Systems & Control Letters 17 (1991) 321-325 North-Holland

Servo-svstems with discrete-variable struck&e control

C.Y. Chan School of Electrical and Electronic Engineering. Nanyang Tech- nological Uniuersi~v, Nanvang Auenue, Singapore 2263

Received 15 February 1991 Revised 11 May 1991

Abstract: This paper is concerned with the computer-based variable structure control of servo-systems. A discrete-variable structure control is presented and a condition is given for stability. The strategy ensures that sliding mode can be ob- tained exponentially fast to keep the system robust. Simulation results show the effectiveness of the proposed technique.

Keywords: Servo-systems; variable-structure control.

the computer- based variable structure control of servo-systems. A discrete-variable structure control is presented

This paper is concerned with

and a condition is given for stability. A feature of the strategy is that sliding mode can be obtained exponentially fast so that the system can be kept robust. The analysis technique used is based on that of [4,5]. Simulation results are included to demonstrate the effectiveness of the proposed technique.

321

The organization of the paper is as follows. Section 2 presents the development of a discrete- variable structure controller for servo-systems, and a stability condition is derived. Some simulation results are presented in Section 3 to illustrate the features of the proposed method.

1. Introduction 2. Discrete-variable structure controller

There has been a remarkable increase in the application of variable structure control based on sliding mode [ll] to practical systems, for exam- ple, servo systems [2,3,6,8]. When in sliding mode, the system is insensitive to parameter variables and disturbances.

Variable structure control designed on the basis of a continuous system is usually implemented using a digital computer. However, the choice of the sampling interval affects the system perfor- mance. The system gives rise to chattering and may become unstable depending on the sampling interval used. In practice, fast sampling is used. But this may give rise to fatigue of system compo- nents. Thus, a digital algorithm should be used for a computer-based control of a practical system.

Some recent studies have been made on dis- crete-variable structure control [1,4,5,9,10]. The results obtained are quite different from those for continuous-time systems. For example, the control input does not switch so frequently that sliding mode may be brought into practice even for a long sampling interval.

In this section, the design of the discrete-varia- ble structure controller is presented for a servo- system.

Consider a single-input system represented by

x(k+l)=Ax(k)+Bu(k >

y(k) = CTx(k)

single-output discrete

3 (la)

w

where u(k) is the input, x(k) is an n x 1 state vector, and y(k) is the output.

Let

e(k)=x(k)-x*(k)

where x*(k) is the reference input. The switching hyperplane is chosen to be

s(k) = GTe(k) (3)

where the elements of G are determined such that e,(k) is stable on the hyperplane s(k) = 0. G is chosen as follows.

0167-691 l/91/$03.50 0 1991 - Elsevier Science Publishers B.V. All rights reserved

322 C. Y. Chan / Servo-systems with discrete-vanable structure control

When in sliding mode, the state satisfies the condition s(k + 1) = s(k) and using (l)-(3), the equivalent control is given by

u(k)= -;GT(A-Z)x(k)+;GTAx*(k+l)

where (Y = GTB and

(4)

Ax*(k+ 1) =x*(k+ 1) -x*(k).

In what is to follow, it is assumed that Ax *(k + 1) = 0. Thus, for a bounded x *, G should be chosen such that the system

e(k+ 1) = [A - iBG’(A -Z)]e(k)

+ [

I-;BGT 1 [A -Z]x*, (5) GTe(k) = 0,

is stable. Now, the control law to transfer the state on

the hyperplane is examined. The control law of the following form is considered:

u(k)=-~GT(A-Z)x(k)+$Te(k)-$,,s(k)

(6)

where 4 and q0 are the feedback gains to be determined later. The additional term &s(k) is included, as was done in the continuous case [7] for robustness improvement.

In the following, a condition is derived for the feedback gains 1c/ and lc/O to ensure that the system is stable. The analysis technique of [4,5] will be used.

The Lyapunov function is chosen as

V(k) =s(k)2

and let

(7)

As(k+l)=s(k+l)-s(k). (8)

In order to ensure the existence of a sliding mode, the condition

V(k + 1) - Y(k)

= s(k + 1)’ - s(k)2

= 2s( k)As( k + 1) + As( k + l)*

-=o (9)

must be satisfied. Let

P = $,(GTB).

Using (l), (3), (6) and the condition Ax*(k + l)=O gives

s(k+l)=(GT+aGT)e(k)-ps(k)

and

(10)

As(k+l)=s(k+l)-s(k)

= (GT + aGT)e(k) - ps(k) - GTe( k)

Thus

= c N,e,(k) -w(k). I=1

(11)

2s(k)As(k+l)+As(k+l)*

= [!I N.e.(k))’

n + 2(1 -P) c @,e,(k)s(k)

I=1

+(p*-2p)s(k)2. (12)

The following result gives a condition for the feedback gains J/ and #0 to make the system stable.

Theorem. The system will be stable if the following conditions are met: (i) 0 < p < 1, and (ii)

if ae,(k)s(k) < --a,,

if - 6, 5 ae,(k)s(k) 5 6,, (13) -F, ifae,(k)s(k) > a,,

fori=l,...,n where

Foa2 &= 2(1-p) ’ le (k) I i le,(k) I J=l

(14)

and F, > 0 is constant for all time.

Proof. If $, is chosen as F, or -F,, then from (1%

V( k + 1) - (1 - p)2V(k)

I-2(1-p)5F08,. r=l

C. Y. Chan / Servo-sptems with discrete-variable siructure control 323

Thus, if 6, is chosen as (14) and 0 < p ( 1, .

V(k+l)-(1-p)2v(k)<0.

Also, if 1 ae,(k)s(k) 1 -c 8, and +, = 0,

V(k+1)-(1-p)2V(k)=0.

Thus, the control law (6) ensures that

V(k+1)s(1-p)2V(k).

Since (1 - p) < eCp,

(1 - p)' < e-m'P.

Hence

V( k + 1) I eeZPV( k)

and

V(N) 4 V/(O) eWZNP.

Thus from the definition of V(k), s(k) con- verges to zero exponentially fast.

In the above theorem, it is assumed that F, exists satisfying (13) i.e.

lae,(k)s(k) 1’6,. (15)

Thus, substituting (14) into (15) gives

O<F < 2(1-p)ls(k)l 0 Ial t (e,(k) I .

(16)

/=1

Let

e(k)=[t, tz ... t,l4k)

where G rt, = 1, GTt, = 0, t, is linearly indepen- dent of t, (i=2 ,.... n). Thus,

s(k) = GTe(k) -q(k).

It follows that e,(k) + t,jq(k) (k-+ a), j= 1, . . , n. The upper limit of F, is now given by

O<F < 2(1-P)

0 Ial 2 l4,I j=l

(17)

and

2(1- P>

lal Ii It1,I

~ a1 -P>

la1 max I g, I (18)

J=l

where g, # 0 (j = 1,. . . , n) are the elements of G. To achieve fast convergence of s(k) to 0, the

upper limit of F, should be reduced.

3. Examples

In this section, simulation results are given to illustrate the features of the proposed design method.

Consider the following discrete system [4]:

x(k + 1) = ( _;:;;;;;;;;18 ;:;;;;;;;;;;)x( k)

+ 0.00467884016 0.0904837418 u(k),

y(k) = [l 01x(k), which has been obtained by sampling the continu- ous system with a sampling interval of 0.1 s.

The switching hyperplane used is

s(k) = [l 0.5]e(k).

Figure 1 shows the corresponding step re- sponse, control signal, s(k) and phase plane for a command x * (k ) = [5, 0] T and negligible value for #a. The upper limit of F, is approximately 40 and a value of 3 has been used.

Figure 2 shows the step response, control sig- nal, s(k) and phase plane for a command x*(k) = [5, OIT and using values of $0 = 8 and p = 0.4. The upper limit of F0 is 24 and a value of 3 has been used. It is seen that the transient perfor- mance is better than the previous case for ap- propriate values of q0 and F,.

4. Conclusion

A discrete-variable structure control for servo- systems has been presented. A condition for sta- bility is given. A key feature of the strategy is that it ensures that sliding mode can be obtained ex- ponentially fast so that the system can be kept robust. The effectiveness and features of the pro- posed controller have been demonstrated using examples.

324 C. Y. Chan / Seroo-systems with discrete-oariable structure control

68,

-r-mm-- % 18

1

8

6

4

2.

e -_B -k -b

Fig. 1. Step response, control signal, s(k), and phase plane for & negligible.

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a

6

Fig. 2. Step response, control signal, s(k), and phase plane for Go = 8 and p = 0.4.

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[3] Y. Dote. Servo Motor and Motion Control Using Digital Signal Processors (Prentice-Hall, Englewood, NJ, 1990).

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