 2013. He is working towards the M.E. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China. His current research interests include fuzzy control, adaptive control, and fault-tolerant control.
Shaocheng Tong, received the B.S. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.S. degree in fuzzy mathematics from Dalian Marine University, PRC, and the Ph.D degree in fuzzy control from Northeastern University, PRC, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PRC. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control. Yongming Li, received the B.S. degree and the M.S. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively. He received the Ph.D degree in Transportation Information Engineering & Control from Dalian Maritime University, Dalian, China in 2014. He is currently an Associate Professor in the Department of Basic Mathematics, Liaoning University of Technology. His current research interests include adaptive control, fuzzy control and neural networks control for nonlinear systems.
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PII: DOI: Reference:
S0925-2312(15)02053-6 http://dx.doi.org/10.1016/j.neucom.2015.12.080 NEUCOM16604
To appear in: Neurocomputing Received date: 18 November 2015 Revised date: 20 December 2015 Accepted date: 22 December 2015 Cite this article as: Zhiyao Ma, Shaocheng Tong and Yongming Li, Fuzzy Adaptive State-feedback Fault-Tolerant Control for Switched Stochastic Nonlinear Systems with Faults, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.12.080 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.
Fuzzy Adaptive State-feedback Fault-Tolerant Control for Switched Stochastic Nonlinear Systems with Faults Zhiyao Ma*, Shaocheng Tong, Yongming Li Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, Liaoning, 121001, P. R. China Abstract: In this paper, the fault-tolerant control (FTC) problem is investigated for a class of uncertain switched stochastic nonlinear systems. The controlled system contains unknown nonlinear functions and actuator faults, which are modeled as both lock-in-place and loss of effectiveness. With the help of fuzzy logic systems to approximate the unknown nonlinear functions and by utilizing the common Lyapunov function (CLF) method, a fuzzy adaptive state-feedback fault-tolerant tracking control approach is developed. It is shown that all the variables of the closed-loop system are semi-globally uniformly bounded (SGUUB) in probability, the tracking error converges to an arbitrary small neighborhood of the origin. A simulation is given to demonstrate the effectiveness of the proposed approach. Keywords: Nonlinear switched stochastic systems; common Lyapunov function; fault-tolerant control; backstepping technique 1. Introduction In the past decades, adaptive control design for stochastic nonlinear systems has attracted more and more attention and many important control methods have been developed, see [1-6] and references therein. Among them, [1, 2] proposed the backstepping control design method for stochastic single-input-single-output (SISO) nonlinear systems by utilizing Quartic Lyapunov function. In [3, 4], according to changing supply function and small-gain theorem, the adaptive fuzzy backstepping design schemes were developed for nonlinear stochastic SISO systems in the presence of unmodeled dynamics. Then, [5, 6] proposed the adaptive fuzzy output-feedback decentralized control methods for stochastic nonlinear large-scale systems. However, the main
This work was supported by the National Natural Science Foundation of China (Nos. 61374113, 61573175). †Corresponding author. Tel.: +86-416-4199101; Fax: +86-416-4199415. E-mail: [email protected].
1
limitation in [1-6] is that these approaches are only suitable for those stochastic nonlinear systems, in which the nonlinear uncertainties must be linearly parameterized. To cope with the above limitation, many novel adaptive control methods have been developed, see [7-14]. [7-10] proposed some adaptive state-feedback control methods for the nonlinear stochastic systems that the states can be directly used, and [11-14] proposed several adaptive output-feedback control schemes for the nonlinear stochastic systems in the presence of the unmeasured states. On the other hand, switched systems have gained considerable interest in the past decades and many physical and engineering systems can be constructed as nonlinear switched systems [15-17]. As we know that the stability of the switched system can be guaranteed if a CLF exists under arbitrary switching (see [18]). In recently years, some backstepping control schemes have been investigated [19-21] for the nonlinear switched systems on the basis of the CLF stability theory, but the nonlinear functions of considered systems are both known. Furthermore, due to the existence of stochastic disturbances in practical systems, some researchers begin to put more time and energy into the investigations of switched stochastic systems. Many remarkable consequences of stability analysis (see [22-24]) and controller design (see [25-27]) have been acquired. Among them, [24] proposed an adaptive control scheme for a class of stochastic nonlinear systems in the presence of irreducible homogenous Markovian switching. Based on the common Lyapunov function method, [27] proposed the state feedback and output feedback control approaches for a class of stochastic nonlinear systems with measured and unmeasured states, respectively. However, the above control schemes assume that all the components of the considered systems are in good operating conditions, and they did not consider the problem of faults. In general, there always exist some faults [28] in the practical control systems, like actuator and sensor faults, these faults often reduce the control characteristics and influence the stability of the considered system, and more serious result is the catastrophic accidents. Thus, many FTC design methods have been proposed to accommodate the actuator faults involved in the nonlinear systems [29-34]. The works in [29, 30] investigated the adaptive fuzzy FTC schemes for a class of unknown SISO nonlinear strict-feedback systems in the presence of actuator failures. Based on their results, a fuzzy adaptive FTC design method was proposed in [31] for a class of nonlinear multi-input-multi-output (MIMO) systems. To handle the unmeasured states problems, [32] proposed a fuzzy adaptive output-feedback tracking FTC method for large-scale nonlinear systems. 2
The fuzzy adaptive output feedback tracking FTC approaches were proposed in [33, 34] for the stochastic nonlinear MIMO and large-scale systems. However, the aforementioned methods are only applied to the nonlinear non-switched systems. Inspired by the above observations, in this paper, a fuzzy adaptive state-feedback fault-tolerant tracking control scheme is developed for a class of switched stochastic nonlinear systems, under arbitrary switching. It is shown that the proposed control approach can guarantee that all the variables of the closed-loop switched stochastic system are SGUUB in probability and the tracking error regulates to a small neighborhood of the origin regardless of actuator faults. Compared with previous results, this paper owns two contributions: (1) This paper proposed a fuzzy adaptive FTC design scheme for a class of switched stochastic nonlinear systems. The proposed adaptive control method has solved the actuator faults problem. Note that the previous References [19-27] assumed that all the components in the considered systems are in good operating conditions. Thus, they can not be used for dealing with the switched stochastic nonlinear systems with actuator faults. (2) The considered nonlinear systems of this paper contain stochastic disturbance, actuator faults and switched signals. Consequently, based on the common Lyapunov function stability theory and FTC design method, the proposed adaptive controller can not only have the robustness to stochastic disturbance and accommodate the actuator faults, but also guarantee the stability of the whole controlled systems, thus the existing results in [33-38] can not be applied to control the switched stochastic nonlinear systems considered in this paper. 2. Problem Formulations and Preliminaries 2.1 System descriptions and assumptions Consider a class of uncertain switched stochastic nonlinear systems:
dxi ( f (t ), i ( xi ) xi 1 )dt gT (t ), i ( xi )dw 1 i n 1 T T n2 dxn ( f (t ), n ( xn ) qn ( xn )u )dt g (t ), n ( xn )dw y x1
(1)
where xi [ x1 ,, xi ] R , i 1,, n are the state vectors, u [u1 , u2 ,, um ] R T
i
T
def
m
is
the input vector of the system, y R is the system output. (t ) : [0, ) {1,2,, N } 3
is the switching signal, which takes its values in the compact set . Moreover, (t ) p ,
p 1,, N means that p -th subsystem is active. qn ( xn ) [qn1 ( xn ),, qnm ( xn )]T Rm ,
f p ,i , qnj and g p ,i for i 1,, n , j 1,, m are unknown nonlinear smooth functions. w R is an independent r -dimension standard Wiener process. The actuator faults involved in this paper are both lock-in-place and loss of effectiveness are defined by [29-34] as follows. Lock-in-place fault model:
u j (t ) u j , t t j , j { j1 , j2 ,, jh } {1,2,, m}
(2)
Loss of effectiveness fault model:
ui (t ) i vi (t ) t ti , i { j1 , j2 ,, jh } {1,2,, m}, i [ i ,1] , 0 i 1
(3)
where u j is the unknown constant where the j -th actuator stuck at, vi (t ) is the applied control signal which experiences loss of effectiveness fault, t j and t i are the time instants at which the j -th lock-in-place and i -th loss of effectiveness fault occurs respectively. effective proportion of the actuator after losing some effectiveness, In particular,
i is still
i is the lower bound of i .
i is 1 means that the actuator u (t ) is in the failure-free case.
Taking the actuator faults (2) and (3) into account, the input vector u (t ) can be expressed as
u(t ) v(t ) s(u v(t ))
(4)
where u [u1, , um ] , v(t ) [v1 (t ),, vm (t )] , and T
T
diag{1 ,2 ,,m } s diag{s1 , s2 ,, sm }
(5)
1, if the j th actuator fails as (2) i.e. u j u j , sj 0, otherwise. In this paper, our control objective is to design a fuzzy adaptive state-feedback fault-tolerant tracking controller
u0
with parameter adaptive laws for the systems (1) with actuator faults (2) 4
and (3) to ensure all the variables of the closed-loop system are bounded in probability and the output y (t ) follows the ideal reference signal yr under arbitrary switching. In order to accomplish the control purpose, the nominal plant is proposed in the following form:
dxi ( f (t ), i ( xi ) xi 1 )dt gT (t ), i ( xi )dw 1 i n 1 T n2 dxn ( f (t ), n ( xn ) qn ( xn )u0 )dt g (t ), n ( xn )dw y x1
(6)
where qn ( xn ) is an unknown smooth nonlinear function. u0 is the designed controller by the following backstepping proof procedures. Starting from the control target, it is required that at least one actuator does not stuck at, of course, the remaining actuators can lose effectiveness or be normal, Supposed that all but one actuator have
been
stuck
and
the
values
are
equal
to
zero,
i.e.,
u j (t ) u j (t ) 0 ,
j 1,2,, i 1, i 1,, m , the considered system can still match the nominal system (6), i.e.,
qni ( xn )ui qn ( xn )u0 This will require such constants
(7)
1*j such that 1*j qnj ( xn ) qn ( xn ) , j 1,, m . The values
of 1 j are unknown, but it is necessary to design parameters adaptive laws for the sign of *
each 1 j . *
To facilitate the control design, the following assumptions are given. Assumption 1 [29]: The system (1) is constructed that for any up to m 1 actuators stuck at some places and the remaining actuators may lose effectiveness, the controlled system can still accomplish the control purpose. Assumption 2[30]: There exist constants 0 qn 0 qn1 such that qn 0 qn ( xn ) qn1 , qn ( xn ) is given in (6). Assumption 3[30]: There exist constants qnd 0 such that qn ( xn ) qnd , xn i R . n
5
Assumption 4[30]: The known values sign[1 j ] , j {1,2,, m} take the place of the sign of the *
constant
1*j .
2.2 Fuzzy logic systems In this paper, we employ fuzzy logic systems in [35] to approximate the unknown nonlinear functions. The common form of fuzzy logic systems is described as y( x)
T
( x) , where
R N is the estimate parameter vector, and (x) is the vector of the fuzzy basis function. Lemma 1[35,40-42]: Let F (x) be a real smooth function defined on a compact set R N , and for a positive constant , there exists a FLS y( x)
T
( x) such as
sup F ( x) T ( x)
(8)
x
Define the optimal parameter vector as follows:
arg min{sup F ( x) T ( x) } R N
(9)
x
Define the fuzzy minimum approximation error
as:
F ( x) T ( x)
(10)
Lemma 2[2]: (Young’s inequality) For any vectors x, y R , the following inequality must exist: n
xT y
op p 1 x q y p qo
q
(11)
where o 0 , p 1 , q 1 and ( p 1)(q 1) 1. 2.3 Stochastic System and Stability Definition Consider the nonlinear stochastic system
dX (t ) F ( X (t ))dt G( X (t ))d(t ) where X R
n
is the state, F : R R and G : R R n
n
n
which satisfied F (0) 0 and G(0) 0 , and Wiener process.
6
nr
(12) are locally Lipschitz functions,
is an r -dimensional independent standard
Definition 1[36]: For any given V ( X ) C , combined with (12), the differential operator is 2
defined in the following form:
V 1 T 2V V ( X ) F ( X ) Tr G ( X ) 2 G( X ) X 2 X
(13)
Lemma 3[36]: It is assumed that there exists a C function V : R R , two positive constants n
2
A1 and A2 , two functions 1, 2 K , one has
1 ( X ) V ( X ) 2 ( X ) [V ( X )] A1V ( X ) A2 For all X R
n
and t t0 . The system (12) for each X 0 R
(14)
n
has a unique strong solution
and the following inequality is satisfied
E[V ( X )] V ( X 0 ) E A1t
A2 , t t0 . A1
(15)
In addition, if inequality (15) satisfies, the signals of the system (12) are SGUUB in mean square. 3. Adaptive Fuzzy Control Design and Stability Analysis In this section, by utilizing the backstepping design approach [37, 38], and combing the CLF stability theory, an adaptive baskstepping state-feedback FTC method will be developed in detail, and the stability of the controlled system will be proved. With the solution to the nominal plant, the special controller structure is constructed in [30]:
v(t ) 1 1*u0 1 2* where
(16)
1* R m and 2* R m are the special controller parameter vectors, which are meet some
matching conditions for the control objective. Suppose that the lock-in-place fault (2) only occur at time instant t k , k 1,, l , and t0 t1 tl , t0 0 , tl 1 . Up to t k , there are h actuators stuck at some unknown places, that is u j (t ) u j , j j1 , jh , 0 h m 1 , meanwhile, the others may lose effectiveness, i.e., u j (t ) j v j , j j1 ,, jh , as long as 7
j [ j ,1] . From (4) and (16), one has
qnT u qnT ( I s) 1 1*u0 qnT ( I s) 1 2* qnT su qnT ( I s) 1*u0 qnT ( I s) 2* qnT su
(17)
If the switched stochastic nonlinear system (1) can match the nominal plant (6) when all but one actuator take place faults, the following equations can be obtained:
q ( I s) T n
* 1
j j1 j h
q
* 1 j nj
qnT ( I s ) 2* qnT su
j j1 j h
* 2j
2* j q 1*j n
j j1 j h
1*j q qn 1*j n
j j1 j h
j
u
j j1 j h
j
j j1 j h
(18)
j
uj
1*j
qn
(19)
0 The controller parameters
1*j and 2* j ( j {1,2,, m} ) of the matching conditions are
required as follows
j j1 j h
j j1 j h
The choice of the controller parameters
1*j 1 1*j
2* j 1*j
j j1 j h
(20)
u
1*j
0
(21)
1*j and 2* j , j j1,, jh has no relationship with the
, m} , their values can be chosen as considered system, when j { j | j j1 ,, jh } {1,2，
1*j 0 , 2* j 0 . Note that the parameters 1 , *
2* and j are all unknown, therefore the special controller
structure (16) cannot be applied. Let
1 1* 1* and 1 2* 2* , then (16) can be written as v(t ) 1u0 2
where 1 and 2 are the estimate values of
1* and 2* , respectively.
8
(22)
In this paper, the n -step adaptive backstepping [46, 47] state-feedback FTC controller design is based on the following change of coordinates:
z1 y yr
where
(23)
zi xi i 1 , i 2,, n 1
(24)
zn xn n 1
(25)
i 1 are the virtue control functions, which will be obtained in the following design
procedures. Step 1 : From (1), by using z1 y yr , one has
dz1 ( f p,1 ( x1 ) x2 )dt g Tp,1 ( x1 )dw y r dt
(26)
Similar to [39], consider the Lyapunov function candidate as:
V1
1 4 1 ~2 1 ~2 z1 1 1 4 2 1 21
~
(27)
~
where 1 0 and 1 0 are design parameters. 1 1 1 , and 1 1 1 are the *
*
parameters estimation errors, respectively. And 1 and 1 being the estimations of 1 and *
1* , respectively. By Itˆ formula and (26), one has
V1 z13 ( z2 1 f p ,1 ( x1 ) y r )
3 2 T 1 ~ 1 ~ z1 g p ,1 ( x1 ) g p ,1 ( x1 ) 1 1 2 1 1 1 1
(28)
where z2 x2 1 is used. From Lemma 2, the following inequality holds: 4 3 2 T 3 3 z1 g p ,1 ( x1 ) g p ,1 ( x1 ) b1 2 z14 g p ,1 ( x1 ) b12 2 4 4
where b1 0 is a design parameter. By (29), (28) becomes
9
(29)
4 3 3 1 ~ 1 ~ V1 z13 ( z2 1 f p ,1 ( x1 ) y r b1 2 z1 g p ,1 ( x1 ) ) b12 1 1 4 4 1 1 1 1
where
(30)
f p ,1 ( x1 ) and g p ,1 ( x1 ) cannot be straightway applied to construct virtual control
function, then fuzzy logic systems
pT,1 p,1 ( x1 ) and pT,1 p,1 ( x1 , yr ) are used to approximate
4 3 2 b1 z1 g p ,1 ( x1 ) , respectively, i.e., 4 4 3 f p,1 ( x1 ) p*T,1 p,1 ( x1 ) p,1 ( x1 ) , b12 z1 g p ,1 ( x1 ) p*T,1 p ,1 ( x1 , yr ) p ,1 ( x1 , yr ) 4
f p ,1 ( x1 ) and
(31)
By substituting (31) into (30), it produces
V1 z13 ( z2 1 p*T,1 p ,1 ( x1 ) p*T,1 p ,1 ( x1 , yr ) 3 1 ~ 1 ~ d p ,1 () y r ) b12 1 1 4 1 1 1 1
(32)
where d p ,1 () p ,1 ( x1 ) p ,1 ( x1 , yr ) . d p ,1 () is bounded, i.e., an unknown positive constant
d1* will be found such that max d p ,1 () d1* . p
By using equation (11) and the fact that p ,1 p ,1 1 , p ,1 p ,1 1 , we have T
z13 p*T,1 p ,1 ( x1 )
z16 * T z6 1 p ,1 p ,1 1 1* 4 4
z13 p*T,1 p ,1 ( x1 , yr ) where 1 max p ,1 *
*
p
2
T
z16 * T z6 1 p ,1 p ,1 1 1* 4 4
and 1 max *
p
(33)
(34)
2
p*,1 , is a positive designed constant.
By Lemma 2, we can obtain the following inequality: 4
3 z4 d *4 z ( z2 d p ,1 ) 13 z14 2 4 1 4 2 4 1 4 1 3 1
(35)
Substituting inequalities (33)-(35) into (32) produces 4
z13 z3 3 1 1 1 1 13 z1 y r ) 4 4 2 4 *4 z2 d1 3 2 1 ~ 1 z16 1 ~ 1 z16 2 b1 1 (1 ) 1 (1 ) 4 14 4 14 4 1 4 1 4
V1 z13 (
Design the virtue control signal 1 , the adaptive laws of the parameters 1 and 1 as: 10
(36)
4
3 3 z13 z13 1 c1 z1 1 z1 1 1 y r 2 4 4
(37)
1 z1 1 1 1 4
(38)
6 1 z1 1 1 1 4
(39)
6
where c1 0 , 1 0 and 1 0 are design constants. By equations (37)-(39), (36) becomes in the following form:
V1 c1 z14
z24 ~ ~ H1 1 11 1 11 4 4 1 1 1
(40)
d1*4 3 where H1 2 b12 . 4 4 1 4 Step i (2 i n 1) : By zi xi i 1 , it produces
i 1 g p , j ( x j )]T dw j 1 x j
i 1
dzi ( f p ,i ( xi ) xi 1 i 1 )dt [ g p ,i ( xi )
(41)
where
i 1
i 1 i 1 i 1 ( j 1) i 1 i 1 ( f ( x ) x ) y j p, j j j 1 ( j) r j 1 x j j 0 yr j 1 j i 1
1 i 1 2 i 1 T i 1 g p , m ( xm ) g p , k ( xk ) j 2 m, k 1 xmxk j 1 j i 1
(42)
Consider the Lyapunov function candidate as:
Vi Vi 1 where
1 4 1 ~2 1 ~2 zi i i 4 2 i 2i
~
(43)
~
i and i are positive design parameters. i *i i and i *i i are
parameters estimation errors, respectively. By Itˆ formula and (41), it follows
11
i 1 f p, j ( x j ) j 1 x j
i 1
Vi Vi 1 zi3[ zi 1 i f p ,i ( xi )
1 i 1 2 i 1 T 1 ~ 1 ~ g p , m ( xm ) g p , k ( xk ) Pi ] i i 2 m, k 1 xmxk i i i i
i 1 i 1 3 2 zi [ g p ,i ( xi ) i 1 g p , j ( x j )]T [ g p ,i ( xi ) i 1 g p , j ( x j )] 2 j 1 x j j 1 x j
(44)
where i 1 i 1 i 1 i 1 i 1 x j 1 (i j )1 yr( j 1) i 1 j j j 1 x j j 0 yr j 1 j j 1 j
i 1
Pi
(45)
Firstly, by applying Lemma 2 to the last term in (44), one has i 1 i 1 3 2 zi [ g p ,i ( xi ) i 1 g p , j ( x j )]T [ g p ,i ( xi ) i 1 g p , j ( x j )] 2 j 1 x j j 1 x j
3 3 bi2 bi 2 zi4 g p ,i ( xi ) i 1 g p , j ( x j ) 4 4 j 1 x j i 1
4
(46)
where bi 0 is a constant. By substituting (46) into (44), one has i 1 j ~ 1 ~ j j j j j H i 1 zi3 ( zi 4 i 1 j 1 j j 1 j
i 1
i 1
Vi c j z 4j j 1
(47)
1 ~ 1 ~ 3 zi 1 i M i Ni Pi ) i i i bi2 i i i 4 where
i 1 f p, j ( x j ) j 1 x j
i 1
M i f p ,i ( xi )
(48)
4
i 1 3 1 i 1 2 i 1 T Ni bi 2 zi g p ,i ( xi ) i 1 g p , j ( x j ) g p , m ( xm ) g p , k ( xk ) 4 2 m, k 1 xmxk j 1 x j
(49)
and i 1
i 1 3 2 2 ( i 1 ) b j , i 2,, n 1 4 j 1 4 j j 1 4
H i 1 Then,
fuzzy
logic
d *j 4
systems
(50)
pT,i p,i ( xi , yr(i 1) , i 1, i 1 ) , yr(i1) [ yr , y r ,, yr(i1) ]T , 12
i1 [1 ,, i1 ]T , i 1 [1,, i 1 ]T and pT,i p ,i ( xi ) are used to approximate N i and M i , respectively, i.e.,
Ni p*T,i p,i () p,i () , M i p*T,i p,i ( xi ) p ,i ( xi )
(51)
Substituting (51) into (47), one has i 1 j ~ 1 ~ j j j j j H i 1 zi3[ zi 4 i 1 j 1 j j 1 j
i 1
i 1
Vi c j z 4j j 1
(52)
1 ~ 1 ~ 3 zi 1 i p ,i ( xi ) p ,i () d p ,i () Pi ] i i i bi2 i i i 4 *T p ,i
*T p ,i
where d p ,i () p ,i ( xi ) p ,i () . Similar to step 1 , an unknown positive constant d i will be *
found such that max d p ,i () di* . p
Similar to inequality (33), (34), we have
zi6 * T zi6 * z p ,i ( xi ) i p ,i p ,i i 4 4
(53)
zi6 * T z6 i p ,i p ,i i *i 4 4
(54)
3 *T i p ,i
zi3 p*T,i p ,i ()
4
3 z4 d *4 z ( zi 1 d p ,i ) i3 zi4 i 14 i 4 2 4 i 4 i 3 i
where i max p ,i *
*
p
2
, i max *
p
(55)
2
p*,i .
Substituting inequalities (53)-(55) into (52) produces i 1 j ~ ~ j j j j j H i 1 j 1 j j 1 j
i 1
i 1
Vi c j z 4j j 1
4
z3 z3 3 z [ zi i Pi i i i i i3 zi ] 4 i 1 4 4 2 3 i
1
(56)
1 ~ i zi6 1 ~ i zi6 3 2 zi41 di*4 i ( i ) i ( ) bi 2 i i 4 i 4 4 4 i4 4 i4
Design the virtue control function i , the adaptive laws of parameters i and i as:
i ci zi
4
zi3 z3 3 1 i i i i3 zi 4 zi Pi 4 4 2 4 i 1 13
(57)
i z 6 i i i i 4
(58)
i i zi6 i i 4 where ci 0 ,
(59)
i 0 are design constants.
By substituting (57)-(59) into (56), one has i j ~ z4 ~ j j j j j i 14 H i 4 i j 1 j j 1 j
i
i
Vi c j z 4j j 1
(60)
where i
d *j 4
i 3 2 2 i bj 4 j 1 4 j j 1 4
Hi
(61)
Step n : In this step, the actual control input u0 appears. According to (1) and zn xn n 1 , one has
i 1 g p , j ( x j )]T dw x j 1 j
n 1
dzn ( f p , n ( xn ) qnT u n 1 )dt [ g p , n ( xn )
(62)
where
n 1
n 1 n 1 n 1 ( j 1) n 1 n 1 ( f ( x ) x ) y j p, j j j 1 ( j) r j 1 x j j 0 yr j 1 j n 1
1 n 1 2 n 1 T n 1 g p , m ( xm ) g p , k ( xk ) j 2 m, k 1 xmxk j 1 j n 1
(63)
Note that by (4) and (22), u (t ) can be showed as
u ( I s)1u0 ( I s) 2 su
(64)
Substituting (64) into (62), one has
dzn [
q
j j1 j h
u
j nj 1 j 0
q
j j1 j h
2 j
j nj
[ g p , n ( xn ) i 1 g p , j ( x j )]T dw j 1 x j n 1
From (17)-(21), (65) can be expressed as
14
q
j j1 j h
u j f p , n ( xn ) n 1 ]dt
nj
(65)
dzn qn ( xn )[u0 qn1 ( xn )( f p , n ( xn ) n 1 ) n 1 j~1 j j~2 j i 1 u ] dt [ g ( x ) g p , j ( x j )]T dw 0 p,n n * * j j1 j h 1 j j j1 j h 1 j j 1 x j
(66)
* * where ~1 1 1 and ~2 2 2 are the parameter estimate errors.
Take the common Lyapunov function candidate:
Vn Vn 1 where
1 1 ~2 1 ~2 zn4 n n 4qn ( xn ) 2 n 2n
j j1 j h
j~12j j~22j * 2j 1j 2 1*j j j j 2 1 j 1
(67)
h
n , n , 1 j and 2 j are positive design parameters.
From the Itˆ formula and (47), we can easily obtain the following form: n 1 j ~ ~ j j j j j H n 1 j 1 j j 1 j
n 1
n 1
Vn c j z 4j j 1
zn3[
1 4 n 1
z n u0
j j1 j h
j~1 j u 1*j 0
j j1 j h
j~2 j M n Nn ] 1*j
1 ~ 1 ~ 3 i n n bn2 i i n 4 ~ j 1 j1 j j~2 j 2 j qn ( xn ) zn4 1 j 2j 2 4qn ( xn ) 1*j 1*j j j1 j h j j1 j h
(68)
where
M n qn1 ( xn )( f p, n ( xn ) n 1 ) n1 3 N n bn2 zn g p ,n ( xn ) n1 g p , j ( x j ) 4 j 1 x j
Fuzzy logic systems
(69) 4
(70)
pT,n p,n ( xn , yr( n) , n1 , n1 ) and pT, n p, n ( xn , yr( n 1) , n 1, n 1 ) are
used to approximate M n and N n , i.e.,
M n p*T, n p, n () p, n ()
(71)
N n p*T, n p, n () p, n ()
(72)
Substituting (71) and (72) into (68), and using mathematical induction, we have
15
n 1 j ~ ~ j j j j j H n 1 j 1 j j 1 j
n 1
n 1
Vn c j z 4j j 1
zn3[
1 4 n 1
zn u0 p*T, n p , n () p*T, n p , n () d p , n ()] (73)
1 ~ 1 ~ qnd zn4 3 2 nn n n bn n n 4qn20 4 ~1 j (
j j1 j h
j1 j 1 j 1*j
j zn3 j 2 j u0 ) ~2 j ( 2j * 1 j 1*j j j j
j j1 j h
1
h
j j1 j h
j zn3 ) 1*j
where d p , n () p , n () p , n () . We find an unknown positive constant d n such that *
max d p , n () d n* . p
Similar to above step, the following inequalities can be easily obtained:
zn6 * T z6 n p , n p , n n *n 4 4
(74)
zn6 * T zn6 * z p , n () n p , n p , n n 4 4
(75)
zn3 p*T, n p , n ()
3 *T n p,n
4
zn3d p , n 2
where n max p , n , n max *
*
*
p
p
3 3 4 d n*4 n zn 4 2 4n
(76)
2
p*, n .
By the inequalities (74)-(76), (73) becomes: n 1 j ~ ~ j j j j j H n 1 j 1 j j 1 j
n 1
n 1
Vn c j z 4j j 1
z [ 3 n
1 4 n 1
4
z3 z3 3 zn u0 n n n n n3 zn ] 4 4 2 (77)
1 ~ n zn6 1 ~ n zn6 3 2 d n*4 qnd zn4 n ( ) ( ) bn 4 n n n n 4 n 4 4 4 n 4qn20 ~1 j (
j j1 j h
j1 j 1*j
1 j
j j1 j h
j zn3 j 2 j j zn3 ~ u0 ) 2 j ( 2j ) * 1*j 1*j j j j j j j 1 j 1
h
where H n 1 is given in inequality (47) with i n . By selecting real control input and the adaptive laws of parameters as
16
1
h
u0 cn zn lz n
1 4 n 1
4
zn
zn3 z3 3 n n n n3 zn 4 4 2
(78)
n zn n n n 4
(79)
n n zn n n 4
(80)
6
6
1 j sign[1*j ]
(81)
2 j 2 j 2j
(82)
zn3
2 j sign[1*j ] where cn ,
1 j 1 j 1 j
u0 zn3
1 j
2j
n , n , 1 j and 2 j are design positive constants with l qnd /( 4qn20 ) .
Substituting (78)-(82) into (77), one has
j ~2 n j ~2 j j j 1 2 j j 1 2 j
n
n
Vn c j z 4j j 1
j j1 j h
j 1 j~12j 2 1*j
j j1 j h
j 2 j~22j 2 1*j
(83)
Hn
where
Hn
n 3 2 n j *2 n j *2 2 n b j j j 4 j 1 4 j j 1 4 j 1 2 j j 1 2 j
d *j 4
n
j j1 j h
j 1 j 1*j2 2 1*j
j j1 j h
j 2 j 2*2j 2 1*j
and the facts n n j ~ j ~2 j *j2 j j j j 1 j j 1 2 j j 1 2 j n
n n j ~ j ~2 j *j2 j j j 2 2 j 1 j 1 j 1 j j j n
j j1 j h
j 1 j~1 j1 j j1 j~12j 1*j 2 1*j j j j 1
h
17
j j1 j h
j 1 j1*2j 2 1*j
(84)
j j1 j h
j 2 j~2 j 2 j j 2 j~22j 1*j 2 1*j j j j 1
h
j j1 j h
j 2 j 2*2j 2 1*j
are taken into account. For any given positive parameter
, we should choose appropriate parameters such that
1 j 1 j , 2 j 2 j , j 1,, m . Then (83) can be expressed as n 1
z 4j
z 4j
1 ~2 n 1 ~2 j j 4qn 0 j 1 2 j j 1 4 j 1 2 j j~12j j~22j 2j) 1j * * j j1 j h 2 1 j j j1 j h 2 1 j
V (
n
(85)
V where
min{4c1,,4cn 1,4cn qn0 ,1,,n ,1,, n , } and H n . t
Multiplying V by e
and by Itˆ formula, one has
d (etV ) et (V V )dt et M (t )dw
(86)
where
M (t )
n i 1 V T V g1 ( x1 ) ( gi ( xi ) i 1 g j ( x j ))T z1 i 2 zi j 1 x j
From (85) and (86), one has
d (etV ) et dt et M (t )dw
(87)
Integrating (87) over [0, T ] , we get
V (T ) etV (0)
T eT et 1dw(t ) 0
(88)
Taking expectation on (88), it follows that
E[V (t )] V (0)e t
(89)
By Lemma 3 and utilizing the similar conclusions in [8-14], then all the signals of the considered system are bounded in probability.
18
The above control design and analysis are concluded in the following theorem. Theorem 1: For uncertain switched stochastic nonlinear system (1) with actuator faults, under Assumptions 1-4, the controller (78) with the virtue control signals (37), (57) and parameter laws (38), (39), (58), (59), (79), (80), (81) and (82) guarantee that all the signals in the closed-loop system is SGUUB in probability. Moreover, the tracking error can be made to converge to a small neighborhood of zero by selecting the design parameters appropriately. Remark 1: In this paper, we not only consider stochastic nonlinear systems but also consider the switched stochastic nonlinear systems with actuator faults, it is more complexity and difficult than common stochastic systems. We adopt common Lyapunov function stability theory and fault-tolerant control technique to solve the switched problem and actuator faults, respectively. Therefore, it is more challenge than previous investigations in the literatures [10-12]. And also, the literature [43-45] have obtained great achievements on the Takagi-Sugeno model systems, however, they can not applied in the system which considered in this paper. The parameters design guideline of the proposed control scheme is summarized as follows. 1) choose the design parameters
i , i , 1 j , 2 j , ci , i , i , l , 1 j and 2 j
(i 1,, n) , j ( j1,, jh ) in (27), (37-39), (43), (57-59), (67) and (78-82) to satisfied that
i 0 , i 0 , 1 j 0 , 2 j 0 , ci 0 , i 0 , i 0 , l 0 , 1 j 0 and
2 j 0 , respectively. 2) Choose the design parameters i ,
i , 1 j and 2 j (i 1,, n) , j ( j1,, jh ) in
(39), (58), (59) and (79)-(82), which are positive constants and are used for the According to [32]–[34], the appropriate choices of i , parameters
(38),
-modification.
i , 1 j and 2 j can
prevent the
i i 1 j and 2 j to drift.
4. Simulation Study In this section, the effectiveness and the control performances of the design method are 19
showed by the numerical example. Consider the second-order switched stochastic nonlinear systems with actuator faults:
dx1 ( f1,1 ( x1 ) x2 )dt g1,1 ( x1 )dw dx2 ( f1, 2 ( x2 ) q21( x2 )u1 q22 ( x2 )u2 )dt g 2,1 ( x2 )dw y x 1
(90)
dx1 ( f 2,1 ( x1 ) x2 )dt g1, 2 ( x1 )dw dx2 ( f 2, 2 ( x2 ) q21( x2 )u1 q22 ( x2 )u2 )dt g 2, 2 ( x2 )dw y x 1
(91)
where q21( x2 ) 2 cos( x1x2 ) , q22 ( x2 ) 4 sin( x1 ) , f1,1 ( x1 ) 0.6 x1 , f1, 2 ( x2 ) x1x2 , 2
f 2,1 ( x1 ) 0.9 sin( x1 ) , f 2, 2 ( x2 ) 1.1sin( x1x2 ) , g1,1 ( x1 ) x12 cos x12 , g2,1 ( x2 ) x12 sin x2 , g1, 2 ( x1 ) x12 sin x1 /(1 x12 ) , g2, 2 ( x2 ) 5x2e x1 / 2 /(1 x12 ) . The given reference signal is yr (t ) sin(0.2t ) . Parameters in controller and adaptive laws are chosen as c1 20 , c2 30 , l 10 , 1 1 ,
2 1 , 1 , 1 0.1 , 2 0.2 , 1 0.3 , 2 0.5 , 11 0.001 , 12 0.001 ,
21 0.001 , 22 0.001 , 11 0.01 , 12 0.01 , 21 0.01 , 22 0.01 , 1 0.01 , 2 0.01 , r1 0.05 , r2 0.05 . For simulation purpose, the actuator faults involved in this paper are defined as u1 (t ) 1 when t 5 , and u2 (t ) 0.6v2 for t 10 . The initial conditions are chosen as x1 (0) 0.01 , x2 (0) 0.2 , 11(0) 0.3 , 12 (0) 0.6 and the others initial values are chosen as zeros. The simulation results are shown in Figs. 1-5, where Fig. 1 exhibits the trajectories of the output and tracking signal; Fig. 2 expresses the trajectory of x2 ; Fig. 3 shows the trajectories of actuators u1 and u 2 ; Fig. 4 exhibits the trajectory of the actual input u0 ; Fig. 5 exhibits the trajectory of switched signal.
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Fig. 1. The trajectories of x1 (red) and yr (black).
Fig. 2. The trajectory of x2 .
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Fig. 3. The trajectories of u1 (black) and u 2 (red).
Fig. 4. The trajectory of u0 .
Fig. 5. The trajectory of the switched signal. 5. Conclusions In this paper, a fuzzy adaptive state-feedback fault-tolerant tracking control method has been developed for a class of uncertain switched stochastic nonlinear systems in the presence of actuator faults. Based on the baskstepping technique, nonlinear FTC design approach and common Lyapunov function method, a fuzzy adaptive fault-tolerant tracking control scheme has been
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developed. The proposed control scheme can guarantee that all signals of the closed-loop system are SGUUB in mean square and the tracking error converges to an arbitrarily small neighborhood of the origin. Our further research works will concentrate on the switched stochastic nonlinear MIMO and large-scale systems based on this paper.
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Zhiyao Ma, received the B.S. degree in Information and Computing Science from Liaoning University of Technology, Jinzhou, China, in 2013. He is working towards the M.E. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China. His current research interests include fuzzy control, adaptive control, and fault-tolerant control.
Shaocheng Tong, received the B.S. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.S. degree in fuzzy mathematics from Dalian Marine University, PRC, and the Ph.D degree in fuzzy control from Northeastern University, PRC, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PRC. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control. Yongming Li, received the B.S. degree and the M.S. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively. He received the Ph.D degree in Transportation Information Engineering & Control from Dalian Maritime University, Dalian, China in 2014. He is currently an Associate Professor in the Department of Basic Mathematics, Liaoning University of Technology. His current research interests include adaptive control, fuzzy control and neural networks control for nonlinear systems.
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