Proceedings of the 2004 IEEE Conference on Robotics, Automation and Mechatronics Singapore, 1-3 December, 2004

Sliding Mode Neural Controller for Nonlinear

Systems with Higher-Order and Uncertainties

Tri V.M. Nguyen, Hung T. Nguyen and Q.P. Ha Faculty of Engi neering

University of Technology, Sydney Broadway NSW 2007 Australia

E-mail: trnguyen@ eng.uts.edu.au.hung.nguyen@uts.edu.au.quangha@eng.uts.edu.au

Abstract - In this paper, we propose a new ncural controller

architecture which is derived from adaptive sliding mode control

framework for SISO nonlinear system with higher-order and

uncertainties. This neural controller can ovcrcomc some

disadvantages inherent in sliding mode controllers such as

chattering problem, complex calculation of the equivalent control

term and unavailable knowledge of the upper bounds of system

uncertainties. Experimental results for a Coupled Drives CE8

system show that a real-time neural controller has been implemented successfully,

Keywords-sliding mode control; neural networks

L INTRODUCTION

Over the year. considerable attention has been given to the problem of stabilising incompletely modelled or uncertain systems. The variable structure control (VSC) or sliding mode control (SMC) theory has provided effective means to design robust state feedback controllers for uncertain dynamic systems. The main idea in VSC is to force the states of a control system to reach a predefined surface within state space and subsequently continue its motion along the surface. The surface is called sliding surface and such behaviour is known as sliding mode. In the sliding mode. the system is completely insensitive to the parameter variations and external disturbances. Early contributions to this VSC approach can be found in the literature [1-3].

Typically, VSC suffer from the chattering phenomenon because of the discontinuous switching of control laws across the sliding surface. In practical engineering systems, chattering may cause serious damage to system components, and excite unmodelled high frequency plant dynamics [4]. Moreover, there exists appreciable difficulty in the calculation of the equivalent control because it requires complete knowledge of the plant dynamics [5].

To alleviate these difficulties, several modifications to the original sliding control law have been proposed. The most popular solution is the boundary-layer approach which uses a high-gain feedback when the system motion reaches the rp. vicinity of a sliding manifold [6]. But this scheme results in small tracking errors. Another result of sliding mode based chattering-free SMC is proposed in [7] with system uncertainties learnt online by a feed forward neural networks [8]. This neural SMC approach shows that it is able to model nonlinear dynamics and provide on-line adaptation effectively.

0-7803-8645-0/041$20.00 © 2004 IEEE

The combination of neural networks (NNs) and SMC recently has attracted much attention from several researchers because NNs can offer the improvements for VSC through its online learning capability. In [7,9], a neural network controller with the learning rule based on sliding mode algorithm is used to calculate the unknown part of equivalent control in the presence of plant uncertainties. Fang et a!. [10] focussed on a class of nonlinear discrete sliding-mode control with recurrent neural network (RNN) to assure the stability of the control system.

A fuzzy NNs sliding mode controller (FNNSMC) was also developed for a class of large-scale systems with unknown bounds of high-order interconnections and disturbances [11]. The whole system has a parallel structure so that each subsystem has one FNN for control and another FNN for identification. Recently, two parallel NNs are used to realize the equivalent control tenn and the corrective control tenn of the SMC design [12, 13].

In this paper, a novel neural control structure is proposed. This neural controller is based on the proposed continuous

SMC approach and results in smoothed control perfonnance without requiring complex calculation of equivalent control term. In addition, online adaptive updating of the switching gain in this neural controller can eliminate the need for large control signal and avoid the requirement for estimating the bounds on system uncertainties.

II. SLIDING MODE CONTROL DESIGN

A. SMC design Consider a SISO nonlinear system

(1)

where x",, [x], .. . ,xnTf E Rnis the state vector, UE Rthe control input, Y E R the output, and d the unmodelled dynamic part. The control problem is to get the output y to track a desired output Yd in the present of model imprecision onf{x) , b(x) and d(x,r).

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Assumption 1 .. Assume that a known function f max (x);;:: 0 and two constants bo > 0, D > 0 exist such that f{x):s f max ' b{x);;::bo and Id{x,tl:SD.

Define desired state vector and tracking error vector

Xd ;;&d, .. . ,y�n-l)r ;; [XdJ, ... ,XdnY e==x-xd == [ej, ... ,en Y

We first define a time-varying surface S{t) in the state­space Rn by a scalar equation

S(X,/);; H.e

(2)

(3)

where positive constant vector H ;;;;; [hj,h2, ... , hn_1,J] is chosen such that the polynomial

is Hurwitz.

0-1 h n-2 h s + n_js + ... + J (4)

The second step is to find control law u which drives the system trajectory to the sliding surface. By choosing a Lyapunov candidate V ;;;;; Y; s2 ;;:: 0, the control u can be taken as

u == v :::: -K(X )sgn(s)

where sgnO is the signum function. Take the derivative of S and V, we have

S:;;; hfe + ... + hn_Je(n-f) - Xdn + f + d -bK sgn(s) V = SS;;;;; S[hJe + .. . + hn_Je(rI-/) -xdn + f + d J-bKS sgn(s)

:s ]Sllh/e + ... + hn_Je(rI-i) - Xdn + f + dl-bK]s]

Therefore, V :S -11]sl if K is chosen as

K;;:: fmax +D+11+lhfe+ . .. +hn_1e(n-I)-Xdn l bo

with 11 is a strictly positive constant,

(5)

(6)

Consequently, system states starting from any initial position will be forced to the sliding surface in a finite time [5]

(reaching mode) and remained on this surface. On the sliding surface (sliding mode), s :;;; H.e ;; 0 , and Equation (4) implies that the tracking errors asymptotically approach zero as t -'; co .

B. Continuos control laws To eliminate chattering, a saturated function can be used to

replace the signum function in (5)

u ;;;;; -K(X )'sa{ �) (7)

where K(X) is satisfied (6) and

(s) {Sgn(S)' sat - = S

� �' (8)

When lsi> <\l , sat(;{) = sgn(s), and the control law u is chosen as described in (5). This guarantees that all system states starting from an(' initial position will be forced into a boundary layer B(x);; Nsf:s $} in a finite time [5].

When lsi :S <!J, define S =: [e/ ,,,., en-J r ,

where A is a stable matrix. The system is represented by

By choosing a Lyapunov function VAs) == ST Pc;, where P is the solution of the Lyanunov equation PA + AT p;; -/ , it can be verified that

where 0 < e J < 1 . Thus all system state trajectories reach the set

(9)

in finite time. The closed-loop system (1) with control law (7) is globally uniformly ultimately bounded with respect to the set n around the origin of the error space.

However, asymptotic stability of (I) can not be concluded, and limit cycles may exist. We will see that the below method with some assumptions can assure system stability.

We first introduce a saturated proportional-integral function

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(10)

where to is the initi<¥, time }when system states enter the boundary layer B{x) '" 1tI1s1 � «1> '

Then the control law u is chosen as

u '" -K(X \J{s) (11)

where K(X) is satisfied (6),

Lemma J: For the system (1) with feedback control law (II), the reaching mode is always guaranteed.

Proof" When lsi> «1> , this guarantees that all system states starting from any initial position will be forced into a boundary layer B�x) in a finite time.

When lsi � «1>, there are three cases by the value Ti.

If s +.i. s '" 0 , s will tends to zero with a time constant Ti . 11

If s +.i. s < 0 for all s > 0, or s +.i s > 0 for all s < 0 it 11 1";

J 2 . 2 leads to ss + - S < 0 and V < -- V. So the Lyapunov

11 11 theorem is satisfied and s = 0 is the system asymptotically stable point.

If s +.!.... s > 0 for all s > 0, or S +.!.... s < 0 for all s < 0 , this 11 11

means that p increases on all s > 0 and p decreases on all s < 0 [14 J, As a result, the below inequalities can be obtained

after a finite time.

J rf J

rJ s+- sdt2:«1>fors>O,ors+- sdt�-4Jfors<O

T T I 0 I 0

The control law (II) with K as in (6) causes the condition

V � -l1Jsl satisfied, and system states will go to the sliding

surface s = 0 in a finite time.

Assumption 2: The close-loop system (1) with feedback control law (II) satisfied Lemma 1 has an equilibrium point at the origin, e = 0 .

Consequently, application of the invariance principle shows that the equilibrium point e = 0 is asymptotically stable and attracts every trajectories in Q. We summarize our conclusion in the next proposition.

Proposition 1: For system (1), if Assumption 1 and 2 are satisfied, and the control law is chosen as shown in (II), with

K(X) as in (6), the equilibrium point e = 0 of the closed-loop

system is asymptotically stable.

However, the values of K in (6) are often large and unaccepted in real-time system. To reduce the amplitude of K, an equivalent control tenn is conventionally added [5]. However, there exists appreciable difficuHy in the calculation of the equivalent control tenn because a complete knowledge of the plant dynamics is required for this purpose. These disadvantages will be avoided by applying a learning algorithm of K in the next section.

Ill. PROPOSED NEURAL CONTROLLER DESIGN

D .. - s hJ h2 _ 1 ellOe s =- =-el +-e2 + ... +-en

«1> «1> «1> «1>

From (10) +(12), control law u can be modified as

I

U = -K psat(s) - Ki Jsat(s)dt o

A. Strncture and function of the neural controller

(12)

(13)

A feed-forward neural networks is proposed in Figure 1 to replace the controller (13). The NN have 1 input layer, 2 hidden layers and 1 output layer. It contains n neurons X II , ... , X In in input layer, one neuron X s in hidden layer 1,

two neurons X HI' X H 2 in hidden layer 2 and one neuron

Xo in outpullayer.

1) 1nput layer

(14)

where ei(kti= l, .. . ,n are the input signals at sampling

time k; X II (k), i = 1, ... , n the output signal of neuron X Ii of input layer at sampling time k.

2) Hidden layer J Define input and output function of the neuron of hidden

layer I at sampling time k as

figure I: Structure of a NN controller

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n

us(k)= L w/iX/i(k) (15) i=l

(16)

where Wli' i = 1, ... , n are the weight values between hidden layer 1 and input layer. From (12) and (15), it is verified that

U:; =s (7)

3) Hidden layer 2 Define input and output function of the proportional and

integral neurons of hidden layer 2 at sampling time k as

UHI(k)= wH1X .. (k) XHI(k)= UHI(k) UH2(k)= wH2Xs(k) XH2(k)= XHz(k-1)+UH2(k)

(IS)

(19)

(20)

(2l)

where wHi,j =1,2 are the weight values between hidden layer 2 and hidden layer I.

4) Output layer Define input and output function of the output neuron of

outpUllayer at sampling time k as

2

u,,(k)=-L WojXHj{k) j=1

uHAk»umax umin � ua (k) � Umax

Ii HI (k) <: Umin

(22)

(23)

where Woj' j = 1,2 are the weight values between output layer

and hidden layer 2.

B. On-line learning algorithm of the neural controller The back-propagation algorithm ean be applied for this

feed-forward neural network to optimize its weight values.

Define a cost function

(24)

The change of the weight values between output layer and hidden layer 2 can be found in the direction of negative gradient as

where a is the learni ng rate.

Note that X" =u and from (1),(2) and (3), we obtain

as as b Therefore, --= -::;- "" f�t , and

ax" aU ¢'

where So = uUs(k )f�dt . ¢' From Assumption I and (12), we can let

which only reduces the learning speed of NN weights.

Finally,

(25)

(26)

The change of the weight values between hidden layer 2 and hidden layer I can be found as

where

Therefore,

(2S)

The change of the weight values between hidden layer I and input layer is founded as

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iJJ aJ as _ as L�wli =-U --=-U.---=-U.s--aWli as aWli aWl;

From (12), (7) and (24), we have

i=1, ... ,n

IV. EXPERIMENT RESULTS AND DISCUSSION In this Section, a neural controller for a Coupled Drives

CE8 system is designed and implemented to provide validation. In the coupled drives apparatus shown in Fig. 2, an elastic belt loops around the two pulleys and an intermediate jockey pulley. Two pulleys are driven by two DC motors while the jockey one is suspended venically by a spring. The tension of the elastic belt is given by the extension of the spring and it is measured by a potentiometer. A pulse sensor on the jockey pulley produces a signal that is proportional to the magnitude of the velocity. The main object of this project is to be able to control both tension and velocity effectively.

The dynamic equations for the coupled drive system are developed in [15], and a pre-compensator is designed to decouple jockey pulley's speed subsystem and tension subsystem [16]. To find the parameters of the coupled drives' model, frequency response identification is performed on the Coupled Drive system.

The speed subsystem's dynamics are obtained as

<i:J=-O.58w+O.217u1

Yl =(1)

The tension subsystem's dynamics are

X(4) = -3.46 xl3) -I2.8x(2) -256.7 x -1831.61x+I353u2 +d

Y2 =x

(30)

(31)

where x is the spring extension, (J) the jockey pulley velocity, d the unmodelled nonlinear pan, Ill> Il2 the control inputs of the velocity and tension subsystems respectively.

For the first-order system (30), jf the control input UI is designed as (7) which is actually a proportional control law, then a static error always exists. This' continuous control does

Figure 2: Coupled Drives CE8 with V,. V2 are the control voltages to motor 1 and 2, respectively.

not guarantee asymptotic stability but rather uniform ultimate boundedncss.

For the tension subsystem, state estimation is an imponant consideration and appropriate observer is constructed. State estimation for this application is achieved through

l"'(A-LC�+Bu2+LY2 (32)

where x denote the state estimate, ;",B,C are drawn from (31), and L is chosen so that all eigenvalues Aj of A - LC satisfy Re [Ai I < 0 .

Choosing the observer eigenvalues as [-5,-10,-15,-20], L can be calculated as

L=[-6196S.26 -1336.33 601.17 46.54f

Two neural controllers are designed for the two subsystems. The NN initial weights are chosen arbitrarily small, but Wli (0), i =:0 1, ... ,n are determined under the' condition of stability (4). In the simulation, in order to represent the change of sampling time, controllers and observer are drawn after n iterations of the plant states' calculation. Fig. 3 shows the closed-loop step response when a step of 2 is applied to velocity reference input and tension reference input is a step of 2. As in Fig. 3, system outputs track to reference inputs well. Moreover, the simulation results can show that the change of the sampling time affects output performances.

Based on the simulation study, two real-time neural controllers are implemented for the actual Electrical Coupled Drives CE8 system. The controllers and observer structure are left exactly as in the simulation. A program in Pascal language is written and complied for use with a high performance data acquisition card, peL-SIS. And a 32-bit personal computer, Pentium 11-350 is used as a PC-based controller. Fig. 4 indicates the step response of the velocity control task under neural controller and traditional PID controller while tension reference input is set to zero.

Figure 3 : The closed· loop step response with different n iterations.

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Figure 4: The actual closed-loop step response

The neural controller is carried out with a sampling lime T = O_04s to attain an appropriate overshoot. The pm controller using below parameters is carried out with a sampling time T = 0_01 s _

Kro =O.ll;Kiro =0.I7;Kdm =0; Kx = 0.3; Kix = 0.2;Kdx "",0.07 As seen in Fig. 4, both control system outputs can track

their setting points after 4s. While the PID controller can not respond to the unmodclled nonlinear parts in tension subsystem, the neural controller can keep system outputs to their reference inputs exactly. These results can validate robust property of the proposed method.

V. CONCLUSION In this paper, a sliding mode based neural controller has

been proposed and implemented. This new controller is first designed based on a continuous sliding mode controller with the chattering problem eliminated and the tracking error minimised. To. avciid large amplitude of control input, an online adaptive updating of controller parameters is based on neural network framework. The structure of this neural controller is a feed-forward neural network with the on-line back-propagation learning algorithm. Experimental results for a real-time Coupled Drives CE8 system show that a real-time neural controller has been designed and implemented successfully.

ACKNOWLEDGEMENT This work is supported by Vietnamese Ministry of Education and Training, the Key UTS Research Centre for Health Technologies and the ARC Centre of Excellence for Autonomous System.

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