Adaptive neural network control of nonlinear systems with ... - NUS
2004
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
[18] O. J. M. Smith, “Closer control of loops with dead time,” Chem. Eng. Progr., vol. 53, no. 5, pp. 217–219, 1957. [19] M. Morari and E. Zafiriou, Robust Process Control. Upper Saddle River, NJ: Prentice-Hall, 1989. [20] L. Mirkin, “Are distributed-delay control laws intrinsically unapproximable?,” presented at the 4th IFAC Workshop Time Delay Systems, Rocquencourt, France, 2003. [21] C. N. Nett, C. A. Jacobson, and M. J. Balas, “A connection between state-space and doubly coprime fractional representations,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 831–832, Sept. 1984.
Adaptive Neural Network Control of Nonlinear Systems With Unknown Time Delays Shuzhi Sam Ge, Fan Hong, and Tong Heng Lee
Abstract—In this note, adaptive neural control is presented for a class of strict-feedback nonlinear systems with unknown time delays. Using appropriate Lyapunov–Krasovskii functionals, the uncertainties of unknown time delays are compensated for such that iterative backstepping design can be carried out. In addition, controller singularity problems are solved by using the integral Lyapunov function and employing practical robust neural network control. The feasibility of neural network approximation of unknown system functions is guaranteed over practical compact sets. It is proved that the proposed systematic backstepping design method is able to guarantee semiglobally uniformly ultimate boundedness of all the signals in the closed-loop system and the tracking error is proven to converge to a small neighborhood of the origin. Index Terms—Adaptive control, neural networks, nonlinear time-delay systems.
[13], the system considered was a class of nonlinear time-delay systems with a so-called “triangular structure”, which was later commented that it could not be “constructively obtained” in [14]. The uncertainties from unknown parameters or unknown nonlinear functions are yet to be discussed, especially when the virtual control coefficients are unknown nonlinear functions of states with known signs. With the aid of neural networks parameterization, stable controllers have been constructed in [15] and [16] (to name just a few) and a family of novel integral Lyapunov functions have been developed in [17] and [18] to avoid controller singularity problem commonly encountered in adaptive feedback linearization control. In this note, we present an adaptive neural controller for a class of strict-feedback nonlinear systems with unknown time delays and unknown virtual control coefficients. Both integral functions and Lyaponov–Krasovskii functionals are used to construct the Lyapunov function candidates such that the controller singularity problem is avoided and the uncertainties from unknown time delays are removed. Semiglobally uniformly ultimate boundedness (SGUUB) of all the signals in the closed-loop system can be guaranteed and the control performance can be improved by appropriately choosing the design parameters. The proposed method expands the class of nonlinear systems that can be handled using adaptive control. The main contributions of the note lie in: 1) the use of integral Lyapunov functions to avoid controller singularity problem commonly encountered in feedback linearization control; 2) the introduction of appropriate Lyaponov–Krasovskii functionals to compensate for the unknown time delays for a time-delay independent controller design; 3) the introduction of practical robust control to avoid possible singularity problem due to the appearance of 1=zi in the controller; and 4) the use of neural networks as function approximators with its feasibility being guaranteed over the practical compact sets for the first time in the literature.
I. INTRODUCTION Recent years have witnessed great progress in adaptive control of nonlinear systems, which plays an important role due to its ability to compensate for parametric uncertainties. To obtain global stability, some restrictions had to be made to system nonlinearities such as matching conditions [1], extended matching conditions [2], or growth conditions [3]. To overcome these restrictions, recursive and systematic backstepping design was developed in [4]. The overparametrization problem was then removed in [5] by introducing the concept of tuning function. Several adaptive approaches for nonlinear systems with triangular structures have been proposed in [6], [7]. Robust adaptive backstepping control has been studied for uncertain nonlinear systems with unknown nonlinear functions in [8] and [9], and among others. Recently, stabilization of nonlinear systems with time delays is receiving much attention [10], [11]. The existence of time delays may make the task more complicated and challenging, especially when the delays are not perfectly known. One way to ensure stability robustness with respect to this uncertainty is to seek for delay-independent solutions. Lyapunov–Krasovskii functionals have been used early as checking criteria for time-delay systems’ stability in [12] and [13]. In
Manuscript received January 30, 2003; revised March 25, 2003. Recommended by Associate Editor P. Tomei. The authors are with Department of Electrical and Computer Engineering, National University of Singapore, 119260 Singapore (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.819287
II. PROBLEM FORMULATION AND PRELIMINARIES Consider a class of single-input–single-output (SISO) nonlinear time-delay systems x_ i (t) =gi ( xi (t))xi+1 (t) + fi ( xi (t)) + +hi ( xi (t
1 i n 0 1
xn (t))u + fn ( xn (t)) + hn ( xn (t x_ n (t) =gn (
0 i ))
0 n)) 2
(1)
where x i = [x1 ; x2 ; . . . ; xi ]T , x = [x1 ; x2 ; . . . ; xn ]T Rn and u R are the state variables and system input respectively, gi ( ), fi ( ), hi ( ) are unknown smooth functions, and i are unknown time delays of the states, i = 1; . . . ; n. The control objective is to design an adaptive controller for (1) such that the state x1 (t) follows a desired reference signal yd (t), while all signals in the closed-loop system are bounded. Define the desired trajectory x d(i+1) = [yd ; y_ d ; . . . ; yd(i) ]T , i = 1; . . . ; n 1, which is a vector of yd up to its ith time derivative (i) yd . We have the following assumptions for the system’s signals, un-
2 1
1
1
0
known functions and reference signals. Assumption 2.1: The system states x(t) and their partial time derivatives, x _ n01 (t), are all available for feedback. Remark 2.1: Note that the requirement for x _ n01 (t) is a constraint but realistic for many physical systems as we are not requiring x_ n which is directly influenced by the control. Assumption 2.2: The signs of gi ( xi ) are known, and there exist constants gi0 and known smooth functions gi ( xi ) such that gi0 jgi (xi )j gi (xi )1, 8xi 2 Ri .
0018-9286/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
Assumption 2.3: The desired trajectory vectors x di , i = 2; . . . ; n are continuous and available, and x di 2 di Ri with di known compact sets. Assumption 2.4: The unknown smooth functions hi ( xi (t)) satisfy the following inequality jhi ( xi (t))j ij =1 jxj (t)j%ij (xi (t)) where %ij (1) are known smooth functions. Assumption 2.5: The size of the unknown time delays is bounded by a known constant, i.e., i max , i = 1; . . . ; n. Remark 2.2: There are many physical processes which are governed by nonlinear differential equations of the form (1). Examples are recycled reactors, recycled storage tanks and cold rolling mills [19]. In general, most of the recycling processes inherit delays in their state equations. A function approximator shall be used to approximate the unknown nonlinear functions. Among the two types of artificial neural networks, i.e, 1) linearly parametrized neural networks (LPNNs) [18] and 2) multilayer neural networks (MNN’s), for simplicity of demonstrating the main idea, we shall use radial basis function (RBF) neural network (NN), a kind of LPNN, as an example in this note to approximate the continuous function h(Z) : Rq ! R as
hnn (Z) = W T S(Z)
(2)
where the input vector Z 2 Z Rq , weight vector W 2 Rl , and basis function vector S(Z) = [s1 (Z); s2 (Z); . . . ; sl (Z)]T 2 Rl , with l being the NN node number and si (Z) chosen as the commonly used Gaussian functions, which have the form si (Z) = exp[0(Z 0 i )T (Z 0 i )=i2 ], i = 1; . . . ; l where i = [i1 ; i2 ; . . . ; iq ]T is the center of the receptive field and i is the width of the Gaussian function. Universal approximation results in [20] and [21] indicate that, if l is chosen sufficiently large, W T S(Z) can approximate any continuous function to any desired accuracy over a compact set Z Rq to arbitrary any accuracy as
h(Z) = W 3T S(Z) + (Z); 8Z 2 Z Rq
(3)
where W 3 is the ideal constant weight vector, and (Z) is the approximation error which is bounded over the compact set, i.e., j(Z)j 3 , 8Z 2 Z3 where 3 > 0 is an unknown constant. The ideal weight vector W is an “artificial” quantity required for analytical purposes. W 3 is defined as the value of W that minimizes jj for all Z 2 Z Rq , i.e., W 3 := arg minW 2R supZ 2 jh(Z) 0 W T S(Z)j . The following even function pi (1) : R ! R is introduced for the purpose of the practical controller design in Section III:
pi (x) =
1; jxj cz 0; jxj < cz
8x 2 R:
(4)
III. ADAPTIVE NN CONTROLLER DESIGN In this section, adaptive neural control is proposed for system (1) and the stability results of the closed-loop system are presented. The backstepping design procedure contains n steps. The design of adaptive control laws is based on the following change of coordinates: z1 = x1 0 yd , zi = xi 0 i01 , i = 2; . . . ; n, where i (t) is an intermediate control which shall be developed for the corresponding ith subsystem based on an appropriate Lyapunov function Vi (t). The control law u(t) is designed in the last step to stabilize the whole closed-loop system based on the overall Lyapunov function Vn , which is the sum of the ^ i 0 Wi3 , ~i = W previous Vi (t), i = 1; . . . ; n 0 1. For clarity, let W 2 3 0 3 ^ constant ci := 1=2i Wi 0 Wi + 1=2z , where Wi 2 Rl is the estimate of ideal NN weight Wi3 2 Rl , Wi0 2 Rl is a design
2005
constant, 3z is the upper bound of the NN approximation error, i.e., ki (Zi )k 3z with Zi being the corresponding inputs to be defined later, and small constant i > 0 is to introduce -modification in adaptive control to be developed later. In this note, the following inequalities play an important role with i = 1; . . . ; n:
^ i 0 Wi3 i W
T
2
21 i W^ i 0 Wi3 0 21 i Wi3 0 Wi0 i (Zi )zi (t) 1 3z + 1 zi2 (t): 2 2
^ i 0 Wi0 W
2
(5) (6)
Define the integral Lyapunov function Vz (t) [17], [18], the Lyapunov–Krasovskii functional VU (t) with the positive scalar function Ui (1), and the Lyapunov function candidates Vi (t) as
Vz (t) = Vz (t) = VU (t) =
z
0 0
g10 1 ( + yd )d
z
01 (xi01 ; + i01 )d; gi
t
t0
(7)
i = 2; . . . ; n
Ui (xi ( ))d
(8) (9)
Vi (t) =Vi01 (t) + Vz (t) + VU (t) ~ T (t)0i01W ~ i (t) (10) + 1W 2 i 01 (xi ) = i (xi )=gi (xi ) with i (xi ) > 0 being a smooth where gi 01 )2 = i2 =gi2 , and weighting function to be defined later and (gi i 2 2 Ui (xi (t)) = 1=2 j =1 xj (t)%ij (xi (t)), i = 1; . . . ; n. The choice of weighting function i (1) plays a key role in the controller design
and results in different controllers with varying control performance [18]. The apparent and convenient choices for i (1) are 1 and gi ( xi ) for general nonlinear systems. Detailed explanations will be given based on i ( xi ) = gi (xi ) in the following, while a remark will be given directly addressing the controller design and relevant stability and performance analysis for i ( xi ) = 1 without derivation for conciseness. 01 (xi ) = gi (xi )=gi (xi ). By choosing i ( xi ) = gi (xi ), we have gi 01 (xi ) are bounded by known From Assumption 2.2, we know that gi 01 (xi ) gi (xi )=gi0 . Clearly, Vz are positive–deffunctions as 1 < gi inite functions, i = 1; . . . ; n. Step 1) Let us first consider the z1 -subsystem as
z_1 (t) = g1 (x1 (t))(z2 (t) + 1 (t)) + f1 (x1 (t)) +h1 (x1 (t 0 1 )) 0 y_d (t):
(11)
By variable change = z1 , we may rewrite Vz in (7) as Vz = z12 01 g10 1 (z1 +yd )d . Noting that 1 g10 1 (z1 +yd ) g1 (z1 + yd )=g10 , we have
z12 2
1
2
Vz gz101
0
g1 (z1 + yd )d:
The time derivative of Vz along (11) is
V_ z (t) =z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t) + g10 1 (x1 (t))f1 (x1 (t)) + g10 1 (x1 (t))h1 (x1 (t 0 1 )) 1 0 y_d (t) g10 1 (z1 + yd )d : 0
(12)
2006
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
The remaining term U1 (x1 (t)) from V_ U (t) is a known function of known variables, which does not introduce any uncertainties to the system. Therefore, the design of intermediate control 1 (t) is free from unknown time delay 1 at present stage. For notation conciseness, we will omit the time variables t and t 0 1 after time-delay terms have been eliminated. Under the assumption of exact knowledge, the certainty equivalent control is
Applying Assumption 2.4, we have
V_ z (t) z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t)
+ g10 1 (x1 (t))f1 (x1 (t)) 1
0 y_d
0 0 g1 1
g10 1 (z1 + yd )d
+ jz1 (t)j (x1 (t))jx1 (t 0 1 )j 2 %1 (x1 (t 0 1)):
13 = (13)
V_ z (t) z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t)
+ g10 1 (x1 (t))f1 (x1 (t))
0 y_d
0
+ 21 z12 (t) g10 1 2 + 21 x21 (t 0 1 ) 2 %21 (x1 (t 0 1)): (14) In standard iterative backstepping design, 1 (t) is usually designed to stabilize the z1 -subsystem except for the coupling term g1 z1 z2 to be dealt with in the next step. In doing so under the assumption of known functions, one more difficulty exists in the new problem setting. Although %1 (1) is a known function, it cannot be utilized in designing 1 (t) as x1 (t 0 1 ) is undetermined because of unknown time delay 1 . Intuitively, approximation methods such as neural networks can be used to approximate the unknown functions. The unknown functions g1 (1) and f1 (1) can be dealt with in this way without any problem. However, due to the existence of the unknown time delay 1 , functions of x1 (t 0 1 ) are hard to be approximated using neural networks since the input x1 (t 0 1 ) is undetermined because of the uncertain 1 . To overcome the design difficulties from the unknown time delay 1 , we consider the Lyapunov–Krasovskii functional VU (t) in (9) with its time derivative being (15)
which can be used to cancel the time-delay term on the right-hand side of (14) and thus eliminate the design difficulty from the unknown time delay 1 without introducing any uncertainties to the system. Accordingly, we obtain
V_ z (t) + V_ U (t) z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t)
+ g10 1 (x1 (t))f1 (x1 (t)) 1 0 y_d g10 1 (z1 + yd )d 0 1 + 2 z1 (t) g10 1 2 + 2z 1(t) x21 (t)%12(x1 (t)) : 1
where 1 0
g10 1 (z1 + yd )d
+ 12 z1 g10 1
2
Z1 = [x1 ; yd ; y_d ]T 2 Z R3 and Z := fz1 ; xd2 jz1 2 R; xd2 2 d2 g. It is apparent that controller singularity may occur. In addition, it is certain that 31 is not an admissible control, since 31 is not well-defined when z1 = 0 as limz !0 2z1 = 0, limz !0 x21 %21 (x1 ) 6= 0 and L’Hopital’s rule is not applicable to obtain the limit limz !0 (x12 %21 (x1 )=(2z1 )). Therefore, care must be with
g10 1 (z1 + yd )d
V_ U (t) = U1 (x1 (t)) 0 U1 (x1 (t 0 1 ))
0k1 (t)z1 0 Q1 (Z1 ) 0 21z1 x21 %21 (x1 )
Q1 (Z1 ) = g10 1 (x1 )f1 (x1 ) 0 y_ d
By using Young’s Inequality, (13) becomes
1
1
g1 (x1 )
(16)
Comparing (16) with (14), it is found that the difficulty from the unknown time delay 1 has been eliminated by introducing the Lyapunov–Krasovskii functional VU (t). By differentiating VU (t) with respect to time, the unknown time delay term U1 (x1 (t 0 1 )) = 1=2x12 (t 0 1 )%21 (x1 (t 0 1 )) appeared in (15) can be used for exact cancellation on the right-hand side of (14).
taken to guarantee the boundedness of the controller. It is noted that point z1 = 0 is not only an isolated point in Z , but also the case that the system reaches the origin at this point. From a practical point of view, once the system reaches its origin, no control action should be taken for less power consumption. For ease of discussion, let us define sets c Z and 0Z as follows:
c :=fz1 j jz1 j < cz g
0Z := Z 0 c
(17) (18)
where cz is a constant that can be chosen arbitrarily small and “ 0” in (18) is used to denote the complement of set B in set A as A 0 B := fxjx 2 A and x 2= Bg. Accordingly, the following practical control law is proposed:
31 = p1 (z1 ) g1 (x1 )
0k1 (t)z1 0 Q1 (Z1 ) 0 21z1 x21 %21 (x1)
(19)
where pi (1) is defined in (4). Since f (1) and g (1) are unknown smooth functions, the desired practical control 31 in (19) cannot be implemented in practice. Neural networks can be used to approximate the unknown function Q1 (Z1 ). Note that control action is only activated when z1 2 0Z , which means unknown function Q1 (Z1 ) is approximated by neural networks over the set 0Z . According to the main result stated in [22], any real-valued continuous function can be arbitrarily closely approximated by a network of RBF type over a compact set. The compactness of set 0Z is a must to guarantee the feasibility of neural networks approximation, which is shown in the following lemma. Lemma 3.1: Set 0Z is a compact set. Proof: First, we show that 0Z is a closed set. From (18) and applying De Morgan’s laws, we have
Z0c = cZ [ c
(20)
0c where Z and cZ denote the complements of 0Z and Z respectively. Since Z is a compact set, i.e., it is closed and bounded, cZ is an open set. In addition, c is also an open set from its defini0c tion. From (20), we know that Z is an open set, which means that 0 its complement Z is a closed set. Second, from (18), we know that
0Z Z . Since a closed subset of a compact set is compact, we can conclude that 0Z is a compact set.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
Based on Lemma 3.1, it is known that Q1 (Z1 ) is continuous and well-defined over compact set 0Z and can be approximated by neural networks to arbitrary any accuracy as follows Q1 (Z1 ) = W13 T S(Z1 ) + 1 (Z1 ) where 1 (Z1 ) is the approximation error. As the ideal weight W13 is unknown, we shall use its estimate ^ 1 instead, which forms the intermediate control 1 as W
1 (t) = p1 (z1 ) g1 (x1 )
0k1 (t)z1 0 W^ 1T S(Z1 ) 0 2z11 x21 %21 (x1 ) : Z
For uniformity of notation, we define sets c 2; . . . ; n as
c := fzi j jzi j < cz
0Z := Z 0 c :
(21)
and 0Z , i
g
=
(22) (23)
Note that the control objective is to show that certain compact set S is domain of attraction in the sense that for all bounded initial conditions, there exists S such that all closed-loop signals will eventually converge to S . i.e., all Zi (t) starting from within 0Z will enter into
S and will stay within S thereafter. In the following steps, the unknown functions Qi (Zi ), i = 2; . . . ; n will be approximated by neural networks as
Qi (Zi ) = Wi3 T S(Zi ) + i (Zi )
8Zi 2 0Z :
(24)
Consider the Lyapunov function candidate V1 (t) in (10), whose time derivative along (16), (21) and (24) for z1 2 0Z is
^ 1 0 W13 V_ 1 0k1 (t)z12 + g1 (x1 )z1 z2 0 W
(25)
(26)
For z1
2 0z
+g1 (x1 )z1 z2 + c1 :
with 0
< "10
V_ 1 g1 (x1 )z1 z2 0 g10 Vz "10
0 maxt; t], we have the inequality t0 1=2x12( )%21 (x1 ( ))d . 0 "110 VU
2
0 12 1 W^ 1 0 W13 + c1
where the coupling term g1 (x1 )z1 z2 will be handled in the next step. Remark 3.1: Applying Young’s inequality, we know that g; z1 ; z2 1=2 z12 + 1=2 g 21 z22 . The choice for "10 is to guarantee that 0(1="10 0 1=2)z12 0, so that the undesired destabilizing term 1=2 z12 can be suppressed. Step i (2 i n 0 1)) Similar procedures are taken for i = 2; . . . ; n 0 1 as in Step 1). The dynamics of zi -subsystem is given by
z_i (t) = gi (xi (t))[zi+1 (t) + i (t)] + fi (xi (t)) +hi (xi (t 0 i )) 0 _ i01 (t): The time derivative of Vz (t) in (8) is given by V_ z (t) = zi (t) gi (xi (t))(zi+1 (t) + i (t)) 01 (xi (t))fi (xi (t)) + gi 01 (xi (t))hi (xi (t 0 i )) + gi 1 @g01 (xi01 ; zi + i01 ) + x_ i01 zi (t) i d @xi01 0 1 0 _ i01 g01 (xi01 ; zi + i01 )d : 0
i
01 (xi (t))fi (xi (t)) + 1 zi (t)(g01)2 + gi i 2 01 (xi01 ; zi + i01 ) 1 @gi + x_ i01 zi (t) d @xi01 0 1 0 _ i01 g01 (xi01 ; zi + i01 )d
In Step i 0 1), for zj
(27)
(28)
2, we have
V_ 1 0 1 z12 0 g10 Vz "10 "10 t 1 1 x2 ( )%2 (x ( ))d 0 "10 1 1 1 t0 2 2 0 12 1 W^ 1 0 W13 + g1 (x1 )z1 z2 + c1 :
Accordingly, (29) becomes
i
, noting (12), and choosing
1 k1 (t) = 1 1 + g1 (z1 + yd )d "10 0 t 1 x2 ( )%2 (x ( ))d + 12 1 1 z1 t0 2 1
[t 0 1 ; t] [t 1=2x12 ( )%21 (x1 ( ))d
0
i
+ 1 x2j (t 0 i )%2ij (xi (t 0 i )): 2 j =1
Substituting (26) into (25), and using (5) and (6), we have
^ 1 0 W13 2 V_ 1 0k1 (t)z12 0 1 1 W 2
t t0
V_ z (t) =zi (t) gi (xi (t))(zi+1 (t) + i (t))
The following practical adaptive law is given for online tuning the NN weights
^_ 1 = p1 (z1 )01 S(Z1 )z1 0 1 W ^ 1 0 W10 : W
Since
Noting Assumption 2.4, we have
T
S(Z1 )z1 ^_ 1 : ^ 1 0 W13 T 001 1 W +z1 z + W
2007
(29)
(30)
2 0Z , j = 1; . . . ; i 0 1, it has been obtained that
V_ i01 gi01 (xi01 )zi01 zi i01 + 0 g"jj00 Vz 0 "1j0 VU j =1 2 0 21 j W^ j 0 Wj3 + cj : (31) Consider Vi (t) given in (10). For zj 2 0Z , j = 1; . . . ; i, its time derivative along (30) and (31) is
V_ i gi (xi )zi zi+1 + zi [gi01 (xi01 )zi01 + gi (xi ) i + Qi (Zi )] ^_ i ^ i 0 Wi3 )T 0i01W + (W i01 + 0 g"jj00 Vz 0 "1j0 VU j =1 2 0 21 j W^ j 0 Wj3 + cj
(32)
2008
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
where
In Step n 0 1), for zi that
01 (xi )fi (xi ) + 1 zi (g01)2 Qi (Zi ) =gi 2 i 01 (xi01 ; zi + i01 ) 1 @gi + x_ i01 zi d @xi01 0 1 0 _ i01 g01 (xi01 ; zi + i01 )d 0
Zi (t) = [xi ; x_ i01 ; i01 ; @ i01 =@x1 ; @ i01 =@x2 ; . . . ; @ i01 =@xi01 ; !i01] 2 0Z R3i , where i01
@ i01 x_ + ! ; ! i01 i01 @xj j j =1 i01 @ i01 W ^_ j : = @ i01 x_ di + ^ @xdi W @ j j =1
_ i01 =
V_ n01 gn01 (xn01 )zn01 zn n01 + 0 g"jj00 Vz 0 "1j0 VU j =1 0 21 j kW^ j 0 Wj3 k2 + cj :
i
with
Consider Vn (t) given in (10). For jzi j derivative along (36) and (37) is
0gi01 (xi01 )zi01 0 ki (t)zi 0 W^ iT S(Zi ) i 0 2z1 i x2j %2ij (xi ) j =1
^_ i = pi (zi )0i S(Zi )zi 0 i W ^ i 0 Wi0 W
01 (xn )fn (xn ) + 1 zn g01 2 Qn (Zn ) =gn n 2 01 (xn01 ; zn + n01 ) 1 @gn + x_ n01 zn d @xn01 0 1 0 _ n01 g01 (xn01 ; zn + n01 )d
(33) (34)
1 i x2 ( )%2 (x ( ))d ; z 2 0 i ij i Z 2 j =1 j (35)
with 0 < "i0 2. Substituting (33)–(35) into (32), and using (5), (6), and (12), we have
V_ i gi (xi )zi zi+1 +
i
j =1
2 0 g"jj00 Vz 0 "1j0 VU 0 21 j W^ j 0 Wj3 + cj
where the coupling term gi ( xi )zi zi+1 will be handled in the next step. Step n) This is the final step, since the actual control u appears in the dynamics of zn -subsystem as given by
z_n = gn (xn (t))u + fn (xn (t)) + hn (xn (t 0 n )) 0 _ n01 (t): Consider Vz tive is
(t) given in (8). Noting Assumption 2.4, its time deriva-
01 (xn (t))fn (xn (t)) V_ z (t) =zn (t) gn (xn (t))u(t) + gn
n
n
Zn (t) = [xn ; x_ n01 ; n01 ; @ n01 =@x1 ; @ n01 =@x2 ; . . . ; @ n01 =@xn01 ; !n01 ] 2 0Z R3n , where n01 @ n01 x_ + ! _ n01 = n01 @xj j j =1 n01 @ n01 W ^_ j : !n01 = @ n01 x_ dn + ^ @xdn @ W j j =1
with
We construct the following adaptive neural control law:
u(t) = pn (zn ) gn (xn )
0gn01 (xn01)zn01 0 kn (t)zn 0 W^ nT S(Zn ) 0 2z1n
n j =1
x2j %2nj (xn ) (39)
^_ n = pn (zn )0i S(Zn )zn 0 i W ^ n 0 Wn0 W
(40)
1 kn (t) = 1 1 + gn (xn01 ; zn + n01 )d "n0 0
1 n x2 ( )%2 (x ( ))d ; z 2 0 n nj n Z 2 j =1 j (41)
with 0 < "n0 2. Substituting (39)–(41) into (38), and using (5), (6), and (12), we have
V_ n (t)
n
+ 1 x2j (t 0 n )%2nj (xn (t 0 n )): 2 j =1
0
t + 12 zn t0
01)2 + x_ n01 zn (t) + 1 zn (t)(gn 2 01 1 2 @gn (xn0@1x; nz0n1 + n01 ) d 0 1 0 _ n01 g01 (xn01 ; zn + n01 )d 0
(38)
where
1 ki (t) = 1 1 + gi (xi01 ; zi + i01 )d "i0 0 t + 12 zi t0
(37)
2 0Z , i = 1; . . . ; n, its time
V_ i gi (xi )zi zi+1 + zi [gi01 (xi01 )zi01 + gi (xi ) i + Qi (Zi )] ^_ i ^ i 0 Wi3 )T 0i01W + (W i01 + 0 g"jj00 Vz 0 "1j0 VU j =1 2 0 21 j W^ j 0 Wj3 + cj
Similarly, we have the following intermediate control law:
i = pi (zi ) gi (xi )
2 0Z , i = 1; . . . ; n 0 1, it has been obtained
(36)
n j =1
0 g"jj00 Vz 0 "1j0 VU
2
0 12 j W^ j 0 Wj3 + cj :
(42)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
Theorem 3.1: Consider the closed-loop system consisting of the plant (1) under Assumptions 2.1–2.5, the controller (39) and the NN weight updating law (40). For bounded initial conditions, the following properties hold. i) All signals in the closed-loop system remain SGUUB and the vector Z = [Z1T ; . . . ; ZnT ]T remains in a compact set 0Z :=
0Z [ 1 1 1 [ 0Z specified as n
0Z = Z j
i=1
n
zi2 2C0 ;
i=1
2009
Case 3) Some zi 2 0Z and some zj 2 c . In this case, the ^ i will corresponding i or u and the adaptation law for W 0 be invoked for zi 2 Z while j = 0 or u = 0 and
^_ j = 0 for zj 2 c . Let us define VI (t) = i (Vz + W ~ iT 0i01W ~ i ) and VJ (t) = j (Vz + VU + VU + 1=2W 0 1 T ~ j 0j W ~ j ). For zj 2 c , we obtain that VJ (t) 1=2W is bounded, i.e., VJ (t) CJ with CJ being finite, and zi 2 0Z , we have that V_ I (t) 0C1I VI (t) + C2I , i.e.,
kW~ i k 2C(00 ) ; i 0
2
min
VI (t) [VI (0) 0 I ]e0C t + I
1
xdi 2 di ; i = 2; . . . ; n; zi 2= c ; i = 1; . . . ; n
(43)
I
C2
n i=1
~ T 001 W ~ Vz (t) + VU (t) + 1 W 2 i i i
(45)
where Vz (t) and VU (t) are defined in (8) and (9), respectively, and (~1) = (^1) 0 (1). The following three cases are considered. Case 1) zi 2 c , i = 1; . . . ; n. In this case, the controls i = 0, ^_ i = 0, i = 1; . . . ; n. i = 1; . . . ; n 0 1, u = 0 and W Since z1 = x1 0 yd and yd is bounded, x1 is bounded. For i = 2; . . . ; n, xi is bounded as xi = zi + i01 and ^ i is kept unchanged in a bounded i01 = 0. In addition, W value, i = 1; . . . ; n. Observing the definition for Vz (t) and VU (t), and noting that gi (1), %ij (1) are smooth functions, ^ i , Vz (t) and VU (t) we know that for bounded xi , zi and W are bounded, i.e., there exists a finite CB such that
Vn (t) CB : Case 2) zi
(46)
2 Z , i = 1; . . . ; n. From n(42), we have V_n (t) 0
0C Vn(t) + C 1
2
where C1
=
i=1 ci and
C2 = min g10 ; . . . ; gn0 ; 1 ; . . . ; "10 "n0 "10 1 n 1 ; ;...; "n0 min 0101 min 0n01 = C2 =C1 , it follows that 0 Vn (t) [Vn (0) 0 ]e0C t + Vn (0) +
:
Let
(47)
where constant
Vn (0) =
n i=1
z 0
(0)
fore, it can be obtained that
01 (xi01 (0); + i01 (0))d gi
~ T (0)0i01W ~ i (0) +1W 2 i 01 (xi01 (0); + i01 (0)) = g01 () for i = 1. with gi 1
(48) i ci and
g.
There-
Vn (t) = VI (t) + VJ (t) VI (0) + I + CJ :
(49)
Thus, from (46), (47), and (49) for Cases 1)–3), we can conclude that
(44)
where > 0 is a constant related to the design parameters and will be defined later in the proof, and S can be made as small as desired by an appropriate choice of the design parameters. Proof: Consider the following Lyapunov function candidate:
Vn (t) =
I = C2I =C1I with C1I = = mini fgi0 ="i0 ; 1="i0 ; i =min 0i01
where
where C0 > 0 is a constant whose size depends on the initial conditions (as will be defined later in the proof). ii) The closed-loop signal z(t) = [z1 ; . . . ; zn ]T 2 Rn will eventually converge to a compact set defined by
S := fz j kz k2 g
VI (0) + I
Vn (t) C0
(50)
where C0 = maxfCB ; Vn (0) + ; VI (0) + I + CJ g. ^ i , i = 1; . . . ; n, From (50), we know that Vn (t), zi and W are bounded. Since z1 = x1 0 yd and yd is bounded, x1 is bounded. For x2 = z2 + 1 , since 1 is function of bounded ^ 1 , 1 is thus bounded, which in turn leads signals z1 , Z1 , W to the boundedness of x2 . Following the same way, we can prove one by one that all i01 and xi , i = 3; . . . ; n are bounded. Therefore, the systems’ states xi , i = 1; . . . ; n are bounded. Considering (45) and the property for Vz (t) that
1 z 2 V (t) zi2 z 2 i gi0
1 0
gi (xi01 ; zi + i01 )d
we know that n i=1
zi2 2
n i=1
n
Vz (t) 2Vn (t)
i=1
kW~ i k 2Vn0(t)0 : i 2
min
1
(51) From (50) and (51), we readily have the compact set 0Z defined in (43) over which the NN approximation is carried out with its feasibility being guaranteed. In addition, in Case 1), as zi 2 c , i = 1; . . . n, we know that kzk2 = ni=1 zi2 ni=1 cz2 . In Case 2), from (47) and (51), we have that limt!1 kz k2 = 2. In Case 3), from (48) and (51), we have that limt!1 i zi2 = 2I and j zj2 j cz2 . Therefore, as t ! 1, 2 we can conclude that kz k2 where = max 2; 2I ; n i=1 cz , i.e., the vector z will eventually converge to the compact set S defined in (44). This completes the proof. Remark 3.1: Note that the choices of i ( xi ) are not unique. By 01 (xi ) = 1=gi (xi ) [17] and Vz = choosing i ( xi ) = 1, we have gi z =gi (xi01 ; + i01 )d , i = 1; . . . ; n. By the mean value the0 orem, Vz can be rewritten as Vz = s zi2 =gi ( xi01 ; s zi + i01 ), s 2 (0; 1). From Assumption 2.2, 0 gi0 gi (xi ), we know that 0 < Vz s =g10 zi2 . The adaptive control laws are given by
i = pi (zi ) gi (xi )
0gi0 (xi0 )zi0 0 ki (t)zi 0 W^ iT S(Zi ) 1
0 2z1 i
1
i j =1
1
x2j %2ij (xi )
2010
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
u=
pn (zn ) gn (xn )
0gn01 (xn01)zn01 0 kn (t)zn 0 W^ nT S (Zn ) 0 2z1n
n j =1
x2j %2nj (xn )
^_ i = pi (zi )0i S (Zi )zi 0 i W ^ i 0 Wi0 W ki (t) =
1 1+ + 1 s 2
"i0
zi
t
0
t
1 i x2 ( )%2 (x ( ))d ij i 2 j =1 j
where 0 < "i0 2. For bounded initial conditions, all closed-loop signals remain bounded and the tracking error converges to a small neighborhood around zero by appropriately choosing design parameters. Remark 3.2: Note that the size of the compact set 0Z is characterized by C0 , which depends on system initial conditions xi (0) and ^ i (0) as well as the design parameters i , 0i , Wi0 and "i0 , i = W 1; . . . ; n. For the compact set S to which the closed-loop signals eventually converge, its size only depends on the design parameters. ~ i (0), Therefore, it can be seen that large initial errors zi (0) and W i = 1; . . . ; n may lead to a large transient tracking error during the initial period of adaptation, but will not affect the final convergence of the closed-loop signals. Remark 3.3: Since the function approximation property (3) of neural networks is only guaranteed within a compact set, the stability result proposed is semiglobal in the following sense: Given ~ i (0) 2 I , the any bounded initial compact set such that zi (0); W proposed NN controller with sufficiently large number of nodes guarantees that all the closed-loop signals will stay within the compact set, i.e., 0Z in the note, if the compact set 0Zc , over which the neural network approximation is constructed, satisfies that 0Z 0Zc , and eventually all the closed-loop signals will converge to the steady state compact set, i.e., S in the note. The relationships among the sets are as: I ; S 0Z 0Zc . It is apparent that the larger the compact set 0Zc over which the NN controller is built upon, the more relaxed the initial compact set I is. IV. CONCLUSION An adaptive neural-based control has been addressed for a class of strict-feedback nonlinear systems with unknown time delays. The unknown time delays has been compensated for through the use of appropriate Lyapunov–Krasovskii functionals. As a result, the iterative backstepping design can be carried out. In addition, the controller is free from singularity problem by using the integral Lyapunov function and practical robust neural network control. The proposed systematic backstepping design method has been proven to be able to guarantee SGUUB of closed-loop signals and the output of the system has been proven to converge to an arbitrarily small neighborhood of the origin. REFERENCES [1] D. G. Taylor, P. V. Kokotovic´, R. Marino, and I. Kanellakopoulos, “Adaptive regulation of nonlinear systems with unmodeled dynamics,” IEEE Trans. Automat. Contr., vol. 34, pp. 405–412, Apr. 1989. [2] I. Kanellakopoulos, P. V. Kokotovic´, and R. Marino, “An extended direct scheme for robust adaptive nonlinear control,” Automatica, vol. 27, no. 2, pp. 247–255, 1991. [3] S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 1123–1131, Nov. 1989. [4] I. Kanellakipoulos, P. V. Kokotovic´, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, Nov. 1991. [5] M. Krstic´, I. Kanellakopoulos, and P. V. Kokotovic´, “Adaptive nonlinear control without overparsmetrization,” Syst. Control Lett., vol. 19, no. 3, pp. 177–185, 1992.
[6] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of nonlinear systems with a triangular structure,” IEEE Trans. Automat. Contr., vol. 39, pp. 1411–1428, July 1994. [7] S. S. Ge, C. C. Hang, and T. Zhang, “Stable adaptive control for nonlinear multivariable systems with a triangular control structure,” IEEE Trans. Automat. Contr., vol. 45, pp. 1221–1225, June 2000. [8] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 32, no. 3, pp. 423–427, 1996. [9] Z. Pan and T. Basar, “Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 1066–1083, Aug. 1998. [10] A. M. Annaswamy, S. Evesque, S. Niculescu, and A. P. Dowling, “Adaptive control of a class of tinse-delay systems in the presence of saturation,” in Adaptive Control of Nonsmooth Dynamic Systems, G. Tao and F. Lewis, Eds. New York: Springer-Verlag, 2001, pp. 289–310. [11] F.-H. Hsiao and J.-D. Hwang, “Stabilization of nonlinear singularly perturbed multiple time-delay systems by dither,” J. Dyna. Syst., Measure., Control, vol. 118, no. 1, pp. 176–181, 1996. [12] J. Hale, Theory of Functional Differential Equations, 2nd ed. New York: Springer-Verlag, 1977. [13] S. K. Nguang, “Robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 756–762, Apr. 2000. [14] S. Zhou, G. Feng, and S. K. Nguang, “Comments on robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 1586–1586, Sept. 2002. [15] F. L. Lewis, S. Jagannathan, and A. Yesilidrek, Nerual Network Control of Robot Manipulators and Nonlinear Systems. Philadelphia, PA: Taylor & Francis, 1999. [16] K. S. Narendra and K. Parthasanithy, “Identification and control of dynamic systems using neural networks,” IEEE Trans. Neural Networks, vol. 1, pp. 1–23, Feb. 1990. [17] S. S. Ge, C. C. Hang, and T. Zhang, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, no. 4, pp. 74l–747, 1999. [18] S. S. Ge, C. C. Hang, T. H. Lee, and I. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2002. [19] S.-L Niculescu, Delay Effects on Stability: A Robust Control Approach. London, U.K.: Springer-Verlag, 2001. [20] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Networks, vol. 3, pp. 837–863, Dec. 1992. [21] E. B. Kosmatopoulos, M. M. Polycarpou, M. A. Christodoulou, and P. A. Ioannou, “High-order neural network structures for identification of dynamical systems,” IEEE Trans. Neural Networks, vol. 6, pp. 422–431, Apr. 1995. [22] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, no. 5, pp. 359–366, 1989.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
[18] O. J. M. Smith, “Closer control of loops with dead time,” Chem. Eng. Progr., vol. 53, no. 5, pp. 217–219, 1957. [19] M. Morari and E. Zafiriou, Robust Process Control. Upper Saddle River, NJ: Prentice-Hall, 1989. [20] L. Mirkin, “Are distributed-delay control laws intrinsically unapproximable?,” presented at the 4th IFAC Workshop Time Delay Systems, Rocquencourt, France, 2003. [21] C. N. Nett, C. A. Jacobson, and M. J. Balas, “A connection between state-space and doubly coprime fractional representations,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 831–832, Sept. 1984.
Adaptive Neural Network Control of Nonlinear Systems With Unknown Time Delays Shuzhi Sam Ge, Fan Hong, and Tong Heng Lee
Abstract—In this note, adaptive neural control is presented for a class of strict-feedback nonlinear systems with unknown time delays. Using appropriate Lyapunov–Krasovskii functionals, the uncertainties of unknown time delays are compensated for such that iterative backstepping design can be carried out. In addition, controller singularity problems are solved by using the integral Lyapunov function and employing practical robust neural network control. The feasibility of neural network approximation of unknown system functions is guaranteed over practical compact sets. It is proved that the proposed systematic backstepping design method is able to guarantee semiglobally uniformly ultimate boundedness of all the signals in the closed-loop system and the tracking error is proven to converge to a small neighborhood of the origin. Index Terms—Adaptive control, neural networks, nonlinear time-delay systems.
[13], the system considered was a class of nonlinear time-delay systems with a so-called “triangular structure”, which was later commented that it could not be “constructively obtained” in [14]. The uncertainties from unknown parameters or unknown nonlinear functions are yet to be discussed, especially when the virtual control coefficients are unknown nonlinear functions of states with known signs. With the aid of neural networks parameterization, stable controllers have been constructed in [15] and [16] (to name just a few) and a family of novel integral Lyapunov functions have been developed in [17] and [18] to avoid controller singularity problem commonly encountered in adaptive feedback linearization control. In this note, we present an adaptive neural controller for a class of strict-feedback nonlinear systems with unknown time delays and unknown virtual control coefficients. Both integral functions and Lyaponov–Krasovskii functionals are used to construct the Lyapunov function candidates such that the controller singularity problem is avoided and the uncertainties from unknown time delays are removed. Semiglobally uniformly ultimate boundedness (SGUUB) of all the signals in the closed-loop system can be guaranteed and the control performance can be improved by appropriately choosing the design parameters. The proposed method expands the class of nonlinear systems that can be handled using adaptive control. The main contributions of the note lie in: 1) the use of integral Lyapunov functions to avoid controller singularity problem commonly encountered in feedback linearization control; 2) the introduction of appropriate Lyaponov–Krasovskii functionals to compensate for the unknown time delays for a time-delay independent controller design; 3) the introduction of practical robust control to avoid possible singularity problem due to the appearance of 1=zi in the controller; and 4) the use of neural networks as function approximators with its feasibility being guaranteed over the practical compact sets for the first time in the literature.
I. INTRODUCTION Recent years have witnessed great progress in adaptive control of nonlinear systems, which plays an important role due to its ability to compensate for parametric uncertainties. To obtain global stability, some restrictions had to be made to system nonlinearities such as matching conditions [1], extended matching conditions [2], or growth conditions [3]. To overcome these restrictions, recursive and systematic backstepping design was developed in [4]. The overparametrization problem was then removed in [5] by introducing the concept of tuning function. Several adaptive approaches for nonlinear systems with triangular structures have been proposed in [6], [7]. Robust adaptive backstepping control has been studied for uncertain nonlinear systems with unknown nonlinear functions in [8] and [9], and among others. Recently, stabilization of nonlinear systems with time delays is receiving much attention [10], [11]. The existence of time delays may make the task more complicated and challenging, especially when the delays are not perfectly known. One way to ensure stability robustness with respect to this uncertainty is to seek for delay-independent solutions. Lyapunov–Krasovskii functionals have been used early as checking criteria for time-delay systems’ stability in [12] and [13]. In
Manuscript received January 30, 2003; revised March 25, 2003. Recommended by Associate Editor P. Tomei. The authors are with Department of Electrical and Computer Engineering, National University of Singapore, 119260 Singapore (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.819287
II. PROBLEM FORMULATION AND PRELIMINARIES Consider a class of single-input–single-output (SISO) nonlinear time-delay systems x_ i (t) =gi ( xi (t))xi+1 (t) + fi ( xi (t)) + +hi ( xi (t
1 i n 0 1
xn (t))u + fn ( xn (t)) + hn ( xn (t x_ n (t) =gn (
0 i ))
0 n)) 2
(1)
where x i = [x1 ; x2 ; . . . ; xi ]T , x = [x1 ; x2 ; . . . ; xn ]T Rn and u R are the state variables and system input respectively, gi ( ), fi ( ), hi ( ) are unknown smooth functions, and i are unknown time delays of the states, i = 1; . . . ; n. The control objective is to design an adaptive controller for (1) such that the state x1 (t) follows a desired reference signal yd (t), while all signals in the closed-loop system are bounded. Define the desired trajectory x d(i+1) = [yd ; y_ d ; . . . ; yd(i) ]T , i = 1; . . . ; n 1, which is a vector of yd up to its ith time derivative (i) yd . We have the following assumptions for the system’s signals, un-
2 1
1
1
0
known functions and reference signals. Assumption 2.1: The system states x(t) and their partial time derivatives, x _ n01 (t), are all available for feedback. Remark 2.1: Note that the requirement for x _ n01 (t) is a constraint but realistic for many physical systems as we are not requiring x_ n which is directly influenced by the control. Assumption 2.2: The signs of gi ( xi ) are known, and there exist constants gi0 and known smooth functions gi ( xi ) such that gi0 jgi (xi )j gi (xi )1, 8xi 2 Ri .
0018-9286/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
Assumption 2.3: The desired trajectory vectors x di , i = 2; . . . ; n are continuous and available, and x di 2 di Ri with di known compact sets. Assumption 2.4: The unknown smooth functions hi ( xi (t)) satisfy the following inequality jhi ( xi (t))j ij =1 jxj (t)j%ij (xi (t)) where %ij (1) are known smooth functions. Assumption 2.5: The size of the unknown time delays is bounded by a known constant, i.e., i max , i = 1; . . . ; n. Remark 2.2: There are many physical processes which are governed by nonlinear differential equations of the form (1). Examples are recycled reactors, recycled storage tanks and cold rolling mills [19]. In general, most of the recycling processes inherit delays in their state equations. A function approximator shall be used to approximate the unknown nonlinear functions. Among the two types of artificial neural networks, i.e, 1) linearly parametrized neural networks (LPNNs) [18] and 2) multilayer neural networks (MNN’s), for simplicity of demonstrating the main idea, we shall use radial basis function (RBF) neural network (NN), a kind of LPNN, as an example in this note to approximate the continuous function h(Z) : Rq ! R as
hnn (Z) = W T S(Z)
(2)
where the input vector Z 2 Z Rq , weight vector W 2 Rl , and basis function vector S(Z) = [s1 (Z); s2 (Z); . . . ; sl (Z)]T 2 Rl , with l being the NN node number and si (Z) chosen as the commonly used Gaussian functions, which have the form si (Z) = exp[0(Z 0 i )T (Z 0 i )=i2 ], i = 1; . . . ; l where i = [i1 ; i2 ; . . . ; iq ]T is the center of the receptive field and i is the width of the Gaussian function. Universal approximation results in [20] and [21] indicate that, if l is chosen sufficiently large, W T S(Z) can approximate any continuous function to any desired accuracy over a compact set Z Rq to arbitrary any accuracy as
h(Z) = W 3T S(Z) + (Z); 8Z 2 Z Rq
(3)
where W 3 is the ideal constant weight vector, and (Z) is the approximation error which is bounded over the compact set, i.e., j(Z)j 3 , 8Z 2 Z3 where 3 > 0 is an unknown constant. The ideal weight vector W is an “artificial” quantity required for analytical purposes. W 3 is defined as the value of W that minimizes jj for all Z 2 Z Rq , i.e., W 3 := arg minW 2R supZ 2 jh(Z) 0 W T S(Z)j . The following even function pi (1) : R ! R is introduced for the purpose of the practical controller design in Section III:
pi (x) =
1; jxj cz 0; jxj < cz
8x 2 R:
(4)
III. ADAPTIVE NN CONTROLLER DESIGN In this section, adaptive neural control is proposed for system (1) and the stability results of the closed-loop system are presented. The backstepping design procedure contains n steps. The design of adaptive control laws is based on the following change of coordinates: z1 = x1 0 yd , zi = xi 0 i01 , i = 2; . . . ; n, where i (t) is an intermediate control which shall be developed for the corresponding ith subsystem based on an appropriate Lyapunov function Vi (t). The control law u(t) is designed in the last step to stabilize the whole closed-loop system based on the overall Lyapunov function Vn , which is the sum of the ^ i 0 Wi3 , ~i = W previous Vi (t), i = 1; . . . ; n 0 1. For clarity, let W 2 3 0 3 ^ constant ci := 1=2i Wi 0 Wi + 1=2z , where Wi 2 Rl is the estimate of ideal NN weight Wi3 2 Rl , Wi0 2 Rl is a design
2005
constant, 3z is the upper bound of the NN approximation error, i.e., ki (Zi )k 3z with Zi being the corresponding inputs to be defined later, and small constant i > 0 is to introduce -modification in adaptive control to be developed later. In this note, the following inequalities play an important role with i = 1; . . . ; n:
^ i 0 Wi3 i W
T
2
21 i W^ i 0 Wi3 0 21 i Wi3 0 Wi0 i (Zi )zi (t) 1 3z + 1 zi2 (t): 2 2
^ i 0 Wi0 W
2
(5) (6)
Define the integral Lyapunov function Vz (t) [17], [18], the Lyapunov–Krasovskii functional VU (t) with the positive scalar function Ui (1), and the Lyapunov function candidates Vi (t) as
Vz (t) = Vz (t) = VU (t) =
z
0 0
g10 1 ( + yd )d
z
01 (xi01 ; + i01 )d; gi
t
t0
(7)
i = 2; . . . ; n
Ui (xi ( ))d
(8) (9)
Vi (t) =Vi01 (t) + Vz (t) + VU (t) ~ T (t)0i01W ~ i (t) (10) + 1W 2 i 01 (xi ) = i (xi )=gi (xi ) with i (xi ) > 0 being a smooth where gi 01 )2 = i2 =gi2 , and weighting function to be defined later and (gi i 2 2 Ui (xi (t)) = 1=2 j =1 xj (t)%ij (xi (t)), i = 1; . . . ; n. The choice of weighting function i (1) plays a key role in the controller design
and results in different controllers with varying control performance [18]. The apparent and convenient choices for i (1) are 1 and gi ( xi ) for general nonlinear systems. Detailed explanations will be given based on i ( xi ) = gi (xi ) in the following, while a remark will be given directly addressing the controller design and relevant stability and performance analysis for i ( xi ) = 1 without derivation for conciseness. 01 (xi ) = gi (xi )=gi (xi ). By choosing i ( xi ) = gi (xi ), we have gi 01 (xi ) are bounded by known From Assumption 2.2, we know that gi 01 (xi ) gi (xi )=gi0 . Clearly, Vz are positive–deffunctions as 1 < gi inite functions, i = 1; . . . ; n. Step 1) Let us first consider the z1 -subsystem as
z_1 (t) = g1 (x1 (t))(z2 (t) + 1 (t)) + f1 (x1 (t)) +h1 (x1 (t 0 1 )) 0 y_d (t):
(11)
By variable change = z1 , we may rewrite Vz in (7) as Vz = z12 01 g10 1 (z1 +yd )d . Noting that 1 g10 1 (z1 +yd ) g1 (z1 + yd )=g10 , we have
z12 2
1
2
Vz gz101
0
g1 (z1 + yd )d:
The time derivative of Vz along (11) is
V_ z (t) =z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t) + g10 1 (x1 (t))f1 (x1 (t)) + g10 1 (x1 (t))h1 (x1 (t 0 1 )) 1 0 y_d (t) g10 1 (z1 + yd )d : 0
(12)
2006
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
The remaining term U1 (x1 (t)) from V_ U (t) is a known function of known variables, which does not introduce any uncertainties to the system. Therefore, the design of intermediate control 1 (t) is free from unknown time delay 1 at present stage. For notation conciseness, we will omit the time variables t and t 0 1 after time-delay terms have been eliminated. Under the assumption of exact knowledge, the certainty equivalent control is
Applying Assumption 2.4, we have
V_ z (t) z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t)
+ g10 1 (x1 (t))f1 (x1 (t)) 1
0 y_d
0 0 g1 1
g10 1 (z1 + yd )d
+ jz1 (t)j (x1 (t))jx1 (t 0 1 )j 2 %1 (x1 (t 0 1)):
13 = (13)
V_ z (t) z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t)
+ g10 1 (x1 (t))f1 (x1 (t))
0 y_d
0
+ 21 z12 (t) g10 1 2 + 21 x21 (t 0 1 ) 2 %21 (x1 (t 0 1)): (14) In standard iterative backstepping design, 1 (t) is usually designed to stabilize the z1 -subsystem except for the coupling term g1 z1 z2 to be dealt with in the next step. In doing so under the assumption of known functions, one more difficulty exists in the new problem setting. Although %1 (1) is a known function, it cannot be utilized in designing 1 (t) as x1 (t 0 1 ) is undetermined because of unknown time delay 1 . Intuitively, approximation methods such as neural networks can be used to approximate the unknown functions. The unknown functions g1 (1) and f1 (1) can be dealt with in this way without any problem. However, due to the existence of the unknown time delay 1 , functions of x1 (t 0 1 ) are hard to be approximated using neural networks since the input x1 (t 0 1 ) is undetermined because of the uncertain 1 . To overcome the design difficulties from the unknown time delay 1 , we consider the Lyapunov–Krasovskii functional VU (t) in (9) with its time derivative being (15)
which can be used to cancel the time-delay term on the right-hand side of (14) and thus eliminate the design difficulty from the unknown time delay 1 without introducing any uncertainties to the system. Accordingly, we obtain
V_ z (t) + V_ U (t) z1 (t) g1 (x1 (t))z2 (t) + g1 (x1 (t)) 1 (t)
+ g10 1 (x1 (t))f1 (x1 (t)) 1 0 y_d g10 1 (z1 + yd )d 0 1 + 2 z1 (t) g10 1 2 + 2z 1(t) x21 (t)%12(x1 (t)) : 1
where 1 0
g10 1 (z1 + yd )d
+ 12 z1 g10 1
2
Z1 = [x1 ; yd ; y_d ]T 2 Z R3 and Z := fz1 ; xd2 jz1 2 R; xd2 2 d2 g. It is apparent that controller singularity may occur. In addition, it is certain that 31 is not an admissible control, since 31 is not well-defined when z1 = 0 as limz !0 2z1 = 0, limz !0 x21 %21 (x1 ) 6= 0 and L’Hopital’s rule is not applicable to obtain the limit limz !0 (x12 %21 (x1 )=(2z1 )). Therefore, care must be with
g10 1 (z1 + yd )d
V_ U (t) = U1 (x1 (t)) 0 U1 (x1 (t 0 1 ))
0k1 (t)z1 0 Q1 (Z1 ) 0 21z1 x21 %21 (x1 )
Q1 (Z1 ) = g10 1 (x1 )f1 (x1 ) 0 y_ d
By using Young’s Inequality, (13) becomes
1
1
g1 (x1 )
(16)
Comparing (16) with (14), it is found that the difficulty from the unknown time delay 1 has been eliminated by introducing the Lyapunov–Krasovskii functional VU (t). By differentiating VU (t) with respect to time, the unknown time delay term U1 (x1 (t 0 1 )) = 1=2x12 (t 0 1 )%21 (x1 (t 0 1 )) appeared in (15) can be used for exact cancellation on the right-hand side of (14).
taken to guarantee the boundedness of the controller. It is noted that point z1 = 0 is not only an isolated point in Z , but also the case that the system reaches the origin at this point. From a practical point of view, once the system reaches its origin, no control action should be taken for less power consumption. For ease of discussion, let us define sets c Z and 0Z as follows:
c :=fz1 j jz1 j < cz g
0Z := Z 0 c
(17) (18)
where cz is a constant that can be chosen arbitrarily small and “ 0” in (18) is used to denote the complement of set B in set A as A 0 B := fxjx 2 A and x 2= Bg. Accordingly, the following practical control law is proposed:
31 = p1 (z1 ) g1 (x1 )
0k1 (t)z1 0 Q1 (Z1 ) 0 21z1 x21 %21 (x1)
(19)
where pi (1) is defined in (4). Since f (1) and g (1) are unknown smooth functions, the desired practical control 31 in (19) cannot be implemented in practice. Neural networks can be used to approximate the unknown function Q1 (Z1 ). Note that control action is only activated when z1 2 0Z , which means unknown function Q1 (Z1 ) is approximated by neural networks over the set 0Z . According to the main result stated in [22], any real-valued continuous function can be arbitrarily closely approximated by a network of RBF type over a compact set. The compactness of set 0Z is a must to guarantee the feasibility of neural networks approximation, which is shown in the following lemma. Lemma 3.1: Set 0Z is a compact set. Proof: First, we show that 0Z is a closed set. From (18) and applying De Morgan’s laws, we have
Z0c = cZ [ c
(20)
0c where Z and cZ denote the complements of 0Z and Z respectively. Since Z is a compact set, i.e., it is closed and bounded, cZ is an open set. In addition, c is also an open set from its defini0c tion. From (20), we know that Z is an open set, which means that 0 its complement Z is a closed set. Second, from (18), we know that
0Z Z . Since a closed subset of a compact set is compact, we can conclude that 0Z is a compact set.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
Based on Lemma 3.1, it is known that Q1 (Z1 ) is continuous and well-defined over compact set 0Z and can be approximated by neural networks to arbitrary any accuracy as follows Q1 (Z1 ) = W13 T S(Z1 ) + 1 (Z1 ) where 1 (Z1 ) is the approximation error. As the ideal weight W13 is unknown, we shall use its estimate ^ 1 instead, which forms the intermediate control 1 as W
1 (t) = p1 (z1 ) g1 (x1 )
0k1 (t)z1 0 W^ 1T S(Z1 ) 0 2z11 x21 %21 (x1 ) : Z
For uniformity of notation, we define sets c 2; . . . ; n as
c := fzi j jzi j < cz
0Z := Z 0 c :
(21)
and 0Z , i
g
=
(22) (23)
Note that the control objective is to show that certain compact set S is domain of attraction in the sense that for all bounded initial conditions, there exists S such that all closed-loop signals will eventually converge to S . i.e., all Zi (t) starting from within 0Z will enter into
S and will stay within S thereafter. In the following steps, the unknown functions Qi (Zi ), i = 2; . . . ; n will be approximated by neural networks as
Qi (Zi ) = Wi3 T S(Zi ) + i (Zi )
8Zi 2 0Z :
(24)
Consider the Lyapunov function candidate V1 (t) in (10), whose time derivative along (16), (21) and (24) for z1 2 0Z is
^ 1 0 W13 V_ 1 0k1 (t)z12 + g1 (x1 )z1 z2 0 W
(25)
(26)
For z1
2 0z
+g1 (x1 )z1 z2 + c1 :
with 0
< "10
V_ 1 g1 (x1 )z1 z2 0 g10 Vz "10
0 maxt; t], we have the inequality t0 1=2x12( )%21 (x1 ( ))d . 0 "110 VU
2
0 12 1 W^ 1 0 W13 + c1
where the coupling term g1 (x1 )z1 z2 will be handled in the next step. Remark 3.1: Applying Young’s inequality, we know that g; z1 ; z2 1=2 z12 + 1=2 g 21 z22 . The choice for "10 is to guarantee that 0(1="10 0 1=2)z12 0, so that the undesired destabilizing term 1=2 z12 can be suppressed. Step i (2 i n 0 1)) Similar procedures are taken for i = 2; . . . ; n 0 1 as in Step 1). The dynamics of zi -subsystem is given by
z_i (t) = gi (xi (t))[zi+1 (t) + i (t)] + fi (xi (t)) +hi (xi (t 0 i )) 0 _ i01 (t): The time derivative of Vz (t) in (8) is given by V_ z (t) = zi (t) gi (xi (t))(zi+1 (t) + i (t)) 01 (xi (t))fi (xi (t)) + gi 01 (xi (t))hi (xi (t 0 i )) + gi 1 @g01 (xi01 ; zi + i01 ) + x_ i01 zi (t) i d @xi01 0 1 0 _ i01 g01 (xi01 ; zi + i01 )d : 0
i
01 (xi (t))fi (xi (t)) + 1 zi (t)(g01)2 + gi i 2 01 (xi01 ; zi + i01 ) 1 @gi + x_ i01 zi (t) d @xi01 0 1 0 _ i01 g01 (xi01 ; zi + i01 )d
In Step i 0 1), for zj
(27)
(28)
2, we have
V_ 1 0 1 z12 0 g10 Vz "10 "10 t 1 1 x2 ( )%2 (x ( ))d 0 "10 1 1 1 t0 2 2 0 12 1 W^ 1 0 W13 + g1 (x1 )z1 z2 + c1 :
Accordingly, (29) becomes
i
, noting (12), and choosing
1 k1 (t) = 1 1 + g1 (z1 + yd )d "10 0 t 1 x2 ( )%2 (x ( ))d + 12 1 1 z1 t0 2 1
[t 0 1 ; t] [t 1=2x12 ( )%21 (x1 ( ))d
0
i
+ 1 x2j (t 0 i )%2ij (xi (t 0 i )): 2 j =1
Substituting (26) into (25), and using (5) and (6), we have
^ 1 0 W13 2 V_ 1 0k1 (t)z12 0 1 1 W 2
t t0
V_ z (t) =zi (t) gi (xi (t))(zi+1 (t) + i (t))
The following practical adaptive law is given for online tuning the NN weights
^_ 1 = p1 (z1 )01 S(Z1 )z1 0 1 W ^ 1 0 W10 : W
Since
Noting Assumption 2.4, we have
T
S(Z1 )z1 ^_ 1 : ^ 1 0 W13 T 001 1 W +z1 z + W
2007
(29)
(30)
2 0Z , j = 1; . . . ; i 0 1, it has been obtained that
V_ i01 gi01 (xi01 )zi01 zi i01 + 0 g"jj00 Vz 0 "1j0 VU j =1 2 0 21 j W^ j 0 Wj3 + cj : (31) Consider Vi (t) given in (10). For zj 2 0Z , j = 1; . . . ; i, its time derivative along (30) and (31) is
V_ i gi (xi )zi zi+1 + zi [gi01 (xi01 )zi01 + gi (xi ) i + Qi (Zi )] ^_ i ^ i 0 Wi3 )T 0i01W + (W i01 + 0 g"jj00 Vz 0 "1j0 VU j =1 2 0 21 j W^ j 0 Wj3 + cj
(32)
2008
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
where
In Step n 0 1), for zi that
01 (xi )fi (xi ) + 1 zi (g01)2 Qi (Zi ) =gi 2 i 01 (xi01 ; zi + i01 ) 1 @gi + x_ i01 zi d @xi01 0 1 0 _ i01 g01 (xi01 ; zi + i01 )d 0
Zi (t) = [xi ; x_ i01 ; i01 ; @ i01 =@x1 ; @ i01 =@x2 ; . . . ; @ i01 =@xi01 ; !i01] 2 0Z R3i , where i01
@ i01 x_ + ! ; ! i01 i01 @xj j j =1 i01 @ i01 W ^_ j : = @ i01 x_ di + ^ @xdi W @ j j =1
_ i01 =
V_ n01 gn01 (xn01 )zn01 zn n01 + 0 g"jj00 Vz 0 "1j0 VU j =1 0 21 j kW^ j 0 Wj3 k2 + cj :
i
with
Consider Vn (t) given in (10). For jzi j derivative along (36) and (37) is
0gi01 (xi01 )zi01 0 ki (t)zi 0 W^ iT S(Zi ) i 0 2z1 i x2j %2ij (xi ) j =1
^_ i = pi (zi )0i S(Zi )zi 0 i W ^ i 0 Wi0 W
01 (xn )fn (xn ) + 1 zn g01 2 Qn (Zn ) =gn n 2 01 (xn01 ; zn + n01 ) 1 @gn + x_ n01 zn d @xn01 0 1 0 _ n01 g01 (xn01 ; zn + n01 )d
(33) (34)
1 i x2 ( )%2 (x ( ))d ; z 2 0 i ij i Z 2 j =1 j (35)
with 0 < "i0 2. Substituting (33)–(35) into (32), and using (5), (6), and (12), we have
V_ i gi (xi )zi zi+1 +
i
j =1
2 0 g"jj00 Vz 0 "1j0 VU 0 21 j W^ j 0 Wj3 + cj
where the coupling term gi ( xi )zi zi+1 will be handled in the next step. Step n) This is the final step, since the actual control u appears in the dynamics of zn -subsystem as given by
z_n = gn (xn (t))u + fn (xn (t)) + hn (xn (t 0 n )) 0 _ n01 (t): Consider Vz tive is
(t) given in (8). Noting Assumption 2.4, its time deriva-
01 (xn (t))fn (xn (t)) V_ z (t) =zn (t) gn (xn (t))u(t) + gn
n
n
Zn (t) = [xn ; x_ n01 ; n01 ; @ n01 =@x1 ; @ n01 =@x2 ; . . . ; @ n01 =@xn01 ; !n01 ] 2 0Z R3n , where n01 @ n01 x_ + ! _ n01 = n01 @xj j j =1 n01 @ n01 W ^_ j : !n01 = @ n01 x_ dn + ^ @xdn @ W j j =1
with
We construct the following adaptive neural control law:
u(t) = pn (zn ) gn (xn )
0gn01 (xn01)zn01 0 kn (t)zn 0 W^ nT S(Zn ) 0 2z1n
n j =1
x2j %2nj (xn ) (39)
^_ n = pn (zn )0i S(Zn )zn 0 i W ^ n 0 Wn0 W
(40)
1 kn (t) = 1 1 + gn (xn01 ; zn + n01 )d "n0 0
1 n x2 ( )%2 (x ( ))d ; z 2 0 n nj n Z 2 j =1 j (41)
with 0 < "n0 2. Substituting (39)–(41) into (38), and using (5), (6), and (12), we have
V_ n (t)
n
+ 1 x2j (t 0 n )%2nj (xn (t 0 n )): 2 j =1
0
t + 12 zn t0
01)2 + x_ n01 zn (t) + 1 zn (t)(gn 2 01 1 2 @gn (xn0@1x; nz0n1 + n01 ) d 0 1 0 _ n01 g01 (xn01 ; zn + n01 )d 0
(38)
where
1 ki (t) = 1 1 + gi (xi01 ; zi + i01 )d "i0 0 t + 12 zi t0
(37)
2 0Z , i = 1; . . . ; n, its time
V_ i gi (xi )zi zi+1 + zi [gi01 (xi01 )zi01 + gi (xi ) i + Qi (Zi )] ^_ i ^ i 0 Wi3 )T 0i01W + (W i01 + 0 g"jj00 Vz 0 "1j0 VU j =1 2 0 21 j W^ j 0 Wj3 + cj
Similarly, we have the following intermediate control law:
i = pi (zi ) gi (xi )
2 0Z , i = 1; . . . ; n 0 1, it has been obtained
(36)
n j =1
0 g"jj00 Vz 0 "1j0 VU
2
0 12 j W^ j 0 Wj3 + cj :
(42)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
Theorem 3.1: Consider the closed-loop system consisting of the plant (1) under Assumptions 2.1–2.5, the controller (39) and the NN weight updating law (40). For bounded initial conditions, the following properties hold. i) All signals in the closed-loop system remain SGUUB and the vector Z = [Z1T ; . . . ; ZnT ]T remains in a compact set 0Z :=
0Z [ 1 1 1 [ 0Z specified as n
0Z = Z j
i=1
n
zi2 2C0 ;
i=1
2009
Case 3) Some zi 2 0Z and some zj 2 c . In this case, the ^ i will corresponding i or u and the adaptation law for W 0 be invoked for zi 2 Z while j = 0 or u = 0 and
^_ j = 0 for zj 2 c . Let us define VI (t) = i (Vz + W ~ iT 0i01W ~ i ) and VJ (t) = j (Vz + VU + VU + 1=2W 0 1 T ~ j 0j W ~ j ). For zj 2 c , we obtain that VJ (t) 1=2W is bounded, i.e., VJ (t) CJ with CJ being finite, and zi 2 0Z , we have that V_ I (t) 0C1I VI (t) + C2I , i.e.,
kW~ i k 2C(00 ) ; i 0
2
min
VI (t) [VI (0) 0 I ]e0C t + I
1
xdi 2 di ; i = 2; . . . ; n; zi 2= c ; i = 1; . . . ; n
(43)
I
C2
n i=1
~ T 001 W ~ Vz (t) + VU (t) + 1 W 2 i i i
(45)
where Vz (t) and VU (t) are defined in (8) and (9), respectively, and (~1) = (^1) 0 (1). The following three cases are considered. Case 1) zi 2 c , i = 1; . . . ; n. In this case, the controls i = 0, ^_ i = 0, i = 1; . . . ; n. i = 1; . . . ; n 0 1, u = 0 and W Since z1 = x1 0 yd and yd is bounded, x1 is bounded. For i = 2; . . . ; n, xi is bounded as xi = zi + i01 and ^ i is kept unchanged in a bounded i01 = 0. In addition, W value, i = 1; . . . ; n. Observing the definition for Vz (t) and VU (t), and noting that gi (1), %ij (1) are smooth functions, ^ i , Vz (t) and VU (t) we know that for bounded xi , zi and W are bounded, i.e., there exists a finite CB such that
Vn (t) CB : Case 2) zi
(46)
2 Z , i = 1; . . . ; n. From n(42), we have V_n (t) 0
0C Vn(t) + C 1
2
where C1
=
i=1 ci and
C2 = min g10 ; . . . ; gn0 ; 1 ; . . . ; "10 "n0 "10 1 n 1 ; ;...; "n0 min 0101 min 0n01 = C2 =C1 , it follows that 0 Vn (t) [Vn (0) 0 ]e0C t + Vn (0) +
:
Let
(47)
where constant
Vn (0) =
n i=1
z 0
(0)
fore, it can be obtained that
01 (xi01 (0); + i01 (0))d gi
~ T (0)0i01W ~ i (0) +1W 2 i 01 (xi01 (0); + i01 (0)) = g01 () for i = 1. with gi 1
(48) i ci and
g.
There-
Vn (t) = VI (t) + VJ (t) VI (0) + I + CJ :
(49)
Thus, from (46), (47), and (49) for Cases 1)–3), we can conclude that
(44)
where > 0 is a constant related to the design parameters and will be defined later in the proof, and S can be made as small as desired by an appropriate choice of the design parameters. Proof: Consider the following Lyapunov function candidate:
Vn (t) =
I = C2I =C1I with C1I = = mini fgi0 ="i0 ; 1="i0 ; i =min 0i01
where
where C0 > 0 is a constant whose size depends on the initial conditions (as will be defined later in the proof). ii) The closed-loop signal z(t) = [z1 ; . . . ; zn ]T 2 Rn will eventually converge to a compact set defined by
S := fz j kz k2 g
VI (0) + I
Vn (t) C0
(50)
where C0 = maxfCB ; Vn (0) + ; VI (0) + I + CJ g. ^ i , i = 1; . . . ; n, From (50), we know that Vn (t), zi and W are bounded. Since z1 = x1 0 yd and yd is bounded, x1 is bounded. For x2 = z2 + 1 , since 1 is function of bounded ^ 1 , 1 is thus bounded, which in turn leads signals z1 , Z1 , W to the boundedness of x2 . Following the same way, we can prove one by one that all i01 and xi , i = 3; . . . ; n are bounded. Therefore, the systems’ states xi , i = 1; . . . ; n are bounded. Considering (45) and the property for Vz (t) that
1 z 2 V (t) zi2 z 2 i gi0
1 0
gi (xi01 ; zi + i01 )d
we know that n i=1
zi2 2
n i=1
n
Vz (t) 2Vn (t)
i=1
kW~ i k 2Vn0(t)0 : i 2
min
1
(51) From (50) and (51), we readily have the compact set 0Z defined in (43) over which the NN approximation is carried out with its feasibility being guaranteed. In addition, in Case 1), as zi 2 c , i = 1; . . . n, we know that kzk2 = ni=1 zi2 ni=1 cz2 . In Case 2), from (47) and (51), we have that limt!1 kz k2 = 2. In Case 3), from (48) and (51), we have that limt!1 i zi2 = 2I and j zj2 j cz2 . Therefore, as t ! 1, 2 we can conclude that kz k2 where = max 2; 2I ; n i=1 cz , i.e., the vector z will eventually converge to the compact set S defined in (44). This completes the proof. Remark 3.1: Note that the choices of i ( xi ) are not unique. By 01 (xi ) = 1=gi (xi ) [17] and Vz = choosing i ( xi ) = 1, we have gi z =gi (xi01 ; + i01 )d , i = 1; . . . ; n. By the mean value the0 orem, Vz can be rewritten as Vz = s zi2 =gi ( xi01 ; s zi + i01 ), s 2 (0; 1). From Assumption 2.2, 0 gi0 gi (xi ), we know that 0 < Vz s =g10 zi2 . The adaptive control laws are given by
i = pi (zi ) gi (xi )
0gi0 (xi0 )zi0 0 ki (t)zi 0 W^ iT S(Zi ) 1
0 2z1 i
1
i j =1
1
x2j %2ij (xi )
2010
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003
u=
pn (zn ) gn (xn )
0gn01 (xn01)zn01 0 kn (t)zn 0 W^ nT S (Zn ) 0 2z1n
n j =1
x2j %2nj (xn )
^_ i = pi (zi )0i S (Zi )zi 0 i W ^ i 0 Wi0 W ki (t) =
1 1+ + 1 s 2
"i0
zi
t
0
t
1 i x2 ( )%2 (x ( ))d ij i 2 j =1 j
where 0 < "i0 2. For bounded initial conditions, all closed-loop signals remain bounded and the tracking error converges to a small neighborhood around zero by appropriately choosing design parameters. Remark 3.2: Note that the size of the compact set 0Z is characterized by C0 , which depends on system initial conditions xi (0) and ^ i (0) as well as the design parameters i , 0i , Wi0 and "i0 , i = W 1; . . . ; n. For the compact set S to which the closed-loop signals eventually converge, its size only depends on the design parameters. ~ i (0), Therefore, it can be seen that large initial errors zi (0) and W i = 1; . . . ; n may lead to a large transient tracking error during the initial period of adaptation, but will not affect the final convergence of the closed-loop signals. Remark 3.3: Since the function approximation property (3) of neural networks is only guaranteed within a compact set, the stability result proposed is semiglobal in the following sense: Given ~ i (0) 2 I , the any bounded initial compact set such that zi (0); W proposed NN controller with sufficiently large number of nodes guarantees that all the closed-loop signals will stay within the compact set, i.e., 0Z in the note, if the compact set 0Zc , over which the neural network approximation is constructed, satisfies that 0Z 0Zc , and eventually all the closed-loop signals will converge to the steady state compact set, i.e., S in the note. The relationships among the sets are as: I ; S 0Z 0Zc . It is apparent that the larger the compact set 0Zc over which the NN controller is built upon, the more relaxed the initial compact set I is. IV. CONCLUSION An adaptive neural-based control has been addressed for a class of strict-feedback nonlinear systems with unknown time delays. The unknown time delays has been compensated for through the use of appropriate Lyapunov–Krasovskii functionals. As a result, the iterative backstepping design can be carried out. In addition, the controller is free from singularity problem by using the integral Lyapunov function and practical robust neural network control. The proposed systematic backstepping design method has been proven to be able to guarantee SGUUB of closed-loop signals and the output of the system has been proven to converge to an arbitrarily small neighborhood of the origin. REFERENCES [1] D. G. Taylor, P. V. Kokotovic´, R. Marino, and I. Kanellakopoulos, “Adaptive regulation of nonlinear systems with unmodeled dynamics,” IEEE Trans. Automat. Contr., vol. 34, pp. 405–412, Apr. 1989. [2] I. Kanellakopoulos, P. V. Kokotovic´, and R. Marino, “An extended direct scheme for robust adaptive nonlinear control,” Automatica, vol. 27, no. 2, pp. 247–255, 1991. [3] S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 1123–1131, Nov. 1989. [4] I. Kanellakipoulos, P. V. Kokotovic´, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, Nov. 1991. [5] M. Krstic´, I. Kanellakopoulos, and P. V. Kokotovic´, “Adaptive nonlinear control without overparsmetrization,” Syst. Control Lett., vol. 19, no. 3, pp. 177–185, 1992.
[6] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of nonlinear systems with a triangular structure,” IEEE Trans. Automat. Contr., vol. 39, pp. 1411–1428, July 1994. [7] S. S. Ge, C. C. Hang, and T. Zhang, “Stable adaptive control for nonlinear multivariable systems with a triangular control structure,” IEEE Trans. Automat. Contr., vol. 45, pp. 1221–1225, June 2000. [8] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 32, no. 3, pp. 423–427, 1996. [9] Z. Pan and T. Basar, “Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 1066–1083, Aug. 1998. [10] A. M. Annaswamy, S. Evesque, S. Niculescu, and A. P. Dowling, “Adaptive control of a class of tinse-delay systems in the presence of saturation,” in Adaptive Control of Nonsmooth Dynamic Systems, G. Tao and F. Lewis, Eds. New York: Springer-Verlag, 2001, pp. 289–310. [11] F.-H. Hsiao and J.-D. Hwang, “Stabilization of nonlinear singularly perturbed multiple time-delay systems by dither,” J. Dyna. Syst., Measure., Control, vol. 118, no. 1, pp. 176–181, 1996. [12] J. Hale, Theory of Functional Differential Equations, 2nd ed. New York: Springer-Verlag, 1977. [13] S. K. Nguang, “Robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 756–762, Apr. 2000. [14] S. Zhou, G. Feng, and S. K. Nguang, “Comments on robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 1586–1586, Sept. 2002. [15] F. L. Lewis, S. Jagannathan, and A. Yesilidrek, Nerual Network Control of Robot Manipulators and Nonlinear Systems. Philadelphia, PA: Taylor & Francis, 1999. [16] K. S. Narendra and K. Parthasanithy, “Identification and control of dynamic systems using neural networks,” IEEE Trans. Neural Networks, vol. 1, pp. 1–23, Feb. 1990. [17] S. S. Ge, C. C. Hang, and T. Zhang, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, no. 4, pp. 74l–747, 1999. [18] S. S. Ge, C. C. Hang, T. H. Lee, and I. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2002. [19] S.-L Niculescu, Delay Effects on Stability: A Robust Control Approach. London, U.K.: Springer-Verlag, 2001. [20] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Networks, vol. 3, pp. 837–863, Dec. 1992. [21] E. B. Kosmatopoulos, M. M. Polycarpou, M. A. Christodoulou, and P. A. Ioannou, “High-order neural network structures for identification of dynamical systems,” IEEE Trans. Neural Networks, vol. 6, pp. 422–431, Apr. 1995. [22] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, no. 5, pp. 359–366, 1989.
