Systems & Control Letters 23 (1994) 371-374 371 North-Holland

Robust discrete-time sliding mode controller

C.Y. Chan School of Electrwal and Electromc Engineering, Nanyang Technologtcal Unwerstty, Nanyang Avenue, Smgapore 2263, Singapore

Received 31 May 1993 Revised 13 September 1993

Abstract A discrete-time shding mode controller which is robust against slowly varying perturbations is presented The attraction of the technique lies in its s~mpllclty and ease of tmplementauon No knowledge of the upper bounds of the perturbations is required. Also, the estimation process of the perturbations IS avoided as the controller contains an estimate of the perturbations Simulation examples are presented to Illus- trate the features of the proposed technique

Keywords Discrete-time systems; sliding mode, robust perfor- mance, slowly varying perturbations

1. Introduction

Sliding mode control of continuous-time systems has been widely studied over the past 30 years [9]. The method offers the advantage of robustness agamst perturbations. However, sliding mode con- trol of discrete-time systems has attracted attention only recently, e.g. [1-8]. Some of the latest works can be found in the Proceedings of the IEEE Inter- national Workshop on Vanable Structure and Lyapunov Control of Uncertain Systems, Sheffield, UK, 1992 and in the May 1993 issue of the Interna- tional Journal of Control. The important difference between continuous-time and discrete-time sliding mode control is that the latter is not necessarily robust with respect to perturbations, if the design philosophy for continuous-time systems is ex- tended to the discrete-time case. Unless special steps are taken, the existence of sliding mode is not guaranteed in the presence of perturbations.

Correspondence to C Y Chart, School of Electncal and Electromc Englneenng, Nanyang Technological Umverslty, Nanyang Avenue, Singapore 2263, Singapore

Several authors have proposed robust discrete- time sliding mode methodologies. These include switching gain selection [1, 3-5], state augmenta- tion [6], hyperplane design [8], and perturbation estimation [3]. The disadvantages of these methods are as follows. The upper bounds of disturbances and modelling uncertainties need to be known, which may not be easy to obtain. In [6], the system has to be expressed in a canonical form to include an integral term. The selection of an appropriate hyperplane is not easy, and the strategy of [8] does not cope well with step disturbances. In [3], the estimated perturbation is included in the control law to avoid the need of the knowledge of the upper bounds of the perturbations. However, the need to estimate the perturbations still remains

In this paper, a discrete-time sliding mode con- trol strategy which offers robust control against slowly varying perturbations is presented The method does not suffer from any of the afore- mentioned disadvantages. Its attraction lies in its simplicity and ease of implementation. The stability of the equivalent system is the only design specification.

The organization of the paper is as follows Section 2 presents the robust discrete-time sliding mode strategy for slowly varying perturbations Simulation results are given in Section 3 to demon- strate the features of the proposed technique.

2. Discrete-time sliding mode controller

In this section, the robust discrete-time shdlng mode controller for slowly varying perturbations IS presented.

Consider the following discrete-time system:

x(k + l) = Ax(k) + Bu(k) + f(k), (2.1)

where x(k) is the n x 1 state vector, u(k) is the m x 1 control vector andf(k) is the perturbation, assumed

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372 C Y Chan / Robust dt~crete-ttme vhdmg mode ~ontroller

bounded. I t l s f u r t h e r assumed that there exists ~(k) such that

f ( k ) = B~(k). (22)

The sliding surface is defined by

s ( k ) = G x ( k ) = O , (2.3)

where the hyperplane G is a m x n matr ix chosen to ensure that x(k) is asymptot ical ly stable on s(k) = 0. The design procedure for choosing G is given in [5, 8].

The control law of the following form is considered

u(k) = u(k - 1) + (GB)-1 [ _ GA(x(k )

- x ( k - 1)) - (I - P)s(k)] , (24)

where G is chosen such that GB is nonslngular, and P is a user specffied m x m matrix. H o w P is chosen will be gwen later Note that the control law re- quires no knowledge of the per turbat ions. In fact, it will be shown that the control law (2.4) contains an est imate of the per turbat ions.

Cont ro l law (2.4) can be rewrit ten as

u(k) = (GB)-X [ - G A x ( k ) - s(k) + (GB)u (k - 1)

+ G A x ( k - 1) + Ps(k)] . (2.5)

Now, mult iplying (2 1) by G gwes

Gx(k + l) = G A x ( k ) + GBu(k) + Gf(k) . (2.6)

Delaying each term in (2.6) one ume and using s(k) = Gx(k ) leads to

s ( k ) = G A x ( k - 1) + G B u ( k - 1)

+ Gf (k - 1) (2.7)

Substi tut ing (2.7) into (2.5) yields the following control law

u(k) = (GB)- ' ( - G A x ( k )

+ P s ( k ) - Gf (k - 1)) (28)

It has been shown that (2.4) and (2.8) are equivalent. No te that f ( k - 1) provides an est imate of f ( k ) , which is unknown, rhe idea here is qmte similar to that of [3]. However , the control structure is &ffer- ent f rom that of [3] N o addit ional es t imat ion ~s per formed here, as the control law contains an est imate of the per turba t ion

Using (2.2) and (28), (21 ) can be described as

x(k+ 1)=A,qx(k)+B(GB)-lPs(k)

+ f ( k ) - f ( k 1}, (29)

where

Aeq : A -- B ( G B ) - 1GA, (210)

G is chosen such that Aeq lS asymptotmal ly stable. Now, mult iplying (2.9) by G and using

s(k) = Gx(k) yields

s(k + 1) = Ps(k) + G ( f ( k ) - f ( k - 1)) (2.11)

In the absence of per turba tmns , P determines the rate of convergence of s(k) to zero. Shdang mode can be achmved with a P that has elgenvalues at zero However , in the presence of per turbat ions, ~t is necessary to choose P = 0 so that the term in (2.9) containing s(k) will be zero. Thus, with P = 0, (2.9) and (2.11) become

x ( k + 1 ) = A , q x ( k ) + f ( k ) - f ( k - 1), (212)

s ( k + 1 ) = G ( f ( k ) - f ( k - 1)). (2.13)

If the variat ions m the per turba t ions are slow with respect to the sampling frequency, I f (k) - f ( k - 1)[ is expected to be small In this case, Ix(k)l and Is(k)l will be bounded.

For servo applications, the shdmg surface is defined by

s ( k ) = G e ( k ) = O , (2.14)

e(k) = x(k) - r(k), (2.15)

where r(k) is the reference m p u t The control law (2.4) (with P = 0) is mo&fied accordingly to gwe

u(k) = u(k - 1) + (GB) - x [ - GA(x (k ) - x ( k - 1))

+ G(r(k + 1 ) - r ( k ) ) - s(k)] (2.16)

Using (2.1), (2 15) and (2.16) and s(k) = Ge(k) gives the following system:

e(k + l) = A~qe(k) + f ( k ) - f ( k - l)

+ (I - B ( G B ) - ~ G ) (Ar(k) - r(k + 1)),

(2.17)

s(k + 1) = G ( f ( k ) - f ( k -- 1)). (218)

C Y Chan / Robust dzscrete-t~ne shdmg mode controller 373

For small and slowly varying perturbations, the error will be expected to be small. Thus, for servo applications, the perturbations are expected to be much smaller than those for regulation, if system performance is required to be sausfactory.

3. Simulation results

Xl(k) "/ O t~ ~t _

tN

This section presents some simulation examples to illustrate some features of the proposed technique.

xl(k)

°'a I I] nn

.,.,l~-""~. , -- _.~

51 tU

utk)

g4

I,i

i0 t $1 lira

s(k) "t o JIn . n L ._~_ , ,~ ] U ' ? " " " ~ ' ' - - ~ - - - " "

"0'l I 5Q IN

Fig 1 Responses for Example 1

u(k)

0,4

0'| 1

-0,1

°0. 5Q 1•

s(k) "t 5Q

Ftg. 2 Responses for Example 2.

IN

Consider the following dlscrete-ume system I-6]:

o , lr ,(k,7 x2(k + 1) 0.24 o.2JLx2(k)j

+[~]u(k)+f(k) with

f(k) = [0 d(k)] T,

d(k) = 0.5 cos(O.15~zk )xl (k) + 0.53 cos(O.O75nk)x2(k) + 0.2 cos(0.1rck) u(k) + 0.05 cos(0.1 nk) + 0.05 sin(0.05nk).

374 C ~ Chan Robust dt~'¢rew-tmte vluhng mode controller

Xl(k)

and

r 1

u~k)

1 Example 3. F igure 3 shows the responses for x{0) = [0 0] T, r(k) = [1 1] v, G = [0 1] T and the

same d(k) as m Example 2 Note that excellent t rackmg is achteved desptte the presence of per- tu rba t ions Other s~mulatlons showed that t tme- varying pa rame te r var ia t ions should be small ff sat isfactory per formance ts to be achieved

4. Conclusion

Sl iO0

In this paper , a discrete-t~me shding mode cont ro l le r which is robus t agains t slowly varying pe r tu rba t tons has been presented The a t t rac t ion of the techmque hes in tts stmplicity and ease of tmplementatxon N o knowledge of the upper bounds of the p e r t u r b a u o n s ts reqmred An es t imat ton process of the pe r tu rba t ions ts avoided as the contro l ler conta ins an es t imate of the per turba t ions . Slmulataon results have been pres- ented to i l lustrate the features of the p roposed techntque

stk)

_ _ _

Fig 3_ Responses for Example 3

Example 1. F igure 1 shows the responses, for the init ial cond i t ion x ( 0 ) = [0.2 0.2] T and r ( k ) = [0 0] r G has been set to [0 1] T such that the

eigenvalues of Aeq are at 0. No te tha t there exist quas t -s teady state osctllattons.

Example 2. F igure 2 shows the responses for d(k) = 0.5xx(k) + 0.53x2(k) + 0 2u(k) + 0.1, with the same init ial c o n & n o n , zero-reference input and G as m Example 1

References

[-1] C Y Chan, Servo-systems with &screte-vanable structure control, Systems Control Lett 17 (1991) 321-325

[-2] S V Drakunov and VI Utkln, On &screte-tlme shdmg modes, in Proc IFAC on Nonhnear control systems Deslon, Capri, Italy (1989) 273-277

[3] H Elemal and N Olgac, Shdmg mode control with per- turbatlon estimation (SMCPE) a new approach, Internat J Control 56 (1992) 923-941

I-4] K Furuta, Shdmg mode control of a discrete system, Systems Control Lett 14 (1990) 145-152.

[5] C L Hwang, Design ofservo controller via the shdmg mode technique, Proc IEE-D 139 (1992) 439-446

[6] P Myszkorowskl, Robust controller for the hnear discrete- time systems, m Proc 30th IEEE Conf on Dec~ston and Control, Brighton, England (1991) 2972-2973

[7] S.Z Sarpturk, Y Istefanopulos and O. Kaynak, On the stability of discrete-rime shdmg mode control systems, IEEE Trans Automat Control 32 (1987) 930-932

[8] S K Spurgeon, Hyperplane design techmques for &screte- time variable structure control systems, Internat J. Control 55 (1992) 445--456_

[-9] V_I Utkln, Shdmg modes m control optzmzzatton (Spnnger, New York, 1992)_