DSP based integral variable structure control for DC motor servo drivers

DSP based integral variable structure control for DC motor servo drivers T.-L. Chern J.-S. Wong

lndrxiny rerms. DC motor .serro drraer, Microprucessor. Slidiny morion

Abstract: The design and implementation of a DSP microprocessor based D C motor servo driver are presented. The integral variable structure control (IVSC) approach is proposed for the outer loop of the driven system. Conditions are derived which ensure the existence of a nonideal sliding motion for the IVSC approach and also prove that the described motion in the nonideal sliding region will be close to the ideal sliding motion. Simulation and experimental results show that the proposed scheme can achieve an accurate velocity/position servo tracking result and is robust to plant parameter variations and external load disturbance.

approach is a robust and practical control law for motor servo drivers. To suppress the chatterings in the IVSC approach, the sign function sign (U) in the control U is replaced by a modified proper continuous function as described in References 1-3. In this situation, the control U can not satisfy the condition which is to assure the existence of a ideal sliding motion and stated as uu

< 0 for all time

uue

where E is a small positive constant. Thus, a nonideal sliding motion within the range of E will appear. This paper also demonstrate that, if the solution of the ideal sliding motion is asymptotically stable, it will be close to the solution of the nonideal sliding motion within the range of E in infinite time interval starting with the time when the sliding mode begins.

Dositionivelocitv feedback Fig. 1 tl1or.L diayram conrrol . s w e m s

(f

I V S C DC motor c i d o d y posirirrn s ~ r c u

control loop is a current controlled pulse width modulation (PWM) voltage source inverter (VSI). which is widely applied in high performance servo drivers. The outer loop controller is designed to achieve a fast and accurate servo tracking response under load disturbance and plant parameter variation. To achieve these requirements, an IVSC approach is proposed for the outer loop controller. The IVSC approach is composed of an integral controller to achieve a zero steady-state error and a variable structure controller to enhance the robustness. With this special scheme, it yields improved performance when compared to conventional VSC and linear approach 11-31, However, References 1-3 are based on computer simulations to demonstrate the potenial of the IVSC approach. In this paper, simulation and experimental results are presented to explain that the proposed i IEE. 1995 Paper 2047D ICX. CIII. first received 3rd January and in revised form 30th M a y 1995 The author5 are with the Department of Electrical Engineering, National Sun Yat-Sen University. Kaohsiung Taiwan 80424. Republic of China

444

Integral variable structure system (IVSS)

2

The proposed IVSS is described as [ 1-31 : i = l , ..., n

(34

X,= - z a , X , + b U - f

(36)

X,=X,+,

,=I

(34

Z=r-X,

where X , is the output signal, r is the input command,f is the disturbance, u L ,h are the plant parameters. The switching function a is given by U

=

c,(X,

-

K,Z)

+ ccix,

(4)

r=2

in which ci and K , are constant and c, From eqns. 3 and 4, one has

=

1.

n

h = -clK,(r-

XI)+

cci~~,xi i=2

-

c u i X i + hU -f i=1

1b.k Proc-Control Theory Appl., Vol. 142, No. 5. September 1995

Thus, the conditions for satisfying the inequality (eqn. 1) are

Let a L. = a ot + A a i i = 1, ..., n

+

where a? and bo are nominal values, Aai and Ab are the associated variations. Let the control function U be decomposed into U

+

+

-

f o r i = l , ..., n - 1

Ueq + AU

=

+

ai < Inf {[Aai - aPAb/bo ci-,Ab/bO - c,(c,- I - a:)(l Ab/bo)]/b} bi > Sup { [Aai - ayAb/bo ci- ,Ab/bo - u:)(l Ab/bo)]/b}

A b > -bo

bo>O

b=bo+Ab

(6)

here U,,, called the equivalent control, is defined as the solution of the Droblem ir = 0 under ,f = 0.. a;. = U:, b = bo. That is

-

K,Z)

cixi

-

a, + 1 < Inf I- N,/bI = {L?"+l

> SUP C-N,/bl

+ 1, be chosen as

i = 1, . . . , n Let $ L ,

(8)

i=2

(13)

+ a, - c,-,l/b}

and

yn+l

-cl(X1

c o = O (12)

+ a, - c, - Jb}

a, < Inf { [Aa. y n = {a. > SUP {[A..

(7) In the sliding motion, U = 0, one can obtain

and

$, = a, =

-P,

Then, the control function can be represented as

Substitution of eqn. 8 into eqn. 7 yields:

"-1 "-1

clK,(r

+

-

clK,(r

x,)- i = 2 c i + l X i

c a y x , + (c,-

-

XI) -

1 c,-,X,

,=2

"-1

-

I

a:)

i= 1

"-1

x [cl(Xl - K , Z )

+ 1ciXi]}/bo i=2

The function A U is constructed as

jZ)

+ iC= Z c i X i]},/bo n-l

+ (yl I X l - K , Z / + i = Z Til Xil + Yn+l)

+ CYiXi +

AU = Yl(X1 - K j Z )

-K

c1(Xl

(9)

(10)

i=z

x sign (U)

(15)

where

where

+ ci-,Ab/bO

Yi < -Sup I [Aai - aPAb/bo if (XI - K,Z)a < 0

if X , a > 0 i if X i o < 0

,;{

Yi =

=

- cAc.-

-

+ Ab/bo)]/b I

a:)(l

f o r i = l , ..., n - 1

2, . . _ ,n

Y n< -Sup I(Aa.

and

+ a, - cn-J/bI

and

c o = O (16) (17)

and yn+i< -SUP lN,/'bl

(18)

Under ideal sliding motion, the system described by eqn. From eqns. 5-10, one can obtain

3 can be reduced to [ 1-31 :

+ ayAb/bO+ cl(cn- - a:) + Ab/bo) + bYlI(X1 - K , Z ) U

iro = [ -Aal x (1

+

"-1

{[

Xi = xi+, i

I

- Aa,

+ a: Ab/bo - e,- ,Ablho

t=z

+ CAC.-~

-

+ Ab/bo) + bY,]X,a)

a:Xl

+ [ - Aan + (e,+ [NJ +

-

a.")

+ bY',]X,

U

11.

N , = { - K, Z(Aal

-

ayAb/bo)

+ Ab/bo[clKj(r - X l ) l -f) I E E Pror -Control Theory A p p l . Vol 142, No 5 , September l Y Y 5

1, ..., n

-

2

(19)

(11)

x"-l= - i = 1 CiXi + c , K , Z

(20)

i=r-Xl

(21)

The eigenvalues of this system can be set arbitrarily by and K,. Let the choosing the values of clr ..., desired characteristic equation be S"

where

=

"-1

+ r,S"-' +

_ ' '

+ r, = 0

Then c, and K , can be chosen as =

ai f o r i = 1, ,.., n

-

1 445

The solution of eqn. 24 is of the form

and

K,

3

= z,,/z.-

J1'

X(C) = eA'X(0)f

I

(25)

eA'Br(t - 7) dt

In the nonideal sliding motion, the switching function CT is not equal to zero and I U I < E , so that the state X , can be described as

Chattering consideration

Since the control function U described by eqn. 15 contains the sign function sign U, direct application of such a control signal to the plant may give rise to chatterings. To reduce the chatterings, the sign U can be replaced by a smoothing function as [l-3, 81:

-cl(X,

-

K,Z)

1

Substituting eqn. 26 into eqn. 3, X(c) = AX(t)

1

- i = 2 cixi + U

+ Br(t) + H u

(27)

where H = [O

where 6 is chosen as

6

=

+ 6, IX,

6,

-

"-1

cIK,(r- X,) -

+

x(t)= eA'x(0)

II&S

[ + (

-

K,Z)

,=2

When the control function U is generated in according to eqn. 23, the system can not obtain an ideal sliding motion. The conditions assuming the existence and reachability of a nonideal sliding motion is described in inequality (eqn. 2). The control function must be modified to assume that the inequality (eqn. 2) is satisfied, so a nonideal sliding within the range of E will appear, or IuI < E . Now let us demonstrate that, if the solution of the ideal sliding motion is asymptotically stable, it will be close to the solution of the nonideal sliding within the range of infinite time interval starting with the time when the sliding mode begins. While in the ideal sliding motion, the system can be described by eqns. 19-21 and rewritten in matrix form as

A X ( [ )+ Br(t)

(24)

where

0 1 .. . c2

+ Hu(t - z)]

dt

... ... .. . .'.

(28)

dr

l

The switching function U can be chosen so that the system (eqn. 24) is stable. Since, in the nonideal sliding, lu(k)l < E is valid, the //eA*jl and l f o eA'H drII are bounded on an infinite time interval for asymptotically stable system and the initial conditions x(0)and g(0)can always be selected in compliance with

llm) mll

PE

-

where p is a positive number, there exists a positive number N such that