Robust quasi-sliding mode tracking control for discrete input-output systems

ROBUST QUASI-SLIDING MODE TRACKING CONTROL FORDIS...

14th World Congress ofIFAC

Copyright <0 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

G-2e-15-6

ROBUST QUASI-SLIDING MODE TRACKING CONTROL FOR DISCRETE INPUT-OUTPUT SYSTEMS Xinkai Chen* and Toshio Fukuda**

*Department 0.[Electrical and Electronic Engineering, A-lie University, 1515 Kamihama-cho, Tsu 514-8507, Japan (E-nlail: [email protected]./p) **Department of~/echano-Informaticsand ~vstems, NagoJla lJniversity. Chikusa-ku,}\lagoya 464-8603, .Japan ([email protected]_Jp)

Abstract: In this papec, a discrete robust quasi-sliding mode adaptive tracking controller is presented for the input-output systems with unknown parameters) unmodeled dynamics and bounded disturbances. The robust tracking controller is adaptive control in conjunction with a sliding mode based control design. The bounded motion of the system around the sliding surface and the stability of the global system in the sense that all signals remain bounded are guaranteed. In the proposed adaptive algorithmS, the dead-zone Inethod is employed even though the upper and lower bounds of the disturbances are unkno~'n. The effectiveness of the proposed control is verified by simulation. Copyright ~ 1999 IFAC

Key"vords: Discrete quasi-sliding mode; robust tracking control; unmodeled dynamics; bounded disturbances; adaptive algorithm \\i.th dead-zone.

1. INTRODUCTION Discrete sliding mode control has been developed mainly using state-space models (Furuta~ 1990; Utkin~ 1992~ Chen and Fnkud~ 1998). It is ,veil known that the sliding mode method offers excellent robustness and lllvariance properties to matched uncertainties. Recently~ the discrete sliding mode control for the input-output systems has received some attention (Furnta~ 1993; Kaynak and Denker, 1993; Chan, 1995). These approaches involve s\vitching and non-switching types of control la\vs. As the concept '''discrete-time sliding mode':' is differently defined in the above papers, it needs to be clarified. In tllis paper~ ~~discrete sliding mode" means that the sliding surface s(k) --- 0 is reached in a finite time and continues to remain on it thereafter:1 whereas '''discrete quasi... slicling mode" refers to the asymptotic approaching of s(k) to zero. A typical switching adaptive control law is proposed by Furuta (1993) to control the systems, ~rhere only the system pararneters are unkno~n. Kaynak: and Denker (1993) presented a non-

switching control strategy with a predictive corrective measure to deal with both modeling uncertainty and distttrbance. Correspondingly, a simple controller is fonnulated for the systems ",ith model uncertainties and disturbances by Chan (1995), where non-switching~ non-predictive type control la",' is adopted. This paper presents an adaptive quasi-sliding mode controller for a more general class of input-output systems with unknown parameters~ unmodeled dynamics and disturbances. Even though the upper and lower bound of the disturbance are unkno"'ll, the adaptive algorithm can still be constructed. While the upper and lower bounds: \\rhich are not easy to be obtained in practice, are necessary in building the adaptive la~'S traditionally. The adaptive algorithm proposed in this paper is also an important contribution to the adaptive control theory. The formulated controller can achieve robust tracking and quasi-sliding mode. If the umnodeled dynamics and the disturbance are slow varying, the perfonnancc of the controlled system can be improved. This paper is organized as follo"vs. Section 2 gives

3569

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST QUASI-SLIDING MODE TRACKING CONTROL FOR DIS...

14th World Congress of IFAC

the problem fa nnulation. In section 3~ first, the sliding surface is designed. Then, adaptive algorithm l"ith dead-zone is fonnulatcd even though the upper and lower bounds of the disturbance are unkno~. Consequently~ based on the estimates of the unknown parameters, a robust tracking controller is proposed to guarantee the existence of the quasi-sliding mode about the sliding surface. The stability of the controlled system is also guaranteed in the sense that all signal remain

F(q-l) :::::

For the

SYS1CIll

(11)

"vhere ('"7(q-l) and QCq-l) are chosen to make the control system stable satisfying the following lemma (Astrom and Witternnark, 1989). Lemma t. The necessary and sufficient condition that the input making s(k + d) :::::: 0 stable is that all

zeros of A(q-l )Q(q-l) + B(q-l )C(q-l) ;:::; 0

(13)

are inside the lUlit disk and (Q> C)~ (A~ C), (B, Q) have no common zeros outside of the unit disk. Therefore, the sliding surface defined in (4) :yields s(k + d) = G(q- J )u(k) + F(q-l )y(k)

delay operator. d is the time delay. A(q-l) and B(q-l) are coprime polynomials defined by

H.

(10)

= _(G(q-t )rl(~-(q-l)y(k) -C(q-t)r(k +d))(12)

u(k)

where y(k) and u(k) are the output and inpu~ respectively, v(k) is the uncertainty, q-l is the

+

.

is given by

Consider the discrete input-output system described by A(q-I )y(k) == q-d B(q-l )u(k) +v(k) , (1)

-C(q-l)r(k+d)+w(k+d)

+ ... + 0nq-n

(9)

with known parameters and

= 0, the control input to make s (k + d) = 0

v(k)

2. PROBLEM FORlviULATION

B(q-l) ~ b u + b 1q-l

+ ... + J:_rq-n-t-l

G(q-l) = go + glq-I + ... + gm+d_lq-m-d+]

bounded. Section 4 gives an example and its simulation results. Section 5 concludes this paper.

~4(q-j) == 1 +G 1q-l

fa + J;.q-l

(2)

(14)

where w(k-rd) is deIfied as

+ bmq-m ,bo :;t:. 0 , m < n . (3)

If!(k

Further, ,"'e make the folIo\"ing asswnptions. Assumption 1. The time delay d and the plant order n are knoVt.rtl. Assumption 2. The plant is minimum phase~ Assumption 3. The parameters {a;~ bi } of the

+ d):= E(q-I )v(k +

d).

(15)

3.2 Parameter E-stimation

{ai'

As the parameters

bi }

of the polynomials

parameters

A(q-l) and B(q-l) are unkno\vn, the

polynomials A(q-l) and B(q-l) arc unknown.

in G(q-l) and F(q-l) can not be obtained exactly_

The control purpose is to drive the output y(k) to track a uniformly bounded signal r(k) for the

Now~

re""'rite Equation (14) as s(k

uncertain system (1).

where

tj;~

+ d)

:::

t/lT (k)B + w(k + d)

(16)

a and s(k + d) are detmed as

4'(k) == [u(k),u(k - I), ~ .. , u(k ~ m - d

3. THE QUASI-SLIDING MODE CONTROL

+ l)~

_y(k), _y(k -1),···, y(k - n + l)JT

(17)

f

(18)

s(k + d) == s(k + d) + C(q-1 )r(k + d) .

(19)

e~[g 0' g l ' .. ~ '~m+d-l' 0 r f ... r j 0' J I' , J n-l

3.1 Design of/he .Sliding Surface

Let the sliding surface be defined by s(k +d) = C(q-I)(v(k +d)-r(k +d))+Q(q-l)u(k) (4)

where C(q-1) is a stable polynomial defined by C(q-l) :::: 1 + c1q-l

+ ... + cnq-n

(5)

(6)

No"",:. define the polynomials E(q -I) ~ F(q-l) and G ( q - J ) satisfying C(q-l) G(q-l)

where

= A(q-l )E(q-l) + q-d F(q-l)

(7)

E(q-l )B(q-l) + Q(q-L) ,

(8)

:=

satisfying

19om[sfgo{

and gOmgo >0.

loss of generality~ it is assumed that

and tlle polynomial Q(q -I) is defined by Q(q-l) = Qa (q-l )(1- q-l) .

For the parameter go, we assume: Assumption 4.. There exists a knOv,TI constant gOrn

From

(5)-(10)~

gOm

Without >0 .

it can be seen that go is detennined

by b v and Q(q-l). As Q(q-I) is a polynomial ""hich is designed by the designer in advance,

Assumption 4 is equivalent to giving some assumption on b D • For the uncertainty w(k), we assume:

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST QUASI-SLIDING MODE TRACKING CONTROL FOR DIS...

Assumption 5.

14th World Congress of IFAC

prOj{J:J (k)} = J/ (k),

The llllcenainty lil(k) is of the

fol1o~'ing fonn

fOT

j

proj{gjl (k)}== k/ (k) _~ for j I

w(k) =

"'I (k)

+ 14'2 (k)

~

(20)

,"vhere

o{ . .

pro] g,:)

(21)

i(k)}_{kor(k) -

gl/(k)~gDm

The initial conditions

JIW 2 (k)fL~ s p , (22) respectively, i.e. w(k) is composed of unmodeled

(30)

(31) (32)

otherwise

gOm

and

= 0,1,-··, n -1 =t6 .. ~ m +d -1

j; (I)

(i ==O,l,· .. ·~ n-l) and

gj(l) (j==l,···,m+d-l) can be any constants,

while the initial conditions of go(k) should be go(l)~gom for 1==-d+l,···,O.

dynamics and bounded disturbances. Assumption 6. a is a knO\\n positive constant which is Vel}; small; p is an unknown positive constant.

chosenas B~l

recalling the adaptation algorithm defmed in

(27}, it can be seen that the update estimate p(k) of p is too sensitive to the difference

For the practical control systems, the umnodeled dynamics usually exists due to the variety of the tasking environmenl and some unpredictable factors. Also, the presence of bounded disturbance is inevitable, and the bounds can hardly be kno"\\'!l in practice. Thus~ the control problem for the systems formulated above is of great importance~ Remark 1: For the systems with uncertainties

adaptation law of 8 defined in (28)-(32), the update estimate of p in (27) is nonnali7--OO as

satisfying the above assumptions~ adaptive pole placement control lav.'s lnay be considered if p is kno"'n (Good"vin and Sin, 1984~ Lozano and Zhao,

In (28) and (33), gain.

1994).

Define

Let 8(k) denote the estimate of the unkno""n parameter 0 at instant k. Define the estimation

Then, it can be easily seen that

le(k)l- ij(k - d).

~ ~ . £ - )~(k) ~ le(k)l p(k)=p(k-dJ+ 1+t/JT(k-d).fjJ(k-d) -

8(k)

·Thus~

tP T (k

- d)8(k - d) .

(23)

+ ""'2 (k) .(24)

vY(k) -

:;;: 8

Thus~

tT

=

E

can

be

constants. The estimate O(k) of by the next adaptation law (Goodwin and Sin, 1984) O(k)

~ eek - d) +

(for positive

28A.(k) {e(k)1fJT (k - d)8(k - d)

B(k) ==

,

(28) (29)

2

+

£2 .A:

1 + rP

T

2

(k)e ( k ) . (k - d) . rjJ(k ~ d)

(37)

e(k)l~!2 (k - d) -Ie(k)(p ::;; 0 ~

(38)

As

1 + IjJT (k - d) ~ rjJ( k - d)

proJ{e/ (k)},

+ le(t)ip(k - d)}

1 +IjJT (k -d) -t/J(k -d)

can be updated ,vith projection

f}

6A(k)r/J(k - d)e(k)

{2--(k d) P -

M(k)'e(k)l } + 1 +(JT (k - d) ·t/J(k -d)

0

small

}

(27)

A(k) fe(k)1 '

any

cl(k)JeCk)i l+rjJT(k-d)-rjJ(k-d)

< 1, the initial conditions p(l)

, == -d + l~ .0_ ~ 0)

EA(k)tjJ(k - d)e(k)

+

can be intuitively updated by the following adaptation law \\'here 0 <

(k)jj! (k) - jjT (k ~ d)8 (k ~ d)

0

p(k)

0

- d)

1 + tjJT (k - d) tjJ(k - d)

(26)

&

f/~(k

+

otherwise

p(k) ~ p(k - d) +

(35)

.

£A(k)e(k}tfrT(k-d) {W(k-d) 1 +fjJT (k - d) ·r/J(k - d)

le(k)l > i}(k - d)

leo(k)!

{

iJIT (k)B/ (k)

(34)

+ {P(k) + p(k - d) }{P(k) - p(k - d)}

The dead-zone function is defined as :i(k) ==

p(k) ~ p(k) - p .

(J,

Taking the difference of V(k) yields

Because the upper bound p of 1t'2 (k) is unknown, its estilnate p(k - d) is used to construct the deadzone function. Define ii(k - d) ~ a}}t;b(k - d)lt2 + p(k - d) . (25) l - i}(k - d)

O(k) -

NO"7, consider the Lyapunov function candidate J7(k) :: jjT (k)B(k) + p2 (k) . (36)

e(k) can also be expressed as

e(k) == tflT (k - d){O -B(k - d)} + )l)1 (k)

=:

(33)

is the parameter adaptation

0

OT (k)B(k) s;

error as e(k) == s(k) -

In practice, corresponding to the

thc~

by using the definitions of i1(k - d)

and

A(k) , it yields

where pro} is the projection operator such that

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14th World Congress of IFAC

ROBUST QUASI-SLIDING MODE TRACKING CONTROL FOR DIS...

A(k){e(k)~T

:::: C(q-l )r(k + d) +,5 s(k) + e(k + d) - w(k + d) (46) Thus, combining (45) and (46) yields

(k - d)B (k - d) + le(k)lp(k - d)}

""'2 (k -

== 2(k){e(k){liJ1 (k - d) +

p}} d)112 - e

d) - e(k)}

))(Y(k))

C(q-l) q--dQ(q-l ( q -d F(q-I) q-d G(q-l)

+ !e(k),{p(k - d) -

~ A(k){le(k)}a)~(k

-

1

(k) +

~(k)lp(k -

== -A? (k)e 2 (k) , Therefore~ substituting (39) into (37) yields

r/~(k) ~ frT(k _ d) s

2

t:A?(k)e (k)(2 - 8) 1 +(JT (k - d)t/J(k - d)

d)}

=O)C(q-l)r(k) + 0)0 s(k -

(39)

A? (k)e 2 (k)

N

k=i

T

1 +? (k - d)JjJ(k - d)

.(41)

(1). lim k-40OO

(PI). 8(k) and p(k) are bounded for all k 2 0 .

•

A? (k)e (k) T

are constants, it

follo~'s

that

We have the following theorem to describe the stability of the controlled system.

~o.

lim{O(k) - O(k ~ d)}== 0 .

Theorem 1..

lim{p(k) - p(k - d)}:= 0 .

Remark 2: If g Om < 0 ~ the projection operator can be defined simil.3rly. The above results, except (P2) which should be go(k) ~go". , can be derived.

(i). lim A.(k)le(k)/ == 0 ~ i.e. k-""oo

!~ sup {le (k)J - ij(k -I)} S; 0 , (ii). all the signals in the loop remain bounded, for the class of uncertainties (20) satisfying a > Cl . Proof: From (45), we have

3.3 The Quasi-sliding iv/ode Control

F rOll no,-y on~ the control input is determined so that the quasi-sliding mode exists along the sliding surface s(k):= 0 . No,,,', consider the follo,ving adaptive controlla"v

Is(k)1 ~ Cl + C 2 ~~~lle(k)ff2

constant

and

Fk (q-l)

are described by (43)

Fk (q-l) = .10 (k) + ~ (k)q-l + ... + In-l (k)q-n+l . (44) The weighting parameter t5 is introduced in order to modify the system response and to control the

size of the control signal. Using control (42), the sliding surface reduces to s(k + d) := t5 s(k) - tfJT (k)B (k) + rv(k + d) == t5 s(k) + e(k + d) (45) By (24)~ the controller in (42) can be rewritten as -1

)y(k) + G(q-l )u(k)

and positive constants C 3 and C 4 !!~(k)1I2 S C"3 + C~4 ~~lle(k)112

(42)

Ok (q-l) == go (k) + gl (k)q -I + ... + gm~d+l (k)q-m-d-rl ,

F(q

a

such that

where li is the \,reighting factor in the range of

Gk (q-I)

(48)

where CL and C 2 are positive constants. By (47)~ it can be concluded that there exist a small positive

= -(Ck(q_r. )}-I (ftk(q-I )y(k) -C(q-l)r(k+d)-6s(k))

Consider the system (1) satisfying

Assumptions (1-6) controlled by the adaptive controller described in (42). Then, there exists an upper bound Cl > 0 such that

k--+:p

o < t5 < 1 ~

K 2

k ---+co

k----.".cfJ

u(k)

and

Kt

limp(k) == 0 .

fX) •

2

k-4oo1+t;6 (k-d)t(J(k-d)

(P6).

p2 (k) == 0 1 + rpT (k)rp(k) ~

Ilcp(k)IIz = {cpT(k)cp(k)¥ :SKI +Kz~~jfJ(r)l,

where

2

L J} (k)e (k) < }I/~CQk=] l+fjJT(k-d),p(k-d)

. (P4). llm (PS).

(2).

.

<~' " j lID

(P3)

+ (Oe(k) -(~)W(k) (47)

So~ by the fact V(Jv') > 0, the following properties can be obtained (Goodwin and Sin~ 1984).

(P2). go(k) ~ gOm

d)

The above closed loop system equation will be employed to analysis the stability of the system (1) controlled by (42). For this purpose, \VC cite the following Lemma (The Key Teclmical Lenuna in Goodwin and Srn, 1984). Lemma 2.. If the following conditions are satisfied.,

(40)

Summing the both sides of relation (40) from k == 1 to 1\( gives V(AT)-V'(O)s--(2-c:)cL

u(k)

(49)

for the class of uncertainties (20) satisfying a > a "' where Lemma 1, the uniformly boundedness of r(k) and relation (48) are employed. Thus, by using the definition of iJ(k - d) , we can conclude that there exist positive constants Cs and C 6 such that

11?5(k)112 ~ C~ + C ~~~1{1}e(k)112 6

-7}(k - d)}. (50)

Therefore, from the property (P4), the result (i) can be proved by employing Lemma 2~ So, from (50), it can be concluded that all the signals in the loop are uJtimately unifonnIy theorem is proved.

bounded.

Therefore,

the

Remark 3: From (45), by applying Theorem 1, it is clear that the system oscillates in a region near the sliding surface due to the existence of uncertainties.

3572

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ROBUST QUASI-SLIDING MODE TRACKING CONTROL FOR DIS...

14th World Congress ofIFAC

Remark 4: By observing tlle parameter adaptation law (28)-(32), the control input (42) and the equation (45)? it can be seen that in the region le(k)l:s: r](k ~ d) near the sliding surface s(k) = 0 ~

defined in (23). Using control (51)~ the sliding surface reduces to s(k + d) == 0 s(k) _.pT (k)B(k) + w(k + d) - e(k)

the linear feedback coefficients at instant k are same as those at the former instant k - d . Outside of this region~ the feedback coefficients arc changed, Therefore, the proposed adaptive sliding mode control is indeed a s'\vitching one, ,,,'here the deadzone Inethod plays an important role. The Iuechanism of the ne,v controller is someu-'hat similar to that of the controller in (Furuta~ 1993). But the new controller can cope \vith a much more

=: § s(k) + e(k + d) -e(k) (53) By Assumption 7, e(k) - e(k - d) may be very small. By using (24), the controller in (51) can be rewritten as

}?(q-L )~y'(k) + G(q-I )u(k) == C(q-l )r(k +d)

993).

=

If the class of bounded disturbances w 2 (k) is absent, then the tracking error can be controlled to be zero for a class of w 1 (k) .

-(Gx (q -I) )-1 (Fk (q

-1

X(k) y >0

=-:

{ 1

le(k)l> (1 + y)iJ(k - d) I

(52)

othen1.>'ise

is a small positive

constant.~

(55)

4. SlMULATION RESULTS This section presents a simulation example to illustrate the proposed strategy~ Consider the system described by y(k + I) + al~v(k) + a 2 y(k -1) = bou(k) + hI u{k -1) + Vi (k) + V 2 (k) where at =-1,3:> 02 =0.42, bo =1, hI =-0.8, VI (k) and v 2 (k) are respectively the luunodeled dynamics and the bounded disturbance governed by v l (k) = O.1(cos(O.lk)~O.5, 1,1, l}6(k) , V"l

~

(k)

=:;

(, .

{

O.l\sln(O~ lk) 1 +

J

1~(k)rt2 n 11

1 + u9'(k)

'

2

ljJ(k) == (u(k),u(k -l),y(k),y(k -1)Y' , (51)

where is is the weighting factor in the range of o < t5 < 1: Gk (q-l) and Fk (q- 1) are described in (43) and (44)~ 7}(k) is defined as D.

d)

(ii). all the signals in the loop remain bounded, for the class of uncertainties (20) satisfying a $ et. Proof: The proof can be derived by a procedure similar to that ofTIloorem 1.

)y(k)

L~(q-l )r(k + d) - 8 s(k) + 1](k)e(k»)

0)0 s(k -

E~ sup {!e(k)\ - i](k ~ I)} ~ 0 ~

In this section, we make an additional assumption. Assumption 7. The uncertainty w(k) is slow varying \,,'ith respect to the sampling frequency.

-

)r(k) +

Similar to Theorem 1, we have the next theorem. Theorem 2: Consider the system (1) satisfying Assumptions (1-7) controlled by the modified adaptive controller described in (51). Then, there exists an upper bound a > 0 such that (i). lim A(k )le(k)1 = 0 ~ i.e. k~a; f

3.4 Alodification of the Quasi-sliLling It--fode c"'ontrol

u(k) =

(i)C(q-l

u(k)

+O)e(k)-e(k-d)}-(?)W(k)

Corollary 1: Consider the system described in Theorem I if '\le know that the class of disturbance w 2 (k) is absent. Then~ there cxi sts an upper bound a > 0 such that the output tracking error can be controlled to approach to zero asymptotically by using the controller (42) for the unmodeled dynamics w~ (k) satisfying a < a . Proof: By observing the proof of Theorem 1~ the corollary is obvious.

Under the above assumption, the performance of the control may be improved. As the estimated parameters are slo1\" va.rying as k is sufficiently large (see property (P5])~ by observing the representation of the variable s(k) in (45) and the estimation error e(k) in (24)~ we can say that s(k) is mainly dominated by the uncertainty ~'(k) and the vector tP(A~ - d) as k is sufficiently large. Therefore, combining (24)~ (45) and (49), "ve can conclude that s(k}, equivalently e(k}, is IIminly dominated by the uncertainty w(k) as k is sufficiently large~ So, the follo\\ting ·improvcd control law can be considered

(54)

)](J'(k)]

C(q-l) q-d Q(q-l ( q-d F(q-J) q-dG(q-l)

general class of uncertainties than the controller in (Furota~ j

+ e(k + d) - e(k) - w(k + d)

+15 s(k)

Thus, combining (53) and (54) yields

and e(k) is

,,-here ",e kno""n that

II v1 (k)/1 ~ O.2j}{6(k)112 .

The desired trajectory of the controlled output y(k)

is r(k)

= 1.2 +sin(2k%s).

The polynomials are chosen as C(q-l) == 1 + q-l + O.25q-2, Q(q-l) == 1.5(1- q-l) .

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14th World Congress of IFAC

Optimization. Springer, New York.

As d:::;: 1 ~ it can be seen that E(q-t) = 1 . Assume

that we know g Om = 1 . The parameter adaptation gain 6 is set to E = 0.1 . The \veighting factor cS in the control la\v is chosen as is = 0.3 . Figure 1 shows the response of the system controlled by the quasi -sliding mode controller described in (42). Figure 2 shows the response of the system controlled by the modified controller described in (51)~ where the parameter y is set to r := 0.2 . It can be seen the responses in Figure 2 are better than those in Figure 2 when k is large. 'The oscillations are due to the existence of uncertainties.

-2

~--,-_--,-_~

fJ9J

1:::0

150

_ _-"",,

:oJ

25,)

-..._--J

300

],!jO

-4cr:

A5D 5DO Time Stl'!P k

-2 " - . - - - ' - _ - - " - _ - - - ' - - _ . . J - - - - - - - ' ' - - - - - - - - - L . . _ - - ' - - _ . . . . L - - - - - - - . . J ' - - - - - . J o 50 1(1) 150 20J 250 3D[) 350 -4OJ 450 5)) TlmeStep It

5. CONCLUSIONS

1(]r----.-----.---........-------r----.--------~_----.

In this paper~ a discrete adaptive tracking quasisliding mode control1cT~ which is inded a switching one~ is proposed fOT a much more general class of system with unknown parameters, unmodeled dynamics and bounded disturbances. The unknown parameters and the bounds of the disturbances are estimated by using the adaptive algorithms with dead-zone. The stability on the controlled system can be guaranteed in the sense that all signals in the loop remain bounded. If the uncertainty is slow varying with respect to the sampling frequency~ the proposed controller can be modified to improve the control performance. Simulation results sho-v..~ that the proposed control strategy is effective.

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s(k)

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50

~, [)J

I

150

2ffi

250

300

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Am

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500

Time S:tep k

Fig. 1 The responses of the system controlled by the quasi.. sliding mode controller (42).

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REFERENCES

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~~~~

Astrom~ K.J.

and Wittenmark, B. (1989). Adaptive Control. Addison-Wesley, Readin~ MA. Chan~ C.Y. (1995). Robust discrete quasi-sliding mode tracking controller. ~4utom.atica~ 31, 15091511. Chen~ X. and Fukuda, T. (1998). Robust adaptive quasi-sliding mode controller for discrete-time systems_ Lt\;ySl. Contr. Lett.~ 35, 165-173. Furuta, K. (1 990). Sliding mode control of a discrete time systclns. Syst. Contr. Lett. ~ 14~ 145-152. Furuta, K. (1993). VSS type self-tuning control. IEEE Trans. on Ind.Electron.~ 40, 37-44. Goodwin, G.C. and Sin~ K.S. (1984). Adaptive Filtering Prediction and Control. Prentice-Hall, Inc. c Engle\vood Cliffs, NJ. Kaynak, O. and DenkeT~ A. (1993). Discrete-time sliding mode control in the presence of system uncertainty. 1nl- .f. Contr., 57, 1177-1189.

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350

400

450

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u(k)

50

1:0

150

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150

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liDO

k

J

s(k)

_

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Lozano~

R. and Zhao, X. (1994). Adaptive pole placement without excitation probing signals, IEEE Trans. Automat. Contr.~ AC-39, 47-58. Utkin, V.I. (1992). Sliding ~\lodes Control

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300

350

400

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Fig. 2 The responses of the system controlled by the modified quasi-sliding mode control (51).

3574

Copyright 1999 IFAC

ISBN: 0 08 043248 4