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Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems
Fuzzy Sets and Systems 159 (2008) 949 – 967 www.elsevier.com/locate/fss
Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems夡 Min Wanga , Bing Chena,∗ , Xiaoping Liub , Peng Shic a Institute of Complexity Science, Qingdao University, Qingdao 266071, PR China b Department of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1 c School of Technology, University of Glamorgan, Pontypridd CF371DL, UK
Received 26 April 2007; received in revised form 9 December 2007; accepted 20 December 2007 Available online 31 December 2007
Abstract This paper is concerned with the problem of adaptive fuzzy output tracking for a class of perturbed strict-feedback nonlinear systems with time delays and unknown virtual control coefficients. Fuzzy logic systems in Mamdani type are used to approximate the unknown nonlinear functions, then the adaptive fuzzy tracking controller is designed by using the backstepping technique and Lyapunov–Krasovskii functionals. The proposed adaptive fuzzy controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. An advantage of the proposed control scheme lies in that the number of the online adaptive parameters is not more than the order of the original system. Finally, two examples are used to demonstrate the effectiveness of our results proposed in this paper. © 2007 Elsevier B.V. All rights reserved. Keywords: Nonlinear time-delay systems; Fuzzy control; Adaptive control; Backstepping
1. Introduction The early results on adaptive control for nonlinear systems were usually obtained based on the assumption that nonlinearities in systems satisfied matching conditions [13,21]. To control the nonlinear systems with mismatched conditions, backstepping design technique has been developed in [14]. With the assumption that the nonlinear functions contain unknown parameters or are unknown with their upper bounds known, a lot of robust adaptive control results have been reported, for details see [15,5]. So far, backstepping approach has become one of the most popular design methods for a class of nonlinear systems. However, in many real plants, not only the nonlinearities in the systems are unknown but also the prior knowledge of the bounds of these nonlinearities is not available. Therefore, the existing adaptive control approaches cannot be used to control these nonlinear systems. In order to stabilize those nonlinear systems, neural networks and fuzzy logic systems are used to approximate the unknown nonlinear functions in systems, and then Lyapunov stability theory are employed to construct adaptive controllers. Recently, some adaptive neural 夡
This work was supported by the National Natural Science Foundation of China (Nos. 60674055, 60774047), the Taishan Scholar Programme, the Natural Science Foundation of Shandong Province, PR China (No. Y2006G04) and Natural Sciences and Engineering Research Council of Canada. ∗ Corresponding author. Tel.: +86 0532 85953607. E-mail address: [email protected] (B. Chen). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.12.022
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control methods and adaptive fuzzy control approaches have been developed for strict-feedback nonlinear systems via backstepping technique, see, for example, [17,27,24,12,11,20,3,22,28,16], and the references therein. A disadvantage of these adaptive control methods is that the proposed controllers contain a lot of adaptive parameters, so that the learning time tends to be unacceptably large when these controllers are implemented. In view of this disadvantage, authors in [25,26,4] present the novel adaptive fuzzy control schemes with much less parameters required to be updated online. In addition, a novel neural learning control scheme is also presented in [23] based on the deterministic learning mechanism, which can effectively recall and reuse the learned knowledge without any further adaptation of neural weights. However, these results in [25,26,4,23] are limited to delay-free systems. It is well known that time delays are frequently encountered in real engineering systems. It has been shown that the existence of time delays usually becomes the source of instability and degrading performance of systems [18]. Therefore, stability analysis and controller synthesis for nonlinear time-delay systems are important both in theory and in practice [1,2]. More recently, several adaptive neural control schemes have been developed to stabilize nonlinear time-delay systems [6,10,9]. In [6,10], the adaptive neural output tracking problem has been addressed via backstepping approach for strict-feedback nonlinear systems with unknown time delays. However, these control methods proposed in [6,10,9] require a large number of neural weights to be adapted online simultaneously. This results in unacceptably large learning time. Based on the above observation, in this paper, the problem of output tracking is revisited for nonlinear time-delay systems via adaptive fuzzy control method. Inspired by the work in [25,26,4] for nonlinear delay-free systems with strict-feedback structure, a new approach to design the adaptive fuzzy output tracking controller is proposed for a class of perturbed strict-feedback nonlinear time-delay systems. During the controller design process, fuzzy logic systems are employed to approximate unknown nonlinear functions, and then, the adaptive laws are constructed by backstepping technique. The main difficulty encountered in the controller design process is how to deal with the unknown time-delay terms in the systems. To overcome this difficulty, the novel Lyapunov–Krasovskii functionals are used to stability analysis and synthesis, and hyperbolic tangent functions are also introduced to handle the singularity problem encountered in Lyapunov synthesis. The proposed controller guarantees a good tracking performance and the boundedness of all the signals in the closed-loop system. Compared with the existing adaptive neural control methods [6,10,9], the main feature of this paper is that the number of adaptive parameters is independent of the number of rules of fuzzy logic systems and system state variables. As a result, the computational burden of the scheme is alleviated. This makes our design scheme more suitable for practical applications. The rest of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. Section 3 presents an adaptive fuzzy tracking control scheme for a class of strict-feedback nonlinear time-delay systems by using the backstepping approach. In Section 4, two simulation examples are given to demonstrate the effectiveness of the method proposed in this paper. Finally, the conclusion is given in Section 5. 2. Problem formulation and preliminaries 2.1. Problem formulation Consider the following unknown nonlinear time-delay dynamic system: ⎧ ⎨ x˙i (t) = fi (x¯i (t)) + gi (x¯i (t))xi+1 (t) + hi (x¯i (t − i )) + di (t, x), x˙n (t) = fn (x(t)) + gn (x(t))u(t) + hn (x(t − n )) + dn (t, x), ⎩ y(t) = x1 (t),
1 i n − 1, (1)
where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R and y ∈ R denote system state vector, system control input and output, respectively. x¯i = [x1 , x2 , . . . , xi ]T , i = 1, 2, . . . , n − 1. Functions fi (.), gi (.) and hi (.) are unknown smooth functions. i are constant unknown time delays, and di (.) stand for external disturbance inputs, i = 1, 2, . . . , n. Remark 1. As shown in [19], there are many physical processes which are governed by nonlinear differential equations in the form of (1). Since system (1) contains the external disturbance inputs, system (1) is of more general form than the one considered in [6,10,9].
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The main goal of this paper is to design an adaptive fuzzy tracking controller for system (1) such that the system output y(t) follows a desired reference signal yd (t), while all the signals in the closed-loop system remain bounded. (i) (1) (i) To this end, define a vector function as y¯di = [yd , yd , . . . , yd ]T , i = 1, 2, . . . , n, where yd is the ith time derivative of yd . Then, the following assumptions are introduced. Assumption 1. The desired trajectory vectors y¯di are continuous and known, and y¯di ∈ di ⊂ R i+1 with di being known compact sets, i = 1, 2, . . . , n. Assumption 2. For 1 i n, there exist unknown positive smooth functions i (x¯i (t)) such that |di (t, x)| i (x¯i (t)). Assumption 3. For 1 i n, the signs of gi (x¯i (t)) are known, and there exist unknown positive constants b and c such that 0 < b |gi (x¯i (t))| c < ∞, ∀x¯i ∈ R i . Without loss of generality, it is assumed that gi (x¯i (t)) b > 0. Remark 2. It should be emphasized that the constants b and c in Assumption 3 are introduced only for the purpose of analysis and are not used to design controllers. Therefore, the constants b and c can be unknown as well. In addition, unlike the assumption in [6,10,9], the upper bounds of unknown functions gi (x¯i (t)) can be unknown in this paper. This makes the design of controllers more difficult than the work in [6,10,9]. 2.2. Function approximation with fuzzy logic systems Throughout this paper, the following IF–THEN rules are used to develop the adaptive fuzzy controller: Ri : IF x1 is F1i , and · · · and, xn is Fni , THEN y is B i , i = 1, 2, . . . , N. By using the singleton fuzzifier, product inference and center average defuzzifier, the fuzzy logic system can be expressed in the following form: N
i=1 i
y(x) = N
n
n
i=1 [
j =1 Fji (xj )
j =1 Fji (xj )]
(2)
,
where x = [x1 , x2 , . . . , xn ]T ∈ R n , F i (xj ) is the membership function of Fji , and i = maxy∈R B i (y). Define j
= [1 , 2 , . . . , N ]T , and (x) = [1 (x), 2 (x), . . . , N (x)]T with the fuzzy basis function i (x) given by n i (x) = N
j =1 Fji (xj )
n
i=1 [
j =1 Fji (xj )]
.
Then, the fuzzy logic system (2) can be rewritten as y(x) = T (x).
(3)
It has been proven in [16] that when the membership functions are chosen as Gaussian functions, the above fuzzy logic system is capable of uniformly approximating any continuous nonlinear function over a compact set x with any degree of accuracy. This property is shown by the following lemma. Lemma 1. For any given real continuous function f (x) on a compact set x ⊂ R n and scalar > 0, there exists a fuzzy logic system y(x) in the form of (3) such that sup |f (x) − y(x)| .
x∈x
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3. Adaptive fuzzy controller design In this section, we will use the recursive backstepping technique to develop the adaptive fuzzy tracking control laws as follows: ˆ i (t) (4)
i (t) = − 2 Ti (Zi )i (Zi )ei (t) − ki ei (t), 2 i
˙ (5) ˆ i (t) = i2 Ti (Zi )i (Zi )ei2 (t) − i ˆ i (t), 2 i where 1i n, i , i and i are design parameters with i > 0 and i > 0, ˆ i (t) is the estimate of the unknown constant i , ei (t) = xi (t) − i−1 (t), 0 (t) is equal to yd (t), the control gain ki satisfies ki > 1/b with being a positive design parameter, i (Zi ) is a fuzzy basis function vector with Zi being the input vector. Note that when i = n,
n (t) is the true control input u(t). Remark 3. From (5), it can be seen that the function Si (t) = i Ti (Zi (t))i (Zi (t))ei2 (t)/2 2i is nonnegative. This ˙ implies that if ˆ i (t)Si (t)/i , then ˆ i (t)0. Consequently, ˆ i (t) increases until ˆ i (t) = Si (t)/i . So, for any given initial condition ˆ i (t0 ) 0, ˆ i (t)0 holds for all t t0 . For simplicity, let i be the upper bound of the fuzzy approximation error i (Zi ), ˜ i = i − ˆ i , and z denote the Euclidean norm of vector z, i.e., z2 = zT z. In addition, the time variable t is omitted in state variables except the delayed state variables x¯i (t − i ). Then, system (1) can be expressed as ⎧ ⎨ x˙i = fi (x¯i ) + gi (x¯i )xi+1 + hi (x¯i (t − i )) + di (x), 1 i n − 1, x˙n = fn (x) + gn (x)u + hn (x(t − n )) + dn (x), (6) ⎩ y = x1 . Now, we propose the following backstepping-based design procedure. Step 1: Define tracking error as e1 = x1 − yd . Then, its time derivative along the first subsystem of (6) is given by e˙1 = f1 (x1 ) + g1 (x1 )x2 + h1 (x1 (t − 1 )) + d1 (x) − y˙d .
(7)
Choose Lyapunov–Krasovskii functional candidate as t e2 VP1 = 1 + P1 (x1 ()) d, 2 t−1 where the unknown positive function P1 (x1 ) will be specified later. Then, the time derivative of VP1 is V˙P1 = e1 (f1 (x1 ) + g1 (x1 )x2 + h1 (x1 (t − 1 )) + d1 (x) − y˙d ) + P1 (x1 ) − P1 (x1 (t − 1 )).
(8)
By Assumption 2 and the triangular inequality, the following inequality can be obtained: e1 21 (x1 ) e 1 + − y˙d + P1 (x1 ) − P1 (x1 (t − 1 )) V˙P1 e1 f1 (x1 ) + 2 2 2a11 2 h21 (x1 (t − 1 )) a11 + , 2 2 is a design parameter. To cancel the unknown time-delay term in (9), P1 (x1 ) is chosen as
+g1 (x1 )e1 x2 +
where a11
P1 (x1 ) =
(9)
h21 (x1 ) . 2
As a result, the following inequality holds: 2 h21 (x1 ) e1 21 (x1 ) a11 e 1 + + g + − y ˙ (x )e x + . V˙P1 e1 f1 (x1 ) + d 1 1 1 2 2 2 2e1 2 2a11
(10)
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Notice that in (10) h21 (x1 )/2e1 is discontinuous at e1 = 0. Therefore, it cannot be approximated by the fuzzy logic system. This creates a difficulty to adaptive fuzzy control method. To overcome this difficulty, an effective approach is to introduce hyperbolic tangent function tanh(e1 /1 ) to deal with the term h21 (x1 )/2e1 [7]. By this way, (10) becomes
2 a11 2 e1 ˆ ˙ H1 , + 1 − 16 tanh (11) VP1 e1 f1 (Z1 ) + g1 (x1 )e1 x2 + 2 1 where 1 is a positive design parameter, H1 = h21 (x1 )/2, and the function fˆ1 (Z1 ) is defined by
e1 21 (x1 ) 16 e1 e1 + H1 + tanh2 − y˙d , fˆ1 (Z1 ) = f1 (x1 ) + 2 2 e1 1 2a11 2 e1 with Z1 = [e1 , y¯ Td1 ]T ∈ Z1 ⊂ R 3 , and Z1 being some known compact set in R 3 . Note that lime1 →0 16 e1 tanh ( 1 )H1 exists, thus, the nonlinear function fˆ1 (Z1 ) can be approximated by a fuzzy logic system T1 1 (Z1 ) with the input vector Z1 ∈ Z1 such that
fˆ1 (Z1 ) = T1 1 (Z1 ) + 1 (Z1 ).
(12)
Then, it follows from substituting (12) into (11) that V˙P1 g1 (x1 )e1 x2 + e1 T1 1 (Z1 ) + e1 1 (Z1 ) +
2 a11 2 e1 H1 . + 1 − 16 tanh 2 1
By using e1 T1 1 (Z1 )
21 b 1 T 2 (Z ) (Z )e + , 1 1 1 1 1 2 2 21
e1 1 (Z1 )
e12 2 + 1, 2 2
(13)
it can be easily verified that
e12 b 1 T 2 2 e1 ˙ H1 , + d1 + 1 − 16 tanh VP1 g1 (x1 )e1 x2 + 2 1 (Z1 )1 (Z1 )e1 + 2 1 2 1
(14)
2 /2 + 2 /2 + 2 /2. where 1 = b−1 1 2 is an unknown constant, is a positive design parameter, and d1 = a11 1 1 In order to construct the virtual control law 1 , choose a Lyapunov function candidate V1 as
V1 = VP1 +
b ˜2 . 2 1 1
Differentiating V1 and then using (14) give that e12 b ˆ 1 T b ˜ 1 1 T ˙ˆ 2 2 ˙ V1 g1 (x1 )e1 x2 + 2 1 (Z1 )1 (Z1 )e1 + (Z1 )1 (Z1 )e1 − 1 + 2
1 2 21 1 2 1
e1 +d1 + 1 − 16 tanh2 H1 . 1 Choosing the adaptive law ˆ 1 in (5) and defining e2 = x2 − 1 , inequality (15) becomes e2 b ˆ 1 V˙1 g1 (x1 )e1 e2 + g1 (x1 )e1 1 + 2 T1 (Z1 )1 (Z1 )e12 + 1 2 2 1
˜ ˆ e1 b1 1 1 H1 . + d1 + 1 − 16 tanh2 +
1 1
(15)
(16)
Based on Remark 3, for any given bounded initial condition ˆ 1 (t0 ) 0, we have ˆ 1 0. Thus, choose the virtual control law 1 in (4) to get that ˆ1 ˆ 1 g1 (x1 )e1 1 = −g1 (x1 ) k1 e12 + 2 T1 (Z1 )1 (Z1 )e12 − bk1 e12 − b 2 T1 (Z1 )1 (Z1 )e12 . (17) 2 1 2 1
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Noting 1 b ˜ 21 1 b ˜ 21 1 b ˜ 1 ˆ 1 1 b ˜ 1 1 1 b 21 =− + − + ,
1
1
1 2 1 2 1 and substituting (17) into (16) show that
1 b ˜ 21 e1 1 V˙1 − bk1 − H1 , + g1 (x1 )e1 e2 + C1 + 1 − 16 tanh2 e12 − 2 2 1 1
(18)
where C1 = d1 + 1 b 21 /2 1 . The coupling term g1 (x1 )e1 e2 will be handled in the next step, and the last term in (18) will be considered later. Step i: Similarly, for each step i (2 i n − 1), considering ei = xi − i−1 , the dynamics of ei -subsystem is given by e˙i = fi (x¯i ) + gi (x¯i )xi+1 + hi (x¯i (t − i )) + di (x) − ˙ i−1 ,
(19)
where
˙ i−1 =
i−1 j i−1 j =1
with Wi−1 =
jxj
(hj (x¯j (t − j )) + dj (x)) + Wi−1 ,
i−1 j i−1 j =1 jxj
(fj (x¯j ) + gj (x¯j )xj +1 ) +
i−1 j i−1 ˙ˆ j i−1 ˙ j =1 j ˆ j + jy¯d(i−1) y¯ d(i−1) . j
Remark 4. As i−1 contains the variable x¯i−1 , unknown time-delay terms in the previous subsystem will appear again when ˙ i−1 is computed. Therefore, Lyapunov–Krasovskii functionals should be designed to compensate for not only the term containing delay i , but also the terms containing j for j < i. Based on such an idea, we choose the Lyapunov–Krasovskii functional as e2 = i + 2 i
VPi
t
j =1 t−j
Pj (x¯j ()) d,
where the unknown positive function Pj (x¯j ) will be specified later. Differentiating VPi along the trajectory (19) yields that ⎛ ⎞ i−1 j
i−1 (hj (x¯j (t − j )) + dj (x)) − Wi−1 ⎠ V˙Pi = ei ⎝fi (x¯i ) + gi (x¯i )xi+1 + hi (x¯i (t − i )) + di (x) − jxj j =1
+
i
(Pj (x¯j ) − Pj (x¯j (t − j ))).
(20)
j =1
By Assumption 2 and the triangular inequality, the following inequalities can be obtained easily:
ei2 2j (x¯j ) j i−1 2 aij2 e2 h2 (x¯i (t − i )) j i−1 ei hi (x¯i (t − i )) i + i dj (x) + , −ei , 2 2 jxj jxj 2 2aij2 −ei
e2 j i−1 hj (x¯j (t − j )) i jxj 2
j i−1 jxj
2
where aij > 0, 1 j i, are design parameters.
+
h2j (x¯j (t − j )) 2
,
ei di (x)
aii2 e2 2 (x¯i ) + i i2 , 2 2aii
(21)
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Consequently, it follows from substituting (21) into (20) that ⎞ ⎛ 2 2
i−1 e 2 (x¯ )
2 (x¯ ) e j i j
e j
e i i i j i−1 i i−1 i + + − Wi−1 ⎠ V˙Pi ei ⎝fi (x¯i ) + + 2 jx 2 jx 2 2a 2aii2 j j ij j =1 +
i a2 ij j =1
2
+ gi (x¯i )ei xi+1 +
i h2 (x¯ (t − )) j j j j =1
2
+
i
(Pj (x¯j ) − Pj (x¯j (t − j ))).
(22)
j =1
To cancel time-delay terms in (22), choose Pj (x¯j ) = h2j (x¯j )/2. As a result, (22) becomes ⎛ 2 2
i−1 e 2 (x¯ )
i j j j
j
e e i i−1 i i−1 V˙Pi ei ⎝fi (x¯i ) + gi−1 (x¯i−1 )ei−1 + + + 2 jxj 2 jxj 2aij2 j =1 ⎞ i a2 ei 2 (x¯i ) Hi ⎠ ij + i 2 − Wi−1 + + gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei + , ei 2 2aii
(23)
j =1
with Hi = ij =1 h2j (x¯j )/2. Similar to Step 1, by introducing a function tanh(ei /i ) to deal with the discontinuous term Hi /ei , (23) can be expressed as V˙Pi ei fˆi (Zi ) + gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei +
ei Hi , + 1 − 16 tanh2 2 i
i a2 ij j =1
(24)
where i is a positive design parameter, Zi = [e¯Ti , y¯ Tdi ]T ∈ Zi ⊂ R 2i+1 with e¯i = [e1 , e2 , . . . , ei ]T and Zi being some known compact set, and the function fˆi (Zi ) is defined by
2 2 i−1 e 2 (x¯ )
i j j j
j
e e i−1 i−1 i i + fˆi (Zi ) = fi (x¯i ) + gi−1 (x¯i−1 )ei−1 + + 2 jxj 2 jxj 2aij2 j =1
ei 16 tanh2 ei 2i (x¯i ) i + − Wi−1 + Hi . ei 2aii2 e
Since lim such that
tanh2 ( i ) i ei
= 0 when ei → 0, a fuzzy logic system Ti i (Zi ) can be used to approximate the function fˆi (Zi )
fˆi (Zi ) = Ti i (Zi ) + i (Zi ).
(25)
Consequently, by substituting (25) into (24) and then using the following inequalities: ei Ti i (Zi )
2i b i T 2 (Z ) (Z )e + , i i i i i 2 2 2i
ei i (Zi )
ei2 2 + i, 2 2
(26)
one can get
e2 ei b i Hi , V˙Pi gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei + 2 Ti (Zi )i (Zi )ei2 + i + di + 1 − 16 tanh2 2 i 2 i where i = b−1 i 2 is an unknown constant, and di = Now, define a function VWi as VWi = VPi +
b ˜2 . 2 i i
i
2 2 2 j =1 aij /2 + i /2 + i /2.
(27)
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Then, the above discussion implies that its time derivative satisfies e2 b ˆ i V˙Wi gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei + 2 Ti (Zi )i (Zi )ei2 + i 2 2 i
˜
i T ei b i ˙ Hi . (Zi )i (Zi )ei2 − ˆ i + di + 1 − 16 tanh2 +
i 2 2i i i
(28)
For any given bounded initial condition ˆ i (t0 ) 0, choosing the virtual control law i as one defined in (4), then the following inequality holds: gi (x¯i )ei i − bki ei2 − b
ˆ i T i (Zi )i (Zi )ei2 . 2 2i
(29)
Substituting (5) and (29) into (28), and using the following inequality: i b ˜ 2i i b ˜ i ˆ i i b 2i − + ,
i 2 i 2 i we have
i b ˜ 2i ei 1 Hi , ei2 − + gi (x¯i )ei ei+1 −gi−1 (x¯i−1 )ei−1 ei + Ci + 1−16 tanh2 V˙Wi − bki − 2 2 i i
(30)
where the constant Ci is defined by Ci = di + i b 2i /2 i . Furthermore, choose a Lyapunov function candidate Vi = Vi−1 + VWi . Similar to (18), we can get
i−1
i−1
j b ˜ 2j 1 2 2 ej ˙ bkj − + gi−1 (x¯i−1 )ei−1 ei + Cj + 1 − 16 tanh Hj . e + Vi−1 − 2 j 2 j j j =1
j =1
(31) Combining (30) with (31) produces
i
V˙i −
j =1
i
bj ˜ 2j ej 1 2 bkj − + gi (x¯i )ei ei+1 + Cj + 1 − 16 tanh2 Hj . ej + 2 2 j j
(32)
j =1
Step n: In this step, the true control u will be constructed. Defining en = xn − n−1 , then we have e˙n = fn (x) + gn (x)u + hn (x(t − n )) + dn (x) − ˙ n−1 , where
˙ n−1 =
n−1 j n−1 j =1
jxj
(hj (x¯j (t − j )) + dj (x)) + Wn−1 ,
n−1 j n−1 ˙ˆ j n−1 ˙ j =1 j ˆ j + jy¯d(n−1) y¯ d(n−1) . j Similarly, choose the following Lyapunov–Krasovskii functional:
with Wn−1 =
n−1 j n−1 j =1 jxj
en2 + 2 n
VPn =
(fj (x¯j ) + gj (x¯j )xj +1 ) +
t
j =1 t−j
Pj (x¯j ()) d,
(33)
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where the unknown positive function Pj (x¯j ) is defined by Pj (x¯j ) = h2j (x¯j )/2. Its time derivative along (33) is ⎛ ⎞ n−1 j
n−1 V˙Pn = en ⎝fn (x) + gn (x)u + hn (x(t − n )) + dn (x) − (hj (x¯j (t − j )) + dj (x)) − Wn−1 ⎠ jxj j =1
+
n
(Pj (x¯j ) − Pj (x¯j (t − j ))).
(34)
j =1
Using a similar approach in (21), we can obtain ⎛
2 2 n−1 e 2 (x¯ )
n j j j
j
e e n n−1 n n−1 V˙Pn en ⎝fn (x) + + + gn−1 (x¯n−1 )en−1 + 2 2 jxj 2 jxj 2anj j =1 n a2 en 2n (x) Hn nj + g , + − W + (x)e u − g ( x ¯ )e e + n−1 n n n−1 n−1 n−1 n 2 en 2 2ann where Hn = becomes
(35)
j =1
n
2 j =1 hj (x¯ j )/2.
The function tanh(en /n ) is used to deal with the term Hn /en . Then, inequality (35)
V˙Pn en fˆn (Zn ) + gn (x)en u − gn−1 (x¯n−1 )en−1 en +
en Hn , + 1 − 16 tanh2 2 n
n a2 nj j =1
(36)
where n is a positive design parameter, Zn = ∈ Zn ⊂ R 2n+1 with e = [e1 , e2 , . . . , en ]T and Zn being a known compact set, and the function fˆn (Zn ) is defined by 2 2
n−1 e 2 (x¯ )
n j j j
e e j
n n−1 n n−1 fˆn (Zn ) = fn (x) + gn−1 (x¯n−1 )en−1 + + + 2 2 jxj 2 jxj 2a nj j =1
en Hn en 2n (x) . − Wn−1 + 16 tanh2 + 2 en n 2ann [eT , y¯ Tdn ]T
Similar to (25), a fuzzy logic system Tn n (Zn ) is used to approximate fˆn (Zn ) such that fˆn (Zn ) = Tn n (Zn ) + n (Zn ).
(37)
By substituting (37) into (36), and applying the triangular inequality, one can get
en b n e2 V˙Pn gn (x)en u − gn−1 (x¯n−1 )en−1 en + 2 Tn (Zn )n (Zn )en2 + n + dn + 1 − 16 tanh2 Hn , 2 n 2 n b−1
where n = and dn = n Furthermore, define a function VWn = VPn +
2
n
(38)
2 2 2 j =1 anj /2 + n /2 + n /2. VWn as
b ˜2 . 2 n n
˙ Apparently, choosing the adaptive law ˆ n in (5), then the time derivative of VWn along (5) and (38) satisfies b ˆ n e2 V˙Wn gn (x)en u − gn−1 (x¯n−1 )en−1 en + 2 Tn (Zn )n (Zn )en2 + n 2 2 n
en bn ˜ n ˆ n Hn . + dn + 1 − 16 tanh2 +
n n
(39)
By taking the true control law u in (4) and using Remark 3, we have gn (x)en u − bkn en2 − b
ˆ n T (Zn )n (Zn )en2 . 2 2n n
(40)
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Moreover, using n b ˜ n ˆ n n b ˜ 2n bn ˜ n n n b ˜ 2n n b 2n =− + − + ,
n 2 n 2 n
n
n and substituting (40) into (39) show that
en 1 n b ˜ 2n Hn , V˙Wn − bkn − − gn−1 (x¯n−1 )en−1 en + Cn + 1 − 16 tanh2 en2 − n 2 2 n
(41)
where Cn = dn + bn 2n /2 n . Now, choose a Lyapunov function candidate Vn = Vn−1 + VWn . Similar to (31), the following inequality holds:
n−1
n−1
bj ˜ 2j 1 2 2 ej ˙ Vn−1 − bkj − + gn−1 (x¯n−1 )en−1 en + Cj + 1 − 16 tanh Hj . e + 2 j 2 j j j =1
j =1
(42) Thus, from (41) and (42), it can clearly be seen that the time derivative of Vn satisfies
n n
bj ˜ 2j 2 2 ej ˙ ¯ Vn − kj e j + +C+ 1 − 16 tanh Hj , 2 j j
(43)
j =1
j =1
where k¯j = bkj − 1/2 > 1/2 with kj > 1/b, and C = nj=1 Cj with Cj = dj + bj 2j /2 j . The last term in (43) will be considered later. At the present stage, we are in the position to give our main result in the following theorem. Theorem 1. Consider the closed-loop system consisting of system (1) under Assumptions 1–3, the control laws (4) and the adaptive laws (5). Suppose that the packaged uncertain functions fˆi (Zi ), i = 1, 2, . . . , n, can be approximated by fuzzy logic systems in the sense that approximation errors are bounded. Then, the closed-loop system has the following properties for bounded initial conditions with ˆ i (t0 ) 0, i = 1, 2, . . . , n, (i) all the signals in the closed-loop system are bounded; (ii) the error signal e, in the mean square sense, eventually converges to the following set: s := {e ∈ R n |ers s },
1 t
(44)
where e = [e1 , e2 , . . . , en ]T , ers = t 0 e()2 d, and s will be given later. s can be made as small as desired by appropriately choosing design parameters. Before the proof of this theorem, we first introduce the following lemma. Lemma 2 (Ge and Tee [7]). For 1 j n, consider the set j given by j := {ej ej | < 0.2554j }. Then, for ej ∈ / j , the inequality 1 − 16 tanh2 (ej /j ) 0 is satisfied. Proof. (i) Boundedness of all the signals in the closed-loop system can be proven by the following three cases. Case 1: For j = 1, 2, . . . , n, ej ∈ j . In this case, |ej | < 0.2554j with j being the positive design parameter, therefore, ej is bounded. Moreover, according to the choice of the adaptive parameter ˆ j , it is obvious that ˆ j is (n) bounded for any bounded ej . Further, ˜ j is bounded as j is a constant. According to Assumption 1, yd , y˙d , . . . , yd are bounded. Since e1 = x1 − yd , and yd are bounded, thus x1 is also bounded. Using (4) with i = 1, and noting that e1 , ˆ 1 and 1 (Z1 ) are all bounded, we can conclude that 1 is bounded. Consequently, it follows from x2 = e2 + 1 that x2 is bounded. Following the same way, j −1 , u and xj , j = 3, . . . , n, can be proven to be bounded. Thus, all the signals in the closed-loop system are bounded in Case 1. Case 2: All ej ∈ / j . In this case, from the definition of Hj , we know that Hj is nonnegative. It follows from Lemma 2 that
ej 1 − 16 tanh2 Hj 0. (45) j
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959
By using (45), (43) becomes n bj ˜ 2j 2 + C, k¯j ej + V˙n − 2 j
(46)
j =1
where C is a constant and k¯j satisfies k¯j > 1/2 > 0. As shown in [23], (46) shows that all the signals in the closed-loop system are bounded. Case 3: Some ei ∈ i , while some ej ∈ / j . For those ei ∈ i , define I as the subsystem consisting of ei ∈ i . Similar to Case 1, ei , ˆ i , and ˜ i can be proven to be bounded for i ∈ I . For those ej ∈ / j , define J as the subsystem consisting of ej ∈ / j and choose the Lyapunov function candidate as b ˜ 2j . VPj + VJ = 2 j j ∈J
By using (30) and (45), we have bj ˜ 2j 2 k¯j ej + − Cj + [gj (x¯j )ej ej +1 − gj −1 (x¯j −1 )ej ej −1 ]. V˙J − 2 j j ∈J
(47)
j ∈J
The last term of (47) can be expressed as [gj (x¯j )ej ej +1 − gj −1 (x¯j −1 )ej ej −1 ] = j ∈J
gj (x¯j )ej ej +1 −
j +1∈J
j −1∈J
j ∈J
j ∈J
+
gj (x¯j )ej ej +1 −
gj −1 (x¯j −1 )ej −1 ej
j +1∈I
j −1∈I
j ∈J
j ∈J
gj −1 (x¯j −1 )ej −1 ej .
(48)
As shown in [7], the first two terms in (48) can be canceled during backstepping. Therefore, from Assumption 3 and Lemma 2, it can be shown that 2 2 ej ej 2 2 2 2 + c ej +1 + + c ej −1 [gj (x¯j )ej ej +1 − gj −1 (x¯j −1 )ej ej −1 ] 4 4 j ∈J
j +1∈I
j −1∈I
j ∈J
j ∈J
j ∈J
ej2 2
+
(c2 (0.2554j −1 )2 + c2 (0.2554j +1 )2 ).
j −1∈I j +1∈I
Based on the above inequality, (47) can be rewritten as
bj ˜ 2j 1 2 V˙J − k¯j − + C J , e + 2 j 2 j
(49)
j ∈J
where the positive constant CJ is given by Cj + (c2 (0.2554j −1 )2 + c2 (0.2554j +1 )2 ), CJ = j ∈J
(50)
j −1∈I j +1∈I
whose size depends on design parameters. Since k¯j > 1/2, we have k¯j − 1/2 > 0. Similar to Case 2, it can be shown that ej , ˆ j and ˜ j are bounded for j ∈ J . Now, we consider the boundedness of all the signals in the whole
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closed-loop system. According to the above analysis, we know that all ei , ˆ i and ˜ i are bounded, i = 1, 2, . . . , n. From (n) Assumption 1, we have that yd , y˙d , . . . , yd are bounded. Following the similar discussion in Case 1, we can conclude that all the signals in the closed-loop system are bounded under this case. In light of the discussion for Cases 1–3, we can conclude that all the signals in the closed-loop system are bounded. This ends the proof of (i). (ii) In this part, we will prove the error signal e, in the mean square sense, eventually converges to a compact set s . Similar to the proof of (i), the proof is divided into the following three cases. Case 1: For all ej ∈ j , we obtain that |ej | < 0.2554j , j = 1, 2, . . . , n. Let = [1 , 2 , . . . , n ]T and note the definition of ers . It is easy to get that 1 t ers = (51) e()2 d < (0.2554)2 2 . t 0 Case 2: For all ej ∈ / j , From (46) and the definition of k¯j , we have n
1 ˙ bkj − e2 + C. Vn − 2 j
(52)
j =1
Integrating (52) from 0 to t shows that t n
1 1 1 bkj − e2 () d + C. (Vn (t) − Vn (0)) − t t 2 0 j
(53)
j =1
Noting that bkj > 1/ > 0 and defining k = min{bk1 , bk2 , . . . , bkn }, we have 1 t Vn (0)/t + C ers = . e()2 d t 0 k − 1/2
(54)
Case 3: Some ei ∈ i , while some ej ∈ / j . According to Case 3 in the proof of (i), and considering the subsystem I consisting of ei ∈ i , we can get 1 t ers |I = eI 2 d < (0.2554)2 I 2 , (55) t 0 / j , it follows from (49) where eI 2 = i∈I ei2 and I 2 = i∈I 2i . For the subsystem J consisting of ej ∈ and the definition of k¯j that
1 2 ˙ bkj − e + C J , VJ − j j ∈J
where CJ is defined by (50). Applying the similar procedures as (53) and (54), we have V (0)/t + CJ 1 t ers |J = eJ 2 d J , t 0 k − 1/ where eJ 2 = j ∈J ej2 , and k is defined in (54). Therefore, from (55) and (56), it is obtained that V (0)/t + CJ 1 t . e()2 d(0.2554)2 I 2 + J ers = k − 1/ t 0 Finally, from Cases 1–3, we can conclude that VJ (0)/t + CJ 2 2 Vn (0)/t + C 2 2 ers max (0.2554) , , (0.2554) I + , k − 1/2 k − 1/ which means that e eventually converges to the following set: s := {e ∈ R n |ers s },
(56)
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where C J C 2 2 , (0.2554) I + . s = max (0.2554) , k − 1/2 k − 1/
2
2
This completes the proof of Theorem 1.
Remark 5. Recently, the similar results have been proposed in [25] for SISO nonlinear delay-free systems, and [4] for MIMO nonlinear delay-free systems. Unlike [25,4], this paper mainly addresses the tracking control problem for nonlinear time-delay systems with the strict-feedback structure. Although the adaptive fuzzy controller has been constructed based on the same design idea and backstepping technique, the existence of nonlinear time-delay terms hi (x¯i (t − i )) makes controller design more difficult than the case without time delays. To compensate for the effect of delay terms, both the novel Lyapunov–Krasovskii functionals and the hyperbolic tangent functions are employed. As a result, the stability analysis of the closed-loop system in this paper is not only more complex than ones in [25,4] but also different from them. In addition, when removing the time-delay terms from the system, our result will include the one in [25] as a special case. 4. Simulation examples In this section, two examples will be used to illustrate the effectiveness of the scheme presented in this paper. Example 1. Consider a controlled Brusselator model with external disturbances in [26] as follows: ⎧ ⎨ x˙1 = C − (D + 1)x1 + x12 x2 + d1 (t, x¯2 ), x˙ = Dx1 − x12 x2 + (2 + cos(x1 ))u + d2 (t, x¯2 ), ⎩ 2 y = x1 ,
(57)
where x1 and x2 denote the concentrations of the reaction intermediates, C, D > 0 are parameters which describe the supply of “reservoir” chemicals. u is the control input. d1 (t, x¯2 ) and d2 (t, x¯2 ) are the external disturbance terms, which come from the modeling errors and other types of unknown nonlinearities in the practical chemical reactions. As stated in [8], it is assumed that x1 = 0. The Brusselator model is one of the most popular nonlinear oscillatory models of chemical kinetics. As a practical chemical reaction, the existence of time delays is inevitable in the Brusselator model. Therefore, we add some time-delay terms in (57) to get that ⎧ ⎨ x˙1 = C − (D + 1)x1 + x12 x2 + d1 (t, x¯2 ) + h1 (x1 (t − 1 )), (58) x˙ = Dx1 − x12 x2 + (2 + cos(x1 ))u + d2 (t, x¯2 ) + h2 (x¯2 (t − 2 )), ⎩ 2 y = x1 , where h1 (x1 (t − 1 )) and h2 (x¯2 (t − 2 )) are the unknown time-delay functions. In the simulation, we choose C = 1, D = 3, d1 = 0.7x12 cos(1.5t), d2 = 0.5(x12 + x22 ) sin3 (t), h1 (x1 ) = 2x12 , h2 (x¯2 ) = 0.2x2 sin(x2 ), 1 = 1s, and 2 = 2s. The control objective is to design an adaptive fuzzy controller such that all the signals in the closed-loop system remain bounded and the system output y follows the given reference signal yd = 3 + sin(0.5t) + 0.5 sin(1.5t). To this end, we define fuzzy membership functions as follows: −0.5(x + 1.5)2 −0.5(x + 1)2 , F 2 = exp , F 1 = exp i i 4 4 −0.5(x + 0.5)2 −0.5x 2 , F 4 = exp , F 3 = exp i i 4 4 −0.5(x − 0.5)2 −0.5(x − 1)2 , F 6 = exp , F 5 = exp i i 4 4 −0.5(x − 1.5)2 F 7 = exp . (59) i 4
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4.5 4 3.5 3 2.5 2 1.5
0
10
20
30 Time(sec)
40
50
60
Fig. 1. System output y(t) (“-’’) and reference signal yd (t) (“- -’’) of Example 1.
3 2.5 2 1.5 1 0.5 0
0
10
20
30 Time(sec)
40
50
60
Fig. 2. Trajectory of state x2 (t) of Example 1.
According to Theorem 1, construct the virtual control law, the true control law and the adaptive laws as follows:
1 = −
u=−
ˆ 1 T 1 (Z1 )1 (Z1 )e1 − k1 e1 , 2 21
ˆ 2 T 2 (Z2 )2 (Z2 )e2 − k2 e2 , 2 22
˙ ˆ i = i2 Ti (Zi )i (Zi )ei2 − i ˆ i , 2 i
(60)
(61) i = 1, 2,
(62)
M. Wang et al. / Fuzzy Sets and Systems 159 (2008) 949 – 967
963
50
25
0
-25
-50
0
10
20
30 Time(sec)
40
50
60
50
60
Fig. 3. The true control input u(t) of Example 1.
25
20
15
10
5
0
0
10
20
30
40
Time(sec) Fig. 4. The adaptive parameters ˆ 1 (t) (“-’’) and ˆ 2 (t) (“- -’’) of Example 1.
where e1 = x1 − yd , e2 = x2 − 1 , Z1 = [e1 , yd , y˙d ]T , Z2 = [e1 , e2 , yd , y˙d , y¨d ]T , and the design parameters are chosen as k1 = 10, k2 = 11, 1 = 0.25, 2 = 0.5, 1 = 50, 2 = 80, 1 = 0.1 and 2 = 0.25. The simulation is run under the initial conditions [x1 (0), x2 (0), ˆ 1 (0), ˆ 2 (0)]T = [2.5, 1, 0, 0]T . And the simulation results are shown in Figs. 1–4. Fig. 1 shows the system output y and the reference signal yd . Fig. 2 shows the response of state variable x2 , Fig. 3 displays the control input signal u, and Fig. 4 shows the boundedness of adaptive parameters ˆ 1 and ˆ 2 . 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.
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1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
10
20
30 Time(sec)
40
50
60
Fig. 5. System output y(t) (“-’’) and reference signal yd (t) (“- -’’) of Example 2.
1 0.5 0 −0.5 −1 −1.5 −2 −2.5
0
10
20
30 Time(sec)
40
50
60
Fig. 6. Trajectory of state x2 (t) of Example 2.
Example 2. To further show the effectiveness of our results, we apply the adaptive fuzzy tracking controller (60)–(62) to the following system which is considered in [6] ⎧ ⎨ x˙1 = (1 + x12 )x2 + x1 e−0.5x1 + 2x12 (t − 1 ), (63) x˙ = (3 + cos(x1 x2 ))u + x1 x22 + 0.2x2 (t − 2 ) sin(x2 (t − 2 )), ⎩ 2 y = x1 , where x1 and x2 denote state variables, u is the system control input and y is the system output. In this example, the fuzzy membership functions (59) are used. The reference signal is yd = 0.5(sin(t) + sin(0.5t)), which is the same as in [6]. Apply the controller (60)–(62) with k1 = 25, k2 = 25, 1 = 2 = 2, 1 = 1000, 2 = 2000, 1 = 0.2, and
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3.5 3 2.5 2 1.5 1 0.5 0 −0.5 −1
0
10
20
30 Time(sec)
40
50
60
50
60
Fig. 7. The true control input u(t) of Example 2.
0.2
0.15
0.1
0.05
0
0
10
20
30 Time(sec)
40
Fig. 8. The adaptive parameters ˆ 1 (t) (“-’’) and ˆ 2 (t) (“- -’’) of Example 2.
2 = 0.15. The simulation is carried out under the initial conditions [x1 (0), x2 (0), ˆ 1 (0), ˆ 2 (0)]T = [0, 0, 0, 0]T and 1 = 2 = 2s. The simulation results are shown by Figs. 5–8. Apparently, the simulation results show that under the action of the suggested controller, a good tracking performance has been achieved. Remark 6. It is evident that the tracking performance depends on the design parameters. Theoretically, a good tracking performance can be achieved by choosing ki large enough or i small enough. Then, how to choose the optimal parameters, such as ki, i , i and so on, to get the optimal tracking performance is still an open problem. In the presented simulations, the design parameters are set using a trial-and-error method. It should be pointed out that the existing adaptive fuzzy control approaches cannot be used to control systems (58) and (63) because of the existence of nonlinear delayed functions. However, the adaptive neural control approaches
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suggested in [6,10] can be applied to design the tracking controllers. Because all the ideal weights are regarded as the estimated parameters, when the controllers in [6,10] are implemented, there will be a large number of adaptive parameters which are required to be tuned online simultaneously. This considerably increases the computational burden. For instance, in the simulation studies of [6], 270 adaptive parameters are required to be tuned online to control system (63). Unlike [6,10], in this paper, a key technique is introduced to reduce the number of adaptive parameters, that is, the unknown constant i = b−1 i 2 is used as an estimated parameter, which results in only one adaptive parameter ˆ i for each virtual control law i . Therefore, for the second-order nonlinear time-delay systems (58) and (63), only two adaptive parameters ˆ 1 and ˆ 2 are required to be updated online. 5. Conclusion In this paper, the adaptive fuzzy control methods proposed in [25,26,4] have been extended to a class of strictfeedback nonlinear systems with unknown time delays. By appropriately choosing Lyapunov–Krasovskii functionals and hyperbolic tangent functions, an adaptive fuzzy tracking control scheme has been presented for a class of perturbed strict-feedback nonlinear time-delay systems. The proposed adaptive fuzzy tracking controller guarantees the boundedness of all the signals in the closed-loop system, while the tracking error eventually converges to a small neighborhood of the origin. Moreover, the suggested adaptive fuzzy controller contains less adaptive parameters. This makes our design scheme easier to be implemented in practical applications. Simulation results have been given to illustrate the effectiveness of the proposed scheme. References [1] E.K. Boukas, N.F. Al-Muthairi, Delay-dependent stabilization of singular linear systems with delays, Internat. J. 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Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems夡 Min Wanga , Bing Chena,∗ , Xiaoping Liub , Peng Shic a Institute of Complexity Science, Qingdao University, Qingdao 266071, PR China b Department of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1 c School of Technology, University of Glamorgan, Pontypridd CF371DL, UK
Received 26 April 2007; received in revised form 9 December 2007; accepted 20 December 2007 Available online 31 December 2007
Abstract This paper is concerned with the problem of adaptive fuzzy output tracking for a class of perturbed strict-feedback nonlinear systems with time delays and unknown virtual control coefficients. Fuzzy logic systems in Mamdani type are used to approximate the unknown nonlinear functions, then the adaptive fuzzy tracking controller is designed by using the backstepping technique and Lyapunov–Krasovskii functionals. The proposed adaptive fuzzy controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. An advantage of the proposed control scheme lies in that the number of the online adaptive parameters is not more than the order of the original system. Finally, two examples are used to demonstrate the effectiveness of our results proposed in this paper. © 2007 Elsevier B.V. All rights reserved. Keywords: Nonlinear time-delay systems; Fuzzy control; Adaptive control; Backstepping
1. Introduction The early results on adaptive control for nonlinear systems were usually obtained based on the assumption that nonlinearities in systems satisfied matching conditions [13,21]. To control the nonlinear systems with mismatched conditions, backstepping design technique has been developed in [14]. With the assumption that the nonlinear functions contain unknown parameters or are unknown with their upper bounds known, a lot of robust adaptive control results have been reported, for details see [15,5]. So far, backstepping approach has become one of the most popular design methods for a class of nonlinear systems. However, in many real plants, not only the nonlinearities in the systems are unknown but also the prior knowledge of the bounds of these nonlinearities is not available. Therefore, the existing adaptive control approaches cannot be used to control these nonlinear systems. In order to stabilize those nonlinear systems, neural networks and fuzzy logic systems are used to approximate the unknown nonlinear functions in systems, and then Lyapunov stability theory are employed to construct adaptive controllers. Recently, some adaptive neural 夡
This work was supported by the National Natural Science Foundation of China (Nos. 60674055, 60774047), the Taishan Scholar Programme, the Natural Science Foundation of Shandong Province, PR China (No. Y2006G04) and Natural Sciences and Engineering Research Council of Canada. ∗ Corresponding author. Tel.: +86 0532 85953607. E-mail address: [email protected] (B. Chen). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.12.022
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control methods and adaptive fuzzy control approaches have been developed for strict-feedback nonlinear systems via backstepping technique, see, for example, [17,27,24,12,11,20,3,22,28,16], and the references therein. A disadvantage of these adaptive control methods is that the proposed controllers contain a lot of adaptive parameters, so that the learning time tends to be unacceptably large when these controllers are implemented. In view of this disadvantage, authors in [25,26,4] present the novel adaptive fuzzy control schemes with much less parameters required to be updated online. In addition, a novel neural learning control scheme is also presented in [23] based on the deterministic learning mechanism, which can effectively recall and reuse the learned knowledge without any further adaptation of neural weights. However, these results in [25,26,4,23] are limited to delay-free systems. It is well known that time delays are frequently encountered in real engineering systems. It has been shown that the existence of time delays usually becomes the source of instability and degrading performance of systems [18]. Therefore, stability analysis and controller synthesis for nonlinear time-delay systems are important both in theory and in practice [1,2]. More recently, several adaptive neural control schemes have been developed to stabilize nonlinear time-delay systems [6,10,9]. In [6,10], the adaptive neural output tracking problem has been addressed via backstepping approach for strict-feedback nonlinear systems with unknown time delays. However, these control methods proposed in [6,10,9] require a large number of neural weights to be adapted online simultaneously. This results in unacceptably large learning time. Based on the above observation, in this paper, the problem of output tracking is revisited for nonlinear time-delay systems via adaptive fuzzy control method. Inspired by the work in [25,26,4] for nonlinear delay-free systems with strict-feedback structure, a new approach to design the adaptive fuzzy output tracking controller is proposed for a class of perturbed strict-feedback nonlinear time-delay systems. During the controller design process, fuzzy logic systems are employed to approximate unknown nonlinear functions, and then, the adaptive laws are constructed by backstepping technique. The main difficulty encountered in the controller design process is how to deal with the unknown time-delay terms in the systems. To overcome this difficulty, the novel Lyapunov–Krasovskii functionals are used to stability analysis and synthesis, and hyperbolic tangent functions are also introduced to handle the singularity problem encountered in Lyapunov synthesis. The proposed controller guarantees a good tracking performance and the boundedness of all the signals in the closed-loop system. Compared with the existing adaptive neural control methods [6,10,9], the main feature of this paper is that the number of adaptive parameters is independent of the number of rules of fuzzy logic systems and system state variables. As a result, the computational burden of the scheme is alleviated. This makes our design scheme more suitable for practical applications. The rest of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. Section 3 presents an adaptive fuzzy tracking control scheme for a class of strict-feedback nonlinear time-delay systems by using the backstepping approach. In Section 4, two simulation examples are given to demonstrate the effectiveness of the method proposed in this paper. Finally, the conclusion is given in Section 5. 2. Problem formulation and preliminaries 2.1. Problem formulation Consider the following unknown nonlinear time-delay dynamic system: ⎧ ⎨ x˙i (t) = fi (x¯i (t)) + gi (x¯i (t))xi+1 (t) + hi (x¯i (t − i )) + di (t, x), x˙n (t) = fn (x(t)) + gn (x(t))u(t) + hn (x(t − n )) + dn (t, x), ⎩ y(t) = x1 (t),
1 i n − 1, (1)
where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R and y ∈ R denote system state vector, system control input and output, respectively. x¯i = [x1 , x2 , . . . , xi ]T , i = 1, 2, . . . , n − 1. Functions fi (.), gi (.) and hi (.) are unknown smooth functions. i are constant unknown time delays, and di (.) stand for external disturbance inputs, i = 1, 2, . . . , n. Remark 1. As shown in [19], there are many physical processes which are governed by nonlinear differential equations in the form of (1). Since system (1) contains the external disturbance inputs, system (1) is of more general form than the one considered in [6,10,9].
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The main goal of this paper is to design an adaptive fuzzy tracking controller for system (1) such that the system output y(t) follows a desired reference signal yd (t), while all the signals in the closed-loop system remain bounded. (i) (1) (i) To this end, define a vector function as y¯di = [yd , yd , . . . , yd ]T , i = 1, 2, . . . , n, where yd is the ith time derivative of yd . Then, the following assumptions are introduced. Assumption 1. The desired trajectory vectors y¯di are continuous and known, and y¯di ∈ di ⊂ R i+1 with di being known compact sets, i = 1, 2, . . . , n. Assumption 2. For 1 i n, there exist unknown positive smooth functions i (x¯i (t)) such that |di (t, x)| i (x¯i (t)). Assumption 3. For 1 i n, the signs of gi (x¯i (t)) are known, and there exist unknown positive constants b and c such that 0 < b |gi (x¯i (t))| c < ∞, ∀x¯i ∈ R i . Without loss of generality, it is assumed that gi (x¯i (t)) b > 0. Remark 2. It should be emphasized that the constants b and c in Assumption 3 are introduced only for the purpose of analysis and are not used to design controllers. Therefore, the constants b and c can be unknown as well. In addition, unlike the assumption in [6,10,9], the upper bounds of unknown functions gi (x¯i (t)) can be unknown in this paper. This makes the design of controllers more difficult than the work in [6,10,9]. 2.2. Function approximation with fuzzy logic systems Throughout this paper, the following IF–THEN rules are used to develop the adaptive fuzzy controller: Ri : IF x1 is F1i , and · · · and, xn is Fni , THEN y is B i , i = 1, 2, . . . , N. By using the singleton fuzzifier, product inference and center average defuzzifier, the fuzzy logic system can be expressed in the following form: N
i=1 i
y(x) = N
n
n
i=1 [
j =1 Fji (xj )
j =1 Fji (xj )]
(2)
,
where x = [x1 , x2 , . . . , xn ]T ∈ R n , F i (xj ) is the membership function of Fji , and i = maxy∈R B i (y). Define j
= [1 , 2 , . . . , N ]T , and (x) = [1 (x), 2 (x), . . . , N (x)]T with the fuzzy basis function i (x) given by n i (x) = N
j =1 Fji (xj )
n
i=1 [
j =1 Fji (xj )]
.
Then, the fuzzy logic system (2) can be rewritten as y(x) = T (x).
(3)
It has been proven in [16] that when the membership functions are chosen as Gaussian functions, the above fuzzy logic system is capable of uniformly approximating any continuous nonlinear function over a compact set x with any degree of accuracy. This property is shown by the following lemma. Lemma 1. For any given real continuous function f (x) on a compact set x ⊂ R n and scalar > 0, there exists a fuzzy logic system y(x) in the form of (3) such that sup |f (x) − y(x)| .
x∈x
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3. Adaptive fuzzy controller design In this section, we will use the recursive backstepping technique to develop the adaptive fuzzy tracking control laws as follows: ˆ i (t) (4)
i (t) = − 2 Ti (Zi )i (Zi )ei (t) − ki ei (t), 2 i
˙ (5) ˆ i (t) = i2 Ti (Zi )i (Zi )ei2 (t) − i ˆ i (t), 2 i where 1i n, i , i and i are design parameters with i > 0 and i > 0, ˆ i (t) is the estimate of the unknown constant i , ei (t) = xi (t) − i−1 (t), 0 (t) is equal to yd (t), the control gain ki satisfies ki > 1/b with being a positive design parameter, i (Zi ) is a fuzzy basis function vector with Zi being the input vector. Note that when i = n,
n (t) is the true control input u(t). Remark 3. From (5), it can be seen that the function Si (t) = i Ti (Zi (t))i (Zi (t))ei2 (t)/2 2i is nonnegative. This ˙ implies that if ˆ i (t)Si (t)/i , then ˆ i (t)0. Consequently, ˆ i (t) increases until ˆ i (t) = Si (t)/i . So, for any given initial condition ˆ i (t0 ) 0, ˆ i (t)0 holds for all t t0 . For simplicity, let i be the upper bound of the fuzzy approximation error i (Zi ), ˜ i = i − ˆ i , and z denote the Euclidean norm of vector z, i.e., z2 = zT z. In addition, the time variable t is omitted in state variables except the delayed state variables x¯i (t − i ). Then, system (1) can be expressed as ⎧ ⎨ x˙i = fi (x¯i ) + gi (x¯i )xi+1 + hi (x¯i (t − i )) + di (x), 1 i n − 1, x˙n = fn (x) + gn (x)u + hn (x(t − n )) + dn (x), (6) ⎩ y = x1 . Now, we propose the following backstepping-based design procedure. Step 1: Define tracking error as e1 = x1 − yd . Then, its time derivative along the first subsystem of (6) is given by e˙1 = f1 (x1 ) + g1 (x1 )x2 + h1 (x1 (t − 1 )) + d1 (x) − y˙d .
(7)
Choose Lyapunov–Krasovskii functional candidate as t e2 VP1 = 1 + P1 (x1 ()) d, 2 t−1 where the unknown positive function P1 (x1 ) will be specified later. Then, the time derivative of VP1 is V˙P1 = e1 (f1 (x1 ) + g1 (x1 )x2 + h1 (x1 (t − 1 )) + d1 (x) − y˙d ) + P1 (x1 ) − P1 (x1 (t − 1 )).
(8)
By Assumption 2 and the triangular inequality, the following inequality can be obtained: e1 21 (x1 ) e 1 + − y˙d + P1 (x1 ) − P1 (x1 (t − 1 )) V˙P1 e1 f1 (x1 ) + 2 2 2a11 2 h21 (x1 (t − 1 )) a11 + , 2 2 is a design parameter. To cancel the unknown time-delay term in (9), P1 (x1 ) is chosen as
+g1 (x1 )e1 x2 +
where a11
P1 (x1 ) =
(9)
h21 (x1 ) . 2
As a result, the following inequality holds: 2 h21 (x1 ) e1 21 (x1 ) a11 e 1 + + g + − y ˙ (x )e x + . V˙P1 e1 f1 (x1 ) + d 1 1 1 2 2 2 2e1 2 2a11
(10)
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Notice that in (10) h21 (x1 )/2e1 is discontinuous at e1 = 0. Therefore, it cannot be approximated by the fuzzy logic system. This creates a difficulty to adaptive fuzzy control method. To overcome this difficulty, an effective approach is to introduce hyperbolic tangent function tanh(e1 /1 ) to deal with the term h21 (x1 )/2e1 [7]. By this way, (10) becomes
2 a11 2 e1 ˆ ˙ H1 , + 1 − 16 tanh (11) VP1 e1 f1 (Z1 ) + g1 (x1 )e1 x2 + 2 1 where 1 is a positive design parameter, H1 = h21 (x1 )/2, and the function fˆ1 (Z1 ) is defined by
e1 21 (x1 ) 16 e1 e1 + H1 + tanh2 − y˙d , fˆ1 (Z1 ) = f1 (x1 ) + 2 2 e1 1 2a11 2 e1 with Z1 = [e1 , y¯ Td1 ]T ∈ Z1 ⊂ R 3 , and Z1 being some known compact set in R 3 . Note that lime1 →0 16 e1 tanh ( 1 )H1 exists, thus, the nonlinear function fˆ1 (Z1 ) can be approximated by a fuzzy logic system T1 1 (Z1 ) with the input vector Z1 ∈ Z1 such that
fˆ1 (Z1 ) = T1 1 (Z1 ) + 1 (Z1 ).
(12)
Then, it follows from substituting (12) into (11) that V˙P1 g1 (x1 )e1 x2 + e1 T1 1 (Z1 ) + e1 1 (Z1 ) +
2 a11 2 e1 H1 . + 1 − 16 tanh 2 1
By using e1 T1 1 (Z1 )
21 b 1 T 2 (Z ) (Z )e + , 1 1 1 1 1 2 2 21
e1 1 (Z1 )
e12 2 + 1, 2 2
(13)
it can be easily verified that
e12 b 1 T 2 2 e1 ˙ H1 , + d1 + 1 − 16 tanh VP1 g1 (x1 )e1 x2 + 2 1 (Z1 )1 (Z1 )e1 + 2 1 2 1
(14)
2 /2 + 2 /2 + 2 /2. where 1 = b−1 1 2 is an unknown constant, is a positive design parameter, and d1 = a11 1 1 In order to construct the virtual control law 1 , choose a Lyapunov function candidate V1 as
V1 = VP1 +
b ˜2 . 2 1 1
Differentiating V1 and then using (14) give that e12 b ˆ 1 T b ˜ 1 1 T ˙ˆ 2 2 ˙ V1 g1 (x1 )e1 x2 + 2 1 (Z1 )1 (Z1 )e1 + (Z1 )1 (Z1 )e1 − 1 + 2
1 2 21 1 2 1
e1 +d1 + 1 − 16 tanh2 H1 . 1 Choosing the adaptive law ˆ 1 in (5) and defining e2 = x2 − 1 , inequality (15) becomes e2 b ˆ 1 V˙1 g1 (x1 )e1 e2 + g1 (x1 )e1 1 + 2 T1 (Z1 )1 (Z1 )e12 + 1 2 2 1
˜ ˆ e1 b1 1 1 H1 . + d1 + 1 − 16 tanh2 +
1 1
(15)
(16)
Based on Remark 3, for any given bounded initial condition ˆ 1 (t0 ) 0, we have ˆ 1 0. Thus, choose the virtual control law 1 in (4) to get that ˆ1 ˆ 1 g1 (x1 )e1 1 = −g1 (x1 ) k1 e12 + 2 T1 (Z1 )1 (Z1 )e12 − bk1 e12 − b 2 T1 (Z1 )1 (Z1 )e12 . (17) 2 1 2 1
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Noting 1 b ˜ 21 1 b ˜ 21 1 b ˜ 1 ˆ 1 1 b ˜ 1 1 1 b 21 =− + − + ,
1
1
1 2 1 2 1 and substituting (17) into (16) show that
1 b ˜ 21 e1 1 V˙1 − bk1 − H1 , + g1 (x1 )e1 e2 + C1 + 1 − 16 tanh2 e12 − 2 2 1 1
(18)
where C1 = d1 + 1 b 21 /2 1 . The coupling term g1 (x1 )e1 e2 will be handled in the next step, and the last term in (18) will be considered later. Step i: Similarly, for each step i (2 i n − 1), considering ei = xi − i−1 , the dynamics of ei -subsystem is given by e˙i = fi (x¯i ) + gi (x¯i )xi+1 + hi (x¯i (t − i )) + di (x) − ˙ i−1 ,
(19)
where
˙ i−1 =
i−1 j i−1 j =1
with Wi−1 =
jxj
(hj (x¯j (t − j )) + dj (x)) + Wi−1 ,
i−1 j i−1 j =1 jxj
(fj (x¯j ) + gj (x¯j )xj +1 ) +
i−1 j i−1 ˙ˆ j i−1 ˙ j =1 j ˆ j + jy¯d(i−1) y¯ d(i−1) . j
Remark 4. As i−1 contains the variable x¯i−1 , unknown time-delay terms in the previous subsystem will appear again when ˙ i−1 is computed. Therefore, Lyapunov–Krasovskii functionals should be designed to compensate for not only the term containing delay i , but also the terms containing j for j < i. Based on such an idea, we choose the Lyapunov–Krasovskii functional as e2 = i + 2 i
VPi
t
j =1 t−j
Pj (x¯j ()) d,
where the unknown positive function Pj (x¯j ) will be specified later. Differentiating VPi along the trajectory (19) yields that ⎛ ⎞ i−1 j
i−1 (hj (x¯j (t − j )) + dj (x)) − Wi−1 ⎠ V˙Pi = ei ⎝fi (x¯i ) + gi (x¯i )xi+1 + hi (x¯i (t − i )) + di (x) − jxj j =1
+
i
(Pj (x¯j ) − Pj (x¯j (t − j ))).
(20)
j =1
By Assumption 2 and the triangular inequality, the following inequalities can be obtained easily:
ei2 2j (x¯j ) j i−1 2 aij2 e2 h2 (x¯i (t − i )) j i−1 ei hi (x¯i (t − i )) i + i dj (x) + , −ei , 2 2 jxj jxj 2 2aij2 −ei
e2 j i−1 hj (x¯j (t − j )) i jxj 2
j i−1 jxj
2
where aij > 0, 1 j i, are design parameters.
+
h2j (x¯j (t − j )) 2
,
ei di (x)
aii2 e2 2 (x¯i ) + i i2 , 2 2aii
(21)
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Consequently, it follows from substituting (21) into (20) that ⎞ ⎛ 2 2
i−1 e 2 (x¯ )
2 (x¯ ) e j i j
e j
e i i i j i−1 i i−1 i + + − Wi−1 ⎠ V˙Pi ei ⎝fi (x¯i ) + + 2 jx 2 jx 2 2a 2aii2 j j ij j =1 +
i a2 ij j =1
2
+ gi (x¯i )ei xi+1 +
i h2 (x¯ (t − )) j j j j =1
2
+
i
(Pj (x¯j ) − Pj (x¯j (t − j ))).
(22)
j =1
To cancel time-delay terms in (22), choose Pj (x¯j ) = h2j (x¯j )/2. As a result, (22) becomes ⎛ 2 2
i−1 e 2 (x¯ )
i j j j
j
e e i i−1 i i−1 V˙Pi ei ⎝fi (x¯i ) + gi−1 (x¯i−1 )ei−1 + + + 2 jxj 2 jxj 2aij2 j =1 ⎞ i a2 ei 2 (x¯i ) Hi ⎠ ij + i 2 − Wi−1 + + gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei + , ei 2 2aii
(23)
j =1
with Hi = ij =1 h2j (x¯j )/2. Similar to Step 1, by introducing a function tanh(ei /i ) to deal with the discontinuous term Hi /ei , (23) can be expressed as V˙Pi ei fˆi (Zi ) + gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei +
ei Hi , + 1 − 16 tanh2 2 i
i a2 ij j =1
(24)
where i is a positive design parameter, Zi = [e¯Ti , y¯ Tdi ]T ∈ Zi ⊂ R 2i+1 with e¯i = [e1 , e2 , . . . , ei ]T and Zi being some known compact set, and the function fˆi (Zi ) is defined by
2 2 i−1 e 2 (x¯ )
i j j j
j
e e i−1 i−1 i i + fˆi (Zi ) = fi (x¯i ) + gi−1 (x¯i−1 )ei−1 + + 2 jxj 2 jxj 2aij2 j =1
ei 16 tanh2 ei 2i (x¯i ) i + − Wi−1 + Hi . ei 2aii2 e
Since lim such that
tanh2 ( i ) i ei
= 0 when ei → 0, a fuzzy logic system Ti i (Zi ) can be used to approximate the function fˆi (Zi )
fˆi (Zi ) = Ti i (Zi ) + i (Zi ).
(25)
Consequently, by substituting (25) into (24) and then using the following inequalities: ei Ti i (Zi )
2i b i T 2 (Z ) (Z )e + , i i i i i 2 2 2i
ei i (Zi )
ei2 2 + i, 2 2
(26)
one can get
e2 ei b i Hi , V˙Pi gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei + 2 Ti (Zi )i (Zi )ei2 + i + di + 1 − 16 tanh2 2 i 2 i where i = b−1 i 2 is an unknown constant, and di = Now, define a function VWi as VWi = VPi +
b ˜2 . 2 i i
i
2 2 2 j =1 aij /2 + i /2 + i /2.
(27)
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Then, the above discussion implies that its time derivative satisfies e2 b ˆ i V˙Wi gi (x¯i )ei xi+1 − gi−1 (x¯i−1 )ei−1 ei + 2 Ti (Zi )i (Zi )ei2 + i 2 2 i
˜
i T ei b i ˙ Hi . (Zi )i (Zi )ei2 − ˆ i + di + 1 − 16 tanh2 +
i 2 2i i i
(28)
For any given bounded initial condition ˆ i (t0 ) 0, choosing the virtual control law i as one defined in (4), then the following inequality holds: gi (x¯i )ei i − bki ei2 − b
ˆ i T i (Zi )i (Zi )ei2 . 2 2i
(29)
Substituting (5) and (29) into (28), and using the following inequality: i b ˜ 2i i b ˜ i ˆ i i b 2i − + ,
i 2 i 2 i we have
i b ˜ 2i ei 1 Hi , ei2 − + gi (x¯i )ei ei+1 −gi−1 (x¯i−1 )ei−1 ei + Ci + 1−16 tanh2 V˙Wi − bki − 2 2 i i
(30)
where the constant Ci is defined by Ci = di + i b 2i /2 i . Furthermore, choose a Lyapunov function candidate Vi = Vi−1 + VWi . Similar to (18), we can get
i−1
i−1
j b ˜ 2j 1 2 2 ej ˙ bkj − + gi−1 (x¯i−1 )ei−1 ei + Cj + 1 − 16 tanh Hj . e + Vi−1 − 2 j 2 j j j =1
j =1
(31) Combining (30) with (31) produces
i
V˙i −
j =1
i
bj ˜ 2j ej 1 2 bkj − + gi (x¯i )ei ei+1 + Cj + 1 − 16 tanh2 Hj . ej + 2 2 j j
(32)
j =1
Step n: In this step, the true control u will be constructed. Defining en = xn − n−1 , then we have e˙n = fn (x) + gn (x)u + hn (x(t − n )) + dn (x) − ˙ n−1 , where
˙ n−1 =
n−1 j n−1 j =1
jxj
(hj (x¯j (t − j )) + dj (x)) + Wn−1 ,
n−1 j n−1 ˙ˆ j n−1 ˙ j =1 j ˆ j + jy¯d(n−1) y¯ d(n−1) . j Similarly, choose the following Lyapunov–Krasovskii functional:
with Wn−1 =
n−1 j n−1 j =1 jxj
en2 + 2 n
VPn =
(fj (x¯j ) + gj (x¯j )xj +1 ) +
t
j =1 t−j
Pj (x¯j ()) d,
(33)
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where the unknown positive function Pj (x¯j ) is defined by Pj (x¯j ) = h2j (x¯j )/2. Its time derivative along (33) is ⎛ ⎞ n−1 j
n−1 V˙Pn = en ⎝fn (x) + gn (x)u + hn (x(t − n )) + dn (x) − (hj (x¯j (t − j )) + dj (x)) − Wn−1 ⎠ jxj j =1
+
n
(Pj (x¯j ) − Pj (x¯j (t − j ))).
(34)
j =1
Using a similar approach in (21), we can obtain ⎛
2 2 n−1 e 2 (x¯ )
n j j j
j
e e n n−1 n n−1 V˙Pn en ⎝fn (x) + + + gn−1 (x¯n−1 )en−1 + 2 2 jxj 2 jxj 2anj j =1 n a2 en 2n (x) Hn nj + g , + − W + (x)e u − g ( x ¯ )e e + n−1 n n n−1 n−1 n−1 n 2 en 2 2ann where Hn = becomes
(35)
j =1
n
2 j =1 hj (x¯ j )/2.
The function tanh(en /n ) is used to deal with the term Hn /en . Then, inequality (35)
V˙Pn en fˆn (Zn ) + gn (x)en u − gn−1 (x¯n−1 )en−1 en +
en Hn , + 1 − 16 tanh2 2 n
n a2 nj j =1
(36)
where n is a positive design parameter, Zn = ∈ Zn ⊂ R 2n+1 with e = [e1 , e2 , . . . , en ]T and Zn being a known compact set, and the function fˆn (Zn ) is defined by 2 2
n−1 e 2 (x¯ )
n j j j
e e j
n n−1 n n−1 fˆn (Zn ) = fn (x) + gn−1 (x¯n−1 )en−1 + + + 2 2 jxj 2 jxj 2a nj j =1
en Hn en 2n (x) . − Wn−1 + 16 tanh2 + 2 en n 2ann [eT , y¯ Tdn ]T
Similar to (25), a fuzzy logic system Tn n (Zn ) is used to approximate fˆn (Zn ) such that fˆn (Zn ) = Tn n (Zn ) + n (Zn ).
(37)
By substituting (37) into (36), and applying the triangular inequality, one can get
en b n e2 V˙Pn gn (x)en u − gn−1 (x¯n−1 )en−1 en + 2 Tn (Zn )n (Zn )en2 + n + dn + 1 − 16 tanh2 Hn , 2 n 2 n b−1
where n = and dn = n Furthermore, define a function VWn = VPn +
2
n
(38)
2 2 2 j =1 anj /2 + n /2 + n /2. VWn as
b ˜2 . 2 n n
˙ Apparently, choosing the adaptive law ˆ n in (5), then the time derivative of VWn along (5) and (38) satisfies b ˆ n e2 V˙Wn gn (x)en u − gn−1 (x¯n−1 )en−1 en + 2 Tn (Zn )n (Zn )en2 + n 2 2 n
en bn ˜ n ˆ n Hn . + dn + 1 − 16 tanh2 +
n n
(39)
By taking the true control law u in (4) and using Remark 3, we have gn (x)en u − bkn en2 − b
ˆ n T (Zn )n (Zn )en2 . 2 2n n
(40)
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Moreover, using n b ˜ n ˆ n n b ˜ 2n bn ˜ n n n b ˜ 2n n b 2n =− + − + ,
n 2 n 2 n
n
n and substituting (40) into (39) show that
en 1 n b ˜ 2n Hn , V˙Wn − bkn − − gn−1 (x¯n−1 )en−1 en + Cn + 1 − 16 tanh2 en2 − n 2 2 n
(41)
where Cn = dn + bn 2n /2 n . Now, choose a Lyapunov function candidate Vn = Vn−1 + VWn . Similar to (31), the following inequality holds:
n−1
n−1
bj ˜ 2j 1 2 2 ej ˙ Vn−1 − bkj − + gn−1 (x¯n−1 )en−1 en + Cj + 1 − 16 tanh Hj . e + 2 j 2 j j j =1
j =1
(42) Thus, from (41) and (42), it can clearly be seen that the time derivative of Vn satisfies
n n
bj ˜ 2j 2 2 ej ˙ ¯ Vn − kj e j + +C+ 1 − 16 tanh Hj , 2 j j
(43)
j =1
j =1
where k¯j = bkj − 1/2 > 1/2 with kj > 1/b, and C = nj=1 Cj with Cj = dj + bj 2j /2 j . The last term in (43) will be considered later. At the present stage, we are in the position to give our main result in the following theorem. Theorem 1. Consider the closed-loop system consisting of system (1) under Assumptions 1–3, the control laws (4) and the adaptive laws (5). Suppose that the packaged uncertain functions fˆi (Zi ), i = 1, 2, . . . , n, can be approximated by fuzzy logic systems in the sense that approximation errors are bounded. Then, the closed-loop system has the following properties for bounded initial conditions with ˆ i (t0 ) 0, i = 1, 2, . . . , n, (i) all the signals in the closed-loop system are bounded; (ii) the error signal e, in the mean square sense, eventually converges to the following set: s := {e ∈ R n |ers s },
1 t
(44)
where e = [e1 , e2 , . . . , en ]T , ers = t 0 e()2 d, and s will be given later. s can be made as small as desired by appropriately choosing design parameters. Before the proof of this theorem, we first introduce the following lemma. Lemma 2 (Ge and Tee [7]). For 1 j n, consider the set j given by j := {ej ej | < 0.2554j }. Then, for ej ∈ / j , the inequality 1 − 16 tanh2 (ej /j ) 0 is satisfied. Proof. (i) Boundedness of all the signals in the closed-loop system can be proven by the following three cases. Case 1: For j = 1, 2, . . . , n, ej ∈ j . In this case, |ej | < 0.2554j with j being the positive design parameter, therefore, ej is bounded. Moreover, according to the choice of the adaptive parameter ˆ j , it is obvious that ˆ j is (n) bounded for any bounded ej . Further, ˜ j is bounded as j is a constant. According to Assumption 1, yd , y˙d , . . . , yd are bounded. Since e1 = x1 − yd , and yd are bounded, thus x1 is also bounded. Using (4) with i = 1, and noting that e1 , ˆ 1 and 1 (Z1 ) are all bounded, we can conclude that 1 is bounded. Consequently, it follows from x2 = e2 + 1 that x2 is bounded. Following the same way, j −1 , u and xj , j = 3, . . . , n, can be proven to be bounded. Thus, all the signals in the closed-loop system are bounded in Case 1. Case 2: All ej ∈ / j . In this case, from the definition of Hj , we know that Hj is nonnegative. It follows from Lemma 2 that
ej 1 − 16 tanh2 Hj 0. (45) j
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By using (45), (43) becomes n bj ˜ 2j 2 + C, k¯j ej + V˙n − 2 j
(46)
j =1
where C is a constant and k¯j satisfies k¯j > 1/2 > 0. As shown in [23], (46) shows that all the signals in the closed-loop system are bounded. Case 3: Some ei ∈ i , while some ej ∈ / j . For those ei ∈ i , define I as the subsystem consisting of ei ∈ i . Similar to Case 1, ei , ˆ i , and ˜ i can be proven to be bounded for i ∈ I . For those ej ∈ / j , define J as the subsystem consisting of ej ∈ / j and choose the Lyapunov function candidate as b ˜ 2j . VPj + VJ = 2 j j ∈J
By using (30) and (45), we have bj ˜ 2j 2 k¯j ej + − Cj + [gj (x¯j )ej ej +1 − gj −1 (x¯j −1 )ej ej −1 ]. V˙J − 2 j j ∈J
(47)
j ∈J
The last term of (47) can be expressed as [gj (x¯j )ej ej +1 − gj −1 (x¯j −1 )ej ej −1 ] = j ∈J
gj (x¯j )ej ej +1 −
j +1∈J
j −1∈J
j ∈J
j ∈J
+
gj (x¯j )ej ej +1 −
gj −1 (x¯j −1 )ej −1 ej
j +1∈I
j −1∈I
j ∈J
j ∈J
gj −1 (x¯j −1 )ej −1 ej .
(48)
As shown in [7], the first two terms in (48) can be canceled during backstepping. Therefore, from Assumption 3 and Lemma 2, it can be shown that 2 2 ej ej 2 2 2 2 + c ej +1 + + c ej −1 [gj (x¯j )ej ej +1 − gj −1 (x¯j −1 )ej ej −1 ] 4 4 j ∈J
j +1∈I
j −1∈I
j ∈J
j ∈J
j ∈J
ej2 2
+
(c2 (0.2554j −1 )2 + c2 (0.2554j +1 )2 ).
j −1∈I j +1∈I
Based on the above inequality, (47) can be rewritten as
bj ˜ 2j 1 2 V˙J − k¯j − + C J , e + 2 j 2 j
(49)
j ∈J
where the positive constant CJ is given by Cj + (c2 (0.2554j −1 )2 + c2 (0.2554j +1 )2 ), CJ = j ∈J
(50)
j −1∈I j +1∈I
whose size depends on design parameters. Since k¯j > 1/2, we have k¯j − 1/2 > 0. Similar to Case 2, it can be shown that ej , ˆ j and ˜ j are bounded for j ∈ J . Now, we consider the boundedness of all the signals in the whole
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closed-loop system. According to the above analysis, we know that all ei , ˆ i and ˜ i are bounded, i = 1, 2, . . . , n. From (n) Assumption 1, we have that yd , y˙d , . . . , yd are bounded. Following the similar discussion in Case 1, we can conclude that all the signals in the closed-loop system are bounded under this case. In light of the discussion for Cases 1–3, we can conclude that all the signals in the closed-loop system are bounded. This ends the proof of (i). (ii) In this part, we will prove the error signal e, in the mean square sense, eventually converges to a compact set s . Similar to the proof of (i), the proof is divided into the following three cases. Case 1: For all ej ∈ j , we obtain that |ej | < 0.2554j , j = 1, 2, . . . , n. Let = [1 , 2 , . . . , n ]T and note the definition of ers . It is easy to get that 1 t ers = (51) e()2 d < (0.2554)2 2 . t 0 Case 2: For all ej ∈ / j , From (46) and the definition of k¯j , we have n
1 ˙ bkj − e2 + C. Vn − 2 j
(52)
j =1
Integrating (52) from 0 to t shows that t n
1 1 1 bkj − e2 () d + C. (Vn (t) − Vn (0)) − t t 2 0 j
(53)
j =1
Noting that bkj > 1/ > 0 and defining k = min{bk1 , bk2 , . . . , bkn }, we have 1 t Vn (0)/t + C ers = . e()2 d t 0 k − 1/2
(54)
Case 3: Some ei ∈ i , while some ej ∈ / j . According to Case 3 in the proof of (i), and considering the subsystem I consisting of ei ∈ i , we can get 1 t ers |I = eI 2 d < (0.2554)2 I 2 , (55) t 0 / j , it follows from (49) where eI 2 = i∈I ei2 and I 2 = i∈I 2i . For the subsystem J consisting of ej ∈ and the definition of k¯j that
1 2 ˙ bkj − e + C J , VJ − j j ∈J
where CJ is defined by (50). Applying the similar procedures as (53) and (54), we have V (0)/t + CJ 1 t ers |J = eJ 2 d J , t 0 k − 1/ where eJ 2 = j ∈J ej2 , and k is defined in (54). Therefore, from (55) and (56), it is obtained that V (0)/t + CJ 1 t . e()2 d(0.2554)2 I 2 + J ers = k − 1/ t 0 Finally, from Cases 1–3, we can conclude that VJ (0)/t + CJ 2 2 Vn (0)/t + C 2 2 ers max (0.2554) , , (0.2554) I + , k − 1/2 k − 1/ which means that e eventually converges to the following set: s := {e ∈ R n |ers s },
(56)
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where C J C 2 2 , (0.2554) I + . s = max (0.2554) , k − 1/2 k − 1/
2
2
This completes the proof of Theorem 1.
Remark 5. Recently, the similar results have been proposed in [25] for SISO nonlinear delay-free systems, and [4] for MIMO nonlinear delay-free systems. Unlike [25,4], this paper mainly addresses the tracking control problem for nonlinear time-delay systems with the strict-feedback structure. Although the adaptive fuzzy controller has been constructed based on the same design idea and backstepping technique, the existence of nonlinear time-delay terms hi (x¯i (t − i )) makes controller design more difficult than the case without time delays. To compensate for the effect of delay terms, both the novel Lyapunov–Krasovskii functionals and the hyperbolic tangent functions are employed. As a result, the stability analysis of the closed-loop system in this paper is not only more complex than ones in [25,4] but also different from them. In addition, when removing the time-delay terms from the system, our result will include the one in [25] as a special case. 4. Simulation examples In this section, two examples will be used to illustrate the effectiveness of the scheme presented in this paper. Example 1. Consider a controlled Brusselator model with external disturbances in [26] as follows: ⎧ ⎨ x˙1 = C − (D + 1)x1 + x12 x2 + d1 (t, x¯2 ), x˙ = Dx1 − x12 x2 + (2 + cos(x1 ))u + d2 (t, x¯2 ), ⎩ 2 y = x1 ,
(57)
where x1 and x2 denote the concentrations of the reaction intermediates, C, D > 0 are parameters which describe the supply of “reservoir” chemicals. u is the control input. d1 (t, x¯2 ) and d2 (t, x¯2 ) are the external disturbance terms, which come from the modeling errors and other types of unknown nonlinearities in the practical chemical reactions. As stated in [8], it is assumed that x1 = 0. The Brusselator model is one of the most popular nonlinear oscillatory models of chemical kinetics. As a practical chemical reaction, the existence of time delays is inevitable in the Brusselator model. Therefore, we add some time-delay terms in (57) to get that ⎧ ⎨ x˙1 = C − (D + 1)x1 + x12 x2 + d1 (t, x¯2 ) + h1 (x1 (t − 1 )), (58) x˙ = Dx1 − x12 x2 + (2 + cos(x1 ))u + d2 (t, x¯2 ) + h2 (x¯2 (t − 2 )), ⎩ 2 y = x1 , where h1 (x1 (t − 1 )) and h2 (x¯2 (t − 2 )) are the unknown time-delay functions. In the simulation, we choose C = 1, D = 3, d1 = 0.7x12 cos(1.5t), d2 = 0.5(x12 + x22 ) sin3 (t), h1 (x1 ) = 2x12 , h2 (x¯2 ) = 0.2x2 sin(x2 ), 1 = 1s, and 2 = 2s. The control objective is to design an adaptive fuzzy controller such that all the signals in the closed-loop system remain bounded and the system output y follows the given reference signal yd = 3 + sin(0.5t) + 0.5 sin(1.5t). To this end, we define fuzzy membership functions as follows: −0.5(x + 1.5)2 −0.5(x + 1)2 , F 2 = exp , F 1 = exp i i 4 4 −0.5(x + 0.5)2 −0.5x 2 , F 4 = exp , F 3 = exp i i 4 4 −0.5(x − 0.5)2 −0.5(x − 1)2 , F 6 = exp , F 5 = exp i i 4 4 −0.5(x − 1.5)2 F 7 = exp . (59) i 4
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4.5 4 3.5 3 2.5 2 1.5
0
10
20
30 Time(sec)
40
50
60
Fig. 1. System output y(t) (“-’’) and reference signal yd (t) (“- -’’) of Example 1.
3 2.5 2 1.5 1 0.5 0
0
10
20
30 Time(sec)
40
50
60
Fig. 2. Trajectory of state x2 (t) of Example 1.
According to Theorem 1, construct the virtual control law, the true control law and the adaptive laws as follows:
1 = −
u=−
ˆ 1 T 1 (Z1 )1 (Z1 )e1 − k1 e1 , 2 21
ˆ 2 T 2 (Z2 )2 (Z2 )e2 − k2 e2 , 2 22
˙ ˆ i = i2 Ti (Zi )i (Zi )ei2 − i ˆ i , 2 i
(60)
(61) i = 1, 2,
(62)
M. Wang et al. / Fuzzy Sets and Systems 159 (2008) 949 – 967
963
50
25
0
-25
-50
0
10
20
30 Time(sec)
40
50
60
50
60
Fig. 3. The true control input u(t) of Example 1.
25
20
15
10
5
0
0
10
20
30
40
Time(sec) Fig. 4. The adaptive parameters ˆ 1 (t) (“-’’) and ˆ 2 (t) (“- -’’) of Example 1.
where e1 = x1 − yd , e2 = x2 − 1 , Z1 = [e1 , yd , y˙d ]T , Z2 = [e1 , e2 , yd , y˙d , y¨d ]T , and the design parameters are chosen as k1 = 10, k2 = 11, 1 = 0.25, 2 = 0.5, 1 = 50, 2 = 80, 1 = 0.1 and 2 = 0.25. The simulation is run under the initial conditions [x1 (0), x2 (0), ˆ 1 (0), ˆ 2 (0)]T = [2.5, 1, 0, 0]T . And the simulation results are shown in Figs. 1–4. Fig. 1 shows the system output y and the reference signal yd . Fig. 2 shows the response of state variable x2 , Fig. 3 displays the control input signal u, and Fig. 4 shows the boundedness of adaptive parameters ˆ 1 and ˆ 2 . 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.
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1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
10
20
30 Time(sec)
40
50
60
Fig. 5. System output y(t) (“-’’) and reference signal yd (t) (“- -’’) of Example 2.
1 0.5 0 −0.5 −1 −1.5 −2 −2.5
0
10
20
30 Time(sec)
40
50
60
Fig. 6. Trajectory of state x2 (t) of Example 2.
Example 2. To further show the effectiveness of our results, we apply the adaptive fuzzy tracking controller (60)–(62) to the following system which is considered in [6] ⎧ ⎨ x˙1 = (1 + x12 )x2 + x1 e−0.5x1 + 2x12 (t − 1 ), (63) x˙ = (3 + cos(x1 x2 ))u + x1 x22 + 0.2x2 (t − 2 ) sin(x2 (t − 2 )), ⎩ 2 y = x1 , where x1 and x2 denote state variables, u is the system control input and y is the system output. In this example, the fuzzy membership functions (59) are used. The reference signal is yd = 0.5(sin(t) + sin(0.5t)), which is the same as in [6]. Apply the controller (60)–(62) with k1 = 25, k2 = 25, 1 = 2 = 2, 1 = 1000, 2 = 2000, 1 = 0.2, and
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3.5 3 2.5 2 1.5 1 0.5 0 −0.5 −1
0
10
20
30 Time(sec)
40
50
60
50
60
Fig. 7. The true control input u(t) of Example 2.
0.2
0.15
0.1
0.05
0
0
10
20
30 Time(sec)
40
Fig. 8. The adaptive parameters ˆ 1 (t) (“-’’) and ˆ 2 (t) (“- -’’) of Example 2.
2 = 0.15. The simulation is carried out under the initial conditions [x1 (0), x2 (0), ˆ 1 (0), ˆ 2 (0)]T = [0, 0, 0, 0]T and 1 = 2 = 2s. The simulation results are shown by Figs. 5–8. Apparently, the simulation results show that under the action of the suggested controller, a good tracking performance has been achieved. Remark 6. It is evident that the tracking performance depends on the design parameters. Theoretically, a good tracking performance can be achieved by choosing ki large enough or i small enough. Then, how to choose the optimal parameters, such as ki, i , i and so on, to get the optimal tracking performance is still an open problem. In the presented simulations, the design parameters are set using a trial-and-error method. It should be pointed out that the existing adaptive fuzzy control approaches cannot be used to control systems (58) and (63) because of the existence of nonlinear delayed functions. However, the adaptive neural control approaches
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suggested in [6,10] can be applied to design the tracking controllers. Because all the ideal weights are regarded as the estimated parameters, when the controllers in [6,10] are implemented, there will be a large number of adaptive parameters which are required to be tuned online simultaneously. This considerably increases the computational burden. For instance, in the simulation studies of [6], 270 adaptive parameters are required to be tuned online to control system (63). Unlike [6,10], in this paper, a key technique is introduced to reduce the number of adaptive parameters, that is, the unknown constant i = b−1 i 2 is used as an estimated parameter, which results in only one adaptive parameter ˆ i for each virtual control law i . Therefore, for the second-order nonlinear time-delay systems (58) and (63), only two adaptive parameters ˆ 1 and ˆ 2 are required to be updated online. 5. 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