Conditions for the Existence of Discrete-time Sliding Mode

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

CONDITIONS FOR THE EXISTENCE OF DISCRETE· TIME SLIDING MODE X.H.Yu Department of Mathematics and Computing, University of Central Queensland. Rockhampton, Australia 4702

Abstract. The up-to-date development of design methods for discrete-time variable structure control is reviewed in this paper. A discrete-time controller is developed to eliminate the zigzagging behaviour peculiar to discrete-time sliding mode. Keywords. sliding mode; discrete-time systems; switching surfaces; nonlinear systems; relative degree

INTRODUCTION

vectors of the state trajectory always point towards the switching surface. A well-known sufficient condition for the existence of sliding mode is lim._o ss < o.

The trend of implementing Variable Structure Control (VSC) is towards using digital rather than analog computers due to the availability of low-cost and high-performance microprocessor. However the implementation is not simply a discretization of continuous-time VSC using a "fairly small" sampling period. It is shown that a welldesigned VSC for a continuous-time system may exhibit chaos in the corresponding discretized system even with a quite small sampling period (Yu, 1992). The chaotic phenomena which do not exist in continuous-time systems are partly due to the comparatively low switching frequency in the discrete-time systems. This note reviews and study the conditions proposed during last several years for the existence of discrete-time sliding mode (DSM). The discussion is undertaken associated with computer simulations which show the merits and limitations of the conditions proposed. A new method is proposed to eliminate the zigzagging behaviour peculiar to discrete-time VSC as well as ease the limitations of existing conditions.

2

It is obvious that the definition for the continuoustime sliding mode can not be applied to the discrete-time sliding mode since the tangent of velocity vectors of the system state trajectory is not available. The switching frequency is actually equal to or lower than the sampling frequency. The comparatively low switching frequency causes the discrete-time system state to move about the switching surface in a zigzag manner.

The discrete-time sliding mode was firstly named the "quasi-sliding mode" (Milosavljevic, 1985). However, the similarity between continuous-time systems and discrete-time systems disappears as the sampling period increases with the system trajectory appearing zigzagging within a bounded domain. Therefore "pseudo-sliding mode" is a more precise statement. In the following the single-input discrete-time dynamic system is considered

EXISTENCE OF DSM

x(k

The conventional VSC says that given a sliding surface, denoted by s = 0, design a discontinuous control law such that the system state will finally attain and remain on the surface so that a desired dynamics is obtained. By existence of a continuous-time sliding mode, it is meant that (DeCarlo, Zak and Matthews, 1988) in a vicinity of the switching surface, the tangent of velocity

+ 1) = f(k , x(k)) + b(k , x(k))u(k)

(1)

where x E Rn, and u( k) is the sliding mode control which mayor may not necessarily be discontinuous on the sliding surface defined by s(k) = s(x(k)) cT(k)x(k) 0 where c( k) may be time-varying. The dynamic system is assumed to have relative degree one, I.e. as(f(k, x(k) , u(k)))/au(k) ¥ 0 for all k. 215

250

7 /

200

12

150

/'

100

50

150

100

200

250

11

11

Fig. 1. An unstable system

Fig. 2. A typical pattern of DSM

Definition 1: The pseudo-sliding mode is said to exist if in a neighbourhood of the manifold {X : s( k) = O}, denoted by n" the following condition holds

V's(k)s(k)

<0

where V's(k) = s(k

Condition (3) actually imposes upper and lower bounds on the discrete- time VSC. Kotta (1989) pointed out that the upp er and lower bounds depend on the distance of the system state from the sliding surfaces. This is one of the drawbacks of using (3) to design a discrete-time VSC controller.

(2)

+ 1) -

Definition 2 can also be set up equivalently by replacing the condition (3) by s2(k + 1) < s2(k) (Furuta, 1990) and Js(k)s(k + 1)J < s2(k) (SiraRamirez, 1991).

s(k).

Definition 1 is actually a mild modification of the definition by Milosavljevic (1985) who first proposed a necessary condition for the existence of pseudo-sliding mode by replacing the derivative term in the well-known condition lim.- O. It i~ obvi~us that .the condition s( k) --+ 0 is rarely satisfied III practice, since it is impossible for the system states to approach a switching surface sufficiently closely.

As shown in (Yu and Potts , 1992) and (Spurgeon 1992) the condition (3) and its equivalents are only sufficient conditions for the existence of pseudo-sliding mode. The following example demonstrates it.

Example 2: Consider the system Xl (k + 1) xl(k) + 0.0102x2(k) + O.OOlu(k) ; x2(k + 1) -0.005Xl(k) + 1.03X2(k) + 0.0161u(k); s(k) x2(k) + xl(k) = 0 with XI(O) = 3.4; X2(0) 2.6 . The control is chosen by u(k) -8sgn(s(k))JXI(k)J. The system response is shown in Fig. 2. Apparently it is an asymptotically stable pseudo-sliding mode. However s(7) > s(6) and s(10) < s(l1) < s(12) which do not meet the requirement (3). This kind of motion is quite typical of pseudo-sliding modes. Even with a fairly small sampling period, the pattern of motion remains the same as in Fig. 2 except the scale of zigzagging is smaller. Note that when s(k )s( k + 1) > 0, i.e. the system states x(k) , x(k + 1) on the same side of either s(k) > 0 or s( k) < 0, the condition (3) is satisfied. However for those points satisfying s( k )s( k + 1) < 0 the condition (3) is violated.

Albeit the condition (2) actually guarantees the switching surface to be approached and/or crossed , it has bee shown that the condition (1) allows an unstable zigzagging along s( k) = 0 (SiraRamirez, 1991). This can also be demonstrated by the following example.

Example 1: Consider the system xl(k + 1) x2(k); x2(k + 1) x2(k) + u(k); s(k) x2(k) xl(k) with XI(O) = 0.5 ;X2(0) = 0.6. the control is chosen as u(k) = -0 .1s(k) if s(k) > O;u(k) = -20s(k) if s(k) < O. Apparently V's(k)s(k) = -l.ls2(k), -21 s2(k) respectively according to s > o and s < O. The system is unstable (see Fig. 1).

=

=

To force the system state to approach and converge to the sliding surface , the following definition is proposed by Sarpturk et al (1987).

In fact as pointed out by the author themselves, one may not be able to find a discrete-time VSC which satisfies (3).

Definition 2: A system is said to exhibit a convergent pseudo-sliding mode, if in the neighbourhood n. defined in Definition 1 the following condition holds

The zigzagging behaviour is peculiar to discretetime VSC since the discontinuous control is constant over a sampling interval regardless of whether the system state is close to the sliding surface or not. Therefore it is difficult and strin-

1

Js(k + l)J < Js(k)J

(3) 216

gent to force the system state to reach and stay on the switching surface s = O. In practice, it would be considered to be satisfactory if the system states reach a neighbourhood of the switching surface s = 0 and do not leave it. The definition for the existence of pseudo-sliding mode may be modified to suit discrete-time systems:

its dynamics can be considered to be driven by an equivalent control. However in the discretetime case, since the system states are rarely very close to the sliding surfaces, how to define the system in sliding is an open question. Ideally we can find a control u eq (k), which can be called discretetime equivalent control such that s(k) = 0 and s(k + 1) = O. Since the switching frequency is comparatively low with the control signals remaining constant during each sampling interval, the zigzagging behaviour is unavoidable. It is extremely hard to dri ve the system to reach as well as stay on the sliding surface.

Definition 3: The discrete-time dynamic system (1) is said to exhibit a pseudo-sliding mode, if there exists an integer K, such that for all k > K , x( k) E n., it follows that x( k + 1) E n. where n. is defined in Definition 1. Definition 3 includes the case that the system state may never reside on the switching surface. It also may not be necessary that in the neighbourhood n. the system state always approaches the switching surface as long as it does not leave n•. The neighbourhood n. can be defined in many ways. For example, for the linear feedback with switched gain type of discrete-time YSC, n. could be the set {x: Is(k)1 < t 11 xii, t > O}; for relay type of discrete-time YSC, n. could be the set {x: Is(k)1 < t, t > O}.

3

An apparent solution to eliminating zigzagging is to choose the following control for s > 0 x ~ n. for x E n. for s < 0, x ~ n.

= n.

Spurgeon (1992) further questioned the appropriateness of the application of traditional hyperplane design philosophy to uncertain discrete YSC system, and developed a complex scheme based on linear equivalent control structure to supress a class of uncertainties.

=

The design philosophy is described as follows.

Nevertheless the traditional discontinuous YSC still has its merits in controlling discrete-time systems, in particular nonlinear discrete-time systems. Sira-Ramirez (1991) examined some properties of discrete-time YSC. Theorem 1: Suppose the system (1) has relative degree one, i.e. os(f(k , x(k) , u(k)))jou(k) =F ofor all k, and the discrete VSC structure is chosen as u u+(x), u-(x) for s > 0 and s < 0 respectively. then the equivalent control u eq such that s( k) 0, s( k + 1) 0 exists and satisfies

=

(5)

in which u+, u- can be designed based on (2) to drive the system state to reach n. at a comparatively big pace. UO is the control which is designed to dirve the system state as close as possible to s O. is a given neighbourhood of s O.

In contrast to above methods, U tkin and Drakunov (1989) proposed a different approach which uses contracting mapping to guarantee the existence of pseudo-sliding mode.

=

A DESIGN METHOD

=

(4)

First of all, the discontinuous control law u+, uare designed and constrained so that, in limiting case, the system state driven from the points on the boundary of n. does not overshoot n.. This enables the reachability of n•. On the other hand the selection of n. can not be arbitrary. With a control on one side of s = 0, there may exist a domain of attraction towards s = 0 in the state space such that the tendency of the system trajectory is toward the switching surface. The neighbourhood n. should then be selected to be within the intersection of these domains of attraction. This enables the avoidance of divergence phenomena (Yu , 1993). Note that in continuoustime case, the overshooting of domain of attraction may never happen as the control structure is switched as soon as the system state crosses the switching surface. For the system state in n. , the softening (or soft-landing) control uO is designed as follows. Firstly, since the equivalent control can be solved as ueq(k) = -(cT(k)b(k,x(k))ff(k,x(k)), and alternatively using the control foliation property, ueq(k) can also be solved with a properly chosen A(k) such that ueq(k) = A(k)u+(k) + (1 -

The triple {u-(x) , u eq , u+(x)} is said to exhibit a control foliation property. The equivalent control plays an important role in the theory of YSC. When a system is in sliding, 217

zigzagging behaviour which is the major problem in discrete-time VSC. The softening control is proved to be effective and releases some of the limitations of existing conditions.

5

REFERENCES

.1

DeCarlo , R .A., Zak , S.H. , Matthews, G.P. (1988) . Variable structure control of nonlinear multi variable systems: a tutorial , Proceedings of IEEE, 76 , 212-232.

Fig. 3. The system response with softening control

A(k))u-(k), 0 < A(k) < 1. Ifuo is chosen as

Furuta, K. (1990). Sliding mode control of a discrete system, System Control Letters, 14, 145-152.

(6) where -

A(k)

=

{ (1 - s'i~l))'\(k) (1 - (1 - s'l~l))A(k))

<0 (7) for s > 0 for s

Milosavljevic, C. (1985). General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable structure systems, Automat. Remote Control, 46 , 307-314.

where S- (k), S+ (k) are the functions obtained along the boundaries of n. on sides s( k) < 0, s(k) > 0 respectively. With control (6), one can easily find

s(k + 1)

s(k)

= S±(k) s

±

(k

+ 1)

(8)

where s±(k + 1) = cT(k)(f(k, x(k)) + b(k, x(k))u±(k)). Apparently one can see that in n. x(k + 1) driven from x( k) with uO( k) is always closer to s 0 than that driven by u-(k) or u+(k) . The closer x(k) to s = 0 is, the softer the control is, tending not to overshoot s = 0 too far. When s(k) = 0, then s(k + 1) = O. With the softening control the system states approach s = 0 with litter chattering. Apparently with this control one does not have to have Is(k + 1)1 < Is(k)1 when s(k)s(k + 1) < O.

=

Example 3: Consider the system in Example 2, is chosen as {x: Isl < 0.3 11 x 11}. The initial condition is the same as in Example 2. The system response with control (5) is depicted in Fig.3. The system trajectory appears to approach s = 0 smoothly, indicating the softening effect of uO.

n.

4

Kotta , U. (1989). Comments on the stability of discrete-time sliding mode control systems , IEEE Trans . Auto. Control, 34 , 1021-1022.

Sarpturk , S.Z., lstefanopulos, Y., Kaynak, O . (1987). The stability of discrete-time sliding mode control systems , IEEE Trans. Auto. Control, 32 , 930-932. Sira- Ramirez , H. (1991) . Nonlinear discrete variable structure systems in quasi-sliding mode , Int. J. Control, 54 , 1171-1187. Spurgeon , S.K . (1992) . Hyperplane design techniques for discrete-time variable structure control systems , Int. J. Control, 55 , 445-456. Vu. X.H. and Potts, R.B. (1992) . Analysis of discrete variable structure systems with pseudo-sliding modes , Int. J. Syst . Sci. , 23 , 503-516 . Vu , X.H. (1992). Chaos in discrete variable structure systems , Proc. 31st IEEE CDC, 2 , 1962-1863 ; (1993) . Discrete variable structure control systems , Int . J. Syst . Sci. , in press . Utkin , V.I. and Drukunov , S.V . (1989). On discrete-time sliding mode control Proc. IFAC Symp . on Nonlinear Control Syst ems , 484-489 , Capri , Italy.

CONCLUSIONS

The up-to-date development of design methods for discrete-time variable structure control has been examined. It has been shown a proper chosen discontinuous discrete-time VSC can eliminate the 218