Variable structure systems with chattering reduction: A microprocessor based design

0005-1098/84 $3.00 + 0.00 Pergamon Press Ltd. ~' 1984 International Federation of Automatic Control

Automatica, Vol. 20, No. 1, pp. 133- 134, 1984 Printed in Great Britain

Technical Communique

Variable Structure Systems with Chattering Reduction: A Microprocessor Based Design* MARTIN

D. ESPAiqA,~" ROMEO

S. O R T E G A ~ "

and JUAN

J. E S P I N O ~ :

Key Words--Microprocessor control; d.c. motors; variable structure control.

Almtraet--An alternative approach for the synthesis of variable structure control algorithms with chattering reduction is presented. The proposed method allows the designer to simultaneously satisfy the conflicting requirements of short sliding plane reaching time and transient behaviour degradation due to the effect of chattering. A new algorithm is obtained for single-input nth order systems. The strategy is illustrated with a microprocessor-based design for the control of a d.c. motor. Transient behaviour is shown to be improved and good robustness properties are demonstrated.

with$i=xi, ifxis>0and$i=fli, when lim J < 0;

lim J > 0.

s~O +

cti>~k °>lfli,

i=l ..... n-1

F(Iki,A) = {x~Rnl~,,(A)x~S},

i = 1. . . . . 2~ i

(7)

where **~(t) is the state transition matrix of the linear stationary closed-loop system (1) with the ~ki feedback law. Clearly, the F(~i, A) are linear subspaces of dimension n - 1 since for X ~ F ( O t , A), cTtl)~,i(A)x = 0. Let us now define the sets At(A) =- {X E Rnltl),/,,(~)x E S, 0 ~< ~ ~ A},

i = 1. . . . . 2o-t (1)

(8)

It is easily seen that the boundaries of each A~(A)set are the linear subspaces S and F(#i, A). The ¢°-zone, as a function of the interval A, (whose selection will be discussed later), is now defined as;

(2)

2--

~,,°-zone (A) ~- ~ Ai(A).

It can be proved (Emilyanov, 1967) that S is a S.P. for a control u defined as ~b,-lX,- t

(6)

The proposed new switching law chooses the ~k°-structure defined by (6) near the S.P. The closer to the S hyperplane at which the switch to O°-structure occurs, the more the motions will be similar to the desired ones. In order to achieve this, a new switching zone called '~,°-zone' is defined in the following way Let

and the hyperplane as

u = -~1xl .....

Co = 0.

2. A n e w s w i t c h i n g law .[or V S C A

S x ~ R " [ a = c l x l + "'" + c . x . = cTx = O;

c. = 1.

1 + cian)/b,

The limiting values ~o are the feedback gains which make the subspace S an invariant subspace of system (1) (Wonham, 1974). Following the definition of the ~bt, the state feedback vector ikT = (~1 . . . . . ~'n-1) ranges over a set of 2~- l values corresponding to the same number of possible structures generically referred as ~k-structures.

1,...,n - 1 - ~ , , x . + bu

(5)

with

VARIABLE structure control algorithms (VSCA) have proven to be, in view of their robustness, useful tools in plants with imprecise knowledge of their parameters [see Klein and Maney (1979); Erscheler and co-workers (1979) for applications, Emilyanov (1967); Utkin (1977) for the main theoretical results]. The main VSCA feature is to drive the state trajectory toward a sliding plane (S.P.) previously determined by computing the feedback control structure. This leads to velocity vectors having a non-zero component orthogonal to the S.P. near it and trajectories continuously crossing the switching hyperplane. It is desirable to simultaneously: (a) reach the sliding plane fast; (b) maintain the trajectory close to it; and (c) reduce the number of switchings between structures (chattering). However, reaching the sliding plane fast implies also a fast departure, unless a frequent switching between structures is allowed. In a digital control scheme the trajectory will depart from the S.P. increasingly with the sampling time. The proposed strategy corlsiders a zone in the vicinity of the S.P. in which the feedback gain is adjusted to obtain a state velocity vector "almost parallel' to it. Let the system be described by

.;c. = - ~ 1 x 1 , . . . ,

(4)

s~O -

~b° = ( c i - 1 - a, - c i c , _

i=

1,

Conditions (4) are satisfied (Emilyanov, 1967) if

1. I n t r o d u c t i o n

-xi = xi+ l,

i f x i s < 0 , i = 1. . . . . n -

(9)

The following control strategy is now proposed: (1) measure x; (2) determine, according to the definition, the corresponding ~,j value; (3) if x e At(A), let ~, = t#° otherwise, ~, = ¢'iTo determine whether or not x(t) lies in the ~k°-zone, we observe that since ~ (A) ~ 1, when A --, 0, for relatively small values of A, the angle between the generating vectors of the linear subspaces S and F,t(A ) (respectively e, ~T#(A)¢) is smaller than n/2. In this way an easily implementable test will be: if sign (cr~,~(A)x) # sign (cTx) then x(t) belongs to Ai and the ~ 0 structure must be chosen. Having in mind that during a sampling interval T, the structure is unchanged, it becomes evident that A ~ T~ ensures tha the switching to ~p0 occurs at the last sampling time before crossing the S.P. In this way, a sampling moment closer to S would not exist before reaching it.

(3)

* Received 11 November 1980; revised 1 May 1982; revised 16 June 1983. This paper was recommended for publication in revised form by associate editor H. G. Kwatry under the direction of editors H. A. Spang, III and A. H. Levis. "t"Instituto de Ingenieria UNAM, C.U. Apdo. Post. 70-472, Coyoachn 04510, Mtxico, D.F. :~lnstituto de lnvestigaciones Eltctricas, Apdo. Post. 475, Cuernavaca, Mor. Mtxico. 133

134

Technical Communique

Two extreme conditions arise: (1) A ---*0, then the probability of falling inside the zone tends to zero, hence the trajectories will be very much like the ones obtained with the conventional strategies; (2) A --, ~ , in this case, for a relatively large T~, there will be trajectories diverging strongly from the sliding plane that will continuously leave the ~°-zone. There is no systematic way of defining the optimal value of A, given that it depends on the trajectories; therefore, its selection implies a trial and error procedure. 3. Application to a microprocessor-based d.c. positioner

I

I

I

I /

t

I

I

I

The proposed algorithm was implemented on a MEK6800D2 microcomputer for the control of a fractional armaturecontrolled d.c. motor with a gear coupling. The plant is modelled by ihe transfer function kw 2 ....... u(s) s(s + 2~w,)

t

i

x(s)

(10)

where x, u and k represent shaft position, armature voltage and loop gain, respectively. The phase equations and the corresponding sliding mode equation are I

~

i

~

I

i

-{:1 = X2

"{~2

=

(11)

- 2 ¢ w , x 2 + kw2u

s = clx 1 + x2 = 0.

I

t

)

I

i

I

I

I

I

I

t

i

)

I

-2 2ii21

(12)

The control takes the form u = - ~kxl and the ~ and/3 values of qJ, according to (5) and (6), must satisfy ~> cl

(2~w, - q ) kw2

~b° ~> fl

(13)

The F~,~(A) linear hypersurfaces introduced in section 2 are defined by their orthogonal vectors: V ( ~ ) = ~0~(AJc,T i = 1,2 with ~b1 = ~, ~k2 = fl F,(A)

=

I.',:6 R2[VI(x)x1

+

V2(~)x

2 = 0t~

G(A) = {x~ R21V,(fl)xl + V2(fl)x2 = 0 I.

(15)

4. Experimental results An off-line identification resulted in the following numerical values: w2 = 320 rad 2 s - 2, 2¢w, = 4.76 rad s - ~, k = 0.25 V V - 1. A 2.5 s ~ value for the sliding mode slope c was chosen leading to the value $0 = 0.076; :t and fl were chosen to be; ~ = 1 ; fl = - 1. With L = 50ms; A = L/2 = 25ms one has

Vx( -- 1) = (4.46, 0.97)

as the orthogonal vectors for F,(A) and ra(A) surfaces obtained in the following way vT(~/i)

= cTeA(0i)A,

~1

=

1, I]/2 =

(bl

(14)

In order to follow a setpoint (SP), the error variable e = x - SP and its derivative g' = .'~are used in the corresponding algorithm.

vT(1 ) = (0.575, 0.92);

(a)

FIG. 1. Velocity, control and position for the step input response without (a) and with (b) the new zone (K = 0.25).

-- l

with A0,O~)the state closed-loop system matrix corresponding to the ¢~ structure and cT = (2.5, 1 ). In order to avoid unnecessary control action and possibly limit cycling (because of the conversion errors and plant nonlinearities), one addition is, when the error vector is within a prescribed neighborhood of the origin, the feedback gain is set to a constant value chosen such that the motor overcomes the static friction (zero steady-state error). The algorithm was tested for a step-input and the results are shown in Fig. 1. (a) without the qJ°-zone and (b) with the ~k°-zone. The behaviour of the control signal in Fig. 1(b) clearly indicates that a sliding motion occurs as designed.

A considerable improvement in the reduction of chattering is obtained with the proposed scheme. Also, the quality of the transient behaviour is evidently improved. From the figures, it is clear that the time required to reach the. sliding mode with the • . x conventional VSCA ~s larger than the t~me needed to enter the tp°-zone with the new control. To test the parameter of variation insensitivity in the proposed strategy, the gain k changed to k = 1 with no appreciable deterioration in the transient behaviour due to the fact that the reaching time to the new zone was very small. Acknowledgements-- The authors wish to express their gratitude to Dr Roberto Canales for insightful and helpful discussions. ReJerences Klein, C. A. and J. H. Maney (1979). Real-time control of a multiple-element mechanical linkage with a microcomputer. I E E E Trans Ind. Elect. & Cont. Inst., IECI-2Ai, 227. Erschler, J. and co-workers (1974). Automation of hydroelectric power station using variable-structure control system• Automatica, 10, 31. Utkin, V. I. (1977). Variable structure systems with sliding modes. I E E E Trans Aut. Control, AC-22, 212. Emilyanov, S. V. (1967). Variable Structure Control Systems. Nauka, Moscow. Wonham, W. M. (1974)• Linear multivariable control: a geometric approach. Springer, Berlin. Ortega, R. (1960). Variable structure systems, sliding modes and control. Research and Advanced Studies Center, IPN technical report M6xico, D.F. (in Spanish).