A control scheme based on discrete time-varying sliding surface

8/7/2019 A control scheme based on discrete time-varying sliding surface

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Proceedings of the 5" World Cong ress on Intelligent Control

and Automation, June 15-19,2004,Hangzhou, P.R. China '

A Control Scheme Based on Discrete Time-Varying

Sliding Surface for Position Control Systems*

Zhang Jinggang Zhang Yibo Chen Zhimei Zhao Zhicheng(Dept. ofAutomation, Ta ip an Heavy Machinery Institute,030024)

ip zhantz65@,163.com

Abstract-A sliding mode control scheme for second-orderdiscrete system is proposed using the time-varying sliding surfaceinstead of the fixed sliding surface. The time-varying slidingsurface is designed first to pass arbitrary initial states, andsubsequently move towards a final sliding surface by rotating.The sliding surface rotating procedures are presented in detail.The proposed control scheme is applied to a position controlsystem. Simulation results show that the scheme provides highperformance dynamic characteristics, and is insensitive touncertainties including parameter variations and external

disturbance in the whole control process.

Index Terms aliding mode control, time-varying sliding surface,position control system

I . INTRODUCTION

Since the sliding mode can offer many good properties,such as insensitivity to parameter variations, extemaldisturbance rejection, and fast dynamic response, manyresearchers have studied the sliding mode control in the past20 years. The motion of the control system employing slidingmode control can be described as two phases: reaching andsliding phase. The system state trajectory in the period oftime before reaching the designed sliding surface is called the

reaching phase, and in which there is a control action towardthe sliding surface. Once the system state trajectory reachesthe sliding surface, it stays on it and slides along it to stateorigin. The system state trajectory sliding along the slidingsurface to origin is the sliding phase. The insensitivity of thesystem to uncertainties including parameter variations andextemal disturbance exists in the sliding phase, but not duringthe reaching phase. In order to solve this problem, the

time-varying sliding surfaces are introduced [11-[SI.However,the characteristics usually cannot be guaranteed when thesliding mode control algorithm is implemented on a digitalcomputer because the switching frequency is limited by thefinite sampling rate. So it is necessary to design a discrete timesliding mode controller for a digitally controlled system.

In this paper, the sliding mode control scheme based on

time-varying sliding surfaces is presented for second-orderdiscrete-time systems. On 'the basis of studying a kind ofposition control system [9][IO], the modified exponentreaching law is adopted to realize a step response. In thisscheme, the sliding surface is initially designed to passarbitrary initial states, and subsequently move towards a

* The research work is supported by Shanxi Nature Science Fund.

predetermined sliding surface by rotating. This robustness andthe tracking performance are obtained. The result ofsimulation is ensured.

11.EXPO NENTEACHINGAWFORSECO ND- O RDERISCRETE

SYSTEM

Consider the following second-order linear system:

x ( k +1) = h ( k ) b u ( k )+d ( k ) ( 1 )

where u ( k ) represents system input signal, d ( k ) epresents

extemal disturbance.The sliding surface is chosen as:

s ( k )= cx(k) (2 )

where c E RIx2 is the parameter of the sliding surface.

The improved exponent reaching law is adopted to ensure

the asymptotic stable of system [111. Suppose E = pls (k) l ,

then the exponent reaching law tums into:

(3 )

It is obviously that the sliding function s ( k ) will

asymptotically stable when the condition -1 < 1 - 6z - p r <1

is satisfied.

s(k + 1) = 1 - Sr - pr ) s (k )

Compared with ( I ) , equivalent control is obtained:

The bound of disturbance is supposed known as U and

, M = - , the additive control is+ D

D . Let V=-

chosen as: -V -A4 sgn[s(k)] nd the control arithmetic tums

into:

u ( k ) = (cb)-'[(l- 6z - p r ) s ( k )- Ax(k)] (4)

2 2

u ( k )= ( c b ) - l [ ( l - S z - p r ) s ( k ) - c A x ( k ) ]

-V -M sgn[s(k)] (5)

( 6 )ubstituting (5) nto (1):

B e c a u ~ l f h ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~is dominated by thed;ljs(&)wt.{RkelBasghfhr(PP#f~~totical ly

stable for linear discrete-time system is known as:

( A + b ( ~ b ) - ' ( l - S r - p z ) - b ( c b ) - ~ c A )s smaller than 1, the

If each eigenvalue breadth of

11750-7803-8273-01041$20.00 0 2 0 0 4 IEEE



system is asymptotically stable about the zero state, which isthe only stable state [lo].

111. ROTATING SLIDINGURFACE

Suppose c = [c, 11, and the sliding surface is chosen as:

s ( k )= c (k )x l k )-

2 ( k ), where c ( k ) is a step hctionand it satisfies c (0) = -x 2 ( 0 ) x1 0) and c ( k )= c

( k 2 n , where n is the time that the rotating of slidingsurface stopped).

Then the equivalent control (2) turns into:

u ( k )= ( c b ) - ' [ ( l - s z - p . r ) s ( k ) - c A x ( k ) ]

-v -M sgn[s (k)] (7)

Substituting t into (1);

x(k + 1) = [ A+b(cb)-'c(l- sz -pz)-b(cb)-'

cAIx(k)- V +M s gnb (Q1 - 4 k ) ) (8)

It is obviously that the parameter c should always satisfy

the condition: each eigenvalue breadth of

( A+b(cb)-'(l - Sz- p z ) - b(cb)-'cA) should be smallerthan 1.

It should be especially noticed that the difference in the

choice of c,. between the second-order continues-time

system and the second-order discrete-time system. It should

satisfy the Lipschitzian, that is, c > 0 , in continues-time

system; yet in discrete-time system, c should satisfy the

convergent condition, which means that c is not always

positive.The algorithm of rotating sliding surface is outlined as

following:

(1) The final value c is determined from the stable

condition. And c,(O) could be determined according to the

initial state: c (0)= -x2 (0 ) x1 0) . If c (0) could satisfiedthe condition, then to the next step. Otherwise, that is to say, ifthe initial state is not on the sliding surface, a sliding surfacecould still be chosen as which could satisfy the asymptoticstable condition and is nearer to the initial state, then to thenext step.(2 ) The direction of rotating is determined from the values of

c , ( k ) , .e. if c , ( k ) > c , clockwise, and if c , ( k ) < c ,

counter-clockwise.(3 ) The surface is immediately rotating before the next sampleperiod. The value could either be added a fitted value simply,

or be obtained from the equation I (k )x l k )+x2 k ) l = A

(where A is positive). When the second method is adopted,

the larger value of the two solutions of c ( k ) is chosen as

the slop for clockwise, and the other for counter-clockwise.(4) The rotating stops until c ( k )=ce .

In addition, there may be two or more areas that satisfy

the asymptotic stable condition. If c, and c,(O) are in

different area, c,.(k) may enter the area of unstable (Fig.1).

To avoid this, c should be chosen the same area with

c (0) ;when the first scheme is adopted, a larger value should

be chosen (positive when counter-clockwise and clockwise

instead) to enforce c , ( k ) enters the other stable area.

Moreover, the first scheme is more available when disturbanceis excessive.

Unstable area

Fig.1 Rotating sliding surface

N.&PLICATION EXAMPLE

Consider the linear system as follows:

x ( k + 1)= A x ( k )+bu(k)+ d(k) (9)

Supposed that the states track the command signalXd ( k ).The error are defined as: e ( k )= x ( k ) - x d ( k ) , then the state

equation of warp could be obtained as:e(k+ 1) = x ( k +1)- d ( k+1)

= A x ( k ) + b u ( k ) - X d ( k + l )

= A @ ) +A Xd ( k )+h ( k )- d ( k+1) (IO)

The sliding surface is chosen as:

Then s(k + 1) = ce(k +1)= cAe(k)+CAXd ( k )

The modified exponent reaching law is adopted [9],

u ( k )= ( c b ) - ' [q s ( k ) - c A x ~ ( k )+ c x ~ ( k + l ) - c A e ( k )]13 )

The rotating sliding surface is introduced to enhance the

robustness, then the sliding surface turns into:s ( k ) = c , e ( k ) ,

then the equivalent control is obtained:

s ( k )= ce(k) (11)

+cbu(k)- Xd ( k+1) (12)

s (k + 1) = q s ( k ), hen the equivalent control is obtained:

u ( k )= ( c , b ) - ' [ q s ( k ) - c , A X d ( k ) + C r X d ( k + 1 ) ]

- ( ~ , b ) - ' c , [ A e ( k ) + V + M ~ g n s ( k ) ] (14)

The above developed control strategy will be tested on thereal time control of a position control system. The structure of

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the system is given in Fig.2. And Fig.3 shows the equivalentblock diagram.

Then the state space representation of the system can bewritten as follows:

P

w

M

Industrial

control

computer

where, mechanical time constant: TM = 0.039s ; Motor

transfer coefficient with null speed: K, = 1.25deg - l JV ,1 I4 .3V.

Suppose T is the sample period, then the discrete formof the state equation can be written as:

Torque

motor

[ =

The tracking simulation results are given in Fig.4 and Fig.5.When the strategy based on modified exponent law with the

fixed sliding surface is adopted (5), and TIM turns into five

times, the simulation results are shown in Fig.5. When themethod of this paper is adopted, the results are shown in Fig.4.We could see that the curves in Fig.4 are almost superposition,which shows that the robustness of the system is obtained.

I I

~~~

Fig.3 System Block diagram

V . CONCLUSIONS a position control system. The simulation results have itseffectiveness.

In this paper, a discrete sliding mode control scheme withtime-varying sliding surface is proposed. The time-varyingsliding surface is obtained by rotating initial sliding surface.

REFERENCES

Thethe reaching phase. The proposed control scheme is applied to

of the system is enhanced due to [I ] S.-B. Choi et U l, “Moving switching Surfaces for robust Control O f

second-order variable structure systems”, Inr. J. Control, vol. 58 , no.1,pp.229-245, 1993

1177



[2] A. Bartoszewicz, “A comment on ‘a time-varying sliding surface for fastand robust tracking control of second-order uncertain systems”’,Automatica,vol. 31, no.12, pp.1893-1895, 1995

[3] A. Bartoszewicz, “Time-varying sliding modes for second-ordersystems”,IEEparr-D,vo1.143,n0.5, pp.455-462, 1996

[4] H.-X. uang et al , “A type of time-varying sliding mode variablestructure control for non-linear system”, Conhol theory and applications,v01..17,no.5,2000

[5] S.-B. Choi et al, “A time-varying sliding surface for fast and robust

tracking control of second-order uncertain Systems”, Automatica,vo1.30,no.5,pp. 899-904, 1994

[6] Y.-D. Chi er al , “Method to design time-vaned sliding surface”, Journalof Shanghai Jiaotong Unive rsity,~01.32, o.6, pp. 70-73, 1998

[7] S.-B. Choi et al, “Moving sliding surfaces for fast tracking control ofsecond-order dynamical systems”, DSMC, vol. 11 6, pp. 154-158,1994

[8] Y.-S. Lu er al, “Design of a global sliding-mode controller for a motordrive with bounded control”, Inr. J. Control, vol. 62, no.5, pp. 1001-1019,1995

[9] F.-H. Sun et al , “Sliding mode control for discrete-time Systems and ItsApplication to the Control of a Servo-Control system”, Control TheoryandApplications, vol. 14,110.4, pp. 466472, 1997

[lo] S.Cong er al, “Design of the sliding mode controller for position controlsystem”, Control Theory and Applications, vo1.14, no.5, pp. 716-721,1997

[l l] C.-L. Zhai , 2.-M. Wu. “Variable structure control method fordiscrete-time systems”, Journal of Shanghai Jiaotong University, 34(5):

[12] D.Z. Zheng et al, “Theory for linear systems”, Tsinghua Universitypp. 7 19-722,2000

Press.

1.2

--- without perturbation

- - with perturbation

06

I , I I , . I I I

--- without perturbation Iiith perturbation

‘(s)

Fig.5

with the time-varying sliding surface

Position response curve

t(s)

Fig. 4

Position response curve

with the fixed sliding surface

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A control scheme based on discrete time-varying sliding surface

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