Neural Comput & Applic (1995)3:157-163 ~) 1995 Springer-Verlag London Limited Neural Computing & Applications A Robust Adaptive Tracking Controller Using Neural Networks for a Class of Nonlinear Systems Man Zhihong Department of Computer and Communication Engineering, Edith Cowan University, WA, Australia A neural network-based robust adaptive tracking control scheme is proposed for a class of nonlinear systems. It is shown that, unlike most neural control schemes using the back-propagation training technique, one hidden layer neural controller is designed in the Lyapunov sense, and the parameters of the neural controller are then adaptively adjusted for the compensation of unknown dynamics and nonlinearities. Using this scheme, not only strong robustness with respect to unknown dynamics and nonlinearities can be obtained, but also asymptotic error convergence between the plant output and the reference model output can be guaranteed. A simulation example based on a one-link non-linear robotic manipulator is given in support of the proposed neural control scheme. Keywords: Asymptotic error convergence; Lyapunov stability; Neural controller; Nonlinearities; Robustness; Uncertain dynamics 1. Introduction In recent years, neural network techniques have been used widely in control engineering. The most useful property of neural networks in control is their ability to approximate arbitrary nonlinear mapping. It is because of the above property that Received for publication 27 April 1994 Correspondence and offprint requests to: Man Zhihong, Department of Computer and Communication Engineering, Edith Cowan University, WA 6027, Australia. many neural network-based controllers have been developed for the compensation of nonlinearities and system uncertainties so that good system performance can be achieved. Although many simulations and experiments have shown that, by selecting a suitable neural network structure and training neural network parameters, good performance can be obtained for neural network-based control systems, many theoretical and practical problems have not been solved. For example, stability and robustness in many neural network-based control systems have not been proved. The relationship between plants and network structures (i.e. number of layers, number of nodes, and types of nonlinear activation functions) still needs further investigation [1]. The recent development in [2] using a Gaussian network for direct adaptive control has made considerable progress in solving the above problems. For example, asymptotic error convergence can be guaranteed within a boundary layer, and the number of nodes is determined by the number of sampling lattices on the approximation area. However, in some practical situations where signals have a wide bandwidth, the sampling interval will be very small, and therefore the number of nodes will be very large. In particular, if the error spectrum is beyond the assumed signal frequency bandwidth, good system performance cannot be guaranteed and the controller design needs to be reconsidered. In this paper, we propose a new robust tracking controller using a general neural network for a class of nonlinear systems considered in [2]. It is shown that a single hidden layer neural network with hard line nonlinear activation functions is used as a controller; the number of nodes is determined only M. Zhihong 158 by the number of input variables of the neural controller. Unlike most neural network-based controllers, the neural network in this scheme is not used directly to approximate the uncertain nonlinearity, but it is a general robust adaptive controller in a Lyapunov sense, i.e. the network parameters are adaptively adjusted based on the Lyapunov stability theory such that the output tracking error between the system output and the model output can asymptotically converge to zero. In addition, because the sliding mode technique is used in the neural controller design, strong robustness and fast error convergence can be guaranteed [3-6, 9-14]. The Lyapunov stability theory has been widely used by scientists and engineers in the area of control systems since the 1960s. The popularity of the Lyapunov stability theory is that it provides a powerful tool for the analysis and design of control systems. For example, Lyapunov functions can be used to judge the stability of linear or nonlinear systems, and Lyapunov controllers can guarantee the stability of closed loop control systems or good tracking performance of the model reference control systems. On the other hand, Lyapunov stability theory can be treated as an optimisation technique because a Lyapunov function is a scalar 'energy-like' function. If the controller is designed such that the derivative of the Lyapunov function is negative, the 'energy' will converge to zero according to the Lyapunov stability theory. This can be seen in a model reference control system, where a Lyapunov controller is designed to guarantee that the output tracking error between the plant and its model converges to zero, which means that the output tracking error is minimised by the use of the Lyapunov stability theory. It has already been seen [2, 8] that Lyapunov stability theory is playing a very important role in the design of adaptive neural control systems. This paper together, with [2] and [8], are pioneering research in this area. The paper is organized as follows. In Section 2, the system model and proposed neural network controller are formulated. In Section 3, controller parameter design, stability and error convergence analysis are discussed in detail. In Section 4, a simulation example using a one-link rigid robotic manipulator is presented in support of the proposed neural control scheme. Section 5 draws conclusions. for a class of single input and single output nonlinear systems whose dynamical equations can be expressed in the following form: x('O(t) + f(x(t), k ( t ) , . . . x(n--1)(t)) = b u(t) (1) where t is the time, x(t) is the output variable, x (i) (i = 1, . . . , n) is the ith derivative of x(t), u(t) is the control input, f ( x , k , . . . x (n-l)) is an unknown nonlinear function, and b is an unknown control gain. For general consideration, the followng assumptions are made: AI The nonlinear function f ( x , ~, . . . upper bounded: I f(x ' ~,. [ ( fo 9 9 x(n-1)) (2) where fo is a positive number. A2 The sign of control gain b is known (b > 0), and it is upper and lower bounded: bl < b < b,,, (3) where bl and b,,, are positive numbers. Expression (2.1) can also be expressed into the following state equation: (4) 2 = AX + Bu + F where X = [x, k, . . .x("-l)] r 0 1 0 ... 0 A = 0 B = O... . . . 0 b] T F = [0.. "O-fF The desired reference model for system (4) to follow is given by: (5) J(m = A , , X m + Bmr where X m = [Xm Xm X(~-I)] T, A,,, and B,, are known constant matrices, and r(t) is a reference input. Defining an output tracking error vector: 9 e(t) = X - " 9 Xm = [el.. 9 en] T = [E(0 E ( 0 " " " I~(0(n--1)lT 2. Problem Formulation In this paper, we focus on the design of a neural network-based robust adaptive tracking controller x (n-I)) is (6) where e (k) = x (~) - x ~ ), k = 0 . . . n - 1 The dynamics of the output tracking error can then be obtained using expressions (4) and (5) as follows: 159 A Robust Adaptive Tracking Controller = Ae + (A-Am)Sm (7) + F - Bmr + B u For further analysis, a filtered output tracking error is defined as follows: s = C e I 2n+2 (8) where C = [Cl, 9 9 cn] is chosen such that zeros of the polynomial Ce = 0 are in the left half of the complex plane. Usually, s is called the switching plane variable and C e = 0 is called the sliding mode in variable structure control [3-6, 9-14]. The control input u(t) in this paper is generated by the output of a three layered neural network shown in Fig. 1. It can be seen from Fig. 1 that the first and third layers of the neural controller have linear nodes which are able to scale and shift the incoming signals [8], and in the second layer (hidden layer), nonlinear nodes with the following nonlinear activation functions are used (see Fig. 2): gi(x) = 2n+2 for l~=1bt/yt -> ~i [sign(x) Ixl--- t [Xl < ~/ (9) 8i Based on the network structure in Fig. 1 and nonlinear activation function in expression (9) or Fig. 2, the control input of the plant is obtained in the following form: U= i~=lait~}= 2n+2 for ~=1 bt~vt < ~i 2n+2 U= where / 2n+2 \ i=l~a'tS:l~,b'<~Y") [Yl " " 9 y.]7" [Yn+l 9 = e Y2n] T = Xm Y2n+l = r Y2n+2 = fo Remark 1 The nonlinear activation function gi(x) in expression (9) is the approximation of the sigmoid activation function ( ( 1 / ( l + e x p ( - a x ) ) - 1/2). It will be shown in the next section that it is convenient to use nonlinear nodes with an activation function in expression (9) in the hidden layer for the robust tracking controller design and stability analysis. Remark 2 As shown in Fig. 2, fo, the upper bound of nonlinearity f(.), is one of the input variables of the neural controller. Except for fo, no other prior knowledge on the nonlinearity f(.) is used in the proposed neural controller design. ea • ml Remark 3 It can be seen from Fig. 1 that the number of nodes in the first and second layers depends only upon the number of input variables y~ ( k = l , . . . , n+2). Therefore, the structure of the proposed neural controller is simple. The objective of this paper is to use the neural controller shown in Fig. 1 with a suitable network parameter design to control a class of nonlinear systems in expression (1) so that the output tracking error between the plant and its reference model can asymptotically converge to zero, and strong robustness with respect to uncertain dynamics can be guaranteed. u Xma fo b 2a+2,2n+2 Fig. 1. The neural network-based controller. ~ gl(x) 3. Neural Controller Design -8i 1~ 8t Fig. 2. Nonlinear activation function gi(x). x For the controller parameter design, stability and convergence analysis, we have the following theorem: Theorem Consider the error dynamics in 160 M. Zhihong expression (7) with assumptions A1 and A2. If parameters of the neural controller shown in Fig. 1 are designed such that ICk-llsign(yk) (2n+2)btc~ [c'e[sign(yk) (2n+2)blcn bki= s k=l...,n n k=l <0 2n+2 1=1 I,y~l r o, I~ 2n+2 -I ~ btyllsign(s) I Z b,~Y,l >- ~, l=1 =s s < Ctn+2, . . ., C'2n]T Defining a Lyapunov function (14) and differentiating V with respect to time, we have V=sk n 2n k=l k=n+l C F - CBmr + CBu] CBmY2n+l+ cnbu] (15) First we calculate be~ (k, i = 1 . . . . 2n+2) using { '~] } l=1 aibki ~] 1 ~-StI.k=n+, E CtkYk"~-Cnbk=~n+l i~=1 [ ,2n+~ ilYe J ' 1 E btiytl/ J +s{ -CBmy2n+l+cnb aib2n+l"i)]Y2n+l} ([222bliYl[ l=1 { + s -c~ [222{aib2n+2'i~] } + c,,b ~ ~ 1 | Y2n+2 L i=1 \[ E bllYl[/-I l;1 k = n+l,...,2n (18) Y2n+l} (19) " '=* '1 X b.y,I "j i=1 (b/b,)I*1 Ic,,I ly2,,+21 = - ,c.f- (20) I*1.0 Using expressions (17)-(20) in expression (16), we have: f'<0 2n+2 for Is[ 4=0 I E bliyll ~ gi, and (21) i = 1,...,2n+2 2n+2 Ill ~ b,&l < 8,, using the control input in expression i=1 (11), expression (15) can be written in the following form: 1~r = S[~=lC,k--lyk l=1 [27=~? 141 lYk] [2~2(ai b2n+2"it | ~Y2n+2 l } ~ ~ i=1 ~l ~ btyll/ j r2n+2( k=n+l s - c , f + c,,b 1=1 2n c'gyk - (b/bl)is[ ~ < o Isy~lr 0, k = 2n+1 (15) can be written in the following form: 2n " i=1 2n+2 k=l aibki )] } s{-CBmY2n+l+cnb [ (i_1~_l ~ 1 )aib2n+l,i ]2n+2 expression (12). If[~-~ bly l] >-- 8i, then expression En r2n+2( E aibki [ (2n+2 E (17) k=n+l "= I ~ b,,y,I 0 Isyk[4=0, <o =4Ec~_~v~+ ~ c~e--c.f-- ~'=s { k = 1 , . . . ,n = sCB,ny2n+l - (b/bl)Isl Icn,,I ly2,+11 12 V = ~s m + " 2n k=n+l then the output tracking error between the plant and its reference model asymptotically converges to zero. = s[CAe + C ( A - A m ) g bliYA ' a lYel k=l 2n l=1 C ( A - A m ) = [Ctn+I, l=1 I~1lYk~] i=1 I Z b,~y,I< 8, where Co = 0 I ~, 2 Ir C'kYk+Cnb E2n kk=n+l 2n+2 -a/sign(s) '=' = s ~ Ce-aYk- (b/b,)Isl k = 2n+2 (2n+2)b~:,, aibki 2,,w~-- "='" k -- 2n+1 (2n+2) blCn Proof: Ce-lYk+ c n b Z k = n, . . ., 2n ICBmlsign(yk) al = Using expressions (12) and (13), the four terms of expression (15) satisfy the following inequalities: +s ,, /2n+2 \ ] t t2n+2 \ ] [2,c),y~ + Cnbk=,,+a ~ [ ~l ai371bt")Yk] Lk=n+l [2n+2 \ ] + s[- CBmY2n+l+ Cnb~i~=lai~Xb2n+l,iJY2n+lj + src. f[ + (16) \ q cnbt/2,,+2 iZ=,a,~;-eb2,.,+z.,)y.,,,+2J (22) A Robust Adaptive Tracking Controller 161 Similar to the analysis in expressions (16)-(21), we can have: 12<0 2n+2 IE /=1 for[s[=~0 bliYil< and (23) 6i i = 1 . . . . ,2n + 2 Expressions (14), (21) and (23) mean that filtered error s asymptotically converges to zero according to the Lyapunov stability theory. Therefore, on the sliding mode Ce = 0, the output tracking error asymptotically converges to zero [9-14]. Remark 4 Unlike most neural network-based control schemes, the neural network in this paper is not directly used to learn unknown nonlinearity, but it is a general adaptive tracking controller in the Lyapunov sense because network parameters are adaptively adjusted based on a Lyapunov function such that the output tracking error asymptotically converges to zero. Details on Lyapunov stability theory can be found in [5]. Remark 5 Due to the fact that the sliding mode technique is used in control network parameter design, the closed loop system will have strong robustness with respect to large system uncertainties [3-6, 9-141. Remark 6 It can be seen that the sign function sign(s) is used in control parameter adaptation law in expression (13), therefore chatterings may occur in the control input. Based on the principle of boundary layer control technique in [3-5], the control input can be smoothed by using (s/g') to replace sign(s) when Is[ < 6', where ~' is a positive number. As shown in [3-5], the boundary layer controller offers a continuous approximation to the chattering control input signal inside the boundary layer, and guarantees attractiveness to the boundary layer and ultimate boundedness of the output tracking error to within a neighbourhood of the origin, depending on g'. Remark 7 It is easy to show that if the initial value of the output tracking error is bounded, all signals in the closed loop system are bounded. Therefore, the controller parameters in expressions (12) and (13) are bounded [2]. network in Fig. 1, we present a simulation example on a one-link rigid robotic manipulator, shown in Fig. 3. The link is of length l and mass m. The mass is assumed to be concentrated at the point of the end of the link. The position variable is the angle 0. The dynamic equation is given by m/2 ~ + d0 + m g l cos(0) = u Comparing with expression (1), expression (24) can be written in the following standard form: (25) {i = f(0,0) + bu ]a6 + m g/cos(0)] where f(0,iJ) = - (m/2)-1 b = (m F) -1 (26) (27) or the following state equation form: [0~ For simplicity, the parameter values in expression (24) are chosen as follows: m=l=g=d=l The reference model for systems (24) or (28) to follow is chosen as /J, = [0 l][X,] [1 -4 -5 , + r where r = 5 for t >0. Filtered output tracking error is defined as s -= 5 e(t) + ~(t) (30) Uncertain bound parameters in A1 and A2 are chosen as fo -- 5, bt = 0.7, bm = 1.2 (31) The Runge-Kutta method with a sampling interval AT = 0.01 s is used to solve the nonlinear differential equation numerically. Figures 4a-c show the output tracking, tracking error and control input. It is easy to see that good tracking performace has been obtained. However, the control input is a chattering signal and the amplitude of the control input is very large. The effects of chattering and the amplitude of the control input are greatly 1 4. A Simulation Example To illustrate the tracking capability of the proposed robust tracking control scheme using the neural (24) 0 IIIII IIIII Fig. 3. One-link rigid robotic manipulator model. 162 M. 14 1.4 12 1 08 o., .~ [.. ,/-- 1.21 v E Zhihong / 36 / 04 / / / " Or 0r - 0 # --- ! g E- 4 //' i // 02 1t O.8 ) ; (a) ; ; ; 5 --- i iiot}:f// 0 Time t (see) 0 // 4 :~ R ; (a) Timer(see) 0.2 g .~ O.15 '~ L2 01 r 005( o~- "-~-....,._~.~.~~_..~..._.._..__.,_. 0.05 ~ \,_ o ~ -0.05} -0 05 V~ g -0.1 "~ E-- -015 0 (b) -0.2 0 Time t ( s e c ) (b) z 1 2 3 4 5 6 7 Time t (see) 15 20 "~ z i0 0 5 g -20 0 -5 -40 0 (C) 2 3 4 5 8 7 8 ~" Time t (see) -]0 -I.=- Fig. 4. (a) The output tracking of joint 0; (b) output tracking error of joint 0; (c) control input signal u(t). (c) reduced by the use of the boundary layer technique (8' = 0.2), as can be seen in Figs 5a-c. Fig. 5. (a) The output tracking of joint 0 using the boundary layer controller; (b) output tracking error of joint 0 using the boundary layer controller; (c) control input u(t) using the boundary layer controller. Time t (see) 5. C o n c l u s i o n s A neural network-based robust adaptive tracking controller is proposed for a class of nonlinear systems in this paper. Our analysis demonstrates that the structure of the proposed neural networkbased controller is simple, and the network parameters are designed in Lyapunov sense. Using this scheme, asymptotic error convergence can be guaranteed, and further, the closed loop system retains strong robustness with respect to uncertain nonlinearities. Acknowledgement. The author would like to thank the referees for valuable c o m m e n t s and suggestions. A Robust Adaptive Tracking Controller References 1. 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