Sliding Mode Control in Discrete Time Linear Systems

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

SLIDING MODE CONTROL IN DISCRETE TIME LINEAR SYSTEMS Wu-Chung Su, S.V. Drakunov and U. Ozgiiner The Ohio State University. Department of Electrical Engineering. Columbus, OH 43210, USA

Abstract. Due to the sample/hold processes in a discrete time system, a number of properties, which are valid in continuous time systems, no longer hold. For sliding mode control, the discrete time matching condition will not be satisfied in general even if the continuous time version holds. Yet one still can maintain the states in the vicinity of the sliding manifolds up to at least 0(T2) at each sampling instant, where T is the sampling period. Moreover, for disturbances with sufficient smoothness, the accuracy can be promoted to 0(( ~ )2) (r being an integer) if additional switchings in between measured samplings are allowed. It is seen that the system uncertainties due to exogenous disturbances can be rejected within 0(T2) to 0(( ~)2) accuracy and that the effect caused by parameter variations will be attenuated to zero asymptotically.

Key Words. Sliding mode; matching condition; disturbance prediction; chattering reduction; variable structure systems

on that concept with robustness. Our approach allows to keep the state in the O(T2) vicinity of the sliding manifold even in the presence of disturbances. Both parameter variations and exogenous disturbances are considered. Only continuous time matching condition and sufficient smoothness properties are required, which are reasonable constraints in most applications.

1. INTRODUCTION The switching problem in continuous time sliding mode control design has two aspects: switching frequency and magnitude. Although high frequency switching in control is theoretically desirable from the robustness point of view, it is usually hard to achieve in practice due to physical constraints on the system. Even if it is possible, the unmodeled high frequency modes will likely be excited, which in turn deteriorates the performance.

Since the system is discretized in time by the sample/hold processes, the control signal space is shrunk from (L~,Tl)m to Rm, m being the number of control inputs. Therefore, the controller will inherently be less capable then the continuous one. Sliding mode is originally a continuous time concept, and at first glance, it may not be clear that it will retain its properties in discrete time. Nevertheless, the advantage is that we are able to overcome the chattering problem without losing the robustness property.

Sliding mode control for discrete time has been considered for years (Miloslavljevic, 1985; Sarpturk et al., 1987; Furuta, 1990; Sira-Ramirez, 1991). In such systems the switching frequency is limited by T- 1 . The use of a discontinuous control law (typically sign functions) leads to chattering in the O(T) vicinity of the sliding manifold s(z) = O. In order to alleviate chattering it was proposed to use discrete time equivalent control in the prescribed boundary layer, whose size is defined by the restriction applied to the control variable (Utkin and Drakunov, 1992). This approach results in the motion in O(T2) vicinity of the sliding manifold. The main difficulty arising in implementation of the proposed control law is that one needs to know the disturbances for calculating the equivalent control. Lack of such information leads to O(T) boundary layer, the same as with discontinuous control.

2. INVARIANCE CONDITION Consider a continuous time linear, time-invariant system (1) x=Ax+Bu+Ej, where x is an n x 1 state vector, u is an m x 1 control vector, and j is an I x 1 deterministic disturbance vector. A, B, E are constant matrices of appropriate dimensions. Assume u is applied through a zero-order-hold. The discretized version of (1) can be formulated

In this paper we introduce a new control based 267

as

(3)

The above lemma states the necessity of the continuous time matching condition, however, the sufficiency is not confirmed. It is important to note that the sampled linear system does not preserve the disturbance rejection property. This is due to the zero-order-hold which is imposed between consecutive sampling instants to the control channels during the sampling process. As a result, the perfect sliding property is destroyed.

where C is a m x n matrix and is chosen to meet some stability or performance criterion. To maintain the state on the surface (3) at the (k + l)th sampling instant, the equivalent control law is given by

Although perfect sliding mode is not possible, one can still maintain the states in the vicinity of the sliding surface and retain the satisfying disturbance rejection character as stated in the following lemma:

= Il>Xk + rUk + dk , = eAT, r = eA>'d>'B, Xk+l

where Il>

f:

eA>' EJ«k

f:

+ I)T -

(2) dk

>.)d>..

The sliding surface is Sk

=

CXk

= 0,

Lemma 2 If the continuous time matching condition (6) holds and jet) is bounded, there exists a control law Uk such that the discrete time sliding manifold can be reached with O(T2) accuracy.

assuming cr is invertible. Here dk represents the lumped effect of the disturbance J(t) to the system in the time interval kT ~ t < (k + l)T. If the system behavior does not depend on J(t), we say the control system has the property of disturbance invariance and the disturbance is said to be rejectable. Given that J(t) is bounded, the integration of J«k + l)T - >.) multiplied by another bounded function eA>' in the sampling interval results in a magnitude of O(T) in each component of d k . If no further information is provided and we simply implement the controller U with state feedback alone, the next state Xk+l will not be able to reach the sliding surface exactly, but will result in

Proof: With (6), there exists an m x I matrix that E = Bn. Choose Uk = _(Cf)-l nJ(kT) to achieve Sk+l = O(T2) .

n such

3. EXOGENOUS DISTURBANCES The major difficulty in maintaining sliding mode for a discrete time system is that the exogenous disturbance d k is essentially independent of the state Xk; i.e., even if we achieve Sk = 0 at the present moment, it is still not guaranteed that Sk+l = 0 since the unknown d k will force the state out of the sliding manifold. In addition, from Lemma 1 we see that the range of the control matrix r cannot cover the whole signal space of d k in general. Therefore, perfect sliding mode is impossible in spite of the knowledge about the disturbances. However, within a certain tolerance, we are able to achieve satisfying system performance by steering the states as close to the sliding manifold as possible. This includes a predictor for d k and proper choice of the feedback compensator. The rudiments of a ~ne-step disturbance predictor were given by Ozgiiner and Morgan (1985) in the context of a robotics application. Here we rigorously analyze this issue.

(5) We will show later that if d k is known, it is still not rejectable by the control in general unless the discrete time matching condition holds. Unlike continuous time VSC, the matching condition rank[B, E] = rank[B] (6) and the boundedness of disturbances are not sufficient for the controller to maintain sliding mode. Rather, it requires some more knowledge about J(t) so that the current disturbance, dk , can be predicted with accuracy to some extent .

3.1 The Disturbance Predictor

It is advantageous to consider the problem in discrete time because through the measurement of the states, the past values of the disturbances can be determined and this will provide the knowledge about the future ones. However, the sampling process will also result in a requirement or a condition stronger than matching.

The past values of the disturbances d i , i = k 1, k - 2"" can be computed exactly from the state and control history by considering the discrete time system (2). If J(t) possesses some continuity attributes, there will exist a strong correlation between the past and future disturbances. If J(t) is bounded, then dk = O(T).

Lemma 1 For disturbance invariance in the sampled linear system (2), it is necessary that the continuous time matching condition (6) holds.

If furthermore ence

lO(T i ), i = 1,2, ... will apply to all entries in a vector.

268

jet)

is bounded, then the differ-

1:

is of order OCr). In the same manner, for 1(t), we have the prebounded f(t), f(t), ... , diction for die

where fr = for eA>'d>.B. The intersample disturbance ~ can be estimated by

r-

r-l

q-1 die = ~)-1)i-1(qi1)dle_i

d1 = (2: ~1:)-ldle

(7)

j=O

i=1

with O.«~)2) error. Each intermediate state value x~ will be estimated by

with O(Tq) error. If one desires to perform p-step ahead prediction, define

- {d.dJ.

dJ· --

J

if j if j

~

k- 1

~

k

(8)

with z~ = XIe . To reach sliding mode with O«~)2) in the intersample period and O(Tq) at each sampling instant, choose

then

q-l

d1:+p-l

= L(-1)i-l(qi 1)dle_i+P_l + O(Tq)

. ; (Cf r )-lC,y"l. -i (Cf r )-lCd-i/.. 1= 0, ...T u,,=....,rX,,1 1 -2 r-l-; . 1 - . u~- = -(Cfr )- C[~Xk + ~~=O ~-r-frU~- +dk.

i=1

Therefore we are able to predict dj, j accuracy up to O(Tq) if f(t) E C(q-l).

~

(9) k with

We have

Equation (9) is actually an incomplete model for disturbances with accuracy depending on the degree of smoothness of f( t). The past values are to accumulate the knowledge base of the unknowns. If f(t) can be measured exactly, such as a reference input signal to the system, no prediction for die is necessary.

4. PARAMETER VARIATIONS Consider a linear system with parameter variation

3.2 Sliding Mode with O(Tq) Accuracy It is understood that the discrete time sliding surface (3) can not be reached exactly due to the effect of unknown disturbances. Nevertheless, it is possible to reach a higher accuracy if appropriate disturbance prediction process is employed.

:i: = (A + .:lA)x + Bu

The unknown .:lA may be treated as an external disturbance in the sampled system so that

Theorem 1 If f(t) E C(q-l), q ~ 2, and (6) holds, then the control law UIe = _(Cr)-IC~XIe­ (Cf)-ICdle will lead to sliding motion such that S1:+1 = O(Tq) for k ~ q and XIe in the O(T) boundary layer of the sliding surface (3).

where

hie =

As to the inters ample system behavior, an O(T2) error is inevitable. This is an inherent limitation of the discrete time controller since the control signal space does not cover the disturbance range space. To further increase the accuracy in between samples, more degrees of freedom should be added to the control.

iT

eA>'.:lA«k+ l)T - >.)x«k + l)T - >.)d>. (13)

In general, it is impossible to achieve invariance of .:lA for the sampled system (12), nevertheless, the matching condition is still necessary for the controller to reduce the effect of .:lA. Assume .:lA to be smooth, then x(t) is smooth in the time interval kT ~ t < (k + l)T but x(kT) does not exist due to the discontinuity of the control u(t) at each sampling instant t = kT, therefore hie can only be predicted with accuracy O(T2) . This leads to the following lemma.

3.3 Additional Switchings To improve accuracy in between consecutive samples, we insert r additional switchings by letting

u(t)

(11)

= u~, (k+i)T ~ t < (k+ i + l)T, i = O, ... r-l r r

The equations for the expanded system becomes XIe+1

= ~XIe +

Lemma 3 For a linear system with parameter variations (11), the effect of .:lA(t) can be reduced to O(T2) if the following conditions hold:

r-1

2: ~ r-~-; fru~ + die

(10)

i=O

269

sliding modes: first, the chattering of the control variable which excites the neglected high frequency modes and in many cases can not be allowed by the physical nature of the actuator; second, direct discrete time implementation of the switching control law leads to additional chattering caused by the sampling delay. The solution to these issues is obtained by using the disturbance prediction algorithms proposed here. The concept of discrete time sliding mode provides the motion in the 0(T2) vicinity of the manifold. Due to the sample/hold effect in the sampled data system, the disturbance generally can not be rejected completely even the matching condition in discrete time is satisfied; nevertheless, it is shown that for sufficiently smooth disturbances with matching condition in continuous time, the desired 0(T9) of deviation from the sliding surface can be achieved at every sampling moment and 0(T2) in between samples. To obtain higher ~V) we should have additional accuracy s = r - 1 switchings of control during that interval. The results are summarized in Table 1.

1. ~A and ~A are bounded.

e.

the matching condition holds, i. e. rank[~A(kT)e,

B) = rank[B]

e

where is an n x (n-m) matrix with the columns forming a basis of the subspace nul/CC).

9.

is at least in the O(T) vicinity of the sliding surface; or sic ~ O(T).

:Cl:

Although higher order prediction of hlc is not possible, it is not necessary from stability point of view. If one designs the sliding surface for an adequate stability margin, 0(T2) perturbation of the closed-loop poles will still be within the tolerance of the performance criterion unless an illconditioned system is encountered. Another important merit of the system (12) is that the disturbance hlc is a function of states. It attenuates as :Clc approaches zero. In fact we will show that the effect caused by parameter variations tends to zero if appropriate control is applied as stated in the following theorem.

0«

6. ACKNOWLEDGEMENT This research reported in this paper is supported by AFOSR, Grant No. F49620-92-J0460.

Theorem 2 Given the three conditions In Lemma 9, the disturbance hl: resulting from

7. REFRENCES

~A(t) attenuates to zero asymptotically if the fol-

Drakunov, S. V. and V.I. Utkin (1992) Sliding Mode Control in Dynamic Systems. International Journal of Control, Vol. 55, No. 4, pp. 1029-1037.

lowing control is applied:

Drazenovic, B. (1969) The invariance condition in variable structure systems. A utomatica, Vol. 5, pp. 287-295.

5. CONCLUSION

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

The main characteristic feature of systems with sliding modes is the motion on the manifold which can be reached in finite time (Drakunov and Utkin, 1992). During that motion, the continuous time finite dimensional systems with discontinuos control variables possess the property of disturbance rejection. However, implementation proves to be more difficult than expected. We can mention two crucial facts that disappoint one trying to implement real systems with

Known f(t)

Unkown f(t)

Parameter variations

Single switching Sl:+1 0 set) < 0(T2) Slc+l 0(T9) set) < 0(T2) Sic ~ 0('£'1) i 2,3" "

= =

=

Milosavljevic, C. (1985). General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable structure systems. Automatica, Vol. 46. pp.307-314. Morgan, R. G. and U. Ozgiiner (1985) A decentralized variable structure control algorithm for robotic manipulators. IEEE Trans. Robotics and Automation, Vol. RA-I, No. 1, March. pp. 57-65.

Additional switchings Slc+l 0 s~ 0«~)2) Sic+! 0(T9) s~ 0«~)2)

= = = =

N.A.

Sarpturk, S. Z., Y. Istefanopulos, and O. Kaynak (1987) On the stability of discrete-time sliding mode control systems. IEEE Trans. Automat. Contr., Vol. AC-32, No. 10, October. 930-932. Sira-Ramirez, H. (1991). Nonlinear discrete variable structure systems in quasi-sliding mode. Int. J. Control, Vol. 54. pp. 1171-1187. Utkin, V. I. and S.V. Drakunov (1989) On discretetime sliding modes. Preprints of IFAC Workshop on Nonlinear Control, Capri, Italy. Young, K. D. and S.V. Drakunov (1992). Sliding mode control with chattering reduction. Proceedings of the 1992 American Control Conference, Chicago,IL, June. pp . 1291-1292.

Table 1: Sliding surface errors using different disturbance estimations

270