Pergamon PII: sooos-lo98(97)ooool-o

Auromdco. Vol. 33. No. 5, 99!L1002, 1997 pp. 0 1997 Elswier Science Ltd. All rights reserved

Printed in Great Britain 0005.1098/97 $17.00 + 0.00

Discrete Adaptive Sliding-mode Tracking Controller*

C. Y. CHANT

Key Words-Discrete sliding mode; adaptive control; stability; robustness.

Abstract-This paper presents the discrete adaptive sliding-mode control of a minimum-phase plant with a bounded disturbance. The control law is based on an input-output model and is of the non-switching type. The system behaviour in the vicinity of the sliding surface is examined. It is shown that the adaptive controller is robust with respect to the bounded disturbance. Simulation results are presented to illustrate the features of the proposed scheme. 0 1997 Elsevier Science Ltd.

1. Introduction Recently. discrete sliding-mode controllers have received much attention (see e.g. Sarpturk et al., 1987; Drakunov and Utkin, 1989; Furuta, 1990; Sira-Ramiraz, 1991; Kaynak and Denker, 1993; Chan 1995). Most of these studies have concentrated on fixed strategies. However, some studies of the adaptive control of discrete systems based on sliding mode have been carried out (Bartolini et al., 1992: Furuta, 1993). The former considers systems represented by the state space and deals with the problem of regulation using a non-switching law, while the latter considers systems described by a transfer function and deals with tracking using a switching law. The aim is to extend to the discrete systems the desirable features of the sliding-mode controller for the continuous-time systems; that is, while in sliding mode, the system is insensitive to parameter variations and disturbances (Utkin, 1992).

In general, it is not possible to generate the equivalent control to keep the state on the sliding surface if the plant uncertainty is unknown. However, by coupling adaptive control with the sliding-mode control, a motion close to the ideal discrete sliding surface can be achieved. Note that the above-mentioned works do not take account of the presence of disturbances. In the presence of disturbances, the ideal sliding surface may not be achieved even if adaptive control is used. Thus the system behaviour in the vicinity of the sliding surface needs to be studied.

This paper presents the discrete adaptive sliding-mode tracking control of a dynamical system with an input-output representation in the presence of a bounded disturbance. The class of systems considered is minimum-phase. A non- switching control law is adopted. The adaptive control scheme is shown to be robust with respect to the bounded disturbance.

The present work differs from previous works on adaptive sliding-mode control in that it studies the robustness of the discrete adaptive sliding-mode controller. In particular, it differs from the work of Bartolini et al. (1992), which uses the state-space model. There the proof of convergence relies on the satisfaction of the strictly positive reality property. But it is not clear how the scheme can be realized, since it may not

*Received 12 December 1995. Revised 3 June 1996; Received in tinal form 3 December 1996. This paper was not presented at any IFAC meeting. This paper was recom- mended for publication in revised form by Editor Peter Dorato. Corresponding author C. Y. Chan. E-mail ecychan(@ntuvax.ntu.ac.sg.

t School of Electrical and Electronic Engineering, Block Sl, Nanyang Technological University, Nanyang Avenue, Singapore 639798.

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be possible to satisfy that requirement with uncertain plant parameters.

The organization of this paper is as follows. Section 2 presents the discrete sliding-mode control of a known plant. The adaptive control and the stability analysis are presented in Section 3. Simulation results to illustrate the features of the proposed controller are presented in Section 4.

2. Discrete-sliding mode control This section presents the discrete sliding-mode controller

for a known plant. Consider the single-input single-output system described

by A(q-‘b(k) = q-‘B(q-‘)n(k) + d(k), (I)

where A(q-‘) and B(q-‘) are polynomials in the unit-delay operator 4-l defined as

A(q-‘) = 1 + a,q-’ + + a&I’, (2)

B(q-‘) = bo + b,q -’ + + b,,q-“‘, (3)

and y(k), u(k) and d(k) denote the output, input and disturbance respectively. It is assumed that B(q-‘) is asymptotically stable and b,, is a non-zero constant. The disturbance d(k) is assumed to be bounded and such that

Id(k) - d(k - 1)1~ P < = for all k.

Let the sliding surface be defined as follows:

s(k) = C(q-‘)e(k) = 0,

where

e(k) = y(k) - r(k),

(4)

(5)

(6)

r(k) is the bounded reference input and C(q-‘) is a stable polynomial defined as

c(q-‘) = 1 + c,q -’ + + C,+,q-(n+‘! (7)

For the purpose of designing the controller, the following incremental plant model is used:

A*(q-‘)y(k) = q-‘B(q-‘) Su(k) + 6d(k), (8)

where

A*(q-‘) = A(q-‘)(l -q -I), (9)

h(k) = u(k) - u(k - l), (lb)

Sd(k) = d(k) - d(k - 1). (II)

It is assumed that A*(q-‘) and q-‘B(q-‘) are relatively prime. Note that incremental plant models are normally used when dealing with disturbances (Mosca, 1995, Chap. 9).

The control law to be considered is of the form

B(q-‘) h(k) = -F(q-‘)y(k) + C(q-‘)r(k + 1) - M(k + l),

(12) where the polynomial F(q-‘) satisfies

C(q-‘) = A*(q-‘) + q-‘F(q-‘). (13)

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Now, using (8), the control law (12) can be written as

B(q-‘)&4(k) = -F(q_‘)y(k) + C(q-‘)r(k + 1) - Sd(k + 1)

= -F(9-‘)(]l -A*(9-‘)IY(~) + 9-‘B(q-‘) &(k) + &j(k)}

+ C(q_‘)r(k + 1) - 6d(k + 1)

= -F(q-‘)[l - A*(q-‘)y(k)]

- q-‘F(q_‘)B(q_‘) 6u(k) - F(9_‘) Sd(k)

+ C(q_‘)r(k + 1) - 6d(k + 1). (14)

Combining (8) and (14) gives the following system:

A*(9-‘) -9-‘B(q-‘) y(k) I[ 1 F(q-‘)[l -A*(q-‘)] B(9-‘)[l +q~‘F(q~‘)] h(k)

=[C(~qr(k+l)+[ ad(k) 1 -F(q-‘) 6d(k) - Sd(k + 1) (15) The system (15) can be viewed as a linear system with inputs r(k) and &f(k), and outputs h(k) and y(k). The zeros of z~+~+‘B(z-‘)C(Z-‘) are the poles of the closed-loop transfer function. Thus, if B(9-‘) and C(q-‘) are asymptotically stable, the signals h(k) and y(k) will be bounded. The boundedness of u(k) can be shown using the following argument (Mosca, 1995, p. 341).

Since A*(9-‘) and 9-‘6(9-‘) are assumed to be relatively prime, there exist polynomials X(9-l) and Y(q-‘) satisfying the B&out identity:

A*(q-‘)X(9-‘) +9-‘B(9-‘)Y(q-‘) = 1.

Therefore

(16)

u(k) = A*(q-‘)X(9-‘)u(k) + 9-‘B(q-‘)Y(q-‘)u(k)

=A(q-‘)X(9-‘) Wk) + Y(9-‘)[A(9-‘)y(k) -d(k)],

(17)

after using (1). (9) and (10). The boundedness of u(k) follows from the boundedness of y(k), h(k) and d(k).

Also, using the definition of s(k) and (6) gives

s(k) = C(9-‘W) - C(9-‘P(k)

= ]A*(9-‘) + 9-‘F(9-‘)ly(k) - C(9-‘)r(k) (using (13))

= 9-‘[B(q-‘) h(k) + F(9-‘)y(k)

+ 6d(k + 1) - C(9-‘)r(k + l)] (using (8))

= 0 (using (12)). (18)

Equation (18) leads to the discrete sliding mode, and it follows from (5) that e(k) will be asymptotically stable if C(9-‘) is asymptotically stable.

3. Discrete adaptive sliding-mode control This section considers the adaptive control of an unknown

plant. It is assumed that for system (1). the non-zero constant b,, is known while the other system parameters a, and b, are unknown but constant. Knowledge of the upper bound of d(k) is not required, except that E in (4) is known.

3.1. Adaptive control law. If the plant parameters are known, the control law (12) can be written as

&r(k) =;{-F(q-‘)y(k) - [B(9-‘) - b,,] h(k)

:C(r,-‘)r(k + 1) - Sd(k + 1))

=$[-~T(k)B+C(9-‘)r(k+l)-6d(k+1)]. (19)

where

4(k)= [y(k) . y(k -n) 6u(k - 1) . Su(k -m)lTq

(20)

0 = [h f;, b, WT.

The normalized signals are defined as:

(21)

(22)

where n(k) is chosen as (see e.g. Lozano and Goodwin, 1985)

n(k) = max (llW)ll, 1)

and II.11 is the 2-norm. Thus it is clear that

(23)

(24)

In the case where the system is unknown, the following adaptive control law is considered:

h(k) = ;{-[@9-l. k)y(k)

:[b(y-‘, k) -b,,] h(k)] + C(9-‘)r(k + l)}

= k [-@T(k)6(k) + C(q-‘)r(k + l)], (25)

where

h(k) = 6(k - 1) + K(k)&,(k - l)s,(k),

O<cu<2(1-y-l), (26)

K(k) = 1 if Is(k)lzye, 1 <y<=, 0 otherwise. (27)

The dead-zone facility is commonly used to stop the updating of the parameter vector when a certain condition is not met (Mosca, 1995). In this case, if Is(k)1 2 ye is not satisfied, the updating of the parameter vector is stopped.

In the following section, it will be shown that the control law given by (25) leads to a closed-loop adaptive system that yields a linear boundedness condition required for the proof of the stability of the adaptive system.

In the sequel, the polynomial argument 9-l will be dropped for notational brevity.

3.2. Closed-loop adaptive system. Before the stability analysis is carried out, a linear boundedness condition has to be obtained first from the closed-loop adaptive system.

Now, using the definition of s(k) and (6) s(k) can be written as

s(k) = Cy(k) - Cr(k)

= (A* + 9-‘F)y(k) - Cr(k) (using (13))

= [B 6u(k - 1) + ad(k) + Fy(k - l)]

- Cr(k) (using (8))

= b,, Su(k - 1) + 47’(k - 1)0

- Cr(k) + ad(k) (using (20) and (21)). (28)

Let the parameter error be defined as

8(k) = 6(k) - 8. (29)

Thus, using (29) in (28) leads to

s(k) = b,, 6u(k - 1) + d”‘(k - l)[@k - 1)

- 8(k -l)] - Cr(k) + 6d(k)

+ [b,, 6u(k - 1) + &T(k - 1)&k - 1)

- Cr(k)] - +I’(k - 1)8(k - 1) + Sd(k)

= -C(k - l)@k - 1) + ad(k) (using (25)). (30)

or

where

s,,(k) = -4;r(k - l&k - 1) + ad,,(k), (31)

W,(k)=$!$ (32)

Now, let

where

B(k) =&k) - B,

F(k) = i‘(k) - F,

b(k) = b,, + B,(k).

(33)

(34)

(35)

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Using (33)-(35) in (30) gives

s(k) = -F(k - l)y(k - 1) - B,(k - 1) 6u(k - 1) + &f(k)

= -F(k - l)y(k - 1) - B(k - 1) &(k - 1) + 6d(k). (36)

Having obtained (36) the control law (25) will now be studied closely. Using (35), (25) can be written as

h(k) h(k) = -p(k)y(k) + Cr(k + 1). (37)

Substituting (33) and (34) into (37) yields

[B + B(k)] h(k) = -[F + F(k)]y(k) + Cr(k + l), (38)

or

B c%(k) = -Fy(k) - [F(k)y(k) + &k)u(k)] + Cr(k + 1). (39)

Thus, using (36) in (39) gives

B h(k) = -Fy(k) + Cr(k + 1) + s(k + 1) - Sd(k + 1) (40)

It can be seen that the control law (25) is equivalent to (40). Using (8), the control law (40) can be written as

B 6u(k) = -Fy(k) + Cr(k + 1) +s(k + 1) - 6d(k + 1) = -F[(l - A*)y(k) + q-‘B h(k) + &d(k)]

+ Cr(k + 1) + s(k + 1) - 6d(k + 1)

= -F(l - A*)y(k) - q-‘BFh(k) + Cr(k + 1) + s(k + 1) - F 6d(k) - 6d(k + 1). (41)

Combining (8) and (41) yields the following system:

A*(q-‘) -q-‘LI(q-‘) y(k) I[ 1 F(q-‘)[l -A*(q-‘)] B(q-‘)[l +q-‘F(q-‘)] h(k)

+ Wk) -F(q-‘) M(k) - 6d(k + 1) 1 (42)

It is interesting to compare (42) with (15). It can be seen that the use of an adaptation algorithm gives rise to another input s(k + 1) to the system (15). The corresponding state-space model of (42) is given by

X(k) = A,X(k - 1) + B,s(k + 1) + D,(k) + H,(k), (43)

where

X(k)=[y(k) . y(k -2n) h(k) . . Su(k -m -n)]’

A, =

-a: . -a:+, 0 b,, 6, 0

1 0

f,al f,a,*+l M, + 6, _b,fn b,, “’ b,, b,, h,

1 0

T

B,, = 0 0 $ 0 0

0 1 ,

D,(k)= [M(k) 0 . 0

l -- ($)t;Sd(k b,,

-j)+Sd(k+-1)) 0 . . . 01’.

H,(k)= [0 0

l (;I cjr(k -j + 1) + r(k + 1)) 1 T

b, 0 . . . 0 . Note that the a,+ in the matrix A, are the coefficients of A*(q-‘), and A, is an asymptotically stable matrix whose eigenvalues are the zeros of z”+“+‘C(z-‘)B(z-‘). Since A,

is asymptotically stable and r(k) and 6d(k) are bounded. it follows from (43) that

(44)

where O< C, < x, O< C2 < =. Also, since ]lX(k)ll 2 ll4(k)ll, (44) becomes

II+(k - 1111 5 CI + G ,max_ IbG:ll. (45)

This linear boundedness condition will be useful in the following stability analysis.

3.3. Stability analysis. With (45) in place, the stability analysis will now be performed. Let the Lyapunov-function candidate be defined as

V(k) =; aT(k)8(k). (46)

Thus, using (26) (29) and (46) gives

V(k) - V(k - 1) = $?T(k)8(k) - @(k - 1)8(k - l)]

= ${[@k - 1) + K(k)a&(k - l)s,(k)lT

x [6(k - 1) + K(k)@,,(k -- l)s,,(k)]

- eT(k - l&k - l)}

= i [2tr(k)cn$:(k - 1)&k - l)s,(k)

+ K*(k)a*&‘(k - l)&,(k -- 1)$(k)]. (47)

Substituting (31) into (47) leads to

V(k) - V(k - 1) = -2K(k)s;(k) + 2K(k) dd,,(k) s,,(k)

+ K*(k)a IIht(k - l)llzsz,Ck) 5 -K(k){[2 - K(k)a]s;(k-)

- 2 6d,,(k) s,,(k)} (using (24)). (48)

If Is(k)1 < YE then

V(k) - V(k - 1) = 0. (49)

Otherwise,

V(k) - V(k - 1) 5 -(2 - a)s$(k) + 2 ISd,(k)l Is,,(k)1

(- -(2 - a)s*(k) + Zy-‘s;(k)

(using (4).“(22), (32) and Is(k)1 2 ye)

5 -(2 - (Y - 2y-‘)s;(k), (50)

where

O<a<2(1-y-l), l<y<=. (51)

Thus (27) ensures that V(k) - V(k - 1‘1 is negative- semidefinite, and hence ]I g(k) II2 5 11 e(O) /I*.

Now, after a finite time T,, Is(k)1 < YE, and K(k) will be 0. This is shown using the following argument (Mosca, 1995. pp. 325 ff.).

Let S = {t: ~(1) = 1 or Is(t)1 2 ye}.

It is claimed that S is a finite set. If it were not so then there would exist an infinite sequence {k,} in the set S with lim ,_x k, = cc. Then it follows from (50) that

lim $(k,) = 0. (52) 1-1

The next thing is to show that {s(k,)} is bounded and that

lim I(s(k,)l = 0 k,-r (53)

If {s(k,)} is bounded then it follows from (45) that II d(k,)[l is bounded as well. It also follows from (52) that (53) holds. Now, suppose that {s(k,)} is not bounded. Then there exists a subsequence {k,} in S such that lim,,<_, Is( = 5) and Is(k,)lsIs(k,)l, for k,s k,. Then, along the subsequence {km],

Ibn(kJl~ IWJI

1 + l/4@, _ 1)ll (using (“)’

2 Is(

1 + C, + Cz ls(k,)l (“lng ‘45))

d>O. C2

Hence

lim Is,,(k,l)l 2 $ > 0. *,,-= z

-1.

r II 20 66 69 88 leu

Fig. 1. System response for the simulation.

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1.

e.5.s

But (55) contradicts (52) and hence {s(k,)} is bounded. It follows from (45) that /ld(k,)ll is bounded. Also, it follows from (52) that (53) holds.

It is apparent that (53) implies that there is a k, in S such that for all k, in 5, k, > k, and Is( < YE. This leads to K(k,) equals to 0 or k, not in 5, a contradiction. Hence S is a finite set.

disturbance. The control strategy is based on an input- output model, and is of the non-switching type. The system behaviour in the vicinity of the sliding surface has been examined, and the adaptive control scheme has been shown to be robust with respect to the bounded disturbance. Simulation results have also been presented to illustrate the features of the proposed scheme.

Since S is a finite set, then, for all k > T,, Is(k)1 < YE or K(k) = 0. Hence {s(k)} is bounded and

References

8(k) = 8, = lim i(r) for all k > T,. (56) r--r=

As {s(k)] is bounded, it follows from (45) that ll4(k)li is bounded, i.e. y(k) and h(k) are bounded. The boundedness of u(k) can be established by using the B&out-identity argument as in Section 2.

Bartolini, G., Ferrara, A. and Utkin, V. I. (1992) Design of discrete-time adaptive sliding mode control. In hoc. 31st

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Chan, C. Y. (1995) Robust discrete quasi-sliding mode tracking controller. Automatica 31, 1509-1511. -

Drakunov. S. V. and Utkin. V. I. (1989) On discrete-time 4. Simulation results

In this section, simulation results are presented to illustrate the features of the proposed adaptive sliding-mode controller.

Consider the discrete single-output single-input plant described by

y(k) =y(k - 1) - 0.24y(k - 2) + u(k - 1) - 0.5u(k - 2)

+ 0.5 + 0.05 cos (0.06k).

sliding modes. In Proc. IFAC Conf: on’Nonlinear Control

Systems Design, Capri. Italy, pp. 273-277. Furuta, K. (1990) Sliding mode control of a discrete time

system. Syst. Control Lett. 14, 145-152. Furuta, K. (1993) VSS-type self-tuning control+ equivalent

control approach. In Proc. American Control Conf, San Francisco, CA, pp. 980-984.

The following specifications have been used:

f?(O) = [-0.2 -0.2 -0.2 -0.21r.

C(q-i) = 1 - 0.89-1,

Kaynak, 0. and Denker, A. (1993) Discrete-time sliding mode control in the presence of system uncertainty. Int. J.

Conrrol57, 1177-1189.

r=2, E=O.OO$ y=2.5, a=l.l9.

Figure 1 shows the system response. It can be seen that there are quasi-steady-state oscillations in the output, control input and s(k). If the sinusoidal disturbance is removed, the oscillations will disappear, and good tracking can be obtained for a negligible value of e.

5. Conclusions

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Sira-Ramiraz. S. (1991) Non-linear discrete variable structure systems in quasi-sliding mode. Int. J. Control 54, 1171-1187.

This paper has presented a discrete adaptive sliding-mode Utkin, V. I. (1992) Sliding Modes in Control Optimization.

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