Robust Intelligent VSS Control1

ELSEVIER

Copyright © IFAC Large Scale Systems: Theory and Applications. Osaka. Japan. 2004

IFAC PUBLICATIONS www.elsevier.comllocarelifac

ROBUST INTELLIGENT VSS CONTROL

1

Bemardo Rine6n Marquez·, Alexander G. Loukianov·· and Edgar N. Sanehez· • CINJ ESTA I' IPN Unidad Glladalajara, ApartadoPostal31-438, Guadalajara, Jal. 44550, Mexico. e-mails: brincon[[email protected] ··Campus Universitario Los Lagos, Universidad de Guadalajara. Enriqlle Diaz de Leim sin, Col. Paseos de la Montana, Lagos de Moreno, Jalisco, )"iexico, e-mail: [email protected]

Abstract: In this paper, a method of robust control based on sliding mode and fuzzy logic techniques is presented. It combines hierarchical high gain control with fuzzy logic to design a discontinuous control for nonlinear multivariable systems in order to eliminate chattering in presence of disturbances. The proposed approach is then used to design a robust controller for a stepper motor. Simulation results are presented to illustrate the applicability of the approach. Copyright © 2004 IFAC Keywords: Sliding mode control, nonlinear system, disturbance rejection, fuzzy control.

In this paper we propose a new control scheme using combination of block control (Loukianov, 1998), sliding mode control, and fuzzy logic control (Driankov et. a/l. 1996) techniques for a class of nonlinear systems that can be presented in the so called Nonlinear Block Controllable form . The first technique is used to design a nonlinear sliding mode manifold with desired sliding mode dynamics . These dynamics content hierarchic high gains to ensure stability of the sliding mode motion in the presence of unmatched perturbations. The sliding mode controller is then designed to reject the effect of matched perturbations. Finally, based on the fuzzy logic technique, the controller gains values are changed in a such manner the amplitude of chattering in the closed-loop system is reduced . Note that the sliding mode fuzzy logic controller was investigated for the system with matched perturbations only .

I. INTRODUCTION A simple and good technique for robust control is the sliding mode control (Utkin, 1992, Alexik and Vittek, 1994) , since it is composed of two clear steps. First, selection of a sliding surface such that a sliding mode dynamics on this surface is satisfies design specifications, at least is stable and robust in presence of perturbations. Second, design of a discontinuous control which stabilizes the projection motion of the closed loop system on the sliding surface subspace.

It is kno\\-lJ that the sliding mode motion is invariant \\-; th respect to perturbations which sati sty the matching condition (Drajenovic, 1969). There are two cases for controlling systems with unmatched condition: known perturbations, and unkno\\-lJ ones. In the first case, the perturbations can be compensated completely, while in the second case, which is more conunon in the practice, the problem can be solved using high gain control. This control, however, can produce oscillations in the closed-loop system (or "chattering', due to the imperfections in the control devices (Utkin, 1992).

I

2. CONTROL METIIOD Let consider a multiple - input, multiple - output (MIMO) nonlinear system subject to disturbances

Work SIlpported by Conacyt. Me:dco. under grants 3696a4 and 398661'

671

i =f(x)+ a:x)u+ G(x)w y =h(x)

pseudo inverse matrix. The transformation (5aH5b) reduces the system (3) to the following desired form :

(I)

where x EX C Rn is the state vector,

U

E R'" is the

control Y E RP is the regulated output. The vector r(x) and the columns of 8(x) are sufficiently matrix

G(x)

represents

(6a)

i. = -kjzj + Bj(z"ro)Zj_1 +Gj(ij . ro.kj+I)w ,

(6b)

i = 2. ...• r-1

C" , the

bounded smooth vector fields of class unknown

i , = -k,z, +R,(z"oo)z'_1 +G, (z"ro)w

the

model ",. - 1 ~j

uncertainties, w represents an external disturbance, which is unknown but bounded, and rcO) =0 . The

where

output must track a reference signal q(ro) generated

f.Cz .ro.k!) and the columns of B.(zj .ro) Gj(zj,oo.k.+I)w

(2)

roE R', ll(O)=O

continuous

and

Gj(Zj. ro ,w,k. +I ),

i = I. .... r-I in (6) are bounded.

Because of the i'h block in (3) is linear with respect to the quasi control xj+l , i = I... .. r -I , then by transformation (5), the disturbance term G,(z,.ro) in

(3b)

(6a) does not depend on the gain k"

(4)

GJ(ZJ ,ro.k,_I) depends on the gains k, and k, _I' and so on. Therefore, it is naturally to introduce the following assumption:

~=(Xj, ..,X,)T , rankD;=n;, \1'xEX,and r.(~)

and D. (Xj) are sufficiently smooth ,and bounded

AI) There exist positive constants Pi'

the

IIGj(Zj , oo.k.+ I)WII

this disturbance:

1

fJ + OIvj ll(ro) ' &0

solution

k- i =(k;, .. , k,)T , D.+ O=D.T 0 [ DjOD.T (.)

JI

of

ZI = 0

in

(6c)

absence

of the

perturbation, that is. u"I (.)"= B;I (.)~ O . Then

. -

(5b)

k r =k,

k I >0 '

(8)

where the equivalent control u.q is calculated as a

-

-

s = r.(Z,oo. k 2 )-k IDI(Z,OO)11

where z=Czp ... ,Z,)T is a new variables vector,

~ =f + I 'j=i - I 0Iv, Ox;

R; (Z,OO)H 0 otherwise

(Sa)

"'.-1

~ I;=j'Yj)kj+I)~zdl +d j

if

Using the block control technique (Loukianov, 1998) the following recursive transformation is introduced :

Z. _I = x. _1+ D; (X. )[~ (X; , ro) + kj"'j (x., ro, kj +I)] == (x._pro, k.), i =2, ... ,r

(7b)

A control law-which induces a. sliding mode on the manifold, s=O , S=ZI = "'1(x,oo.k 2 ) (5b) with the desired dynamics of the closed-loop system for the case ofunknowll w is selected of the following form:

The aim of this paper is to design a discontinuous control which provides robustness for the closedloop system, and chattering free motion in presence of both matched and unmatched disturbances. The control procedure consists of the following steps: a). transformation to a desired form \l;ith disturbances; b) design of the discontinuous hierarchic control , and c) design of the fuzzy logic sliding mode controL

Z, = x, -q(ro) == ""(x,,ro)

and cl,

IIBj (Zj ,(0)11 ~ pj , i = 1, ... , r - 1

that G(x) = 8(x)A.(x) then the sliding mode motion to

qij

such that

If the disturbance w satisfies the so called matching condition, that is, there exists a matrix A.(x) such is invariant with respect (Drajenovic, 1967).

the term

G, _I(Z' _I,oo,k,) depends only on the gain k"

with

functions. In electromechanical systems following structure frequently is considered

bounded

If (2) is bounded, then, the transformation (5) is and bounded too, therefore Rj (Zj ,(0)

(3a)

y =x,

are

and

functions, and rankR I = n l = m .

We assume that there exists a nonlinear transformation which reduces the system (I) to the Block Controllable Form with disturbances:

i, =f,(x, )+B,(x,)X,_1 +G,(x,)w

-

G. = G j + L..J=I ~GJ • J

by a given exosystem described by w=ll(ro),

T

Z. = (z" ... z,) ,

kl > 0

Bi (z,ro}s 1 +G- (z,oo, -k T

DI (z, ro}s

I

2 )w

(9)

The condition. such that the trajectories converge toward the manifold S = ZI = 0, is derived using the

,

. IS the

672

V( s) = 1sTs

Lyapunov function candidate

2

(13)

and its

derivative along (6c) VI

which are positive if the following condition holds:

=ST(~ +G , -k , 118Tsll

(14)

~ -~Si~(8Trl' (kl -18~'1I~ +GII)

This shows that the solution Z2(t) is ultimately bounded by (12), and will arbitrarily fast decay to zero as k2 ~ 00 for fixed k J . At the second step, follo\\
It

is

possible

region,

V) = 1- z; Zl ' calculated along the second block

PI (L;. lhi(k2)IIZill+dl)::;r,
trajectories of (11), under conditions (7), (12)-( 14), is

s(t)

the solution " IIs( t)~ < Ils( to )11- rh (t - to),

VJ ~ -k J IIz)~2 + II z JII18 J(IJ ,ro)lllI z 211

formulated

to

show

as

that

in

the

+ ~z) ~IIG) (I) ,ro, w, k4)~

111 = ~, ql = kl - rl vanishes in the fmite time PI

~ -(k) -Yn(k 4)-a 2.)(&2)p))(l-9 J )lIzJ ,

tl < to + ~lls(to)lI, and the sliding mode starts on

0<9 J
the manifold s = ZI = 0 after this time. lbis motion is described by the (n - n l )'" order system: i, = -k,z, +8,(z"ro)Z'_1 +G,(z"ro)w

'V

(k) -y))(k 4) -a2,J(&2)PJ )9 J Hence, after a finite time t) the state ultimately enters the domain deflned by

(lIa)

Zi =-kiz i +B,(~,(t»Z i_ 1 + Gi(zi'ro, k, . I)W (lIb) i2 = -k 2z 2 +G 2(1 2, ro, k)w,

i = 3, ... ,r-1

Iz) (t)/I ~ I :=4a J.j (&2 , &) )lIz /t)11 + IlJ (&2 , &J), t > t J > t2

(lIc)

where cJ = l/k J with the parameters a J.i and PJ defined as

To achieve the robustness property with respect to unknown but bounded uncertainty, the controller gains k2, ...,k, have to be chosen in a hierarchically way. In order to establish the required hierarchy for the control gains which ensures stability of the sliding mode motion ( 11), a Lyapunov function candidate ~ = 1-z; Zi' ;=2,

,r

IIz)/I~ I;:4(Y).I(k 4)+a 2.1(&2»)lI z lll+d) +P2(&2)

a Jj (&2 ,&J) = ( A (

is selected step by

) _

f-'J &2 ' &J - (

step from the second block to the rIll block of (lla)(lIc).

and k) -Yn(k 4 )-a 2.J (&2)PJ)9 J

d J +P2(c 2) ) . k J -Yn(k 4 )-a2.l(&2)PJ 9 J

These

parameters are positive if k) > Yn(k 4) +a2.l(&2)PJ

At the flrst step, differentiating the Lyapunov function candidate V 2 = tZ~Z2 along the trajectories of ( 11 c) and using assumption A I , yields

(15)

Proceeding the same way for the i'" block of (3), then the convergence domain for the state Z;(/) is

/lZi(t~1 ~ I;:i ai .j(&2'&J " "' &i>lzj(t)1I

(16)

+P,(&2'&) '''', &i), 'Vt> t, > t i_1 a i./&2 ' .. , &) = ( fJ.(&!, ... ,&,) = (

S2(k 2) =(k2 -Y22(k))(l-9 2) · Hence, there is a finile time 12 such that

IIZ2(t~1 ~ I;:)a2.j(&2)IZi(t~l+p2(&2)'

Y.. j (k"I) +a i_lj (&2" "'&,-1)

) k i - Yij(ki+I)-ai-I.i(&! ""' &i-I)Pi 9,

d, +11,-1(&2 ,· .. , &,- 1) ) k, -Yi.i(kl+ l )-ai-I.,(&2" " '&i-I)P, 9,

0<9, < 1, i = 4 ..... r -I are positive if the condition (17)

'Vt> t2 > tl

holds. And, finally, under the condition

(12)

(18)

the solution z,(t) exponentially converges to

673

to zero and. appropriately ,

Ueq (t)

decreases, the

value of k, decreases smoothly up to k,

=k" nun

full

filling (10) and reducing or eliminating chattering. A schematic diagram for the evolution of s and k1 is shown in Figure I.

0<(;1, < I . The above results can be summarized as

Proposition 1: For (6) satisfying AI (7). if the control (8) is implemented with the gains satisfying conditions (10) . (14). (/5) and (18). then

S, Z"

,1,).

15 (1)11 < 11s(lo)l- rh(t - 10 ) Ilz,(t)I~~Z,(to*-~ (k.X I-I. ) , i =2, .. , r-I

kor_ . . /·· ......... . ..J!.~ 1,

iz,(t)11 ~llz,(to)lle - ~: (k. )(t - tl) , ~,(k,) =(k, - q,., -

(Xr-l" (E 2, .. , E, _I )p, )( 1-

*i

e, )

set) = 0, 'VI> I1

and

IIZj (1)11 ~ L:;=j (X,}E 2, E) , ... ,E j 'Vt > tj > t j_1 , i = 2, ... ,r-1

(1)11 +/\(E 2 ,EJ "

Ej ),

Block Control. This block transfonns state J: to the new coordinate z, TBC : J: -+ Z , such Ihat z = Tsc(J:)

Ilz , (t)11 ~ 13,(E 2, EJ , ... , E,), 'Vt > t, > 1'_1

where the map TBC is defined by (Sa)-(Sb), and

ft.,{oreover the tracking error z , (t) tends to zero as E j ~O.

Fig I : Evolution of sand kl The sliding mode fuzzy logic controller consists of the following parts.

computes the value of Ils(t)~ and ~Ueqll (20).

i=I, ... , r . Fuzzv Controller: This block uses only two inputs; 1nl = ~sll

3. FUZZY SLIDING CONTROL In order to reduce the effect of Ihe unknown disturbances action in (6), that is, 10 ensure inequalities (10), (14), (IS) and (18) for the given bounds (7), and, respectively, to increase Ihe region of the sliding mode stability (10), it is needed to increase the value of the controller gain kl in (8). This high gain, however, can produce chattering due to constraints of the control devices. To solve this problem we propose to adjust the value of the gain k, depending on Ihe values of 15(1)11 and the equivalent control

IUeq ~

Input Normalization , which scales inputs. Fuzzijication. which transforms the crisp input values to fuzzified values F:

Inference Alechanism, following type of rule :

(20)

which

uses

the

where CR(j,k} is the corresponding k, gain value for the rule consequent.

kl . For Ihe case of unknown disturbances the value

Ueq (t)

=LIn(i , j)

Rule m: If(Lin(l , j) and Lin(2, k» Then CR(j , k)

Note thal the equivalenl conlrol value defines the reaching condition (ID), Ihe region and the convergence rate of set) 10 zero for a given value of of

In -+ LIn such that F(InJ

where In, E In is a crisp input value defined on the universe of discourse In, and Lin(i, j) is the corresponding fuzzified input value.

calculated from 5 =0 (9) as

Ueq = _ii;I(Z.Q»(~(Z . Q»+GI(Z.Q), k2)W)

and In] = IIUeq 11 , and detennine the gain kl

to satisty the stability condition (10). The controller consists of the following components:

Defllzzification, which is based on the weighted mean defuzzification method (Driankov et. all ., 19%) to produce a sc1ilar value kd calculated as

can be oblained by filtering Ihe control

signal as 't,U eq (t) + Ueq (t) = u( t)

rrLAnt(j,k) CR(j, k)

kd

with a sufficient small time constanl 't s . The kl gain modification can be done by using a fuzzy logic scheme. For the case of a bounded control , which is natural in an application, the value of kl begins with

(21)

rrLAnt(j, k)

)=1 k=1

where LAnt(j,k) = min(LIn(l, j) , LIn(2, k)) is the

k, = k1,mu Ihat ensures (10), and then, as set) tends

premise quantification of the active rule, and nc; for

674

i=I,2, is the fuzzy set size.

the viscous friction and T, presents the load torque.

DenomlGlizotion, which multiplies normalized fuzzy controller output (21) bv a denorrnalization factor (scale), kl = kd . scale, ~uch that the condition (IO) is always satisfied

Setting

X4

=B,

=m ,

xJ

=in' XI = i b ,

x~

II~

=Un

and

u l = llb ' the system (22) is represented as the Block

Controllable form consisting of tree blocks, and subject to unknown disturbance T, = IVI :

Slidin'l Mode Controller: which implements (8) to obtain u with kl = [kll' .... k ,m ] .

(23a)

Figure 2 shows the gain kd (22) computation for hypothetical inputs (1nl and In~)

_______ _______ .k., .~~

COut" .-----

, --:----9/1. : Inz" ,, , :, 'I :,

COUlu--- ---

~ . -...

:

:

where oJ

. -t-----.~--.

,k.,

: :

I

:k..

I: I:

b)1 (x 4 )

/1

=!i ,

c)

J

= K ", , a l = a,_ = ~L J

= c) cos(N,x4) ,

b)1(X 4)

'

C

~

~-J "

= c) sin(N'X4) '

(x) = -a1 x 2 + clb31 (X4)~ ' I

q =Z J;(x)=-~X; -clb)I(X4)~ ' g)(~,WI)=-C2WI -&1J X3

Figure 2. Fuzzy Gain kd Figure 3 displays the effect of output / input relation on the rate the sliding surface is reached .

Cl

=J/L,

,

g~(x) = -&12x 2 + &lb)2(X4)~ '

gl(x) = -&1 l x l

t-<1lln~<1lln~ , , ,

&lb)I(X4)X) ; ~ , ~~ ,

-

&1)

and &) present variations of the cOlTesponding parameters with the following bounds :

I~I S

ql) '

16021 $ q!! , 1&1)1 S q)),

hw,ISd)

1&)lsql3 =q1) '

(24)

where qj, > 0, ; =1, ,-I and d) > o. Suppose that the output y = -'"4 is required to tollow the reference signal q(ro} = ~ generated by (~ =~COz ~=-al~

Figure 3: Input-Output Relation

(25)

where ro = (~ , w~ >" a l > o . Following the block transformation procedure, first we define the tracking elTor as Z4 = -'"4 -lYt 5'1'4 (X 4 ,w) , then

3. STEPPER MOTOR CONTROL In this section, we apply the proposed scheme to control a permanent magnet stepper motor:

(26) Then a desired dynamics for

Z4

are introduced as (27)

di I ---!!.-=-[-Rin + K.,m sin(N/J} + un) dt L di I

(22)

Solving (26) and (27) for ZJ

_b =-[-Rib -K.,mcos(N,B}+lIb})

dt L where B is the angular position; m is the shaft speed; i a and ib are the cUlTents in phases A and B;

where

Un and lib are the voJtages; J is the moment of inertia; Rand L are the resistance and inductance in each of the phase windings, N, is the number of

where II

rotor teeth, K,. is the motor torque constant, Bv is

=k4 (X4 -~}+xJ

xJ =(x

4 , xJ {

.

ZJ

yields

-aim! 5!pJ(xJ ,w ,k 4 }

Then

JJO =(k4 -a)x)

+(al!wl-k4alm~)

and

=(x!,xll , B1 (x.)=[-b1!(X4 ) bJI (x4 }). Taking

in the account that a J > 0 , the quasi control vector

675

x. in (28) is chosen as

=-Bi (X.)[f3(X 3,W,k.)+(k3 +a3)·'II3(X3,lV,k.)

x.

-E•• z.]

(29 )

In the region defined as

that yields

z, =-(k, + aJz, + EIlz where Ell = [I

1

+ g, (X) , w1 )

i

(30)

=1, 2

(L:;lr,I)IIZjll+di )~ki '

the solutions Z2(t) and zl(I) converge to

zero in a finite time, Is , and sliding mode enforced

0] . Now the transformation (29) is

on the manifold z. = 0 is described by the second

extended by

order system with nonvanishing perturbation

En ] =I~ (31) B3(X4)X I =EI~zl' E IZ =[0 I], [ Ell

Z4 = -k.z 4 + Z3

(34a)

such that the square matrix

-

8 , (x4 )..

[B3(X.)] .l

B3 (x.)

=

[-b)2(X.) - b31 (x.)

b31(x. ) ] -bJ2 (x.)

Using

r; = tzi

IZ3(1)1 ~IZ3 (1,)1 e - ~) (.)X'- '. ) ,

has rank n, = 2 . Using (29) and (31), the new

that IZ31:5a34(c3)lz41+P3(C3)'

Zl = 8 3(X 4 )x 1 !;(x3,W,k.)+(k3 +a3)v!3(X3 ,W,k4)] _

= 1jI. (:I,W,

o

[

where x =(x. , ... , xll and

';3 (k3) = (k2 + a3 - Qn )

x(l- 8 3 ), 0 < 8 3 < 1 . So, there is a finite time 13 such

variables vector z. = (Z2 ,ZI)1 can be obtained of as

+

and (33), it is possible to obtain

where

"7

k3)

defined

63

=

1/k3 '

as

a 3• and P3 a 34 = r34 (k. )/(k, + a, - Q22 )8, and the parameters

A=d, /(k,+a,-Q22)8"

k3 =(k. ,k3l . Then

k, > Q22 -

(32a)

a, .

(35)

't;//>/3 >1,

Finally,

are under

positive the

if

condition

k4 > a]4 (6,) the tracki ng error Z3 =-(k3 +a3 )z3 +EIIZI +g3(13,W, wl ,k 4 ) (32b) i.

IZ4 (1)1 ~IZ4 (/3 )le-~ (k,X '-') ) , Is < I < I"

=~(Z,W, k3)+iil(Z,W)U+gl(Z,w,k3 , wl) (32c)

where

13

=(z., z 3 l ,

~O = Lh(·),l.of

with ';3 (k4 ) = k4 - a 34(63) exponentially converges to the bound Iz,(t)I~p.(C4)' 't;//>I, >1, where

and

g.(.) = [g2(·), gIOf. Using (24) and the inverse

P.(6. ,6,) = P,(6,)/[k4 -a34 (6,)]lJ4 and 6 4 =

transformation, the conditions (7) are presented in the case of the motor as

Ib)2(')1 ~ c3 , I~. 01 ~ c3 ,

Is,

lhO+ g2('~l (33) [11.o+g10I :5 [r24(k. >iz. 1+ r23(k. ,k3>iz31 + r221z21 + r211 z d+d2]

4. CONCLUSIONS The proposed scheme performance is encouraging. For the simulations, we assume that the disturbance can not be measured, which is an extreme situation . However, the proposed hierarchical sliding mode control approach with the fuzzy logic control, improves the system behavior, reducing chattering and guaranteeing stability.

rI 4(k 4 >iz.1 + rn(k.,k3>!z3 1+ r12IZ21 + r1l1Z11 + dl where

4 .

In order to reduce the disturbances influence, we apply the described in the section 2 sliding mode fuzzy logic control scheme for adjusting of the controller gains, k z ' k 3 and k4 .

Ig301 ~ qnl z31 + r34(k. )lz.1 +d3 ,

r23(k. , k,)

I/k

r2. (k. , k3) = q22C3 (k.a 3 + k;) + q23k. , r22 =r21 = q22 c3'

=q22 c , (k. +k3)+q23'

REFERENCES dl

= a l [qllc 3(a l +a,) + q13 J,

d,

Alexik M , Vittek J., (1994). Adaptive sliding mode control of position servo system. IFAC-IPE, Smolenice, Slovakia, 7-10, pp 278-283 . Drajenovic B., (1969) . TIf: invariance conditions in variable structure systems, Aulomalica, 5, 287295. Driankov D, Hellendoorn H, Reinfrank M. , (1996) . An introdllction 10 fuzzy control. SpringerVerlag, USA . Loukianov A.G ., (1998). Nonlinear block control with sliding mode. Aulomalion and Remole Control, 59(7), 916-933 . Utkin Y.I , (1992). Sliding modes in control and optimization. Springer-Verlag, USA .

= d 3 + qnal '

YI.(k. ,k3) = Q11 c3(k.a 3 +ki)+Q13k. , and r 23( k.,k3) =Qllc3(k. +k3) + Q13 ' To enforce a sliding mode motion on the manifold ZI = IjII(x,w,k3) the control is chosen as

._["2] -_1-ii~\)[k2(Z2/'Z21)] kl(ZI/IZII)

u-

UI

where k2 > 0 and

o

if

[:; :~] olherwice

kl > O. Then

676