Robust output tracking of a class of nonlinear time-varying uncertain systems

ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIM...

14th World Congress of IFAC

G-2e-17-2

Copyright © 1999 IF AC 14th Triennial World Congress, Beij ing, P.R. China

ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIME-VARYING UNCERTAIN SYSTEMSl Yi-Sheng Zhong' ,Yutaka Nagai b and TOIDOyoshi TakeuchiQ ~ Dep. of Automation,Tsinghua. University, Beijing, P.R. China

E-mail: [email protected] aDep. of Electronics, the University of Electro-communications, Tokyo~ Japan

Abstract: A new method will be proposed to design a linea.r time-invariant(LTI) robust controller for a nonlinear time-varying plant which is unknown but belongs to a. given set. By this method, a nominal controller is first designed to achieve the desired control properties for an artificial LTI nominal plant, then a robust compensator is designed to restrain the influence on the closed-loop properties of the real plant perturbations from the nominal one. It will be shown tha.t for any plant in the set satisfying some assumptions, the closed-loop control systeIn with a LTI controller designed by the presented method is globally asymptotically stable and the output tracking error can be made as small as desired if the initial conditions are zeros. Copyright@1999 IFAG Keywords:

1~

robust control, tracking, uncertain, nonlinear time-varying systems j

results in [3] to systems with mismatching uncertainties and shown a new design method of highga.in robus t controllers.

INTRODUCTION

Lyapunov-based methods [1 . . . . . 8] are important methods to deal with nonlinear time-varying uncertain systems. In these methods, the concept of matching condition plays an important role [2]. In [3, 6 and 7], robust output tracking problems for nonlinear uncertain systems were considered. In (3], a new condition, generalized matching condition, was proposed; for minimum phase nonlinear systems two types of uncertainties were considered: uncertainties satisfying the generalized matching condition and linear paraIlletric uncertainties, and three control strategies were presented: high-gain control, sliding mode control and adaptive control. In [6], the assumptions on the uncertainties were further relaxed but SOUle additional conditions were required. Li et al. [7] extended the

1

Partially supported by Tsinghua University

In this paper 1 we consider a class of nonlinear time-varying uncertain systems . It is assumed that the plant model is not known exactly, but known to belong to a given set . A method similar to that proposed in [9.--11] will be shown. By this method, we first consider an artificial linear time-invariaIlt nominal plant and design a nominal controller for it to obtain the desired properties. Then a robust compensator is designed to reduce the effects on the closed-loop control properties of the nonlinear and tinle-varying uncertainties. In contrast with the existing results, controllers designed by the method proposed in this paper is linear and tiIIleinvariant . Under some conditions, it will be shown that the closed-loop control system can have (1) uniformasymptotical robust stability, (2) uniform bounded-input bounded-state property and (3) uniform robust tracking property simultaneously.

3617

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ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIM...

14th World Congress of IFAC

where Dm(s)

Notations:

Ilxll=Jx~+x~+... +x~, x En'" IIzlloo = sup IIz(t)ll, z(t) E 'Rn

1

-

... -

ltm.l.

An artificial nominal plant is introduced which is described by

z(t) E Tl'U

t

arnnsn -

Sff -

3. CONTROLLER DESIGN

t~to

IJzl)ioo = m~ IIZi(t)lloc>~

=

n

y(n)(t)

2. PROBLEM STATEMENTS

where g(;.;.) is such that the differential equation has a unique solution corresponding to each initial condition and any given piecewise continuous u(t). g(.;.;.) is not known exactly or is not fixed, and is a.~sumed to satisfy the following assumption. g(y(n-l)(t)~ ...,

yet), yet); u(t); t)

y(n)(t) ==

can be represented as g(y(n-l) (t), ... , y(t)~

y(t); u(t); t)

aiy(i-l)(t)

+ bou(t)

+ q(y(n-l)(t), .. ~, yet), yet); u(t); t) where q( . ;.;.) is any function which makes (2.1) have a unique solution corresponding to each initial condition and any given piecewise continuous u(t), and belongs to the following set

(3.2)

== G(s)u, G(s) == boD;l(S) where Dp(s) = sn - anS n - 1 - .. ~ - a2s - al. Construct t he nominal controller as lVg(s)y+ lVr(s)r (3.3) o tt == Du(s) where Du(s), Ny(s) and Nr(s) are polynomials

+ 1](t)lu(t)l}

(2.2)

;::=:1

The constants aj (i == 1, 2, ... , n) and b o , and the uniformly bounded functions ei(t) 2 O(i =; 1,2, ... , n) and TJ(t) ?: 0 (or the upper bounds of ~i(t)(i == 1, 2, ... , n) and 1](t) ) are known. (;>

of degree n - 1 and determined by the following equations for any given Burwitz monic polynomial H(s) of degree n - 1.

bmH(s) = boNr(s) Dm(s)H(s) :::: Dp(s)Dti(s) - bQNy(s)

For the above plant, we want to design a linear time invariant (LTI) controller such that the closed-loop system is asymptotically stable a.nd its output yet) tracks the desired output Ym(t) in the following sense (In fact we can achieve more; details will be shown later):

Then the closed-loop transfer function froIn T to y for the system consisting of the nonunal plant (3.1) and the nominal controller (3.3) is exactly equal to the one of the reference Illodel after canceling the stable factor H(s). So exact tracking of y( t) to Ym (t) can be achieved if both of the closedloop system consisting of (3.1) and (3.3) and the reference model are of zero initial conditions.

sup (y{t) - Ym(t)1 ~ € (2.3) t2: 0 where lE is a. positive constant a priorj given and Ym(t) is the output of the LT! reference model with reference input r(t):

L et the real control input

n

y~)(t) = LaTniy~-l)(t)

+ b~r(t)

(2.4)

1t

It is assumed that the reference model is stable and the reference input r(t) is uniformly bounded~ The frequency domain description of the reference model is given by

=

Uo

+v

(3.4)

where v is the output of the robust compensator to be designed. Considering q as an external disturbance, we can have the description of the closed-

i:::=l

Wm(s)r,

+ bou(t) + q

y

n

Iql :$ L
:=

aiy(i-l)(i)

The nominal plant (3.1) can be described in frequency domain by

== {q(y(n-l)(t), ~ ... ,y(t),y(t);u(t);t) ,

Ym

E

First a nomina.l controller is design which achieves the desired control properties for the nominal plant, then a robust cornpensator is added to reduce the influence on the closed-loop control properties of the perturbation q.

i=l

Q

(3~1)

i;l

n

== L

+ bou(t)

(Here we say the plant description given by (3.1) is an artificial nominal one because it may he not a model at all in any sense for the real plant (2.1) at any working point.) Then the real plant (2.1) can be considered as one which is obtained from the nominal plant (3.1) with nonlinear time-varying perturbation q. That is, the real plant (2.1) can be described as

== g(yC n - 1 )(t), ~~., y(t)~ yet); u(t); t) (2.1)

AssuInption 2.1

aiy(i-l)(i)

,=1

Consider the following nonlinear time-varying plant: y(n)(t)

=L

loop system:

bo(uo+v)+q

y

Wrn(s) = bm. D ;;.l(s)

= - Dp(s)

3618

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIM...

14th World Congress of IFAC

_ boNr(s)r + Du(s)(bov + q) D p (8)D u (s) - boNy(s) _ ~ () -

m

S

r

Du (s)(bov

+

where Xv == [xv!,

-I

+ q)

Dm(s)H(s)

A.,(f)

If we can set v := -qlbo , then the influence of q on the closed-loop control system can be eliminated. However as shown in (3.2), q involves the differentials of y which are assumed to be unavailable. So we construct the robust compensator as 1 ' v == --F(s)q (3.5)

=

F(s)

4~

C~

=

0

o

xvn]T and

o

0 0

f -f o f

o

f >

o

o

o

-/

0

f

-/

o

o

o

Let X == [X~, x-J]T == [xJ, X'[, X;]T, then we can obtain the state-space description of the whole closed-loop system:

x == AX +Brr+Bqq,

f ) n,

o

-/

o

o

ba

where

::Cv 2 , .... ,

y

== ex

where

0

ROBUST PROPERTIES

Now we show the closed-loop control properties of the system consisting of the plant (2.1), the controller (3.4), (3.3), (3.5) and (3.2). To this end, we first give the state-space descriptions of the plant (2.1), the nominal controller (3.4) and (3.3), and the robust compensator (3.5) and (3.2) respectively.

Denote by P the solution to the following Lyapunov equation:

PA + AT p

It can be shown that the following equations give a state space description of the plant (2.1).

Xp ::=: ApXp + e"bou + enq,

Y =

where X p == [X p l' xpz, ..... , xpn]T, en e 1 == rI, 0, ... , 0, 0JT, and U

er X

1 0

0 1

0 0

o

0

0

1

p

TheoreJrl 4.1 Suppose that

fllPBqllllTJrlco < v'2 -

+ "l(t) lu(t) I

bounded and

11X(t)I\

+ 'l(t)lu(t)'

Uo

AcXc + Bly + B 2 r

== CcXc + kyY + krr

where A c E n.(n-l)x(n-l), vectors El, B 2 and Cc. have appropriate dimensions, and k y and k r are constants. Consider the following state-space description of the robust compensator: . 1 Xv == Av (f)X v + b €lq (4.1a)

v -e; f Xv ::=

o

(4.1b)

----?

0,

t

---i-

00

(2) Input-state stability: For any initial time to and any bounded initial condition X(to), the state X(t}(t ~ to) is bounded by a constant which is independent of f. (3) Output tracking: For any initial time to, if the closed-loop control system and the reference model are of zero initial conditions, then

It is assumed that the nominal controller given by (3~4) and (3~3) has the following state-spa.ce realization:

Xc ==

(4.3)

ties. (1) Global stability: When r == 0, for any initial time to and any bounded initial condition X(t o), the state X(t)(t 2: to) is uniformly

1=1

~ 11';(t)IHIXp(t)11

1

then a sufficiently large f can guarantee that the closed-loop system possesses the following proper-

n

e.(t) IXpi(t)f

(4.2)

section.

Then we have

Iql :S L

-IC3n - 1 )X(3n-l)

Now we are ready to state the control properties of the closed-loop system constructed in the last

= [0, ...... ~ 0, l]T,

o

:=

sup Iy(i}(t) t~to

where

lE

y~}(t)I.::;

€,

i

= 0,1, ..., n -

is any given positive constant.

1

<>

5. OUTLINE OF THE PROOFS The proofs of the main results are some involved~ We will state several lemmas without proofs and

3619

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ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIM...

14th World Congress of IFAC

Define

then give the outline of the proof of Theorem 4.1. The details are omitted for space limitation and can be available from the authors upon request.

== fliP B q IJ, = 2vq (II'lIcv and if f > '0 and

V r =:

Vg

7"1'} = 2vq (l1Jlfoc < 1, then further define fto =:- Vl- Toll, Jlv := ~, !1fJo:= TTJ + Toll Evidently if f is sufficiently large, then all the con-

Consider the following Lyapunov function

V=XTpX Since A is a stable matrix, P is positive Partition P as

p =

definite~

P22

Le.mrna 5.2.. If T 7I so that

where PIl E "R(2n-l)x(2n-l) ~ P12 E ~R.t2n-l)xn), P22 E 'Rn-X" and define

Ao Then

Pl1

:=

[A

+eneJbok y

p

+ A~ Pl1 ==

So the elerrlents of

P11

f

04

~

Re~arks~

=:

IIXo (t)1I ~

0-0

11 Xv (t)1I ~

U

then the inequality (5.3)

X a'o(!, II (t o)ID

(1, IIX(to)H) , v (f~ I1X(to)Il),

==

{to, tt] t E [to, tl]

t E

-11:;

f-r + + Aa IlX(to)1I Amin(P11 ) - 2f-17a{;~,1

(1) Since r = 0, we have

Iql:::; 11<11ex> 11 X o 11 + lltJllco(vu llXo ll + tl\Xv lD == (1lelloo + 1117l1oo v u)II X Qtl + f'll1l1~ "Xv 11

(5.1)

If f is sufficiently large so that (5.2) and (5.3) hold and if 7"'1 < 1, then we have

V=:- UXo IJ2 - lJXv H2 + 2XTP Bqq 2 ~ -flXo ll 2 - IIX., 11 + 2(11xo II + IIX., 11) j Iql :::; -t-'~llxoI12 -

=

4~~~a

b't' 1'b, .A in(Pl1) } 7U

then for t E [to, ill we have that

JlXo{t)JI ::;

2 + 2y'1

where

vn (2 + 2y'1 + Amin(P11)V(X(to) Atn in(Pl1)

1-42

==

2

+ Jl~Qllxv IUlxoJ!

~;:lIX.,lIr - (Jl.2I1X"m

J..t; - (~) 2.

2

(5.4)

From the above in-

equality, we see that for any X we have V < 0, and V vanishes only when IIXII =: o. So for ~ny initial time to and any bounded initial condition X(t o), V(X) hence X(t) a.re uniformly bounded and IIXJI -+ O,t ---+ 00. Since v q , Vu, H~lloo and 11'11100 are bounded by constants which are independent of f and the inequality T'TJ < 1 is ensured by (4.3), so we see that the conclusion (1) of Theorem 4.1 holds.

+ Amin(Pl1)V(X(to) ( )

( ) . 2 From Lennnas 5.1 and 5.2, It follows that when IIX(to)1l > ((2: V(; + (;), we have that for t ~

Amin Pl1

({Xv (t)11 S;

1-J;; Ilxv 11

= - (ILollxolI-

+ 70)

Re.marks. From Lemma 5~1 we see that when [t t] f X v (to) 0 an d V· < _ 0 £or t Eo, 1 1 th en suficiently large f can guarantee that HXo(t)IJ and II X 'V (t)/t are uniformly bounded over [to, ill by constants which are dependent on the initial condition X(to) but independent of f. For example, if 1'b,

TfI

=

2

Aa == (..\min (PI1) - 2f- l 1'a Ov)>"max(P) O"v (f~ ltX(to)ID ~ /-1v7i( 'la do (f, II X (to)fl)

max { 'Ya,

>

Outline of the proof of T.heorem 4.1

j-l"'b 6v

f 2:

TTJ

which is equivalent to (4.3). So we see that (4.3) is a sufficient and necessary condition under which there exists an f making (5.3) hold.

where

:=:

-+ 00,

2y'1-

lie 11 co + lITJII~ Vu Ibo I - 1I11llcc

Lemma 5.1. Suppose that f > 21'aOvA;;'~f1,(Pl1)' (bol > 11111100 and that the initial condition of the robust compensator (4~1) i~ zero, i.e. Xv(to) O. If for t E [to, tl], V ::; 0, then we have

f-y

Let f

becomes

Define

1'a

(5.2)

To

2po/Lv > /-LUG (5.3) then there exists a positive constant ( which is independent of f such that IIXII ~ '" irn.plies iT ~

-I(2n-l)x(2n-l)

is independent of

1 and if f is sufficiently large

and

solution to the Lyapunov equation Pl1 A o

<

f >

enCcbo]

Ac is a positive definite matrix. which is the Btel

'TfI

stants defined above can be bounded by constants which are independent of f.

[Pl,] P12] P12

liP Br 11

+ [I1Jlloovu),

To

+

1) IIX

to, o(t)U :::; (jO(f1IfX(io)ll), IIX~(t)fI S lTv(f, IfX(t o )f1)

3620

Copyright 1999 IFAC

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ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIM...

and when

HXo(t)H $

14th World Congress of IFAC

IIX(to)11 ::; (, then 17o(j, (), IJXv (t)H ~ 0:" (f, (), t 2: t o (5-5)

So for any given €. large f to ensure t~to

(i) _

can be bounded by a constant indepen-

1, 2, ... , n - 1)

degree[Dm(s)H(s)] - degree[si Du(s)] == n - i so in this case KQ(f) is still bounded by a constant independent of f, and "'-j (f)(j =: 1,2, ... , n) become to the forrru5

Since both the closed-loop control system and the reference model are of zero initial conditions 1 we can have

==

:=

Since

dent of f.

Y - Ynl

e

siDu(s)(l - F(s) Dm.(s)H(s) q

(i) _

Ym -

Y

(5.6)

K;j

Du(s)(l- F(s)) Dm(s)H(s) q

S

Now consider y(i) (t) - y~) (t)( i which can be represented as

(3 ) Suppose that f is sufficiently large such that the conclusions (1) and (2) of the theorem hold. In this case, it can be shown that

Ilqlloo

0, we can choose a sufficiently

sup Iy(t) - Ym(t)j

Since for any given (, O"o(!, () and (j-c(f, () can be bounded by constants independent of j, 80 the conclusion (2) of Theorem (4.1) holds.

so

>

(5.7)

(f) = fi rPj(f),

= 1,2, OH, n

j

where c:Pj (f)(j =: 1, 2, ... , n) are bounded by constant.s independent of f Therefore we can have 0

Let ~(f, z)

Iy(i)(t)

==C::- 1 + C:-2Izlf-1 + _4. + c~ lz In-2 f-(n-2) + Iz In-l /-(n-l)

1

+

where C;[' = m!(:~m)~. For any finite z E C and a positive Co E 'R, if trV(f, z) :$ Co and 21z Icn ~ f, then we can have (z + f)n - fn I < ~(f, z )lzrfn - 1 (z + f)1t ) - fn - K(!, z)lzlt n - 1

I

Izl

=::

I\,(f~

- f 1< 2coJzl f When

f

~

Co :=

C:-

1

Hence for any given

=

>

0, choosing a sufficiently i == 1,

€,

H., n

- 1

1

6. AN EXAMPLE

colzlf- 1

Consider the following plant

y(t) == g(y(t), yet); u(t); t)

= -l.Oy(t) + Oo5u(t) + q(iJ{t), y(t); u(t); t)

follows~

where q(-; -;.) belongs to the following set. Q ~ {q(y( t)1 y(t); u(t); t)

1

1

tql fi

n

f1cllloo :s v'2,

(5~8)

If~2floo $ V2~

lI1Jffoo:S

0.1

~ (s)_~4_1_ m - 8 + O~2 s + 1 and H(s) is taken to be s+ 1. The reference input r(t) is assumed to ha.ve an amplitude less than or equal to 1, i.e. 11rll oo ::; 1.

f.

Since all the initial conditions are zeros, from (5.8) it follows tha.t there exists a constant Ko(/) which is bounded by a constant independent of f such tha.t 1 1 fl

r

+ ~l(t)~y(t)1 + 17(t)lu(t)l}

The reference model is given by

where L(s, f) is a polynomial of degree 2n - 2(== degree(Dm(s)H(s)] -1); the coefficients of L(s~ f) and the constant ~j(f)(j = 1, -.., n) are dependent on f, but can be bounded by constants which are

ly(t)-Ym(t)1 :$ 71~o(f)lIqII0<>+

:S ~2(t)ly(t)1

I

Suppose that

DffI(.~)H(s) 7 q + r ~J\Af)(s + f)jq

independent of

€

sup Iy(i)(t) - yi~?(t)1 S

Hence we can rewrite (5.7) as Y-Ym

1 (5.10)

t~ta

+ C;:-Zlzl + ... + G;lz,n-2 + 'zln-l L(s,j)

i

large f ca.n lead to

1 1 an evident candidate for the constant

Co can be as

n

r- E ItPi(f)llIqllo,,, i = 1, ..., n j:=l

z)

7 ·1- K(f,z)lz]t< B. Co

-lI~l(t)1 ~ 71~(f)lIlqIl0<>

It is required that the closed-loop control systell1 is globally asymptotically stable for any q E Q~ and for zero initial conditions,

~ l~j(f)llIqlloo

Iy(i) (t)

- y~) (t)1 S 0.1,

i

== 0 1 1

1

t ~0

1~1

(5.9)

3621

Copyright 1999 IFAC

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ROBUST OUTPUT TRACKING OF A CLASS OF NONLINEAR TIM...

14th World Congress ofIFAC

We repeat the above simulations with the results of which show

Robust Controller Design : The nominal controller can be shown t.o be given by

Xc

:=

-1.2xc

110

==

Xc -

+ O~OBy -

,yet) - Yrn(t)1 ~ 3.0 X 10- 3 , t ~ 0 Iy(t) - YIn(t) I :s 0.02, t ~ 0

O.08r

The design requirements is achieved with

O.4y + O.4r

and the robust compensator is given by V

1 ( f == - 0.5 s + f

= -2 [

2 [

y

+

)2

((1 - 2/)s - f2]y (3 + /)2

0.514]

(2) Plotting the functions f ~ f - To and f t---i' 2J.tvJ.Lo - Jkvo, we see that if f ~ 10~ then f -""'0 > 0, i.e. (5.2) holds, and if f ~ 18, then 2j.toJ1.v > J1.1.10' i.e. (5.3) is satisfied.

(3) From the plots of the functions f t--+ ('laa"o(!, () +"Yb) Jbol a.nd f l--+ [!-l!Ka(f)1 + f i - 2 (14)1(!)1 + (
=

(5.10), (5~6) and (5.5), we see tha.t if f 2: 2000, then ly(t)~Ym(t)1 $ 0.1 and )y(t)-Ym(t)1 ~ O.09~ (All the plots mentioned above are omitted for space limitation~)

REFERENCES

= 2000.

[1] Barmish,B.R. and G. Leitmann (1982). IEEE Trans.. on Aut~ Control1 AC-27, pp.153-158. (2] Barmish, B.R., et al.(1983). SIAM J. Control and Optimization, voL21, pp.246-255 . [3] Behtash, S. (1990). Int. J. Control, voL51~ pp.1381-1407. [4] Chen, Y.H~ and G. Leitmann (1987). Int. J. Control, vo1.45, pp.1527-1544. [5] Corless, M. and G~ Leitmann (1987). IEEE Thans9 on Aut. Control, AC-32, pp.763-771. [6] Liao, T.-L., et al.(1992). Systems and Control Letters, vol.18~ pp~39-47. [7] Li~ Z.-H., et al.(1995). SysterIlS and Control Letters., voL25, pp.53-61~ {S] Zak,S~ H.(1990)4 IEEE TI-ans. on Aut. Control, voL35, pp.604-607. {9] Zhong,Y.-S.(1992). Proc. of the 31th IEEE CDC~ pp.1470-1475. [10] ZhoDg~Y.-S.(1993)~Proc. of the 12th IFAC World Congress, vol.6, pp.47-50. [11] Zhong,Y.-S.(1996). Proc. of the 13th IFAC World Congress, vot G~ pp.459-464.

We know that the design above can be quite conservative. Simulation results will show that a much smaller f can achieve the desired closed-loop control properties.

Simulation Results: Consider the following plant Inodels:

y == -y + O.5u + q 1.4Yy

. ( )

A: q

= 1 + y2 + 1 + Iyl + O.Ism u

B: q

= sin (y)y + 2 +Sin(u.~ )Y+O.lsin(Y)cos(u)u cos y

C: q

= 1 + t iJ + 1 + t sin(lOt)y -

e- O. 1t

t

= 100..

Robust control problem for nonlinear tirne-varying plants of the form (2.1) has been considered. A rnethod was proposed to design linear time invariant robust controller. A controller is designed in two steps. In the first step, a nominal controller is designed for the linear tirne-invariant nominal plant to achieve the desired input-output properties. Then in the second step, a robust compensator is added to reduce the influence on the closed-loop control properties of the perturbations of the nonlinear time-varying true plant from the nominal one. It was shown that for any plant in the given set satisfying Assumption 2 . 1, if the pa.rameter of the robust compensator is chosen sufficiently large 80 that (4.3) is satisfied, then the closed-loop system can be m.ade globally asymptotically stable and the output error can be made as small as desired if the initial conditions are zeros.

(1) Plotting the function f t-+ ~Ul]"OQ' we see that for f > 0, vq ll1Jlloo ~ 0.24 < v2 - 1 < O.5~ hence 1 (4.3) is satisfied and 7"11 < 1.

1.4Y

f

7. CONCLUSIONS

q

(4) So finally we can take f

f == 100,

u

sin(t)

0.1 1 +

tul U

The reference input r is taken to a unit step signal, i. e. r (t) == 1, t 2 o.

The simulations are done in the way that the controller is fixed, the plant takes one of Plant A, Plant B and Plant C and all the initial conditions are set to be zeros. Simulation results show that ~y(t) - Ym(t)1

:5 1.5 X 10- 4 ~ 0.1, 11i(t) - YTn (t)1 S 1.5 x 10- 3 «: 0.1,

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Copyright 1999 IFAC

ISBN: 0 08 043248 4