Automatica 45 (2009) 1554–1560
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Novel adaptive neural control design for nonlinear MIMO time-delay systemsI Bing Chen a,∗ , Xiaoping Liu b , Kefu Liu b , Chong Lin a a
Institute of Complexity Science, Qingdao University, Qingdao, 266071, PR China
b
Faculty of Engineering, Lakehead University, Thunder Bay, On, P7B 5E1, Canada
article
info
Article history: Received 2 April 2008 Received in revised form 24 October 2008 Accepted 22 February 2009 Available online 25 March 2009 Keywords: Nonlinear systems with time-delays Backstepping Adaptive neural control Output tracking
a b s t r a c t In this paper, we address the problem of adaptive neural control for a class of multi-input multi-output (MIMO) nonlinear time-delay systems in block-triangular form. Based on a neural network (NN) online approximation model, a novel adaptive neural controller is obtained by constructing a novel quadratictype Lyapunov–Krasovskii functional, which not only efficiently avoids the controller singularity, but also relaxes the restriction on unknown virtual control coefficients. The merit of the suggested controller design scheme is that the number of online adapted parameters is independent of the number of nodes of the neural networks, which reduces the number of the online adaptive learning laws considerably. The proposed controller guarantees that all closed-loop signals remain bounded, while the output tracking error dynamics converges to a neighborhood of the origin. A simulation example is given to illustrate the design procedure and performance of the proposed method. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction During the past two decades, neural network method has attracted considerable attention because of its inherent capability for modelling and controlling highly uncertain, nonlinear and complex systems. In Zhang, Xie, Wang, and Zheng (2007), neural network method was applied to solve chaotic synchronization problem. The works in Psillakis and Alexandridis (2007) and Zhang and Wang (2008) concerned stochastic control using neural networks. Nonlinear adaptive neural control was considered in Chen and Narendra (2001) and Yang and Calise (2007) for continuous nonlinear systems, and Zhu and Guo (2004) for nonlinear discrete-time systems. In Xu and Tan (2007), wavelet network-based adaptive control approach was developed. Lan and Huang (2007) proposed an approximate output regulation scheme for discrete-time nonlinear systems based on neural networks. The problems of tracking and stabilization for nonlinear systems via neural network method with backstepping have been extensively studied, and some significant results on these control issues have been reported (Farrell (1997), Gao and Selmic (2006), Ge, Hang, and Zhang (2000), Ge, Hong, and Lee (2004), Ge and Wang (2004),
I The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +86 532 85953607; fax: +86 532 85953672. E-mail addresses:
[email protected] (B. Chen),
[email protected] (X. Liu),
[email protected] (K. Liu),
[email protected] (C. Lin).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.02.021
Ho, Zhang, and Xu (2001), Narendra and Lewis (2001), Niu, Lam, and Wang (2005), Psillakis and Alexandridis (2007), Vapnik (2000), Wang and Hill (2006), Zhang and Huang (2000), Zhang, Ge, and Huang (1999), Zhou, Lam, Feng, and Daniel (2005)). Recently, in view of possible time-delays in practical systems, approximation-based adaptive neural network control has been also addressed in Ge, Hong, and Lee (2003) for nonlinear SISO time-delay systems with constant virtual control coefficients. An adaptive neural controller has been developed to guarantee that all the closed-loop signals remain bounded, meanwhile output tracking is achieved. Such a result, with the use of Nussbaumtype functions, has been extended to the case of unknown virtual control coefficients in Ge et al. (2004). In Ho, Li, and Niu (2005), based on a wavelet neural network online approximation model, a state feedback adaptive controller is proposed by constructing an integral-type Lyapunov–Krasovskii functional. More recently, in Ge and Tee (2007), neural control has been addressed for MIMO nonlinear time-delay systems in block-triangular form. The suggested adaptive neural controller guarantees that the tracking errors converge to a small neighborhood of the origin, and at the same time, all other signals in the closed-loop system are bounded. Though various approximation-based neural control design methods have been proposed for delay-free systems or delayed systems, there are still some issues which need to be further addressed. When a neural network is used as an approximator, to get the sufficient approximation accuracy, a large number of NN nodes should be adopted. As a result, a great number of adaptation parameters are required to be adapted online simultaneously. This makes the learning time become unacceptably large. In addition, in the above-mentioned adaptive neural control design,
B. Chen et al. / Automatica 45 (2009) 1554–1560
even though an integral-type Lyapunov function has been used to successfully avoid the controller singularity problem, it has the following drawbacks: The upper bounds for unknown virtual control coefficients must be some known functions and real control coefficients should be independent of some specified state variables and control inputs. This may limit the applicability of the approach to certain practical systems. The above observation motivates the research in this paper. A novel adaptive neural control design procedure is proposed for MIMO nonlinear time-delay systems in block-triangular form. It is shown that the proposed neural adaptive control scheme not only avoids the controller singularity problem, but also reduces the number of adaptation parameters considerably. As a matter of fact, to control an m input system, there are only m adaptive laws. In addition, by adopting a new Lyapunov functional, the existing restrictions on the virtual or real control coefficients are removed. The proposed control scheme guarantees the boundedness of all the signals in the closed-loop system, at the same time output tracking is achieved.
upper bounds g¯j,ij (.) are unknown. Therefore, Assumption 2 in this paper is less restrictive than that in Ho et al. (2005) and Ge and Tee (2007), where g¯j,ij (.) are required to be known for constructing control laws. j,ij
Assumption 3. There exist positive functions Qj,l (xτj,l ) for l = 1, 2, . . . , ij such that hj,ij (¯xτj,i ) ≤
Pij
l =1
j
j,ij
Qj,l (xτj,l ).
In this paper, the following radial basis function (RBF) NNs will be used as an approximator to approximate an unknown continuous function. As pointed out in Sanner and Slotine (1992), for a given ε > 0 and any continuous function f (Z ) defined on ΩZ ⊂ Rn , there exists an NN W T S (Z ) such that f (Z ) = W T S (Z ) + δ(Z ),
|δ(Z )| ≤ ε,
(2)
where Z ∈ ΩZ ⊂ R is the input vector, W = [w1 , w2 , . . . , wl ]T is the weight vector, l > 1 is the number of the NN nodes and S (Z ) = [s1 (Z ), . . . , sl (Z )]T is defined by n
si = exp
2. Problem formulation and preliminaries
1555
−(Z − µi )T (Z − µi ) , φi2
i = 1, 2, . . . , l
Consider the following MIMO nonlinear time-delay system with block-triangular structure.
with µi = [µi1 , µi2 , . . . , µin ]T the center of the receptive field and φi the width of the Gaussian function.
x˙ j,ij = fj,ij (¯xj,ij ) + gj,ij (¯xj,ij )xj,ij +1 + hj,ij (¯xτj,i ), j
3. Adaptive NN control design
x˙ j,mj = fj,mj (X , u¯ j−1 ) + gj,mj (X , u¯ j−1 )uj + hj,mj (Xτ ),
(1)
yj = xj,1
= 1, 2, . . . , n, ij = 1, 2, . . . , mj − 1, where x¯ j,ij = [xj,1 , . . . , xj,ij ]T ∈ Rij is the state vector for the first ij differential equations of the jth subsystem, y = [y1 , y2 , . . . , yn ]T ∈ Rn is the output, u¯ j = [u1 , . . . , uj ]T are the inputs for the first j subsystems, fj,ij (.), gj,ij (.) and hj,ij (.) are unknown smooth nonlinear functions, X = [xT1 , . . . , xTn ]T with xj = [xj,1 , . . . , xj,mj ]T , xτj,i = xj,ij (t − τj,ij ) j denotes the delayed state, and x¯ τj,i and Xτ are defined as j for j
x¯ τj,i = [xτj,1 , . . . , xτj,i ]T , j
j
Xτ = [xτ1,1 , . . . , xτ1,n , . . . , xτn,1 , . . . , xτn,mn ]T 1
and τj,ij is the unknown constant time-delay. For t ∈ [−τj,ij , 0], let xj,ij (t ) = φj,ij (t ), which are assumed to be smooth and bounded. Remark 1. In plant (1), each control gain function gj,mj (.) contains all state variables and the inputs of the previous subsystems. This is apparently different from the case in Ge and Tee (2007), where gj,mj do not contain the control input ul (1 ≤ l < j) and the state variables xk,mk (k = j + 1, . . . , n). Obviously, the system considered here is more general. As usually done, the following assumptions are made for system (1).
This section is devoted to developing a novel adaptive NN control design procedure. The design procedure for the jth subsystem is composed of mj design steps. In each step, the radial basis function NN WjT,ij S (Zj,ij ) will be used to approximate the unknown nonlinear function f¯j,ij (Zj,ij ). Thus, define an unknown constant as
θj =
1
2
gj0
max{ Wj,ij : 1 ≤ ij ≤ mj },
where the constant gj0 is defined as in Assumption 2, function f¯j,ij and vector Zj,ij will be specified in each step. Furthermore, for j = 1, . . . , n and ij = 1, . . . , mj − 1, the virtual control laws αj,ij (.) are chosen as follows:
1 1 zj,ij − αj,ij = − kj,ij + θˆj zj,ij S T (Zj,ij )S (Zj,ij ), 2 2
(3)
2aj,ij
where kj,ij > 0 and aj,ij > 0 are design parameters, θˆj is the estimation of the unknown constant θj , S (.) is the basis function vector, and the variables zj,ij are defined by zj,ij = xj,ij − αj,ij −1 ,
zj,1 = xj,1 − ydj
(4)
˙
for j = 1, . . . , n and ij = 2, . . . , mj . The adaptive laws θˆ j for j = 1, . . . , n are given by m
j X rj 2 T θˆ˙j (t ) = zj,ij S (Zj,ij )S (Zj,ij ) − bj θˆj , 2
(5)
Assumption 1. The desired trajectories ydj , j = 1, 2, . . . , n, and their time derivatives up to the nth order, are continuous and bounded.
where rj > 0 and bj > 0 are design parameters.
Assumption 2. The signs of gj,ij (.) are known and there exist some unknown constant gj0 and unknown smooth functions g¯j,ij (.) such
Step (j.1 (1 ≤ j ≤ n)). For the first differential equation of the jth subsystem, consider the Lyapunov function as follows:
that 0 < gj0 ≤ gj,ij (.) ≤ g¯j,ij (.) < ∞.
ij =1
Remark 2. Apparently, Assumption 2 implies that gj,ij (.) is strictly positive or negative. Without loss of generality, we further assume gj,ij (.) > gj0 > 0. In addition, notice that in Assumption 2 the
Vzj,1 =
1 2
2aj,ij
zj2,1 +
gj0 2rj
θ˜j2 ,
where zj,1 = xj,1 − ydj , θ˜j = θj − θˆj , θˆj is the estimate of θj . Then, the time derivative of Vzj,1 is given by
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B. Chen et al. / Automatica 45 (2009) 1554–1560
V˙ zj,1 = zj,1 fj,1 + gj,1 αj,1 − y˙ dj + hj,1 (¯xτj,1 )
+ zj,1 gj,1 zj,2 −
gj0 rj
Step (j.ij (for ij = 2, . . . , mj − 1)). Consider the following Lyapunov–Krasovskii functional
θ˜j θ˙ˆ j .
(6)
1 2
j
V˙ zj,i = zj,ij fj,ij + gj,ij xj,ij +1 − α˙ j,ij −1 + hj,ij (¯xτj,i ) . j j
2
2
t
Z
t −τj,1
j
(7)
rj
k=1
Differentiating Vj,1 and using (7), the inequality below can be obtained easily. gj0
V˙ j,1 ≤ zj,1 f¯j,1 (Zj,1 ) + gj,1 αj,1 + zj,1 gj,1 zj,2 −
where Zj,1 = [xj,1 , ydj , y˙ dj , θˆj ] , Uj,1 =
1 2
2
2 zj,1
j,1 Qj,1
h
rj
tanh2
(x)
i2
1 z
and
≤
z
η
ij −1 k X X1 k=1 l=1
V˙ zj,i ≤ zj,ij
2
be verified that for any initial conditions θˆj (t0 ) ≥ 0, θˆj (t ) ≥ 0 for all t ≥ t0 . Consequently, it follows that zj,1 gj,1 αj,1 ≤ −
+
(9)
2
gj0 2a2j,1
θˆj zj2,1 S T (Zj,1 )S (Zj,1 ) − kj,1 +
1 2
+
gj0 rj
θ˜j
rj 2a2j,1
1 2
−1 a2j,1 + εj2,1 gj0
.
VUj,i =
ij −1
k
XX 1 2
ij X 1h
+ zj,1 gj,1 zj,2 + 1 − 2 tanh
z j ,1
η j ,1
+
i j −1 k X X1h k=1 l=1
2
ij X 1
2
2
i2
j ,k
Qj,l (xτj,l )
.
(15)
zj,ij
fj,k + gj,k xj,k+1 −
ij −1 X ∂αj,ij −1
zj2,ij
∂αj,ij −1
2 ! −
∂ xj,k
j,ij
i2
+
∂αj,ij −1 ˙ θˆ j ∂ θˆj
ij −1 k X X1h k=1 l=1
(k+1)
ydj
∂ y(djk)
k=0
Qj,k (xτj,k )
k=1
t
1h
t −τj,k
2
i j −1 k Z X X
i2
j ,i j
Qj,k (x(s))
t
1h
t −τj,l
k=1 l=1
2
i2
j ,k
Qj,l (xτj,l )
.
(16)
2
ds
j ,k
Qj,l (x(s))
i2
ds.
Differentiating VUj,i yields j
ij
! V˙ Uj,i = Uj,ij −
X1h
j
k=1 2
2
∂ xj,k
k=1
ij Z X
+
˙
(14)
To compensate the delay terms, define an integral function as follows:
(10)
zj2,1 S T (Zj,1 )S (Zj,1 ) − θˆ j
∂αj,ij −1
∂ x j ,k
k=1
Thus, substituting (9) and (10) into (8) results in V˙ j,1 ≤ −kj,1 gj0 zj2,1 +
2
zj,ij
ij −1 X ∂αj,ij −1
j
2 gj0 zj1
∂αj,ij −1 ˙ θˆ j . ∂ θˆj
+
k=1
+
where aj,1 > 0 is a design parameter. In addition, from (5), it can
hj,k (¯xτj,k )
hj,k (¯xτj,k )
k =1 l =1
+ a2j,1 + gj0 zj2,1 + εj2,1 gj0−1 2
∂ ydj
fj,ij + gj,ij xj,ij +1 +
j
−
1
∂ x j ,k
By utilizing (13)–(15), (12) can be rewritten in the following form.
δj,1 ≤ εj,1 ,
1
(k+1)
ydj
is well defined at
gj0 zj2,1 θj S T (Zj,1 )S (Zj,1 )
1
(13)
i j −1 X ∂αj,ij −1 k=1
(k)
∂ xj,k
k=1
ηj,1
ij −1 X ∂αj,ij −1
ij −1 X ∂αj,ij −1
where δj,1 (Zj,1 ) denotes the approximation error. Furthermore, a straightforward calculation shows that 1
.
Similar to (13), we have
zj,1
tanh2
f¯j,1 (Zj,1 ) = WjT,1 S (Zj,1 ) + δj,1 (Zj,1 ),
2a2j,1
2
i2
j,ij
Qj,k (xτj,k )
fj,k + gj,k xj,k+1 +
∂ x j ,k
k=1
− zj,ij
z = 0 and can be approximated by a neural network. So, the NN WjT,1 S (Zj,1 ) is utilized to approximate f¯j,1 such that for given εj,1 > 0,
zj,1 f¯j,1 (Zj,1 ) ≤
ij X 1h k=1
ij −1 X ∂αj,ij −1
k=0
with ηj,1 being a positive constant. As pointed out by Remark 5 in Ge and Tee (2007), the function
α˙ j,ij −1 =
+ (8)
T
zj,1 +
zj2,ij +
2
k=1
θ˜j θ˙ˆ j
z j ,1 + 1 − 2 tanh2 Uj,1 , η j ,1
1
ij X 1
≤
Notice that α˙ j,ij −1 (Zj,ij −1 ) can be expressed as
i2 1 h j,1 Qj,1 (x(s)) ds. 2
f¯j,1 (Zj,1 ) = fj,1 − y˙ dj +
ij X j,ij zj,i Q (xτ ) j,k j,k j
zj,ij hj,ij (¯xτj,i ) ≤
i2 g j0 j,1 Qj,1 (xτj,1 ) − θ˜j θ˙ˆ j .
To deal with the delay term in (7), consider the Lyapunov– Krasovskii functional as follows: Vj,1 = Vzj1 + VUj,1 with VUj,1 =
(12)
By Assumption 3 and completion of the square, we have
˙Vzj,1 ≤ zj,1 fj,1 + gj,1 αj,1 − y˙ dj + 1 zj,1 + zj,1 gj,1 zj,2 +
Substituting this inequality into (6) yields
1h
zj2,ij .
Differentiating Vzj,i to get that
i2 1 h j ,1 Qj,1 (xτj,1 ) . 2
zj2,1 +
2
j
With Assumption 3, completing the square gives j ,1 zj,1 hj,1 (¯xτj,1 ) ≤ zj,1 Qj,1 (xτj,1 ) ≤
1
Vzj,i =
Uj,1 .
(11)
= zj,ij
2 zj,ij
2
tanh
2
j ,i j
Qj,k (xτj,k )
i2
−
ij −1 k X X1h k=1 l=1
zj,ij
ηj,ij
!
2
" Uj,ij + 1 − 2 tanh
j ,k
Qj,l (xτj,l )
2
zj,ij
ηj,ij
i2
!# Uj,ij
B. Chen et al. / Automatica 45 (2009) 1554–1560
−
ij X 1h
2
k=1
−
ij −1 k X X1h k=1 l=1
h
Pij
1 k=1 2
where Uj,ij =
i2
j,ij
Qj,k (xτj,k )
2
Lyapunov–Krasovskii functional.
i2
j ,k
Qj,l (xτj,l )
(17) Vj,mj =
i2 Pi −1 P h i2 (xj,k ) + kj=1 kl=1 12 Qjj,,lk (xj,l ) .
j,ij Qj,k
1 2
zj2,mj + VUj,m
j
Pn Pmj R t
with VUj,m =
It is clearly seen that by adding (17) to (16) the delay terms are cancelled out. Hence, by utilizing (16) and (17), it can be proved that the derivative of the Lyapunov–Krasovskii functional Vj,ij = Vzj,i + VUj,i satisfies j
∂αj,ij −1 ˙ θˆ j + 1 − 2 tanh2 ˆ ∂ θj f¯j,ij + gj,ij αj,ij + gj,ij zj,ij zj,ij +1 , !
"
zj,ij
ϕj,ij −
V˙ j,ij ≤ zj,ij
+ zj,ij
1 − 2 tanh
2
zj,i
j
ηj,ij
zj,i
j
ηj,ij
j,ij
Uj,ij (x) +
Uj,ij (x) is used, and
X1 k=1
∂ x j ,k
k=1
+
ij −1 k X X1
2 k=1 l=1
2
fj,k + gj,k xj,k+1 −
zj,ij
(k)
" ˙Vj,mj ≤ 1 − 2 tanh2 + zj,mj ϕj,mj
∂αj,ij −1
2 +
∂ x j ,k
∂ ydj
2 zj,ij
tanh
2
zj,ij
ηj,ij
uj =
! Uj,ij − ϕj,ij ,(19)
−1
! ∂αj,mj −1 ˙ − θˆ j + zj,mj (f¯j,mj + gj,mj uj ), ∂ θˆj
+
l=1 p=1
∂αj,k−1 ˙ θˆ j ∂ θˆj
gj0 rj
rj
θ˜j
2a2j,mj
+ zj,mj ϕj,mj
cannot be dealt with as done
Let V = V˙ ≤ −
unknown f¯j,ij such that for given εj,ij > 0, the following holds. f¯j,i = W T S (Zj,i ) + δj,i (Zj,i ), δj,i εj,i j
j
j
2
V˙ j,ij ≤ −kj,ij gj0 zj2,ij +
j
rj
θ˜j
rj 2a2j,ij
+ zj,ij ϕj,ij
" zj2,ij S T (Zj,ij )S (Zj,ij ) + 1 − 2 tanh2
! ∂αj,ij −1 ˙ ˆ − θ j + gj,ij zj,ij zj,ij +1 . ∂ θˆj
+
ηj,ij
Uj,ij (x)
(20)
Step (j.mj ). This is the last step for the jth subsystem to construct the real control law uj . Consider the following
1 2
zj,mj .
(22)
n X gj0
rj
!# Uj,mj (x)
mj n X X 1 j=1 k=1
mj
X rj
θ˜j
mj n X X
mj n X X
zj2,k S T 2 2a j ,k k=1
1 − 2 tanh
zj,k
(23)
Vj,k . Then, (11), (18) and (23) imply that
kj,k gj0 zj2,k +
j =1 k =2
!#
k=1
j =1 k =1
+ zj,ij
mj n X X
j =1
1 2 −1 aj,ij + εj2,ij gj0 2
kj,mj +
! ∂αj,mj −1 ˙ ˆ θj . − ∂ θˆj
Pn Pmj j =1
zj,mj
ηj,mj
j =1 k =1
+
where δj,ij denotes the approximation error. Then, by following a similar line used in the procedure from (9) to (10), we have
zj2,mj S T (Zj,mj )S (Zj,mj )
+ 1 − 2 tanh
which will be specified later.
j
δj,m εj,m . j j
1 2 −1 aj,mj + εj2,mj gj0 2
"
2
In what follows, the NN WjT,ij S (Zj,ij ) is used to approximate the
gj0
(21)
Again repeating the procedure from (9) to (10) gives V˙ j,mj ≤ −kj,mj gj0 zj2,mj +
in Ge and Tee (2007), where it is treated as a part of f¯j,ij . So, this makes the control law design more difficult. To overcome this difficulty, we introduce a function ϕj,ij to compensate the term
+
Uj,mj (x)
ηj,mj
zj,mj θˆj S T (Zj,mj )S (Zj,mj ) −
2a2j,mj
i j −1 l i2 X i2 X 1 h j,l j,ij Qj,l (xj,l ) + Qj,p (xj,p ) .
variable xj . Hence, the term
j,ij
!#
zj,mj
˙
j
l =1
(m )
Remark 3. From (5), θˆ j is evidently a function of the whole state
∂αj,k−1 ˙ θˆ j , ∂ θˆj
k=1
[X , ydj , . . . , ydj j , θˆ1 , . . . , θˆj ]T . The NN WjT,mj S (Zj,mj ) is employed to approximate f¯j,mj such that for given εj,mj > 0, the following
ij X 1h
2
Pmj −1 Pk
where f¯j,mj (Zj,mj ) can be defined by (19) with ij = mj , and Zj,mj =
(k+1)
ydj
Zj,ij = [¯xTj,ij , ydj , . . . , ydj , θˆj ]T ,
l =1
ds +
At the present step, choose the control law uj as
i j −1 X ∂αj,ij −1
(ij )
Uj,ij =
i2
ds and zj,mj = xj,mj −αj,mj −1 . Then, taking (18)
f¯j,mj = WjT,mj S (Zj,mj ) + δj,mj (Zj,mj ),
k=0
zj2,ij
Qj,k (x(s))
expression holds.
f¯j,ij (Zj,ij ) = fj,ij + zj,ij −1 gj,ij −1 +
−
j,k
Qj,l (x(s))
i2
j,mj
h
with ij = mj into account results in that
(18)
ij
ij −1 X ∂αj,ij −1
1 t −τj,l 2
h
Uj,ij (x)
ηj,ij
zj,ij z 2 tanh2
=
where the equality Uj,ij
!#
Rt
1 t −τj,k 2
k=1
j =1
j
j
1557
2
2
−1 a2j,k + εj2,k gj0
(Zj,k )S (Zj,k ) − θ˙ˆ j
z j ,k
ηj, k
!
Uj,k (x)
! ∂αj,k−1 ˙ ˆ θj . ϕj,k − ∂ θˆj
(24)
So far, we have completed the control law design. Remark 4. In the existing neural adaptive control design approaches, each weight vector is just the estimated vector. Therefore, the number of the adaptation parameters depends on the number of the NN nodes. Consequently, if a system contains a large number of unknown nonlinear functions, or more NN nodes are used to improve the approximation precision, there will be a large number of adaptation parameters that need to be updated
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B. Chen et al. / Automatica 45 (2009) 1554–1560
online simultaneously. This makes the learning time become unacceptably large. In this paper, inspired by the idea in Yang, Feng, and Ren (2004), instead of estimating each element in the weight vector W , we estimate the norm of all the weight vectors, so only one adaptation learning law is required to control each subsystem. 4. Stability Analysis In this section, the boundedness of all the signals in the closedloop system will be proved. The main result is summarized in the following theorem. Theorem. Consider system (1) satisfying Assumptions 1–3. Suppose that for 1 ≤ j ≤ m, 1 ≤ ij ≤ mj , the packaged unknown functions f¯j,ij can be approximated by neural network in the sense that the approximating error δj,ij are bounded. Then under the action of control law (22) and the NN adaptation law (5), all the closed-loop trajectories remain bounded. Proof. We first determine the functions ϕj,k such that mj
n X X
ϕj,k −
zj,k
j=1 k=2
∂αj,k−1 ˙ θˆ j ∂ θˆj
5. Simulation example
! ≤ 0.
(25)
˙
In light of the definition of θˆ j and 0 < S T (.)S (.) ≤ L (L is the number of neural network weights), a straightforward calculation shows that mj X ∂αj,k−1 ∂αj,k−1 ˙ zj,k bj θˆj zj,k θˆ j ≤ − ∂ θˆj ∂ θˆj k=2 k =2 mj X
+
k−1 X ∂αj,k−1 rj 2 T zj,l S (Zj,l )S (Zj,l ), ∂ θˆj 2a2j,l l =1
j=1
≤
rj
θ˜j
zj2,k S T 2 2a j ,k k=1
(Zj,k )S (Zj,k ) − θ˙ˆ j
(26)
2rj
mj n X X
kj,k gj0 zj2,k −
j=1 k=1 n
+
n X gj0 bj j =1
mj
XX
1 − 2 tanh2
j=1 k=1
where C
=
Pn Pmj j =1
1 k=1 2
2rj zj,k
θ˜j2 + C
ηj,k a2j,k
+ε
and 2 show the corresponding system outputs and the reference signals. Fig. 3 shows the responses of state variables x1,2 and x2,2 , Fig. 3 displays the control input signals u1 and u2 , and
Fig. 4 shows the boundedness of adaptive parameters θˆ1 and θˆ2 . From the simulation results, it can clearly be seen that the proposed controller guarantees the boundedness of all the signals in the closed-loop system, and also achieves the good tracking performance. In addition, to control this system, our method requires only two adaptation laws. 6. Conclusion
Uj,k (x), −1 2 j,k gj0
(28)
[0, 0]T . The simulation results are shown in Figs. 1–4. Figs. 1
At the present stage, choose Lyapunov functional as V = Vn,mn . Then, combining (24)–(26) results in V˙ ≤ −
+ 1 + sin2 (x1,1 ) + 0.5 cos2 (x2,2 ) u1 + xτ1,2 , = −x2,1 + x2.2 + xτ2,1 ,
for −τj,ij ≤ ϑ ≤ 0, j = 1, 2, ij = 1, 2, and [θˆ1 (0), θˆ2 (0)]T =
!
n X gj0 bj −θ˜j2 + θj2 . j =1
x˙ 1,2 = x1,1 x1,2 + x2,1 + x2,2
where xτj,i = xj,ij (t − τj,ij ), for j = 1, 2, ij = 1, 2, and the timej delays are chosen as τ1,1 = 2, τ1,2 = 1.5, τ2,1 = 0.5, and τ2,2 = 1. Clearly, if remove the terms 0.5 cos2 (x2,2 ) in g1,2 and sin2 (u1 ) in g2,2 , respectively, this system is the same as the one studied in Ge and Tee (2007). Now, because g1,2 contains x2,2 and g2,2 contains u1 , the neural control scheme proposed in Ge and Tee (2007) cannot be used to control this system. Given the reference output signals as yd1 = 0.5 (sin(t ) + sin(0.5t )) , yd2 = 0.5 sin(t ) + sin(0.5t ), for the control law (22) and the NN adaptation law (5) we choose the design parameters as: k1,1 = k1,2 = k2,1 = k2,2 = 30, a1,1 = a1,2 = 3, a2,1 = a2,2 = 1, r1 = r2 = 600, b1 = b2 = 0.075. The simulation is run under the initial conditions xj,ij (ϑ) = 0
k X rj L ∂αj,k−1 ∂αj,l−1 − 2 zj,k zj,l 2aj,k ∂ θˆj ∂ θˆj l =2
mj X rj
− sin(x2,1 x2,2 − x1,1 ))u2 + xτ1,1 xτ2,2 ,
(25) holds. Similarly, the following can be verified easily. n X gj0
x˙ 1,1 = −x1,1 + 1 + sin2 (x1,1 ) x1,2 + x2τ1,1 ,
x˙ 2,2 = (x1,2 + x2,1 )x2,2 − x1,1 u1 + (2 + sin2 (u1 )
Thus, by choosing ϕj,k as
ϕj,k
In this section, a simulation example is used to illustrate the effectiveness of the proposed adaptive neural control method. Consider the following nonlinear time-delay system.
x˙ 2,1
! k−1 X ∂αj,k−1 rj 2 T zj,l S (Zj,l )S (Zj,l ) − ∂ θˆj 2a2j,l l =1 ! mj k X X rj L ∂αj,l−1 zj,k + z j ,k zj,l . 2 2aj,k ∂ θˆj k=2 l =2
= −bj θˆj
Fig. 1. System output y1 (t ) (‘‘–’’) and the reference yd1 (t ) (‘‘-.-’’).
(27)
+
Pn
gj0
j=1 2rj
θ
2 j
is a
constant. Thus, by (27) the boundedness follows immediately from following the same line used in the proof of Theorem 1 in Ge and Tee (2007) and Zhou et al. (2005). The proof is thus completed.
A novel adaptive neural network tracking control design scheme has been proposed for a class of MIMO nonlinear timedelay systems with block-triangular structure. The suggested control law guarantees that the tracking errors converge to a neighborhood of the origin and all the other signals in the resulting closed-loop system remain bounded. Compared with the existing
B. Chen et al. / Automatica 45 (2009) 1554–1560
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Foundation of China (60674055, 60774047), and the‘‘Taishan’’ scholar program and the Natural Science Foundation of Shandong Province of Shandong province, China (No. Y2006G04). References Chen, L. J., & Narendra, K. S. (2001). Nonlinear adaptive control using neural networks and multiple models. Automatica, 37(8), 1245–1255. Farrell, J. (1997). Persistence of excitation conditions in passive learning control. Automatica, 33, 699–703. Gao, W., & Selmic, R. R. (2006). Neural network control of a class of nonlinear systems with actuator saturation. IEEE Transactions on Neural Networks, 17(1), 147–156. Ge, S. S., Hang, C. C., & Zhang, T. (2000). Stable adaptive control for multivariable systems with a triangular control structure. IEEE Transactions on Automatic Control, 45(1), 1221–1225. Fig. 2. System output y2 (t ) (‘‘–’’) and the reference yd2 (t ) (‘‘-.-’’).
Ge, S. S., Hong, F., & Lee, T. H. (2003). Adaptive neural network control of nonlinear systems with unknown time delays. IEEE Transactions on Automatic Control, 48(11), 2004–2010. Ge, S. S., Hong, F., & Lee, T. H. (2004). Adaptive neural control of nonlinear timedelay systems with unknown virtual control coefficients. IEEE Transactions on Systems, Man, and Cybernetics, 34, 499–516. Ge, S. S., & Tee, K. P. (2007). Approximation-based control of nonlinear MIMO timedelay systems. Automatica, 43, 31–43. Ge, S. S., & Wang, C. (2004). Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Transactions on Neural Networks, 15(3), 674–692. Ho, D. W. C., Zhang, P., & Xu, J. (2001). Fuzzy wavelet networks for function learning. IEEE Transactions on Fuzzy Systems, 9(1), 200–211. Ho, D. W. C., Li, J., & Niu, Y. G. (2005). Adaptive neural control for a class o f nonlinearly parametric time-delay systems. IEEE Transactions on Neural Networks, 16(3), 625–635. Lan, W. Y., & Huang, J. (2007). Neural-network-based approximate output regulation of discrete-time nonlinear systems. IEEE Transactions on Neural Networks, 18(4), 1196–1208. Niu, Y., Lam, J., & Wang, X. (2005). Adaptive H control using backstepping and neural networks. Journal of Dynamic Systems, Measurement, and Control, 127(3), 313–326.
Fig. 3. The control inputs u1 (t ) (‘‘–’’) and u2 (t ) (‘‘-.-’’).
Narendra, K. S., & Lewis, E. F. L. (2001). Special issue on neural network feedback control. Automatica, 37(8), 734–748. Psillakis, H. E., & Alexandridis, A. T. (2007). NN-based adaptive tracking control of uncertain nonlinear systems disturbed by unknown covariance noise. IEEE Transactions on Neural Networks, 18(6), 1830–1835. Sanner, R. M., & Slotine, J. E. (1992). Gaussian networks for direct adaptive control. IEEE Transactions on Neural Networks, 3(6), 837–863. Vapnik, V. N. (2000). The neural of statistical learning theory (2nd ed.). New York: Springer-Verlag. Wang, C., & Hill, D. J. (2006). Learning from neural control. IEEE Transactions on Neural Networks, 17(1), 130–146. Xu, J. X., & Tan, Y. (2007). Nonlinear adaptive wavelet control using constructive wavelet networks. IEEE Transactions on Neural Networks, 18(1), 115–127. Yang, B. J., & Calise, A. J. (2007). Adaptive control of a class of nonaffine systems using neural networks. IEEE Transactions on Neural Networks, 18(4), 1149–1159. Yang, Y. S., Feng, G., & Ren, J. S. (2004). A combined backstepping and smallgain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 34(3), 406–420. Zhou, S. S., Lam, J., Feng, G., & Daniel, W. C. (2005). Exponential ε -regulation for multi-input nonlinear systems using neural networks. IEEE Transactions On Neural Networks, 16(6), 1710–1714.
Fig. 4. The adaptive parameters θˆ1 (t ) (‘‘–’’) and θˆ2 (t ) (‘‘-.-’’).
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results, the main advantage of our result is that the restriction on the control gain functions has been removed and the number of adaptation parameters has been considerably reduced.
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Acknowledgements This work is partially supported by the Natural Sciences and Engineering Research Council of Canada, and the Natural Science
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B. Chen et al. / Automatica 45 (2009) 1554–1560 Bing Chen received the B.A. degree in mathematics from Liaoning University, P. R. China, the M.A. degree in mathematics from Harbin Institute of Technology, P. R. China, and the Ph.D. degree in electrical engineering from Northeastern University, P. R. China, in 1982, 1991 and 1998, respectively. Currently, he is a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, P. R. China. His research interests include nonlinear control systems, robust control, and fuzzy control theory.
Xiaoping Liu obtained his B. Sci, M.Sci, and Ph.D. degree in electrical engineering from Northeastern University, P. R. China, in 1984, 1987, and 1989, respectively. He spent more than 10 years in the School of Information Science and Engineering at Northeastern University, P. R. China. In 2001, he joined the Department of Electrical Engineering at Lakehead University, Canada. His research interests are nonlinear control systems, singular systems, and robust control. He is a member of the Professional Engineers of Ontario
Kefu Liu is a Full Professor in the Department of Mechanical Engineering at Lakehead University, Canada. He received his B.Eng. and M.Sc. in Mechanical Engineering from Central South University of Technology, China in 1981 and 1984, respectively, and his Ph.D. degree in Mechanical Engineering from Technical University of Nova Scotia, Canada in 1992. He was assistant professor at St. Mary’s University, Canada, from 1993 to 1995, and at Dalhousie University, Canada, from 1995 to 1998. He joined Lakehead University in 1998. Dr. Liu is a member of Professional Engineers Ontario. His research interests include vibration control, control of nonlinear systems, and mechatronics. Chong Lin (SM’2006) received the B.Sci and the M.Sci in Applied Mathematics from the Northeastern University, P.R.China, in 1989 and 1992, respectively, and the Ph.D. in Electrical and Electronic Engineering from the Nanyang Technological University, Singapore, in 1999. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, in 1999. From 2000 to 2006, he was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore. Since 2006, he has been a professor with the Institute of Complexity Science, Qingdao University, China. He has published more than 60 research papers and coauthored two monographs. His current research interests are mainly in systems analysis and control, robust control and fuzzy control.
