An Adaptive Pole Placement Control System for Plants with Unknown Order Subjected to Deterministic Disturbances

Copyright © IFAC Systcm IdcnlificaLJon. Copenhagcn. Denmark. 1994

AN ADAPTIVE POLE PLACEMEJIIT CONTROL SYSfEM fOR PLAJIITS WITH UNKNOWN ORDER SUBJ ECTED TO DETERMINISfIC D1SfURBANCES

N. MIZUNO* and S. FUJII**

Faculty of Engineering, Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Japan Faculty of Engineering, Nagoya University, Furo, Chikusa, Nagoya, Japan

Abstract. This paper presents a new discrete time adaptive pole placement control system for plant with unknown order subjected to deterministic disturbances. Using additional adaptation algorithms, called 'the common order estimator' and ' the common factor estimator', this method avoids the singularity problem in the recursive parameter estimation and can reject the disturbance described by the output of the stable autonomous system with unknown coefficients. In this paper, first, we design the pole placement controller which decouples the disturbance from the plant output, in the case where the parameters of the plant and the characteristic polynomial of the disturbance are known. Next, we propose an indirect adaptive control system with the common order and the common factor estimator, in the case where the order, parameters of the plant and disturbance are unknown. Finally, the results of computer simulation of the adaptive control applied to a plant with disturbance are included to illustrate the effecllveness of the proposed methods. Key Words. Indirect adaptive control; estimator; deterministic disturbance

pole placement; discrete time system; common factor

I. INTRODUCTION Thus far, many different types of discrete time adaptive control schemes have been proposed for plants with arbitrary zeros and various properties of such schemes are investigated (Koivo, 1980; GoodWin, 1981; Ellioll, 1982; Lozano, 1982; Suzuki, 1983; Samson, 1982; Grimble, 1982; M'sadd ~ ~ 1985). In these allempts, the adaptive pole placement control scheme IS an Important class of the design methods which can deal with non-minimum phase, unstable and unknown dead time of the plants. However, there are still some problems to investigate when an adaptive pole placement method IS applied to a real plant. In most adaptive pole placement design, it is assumed that the order of the unknown plant is exactly known or equivalently that the estimates satisfy the nonsingularity condition in the linear simultaneous equation (Diophantine equation) with respect to the controller parameters. However, the order of the real plant is merely known and the estimated parameters may result in singularity problem. Moreover, in these methods, it is assumed that there are no deterministic disturbance which affect the plant. To overcome the former problem, various modification is proposed in the design of adaptive pole placement schemes (Lozano and Goodwin 1985; Ellioll ~ ~ 1985; Kreisselmeier and Smith, 1986; Giri ~ ~ 1988; Youlal ~ ~ 1988; Konishi and Yoshimura, 1989; Mirkln 1991; Giri ~ ~ 1992; Lozano and Zhao, 1992)). However, the resulting adaptive controllers can only avoids the slngurality problem under some conditions or eliminate the common factors in parameter estimates. If the estimated polynomials based on the model with unknown order have the common factors even If the plant polynomials are coprime, tl].e common factors should be regarded as the uncontrollable but observabl{' mode of the plant. On the other hands, it IS noticed that the periodiC disturbance (e.g. slllusoidal or DC level)

can be modeled as some uncontrollable but observable mode of the plant model. In this case, it is important to design an adaptive controller regarding the presence of deterministic disturbances (Sebakhy, 1982; Goodwin and Chan, 1983; Fujii and Mizuno, 1985a, 1985b; Xianya, 1984) • Under these conditions, we propose a new adaptive pole placement scheme for plants with unknown order subjected to deterministic disturbances as IS modeled by the uncontrollable but observable mode of the plant. First, the structure of the controller which decouples the disturbance from the plant output is discussed. Next, we propose an indirect adaptive control system with the common order and the common factor estimator, in the case where the order, parameters of the plant and disturbance are unknown. Finally, the results of computer simulation of the adaptive control applied to a plant with disturbance are included to illustrate the effectiveness of the proposed methods.

2. SfATEMENT Of PROBLEM Consider a single-input, single-output, invariant plant described by

linear,

time-

(2.1 ) where y(k) and u(k) denote the plant output and input respectively, and w(k) is a bounded deterministic disturbance. Alz- I ) and B(z-I) are polynomials in the unit delay operator z-I as follows. (2.2) B( z -I) ; b I z -I +oo.+bnBz -nB

(2.3)

The disturbance w(k) regarded as the output of the linear autonomous system can be described by :

(2.4)

From this figure, it can be seen that the transfer function Grylz· l ) between r(k) and y(k) is given by:

where Olz-I) is a characteristic polynomial having the following form.

01 z • I) =I+dlz ·1 +•.• +d nOz -nO

(2.5)

Note that Eqs.(2.4) and (2.5) imply that the disturbance w(k) can be modeled as some uncontrollable but observable mode of the plant (Goodwin and Chan, 1983; Fujii and Mizuno, 1985a, 1985b). The following assumptions are made about the plant and the dis' turbance.

13.5) This implies that the desired closed loop poles of the system are given by the zeros of znCC(z- ). In Fig.I, 'K' denotes the feed forward compensator which will cause the plant output y(k) to approach the reference sequence y (k) as k tends to inflnitt. Now, assume that the bounded reference sequence y (k) has the following property. 13.6)

a) ai' b i and d i are the unknown but constant parameters. b)

with 13.7)

The upper bound of the order n=max(nA +nO' nB+nO) is known from a priori knowledge.

where C*(z-I) is a stable polynomial of order n*= nC'

d The polynomials tively

A(z·I), Blz· l ) and O(z-I)are rela·

nB+ I.

prime.

The problem here is how to design an adaptive pole placement control system subjected to deterministic disturbances when the plant is a non-minimum phase system with unknown order. In the next section, the structure of the proposed controller is discussed.

In this case, suppose that the polynomials C*lz'l) and Blz- I ) in Eq.(2. J) are relatively prime, there exist unique polynomials K(z·l) and L(z-l) of order nK =n *-1 and nL =nB-1 respectively, such that: 13.8) where

3. CONTROL OF KNOWN PLANTS L( z -I) =l+ll z -I +... + I nLz -nL First, when all parameters ( and the order) of the plant and the characteristic polynomial of the disturb· ance are assumed to be known, we design a control system which decouples the disturbance from the plant output. Given that A(z-I), B(z-I) and O(z·l) are relatively prime polynomials, there exist unique polynomials Slz-l) and R(z·l) of order nB-I and nA +nO-1 respectively, such that:

-nK K( z-I) =kO+k I z-I +... +k nK z

13.9)

USIng the above polynomial Identity and Eqs.(3.5) and 13.6), it follows that: C(z-I )(y(k)-y * (k))=B(z-1 )r(k)-K(z-I Iy * (k)

(3.101

Thus if the external Input IS generated by: (3.1 ) r(k)=Klz-l)y *(k)

(3.11 )

with the control objectives are achieved. In the case where the reference sequence is restricted to step changes, the above compensator is simplified as follows. (3.2) K(z-I )=K( I )=C( II/BI I)

Using the polynomials R(z·l) and S(z-I) in the above polynomial identity, the Input IS generated by the control law:

(3.12)

Moreover, it can be shown that the relation between the disturbance w(k) and the plant output y(k) is given as: (3.131

O(z -I )u(k)=v(k)

(3.3)

S(Z -I )v(k)=r(k).R( z -I )y(k)

13.4)

where r(k) is an external input described later. The structure of this control system based on the Internal model principle can be written as shown In Fig. I.

r-----l~_--_,

2:.:..r0-~--r~1-~..r41-~J.r~~~ L~_r'L2J I

:

y

~ ,~ - ~71 J i ~

r---l

Plant

I

"--------:' R"\'\-----~. ~_J

Fig.

I

Block u,agram of lhe po'" pl"('<'I11<'1I1 cOl1ll'ol system undcr dctcrmlll,stlc u,sturbal1ces

Regarding Eq.12.4) , the disturbance is asymptotically de coupled from the plant output.

4. ADAPTIVE CONTROL OF UNKNOWN PLANTS Since the parameters and the order of the plant are both unknown, the auapllve controller should recur s,vely determine these values. In the proposed method, we estimate the plant order and the parameters based on the non-minimal representation which includes the descrtptlon of the deterministic disturbance. hrsl, 1<'1 us 1I11rouuu' the l1on-m 1111 m <.I I plwlt model which COllt<.llllS ullcontroll"ble but observ<.lble modes. USll1g [qs.ll.l) "I1U 12.41, the pl
as lGoodwin and Chan. 1983): O(z-I )A(z-I )y(k)=O(z -I )B(z -I )u(k)

(4.1 )

Equation (4.1) can also be represented as : y(k)= eT c. (k-I)

(4.21

A'(k.z· l ) is realized by controllable realization. thE order of the common factors can be determined a~ the number of the linearly dependent vectors in thE controllability matrix based on the Gram-Schmit algorithm as follows IShinnaka and Suzuki. 1983): Step I:Oefine the column matrix

vector of the controllablE

with

(4.10 T e = [a·) •..•• a·n·b·I· .. ··b·nl where (4.3)

C. T(k_1 )=[y(k-I )..... y(k-n).u(k-l )•.•.• u(k-n)!

G(k)= [ - O(k)

o

where

O( z -I )B( z - J I-b' - JZ -I + ... + b,-n nZ

(4.4) Note that the plant model is of the higher order than the basic representation (2.1). and the denominator and the numerator have some common factors which correspond to the over parametrization or the characteristic polynomial of the disturbances. In order to estimate the parameters of the plant model (4.2), the folloWing recursive algorithm with constant trace gain matrix is used (Suzuki ~ £1:. 1983).

e(k)=

e(k-!)+ r (k-I) C. (k-I)e *(k)

(4.5)

°T(k)=la' I (k)

a· n(k)!

8 T(k)=[b' 1(k)

b' n(k)!

14.1 J I

Step 2:Calculate normalized vector (4.12 Step 3:Calculate orthogonal vector v i(k)=xni(k}-(x ni (k)T v n J )vn 1(k}-(x ni (k)T v n21k))vn2(k)-..... -(x ni(k)T v ni-I (k))v ni-I (k)

(4.13)

where v I (k)=x n I (k) Step 4:0etermine the system order

r-I(k)= "I(k) r -I(k_I)+ "2(k) c.(k-I) c. T(k_l)

(4.6)

e*(k)=[y(k)- eT(k-I) c.(k-I)I![I+ C.T(k-llr (k-I) C.(k-I)] (4.7)

if I1 vi(k) I1 < t::.. I» t::. >0 then System order ic=i-I else if i
where

e

T(k)=[a' I (k).... ,a ' n,(k).b' I (k).... ,b' n,(k)! (4.81 parameters in (4.8). the followUSing the estimated Ing estimated polynomials can be reconstructed.

(4.14)

else System order ic=n end if

A' lk. z-I) = I -a' I (k) z·1_ ... -a' n,(k) z-n ' B'(k. z-I )=b' I (k)z -I +... +b' n,(k)z -n'

(4.91

In these polynomials. A'(k.z- J ) .B'lk.z- I ) correspond to Olz-I )Alz- I ). O(z-I )B(z-I) respectively. However. the estimated polynomials A·(k.z· I ). B'(k.z· l ) based on the non-minimal model (4.2) usually have the common factors even if the plant polynomials are coprlme and the disturbance is present or not. In this case. the common factors should be regarded as the uncontrollable but observable mode of the plant. On the other hands. to determine the polynomials used in the control law: Eqs.(3.3) and (3.4). the estimated polynomials of O(z·I)A(z-l) and Blz- I ) are required. For this situation. we treat all common factors as the uncontrollable mode of the plant and propose the adaptive control system having the dual additional estimators: ~ order estimator of ~ CO!!)!!!Q!! factors and ~ com!!)on factors estimator for the disturbance rejection feedback loop.

By using the above algorithm. the order of common factor can be determined as nO=n'i c '

the

4.2 The common factors estimator To estimate the common factors in the estimated polynomials A'(k.z- I ). B'(k.z- I ). consider the follOWing polynomial identity similar to Eq.13.I). C(z-I )O(z-I )=Alz- I )O(z-I )S'(z-l )+B(z-I )O(z-I)R I(z-I) (4.15)

where I , -2 -nS' S'I z -I) =I +sI ,z - +s2 z +... +snS'z ,-2 -nR' R 'l z -I) =rO I HI ,-I z +r2 z +... +rnR'z

14.161

Using this identity. the common factors Olk.z- I ). can be determined for given estimated polynomials A'(k.z-II and B·lk.z- I ) as follows (Shtnnaka and Suzuki. 1983),

4.1 The order estimator of the common factors O(k. z-I )Clz -I )=A' lk,z' I )S'(k. z-I )+B'(k. z-I)R 'lk. z-I) When

the eSllmuted

transfer

function

O·lk.,-l)

(4.17:

B (k,z -I) =b, () k z -I +... +b n. nD (k) z -n-nD

This equation can also be written as follows. 4>(k) 8 c(k)=

f)

(4.18)

lk)

where

(4.26)

Substituting B(k.z· l ) and A'(k.z· l ) in to Eq. (3.1). we can obtain the estimated polynomials R(k.z- I ) and S(k.z· l ) used in the control law. Moreover. the feed forward compensator which ensure the tracking objective can be recursively determined when the characteristic polynomial C· (z·l) of the reference sequence .is unknown. In this case. it is assumed the order n to be known. From Eqs.(3.6) and (3.8). we obtain: C(z·1 )y* (k)=K(z·1 )B(z -I )y. (k)

8?

(4.19)

~ f(k)

(4.27)

where 8c T(k)= Is I (k)•.•• snSlk).r O(k) •••• r nR(k).d I (k).... dnD(k)\ (4.20)

T 8 f = IkO· .. ··knKJ

~fT(k)= IB(z·1 )y*(k)..... B(z-1 )y·(k-nK)1

(4.28)

(4.21 ) The matrix 4>(k) IS nonsingular when the polynomials A'(k,z-I) and B'(k.z- I ) has the common factor of order Since k by lated

nD' it is ensured that 4> (k) is full rank for all time common order estimation, 8c (k) may be calcuas (4.22)

or equivalently calculated by some recursive algorithms (Lozano and Landau. 1982; Suzuki et ~, 1983/. From this procedure, we can get D(k.z~) S'(k,z-) and R'(k.z- I ). However. S'lk,z-I) and R'(k z-l) are not the same polynomials as Slz-I) and Rlz- 1) which are used in Eq.(3.4). In order to calculate S(z-I) and R(z-I). we need the polynomial B(z·I). In this case since D(k.z· l ) and B'(k.z·l)(which corresponds D(z-I)(B(z·I)) are already obtained, the estimated Blk.z- I ) can be given from Eqs.(4.5) and (4.9). Consider the following equation.

Based on the above representation and the estimated polynomial B(k.z- I I. the parameters of the feed forward compensator are recursively estimated. If the reference sequence is given by step changes, the adaptive feed forward compensator is of the form: K(k.z· 1)=K(k.1 )=C( I )/B(k.1 I

(4.29)

If required. the Integral action can be incorporated into the adaptive controller. The main modification is In Eqs.(3.1) and (3.3) which become: C(z-I )=( I-z-I )A'(k, z-I )S(k.z- I )+B(k.z- I )R(k.z· l )

(3.1')

Using the adaptive controller proposed in this section. the control objectives of both tracking and regulation are completely achieved when the estimated parameters converge to their true values. In this method based on the non-minimal plant representation, the persistency of excitation of the reference sequence ensures the convergence of the adaptive system.

(4.23)

5. SIMULATION RESULTS

where

In order to investigate the effectiveness of the proposed methods. computer simulations are performed for a plant with disturbance. The plant parameters in Eq. 12.1) abruptly change In two times as follows. Plant-I (nA=2, nB=2. d=1 : k=I ..... IOO) A(z -I )= 1-1.84 z-I +0.84 z-2. Blz -I )=0.0394z -I +0.0373z- 2 (4.241 Plant-2 (nA=I. nB=3. d=2 In the above equation. the matrix <1: Blk) and the vector 6 (k) are determined using the estimates in Eqs.(4.20) and (4.7) respectively. When the estimated parameters converge to their true values. the equation (4.24) has a unique solution. However. it may not be unique In adaptive situation because the coeffiCient matrix IS not square. In order to determine the estimate B(k.z- I ) at all time k, (k) is computed as (Fujii and Mizuno. 1985b) : T -Slkl 8B T(k)-' .
I'

14.25)

k=101 ..... 200)

Alz -I )= 1-0. 779z -I, B(z -I )=0. 109z -2+ 0 . 252z- 3 Plant-3 (nA=I, nB=3. d=2

k=201 ..... 3001

A(z -I )= I -0.846z -I, B( z-I )=0.0656z -2 +0.24 I z-3 During the above parameter changes, the plant characteristics varies from stable minimum phase system to non-minimum phase system with different order i.lnd dead tI me. III 111,- dl'S1gll 01 UdU»IIVl' cuntrul system, the deSired clused Iou» pules are usslgned b} t he zeros of

The threshold of the common order estimator IS selected as 6=0.0001. The initial values of the estimated parameters are set 8T(0)= [-1.-1.1.1.0.7.0.7.0.9.-0.9.-0.91.

In this example. the dlsturb,lnce w(k)= 1.0

(k=71 ..... 170)

w(k)=O.O

(otherwise)

5 Plant'2iPlant'3

~I

\; I . 0lst - . -+:t-

a step change as

IS

lOO

I

200

300

For this situation. the upper bound of the plant order is set at 4. In the adaptation algorithm (4.5)-(4.7). r (0)= 104 ) and 0 =1.0 are used. The reference sequence y*(k) is given by step changes and the feed forward compensator Eq.(3.12) is used. Figure 2 (aHd) shows the simulation result.

300

3 4 Plant.l_FPlant.2 -/<--Plant'3--

.

t

y(U

:

I~

O.

I

·2 7

OlSt

I

-; 0

I

300

200

OsJ'

~

ir

!~

I

1----,

200

i~

if

100

0

i

\..-1

-....,~ i-i--- II =.=::1~v!=},

oI

Fig. 2

~

I

;

·5

-~

100

0

0

1//:

W

L"'-

-;:

:

200

300

I 300

Simulation result for the first example (a)-plant output. (b)-control input

In the upper figure (a). + shows the plant output y(k) and 0 shows the output y"(k) of the reference model with the same poles as the desired closed loop poles and with unit DC gain. The lower figure (b) shows the control input.

Fig. 3

Simulation result for the second example (a)plant output. (b)control input, (c)estimated param eters. (d)estlmated order

:le:,' ~

Plant-l ,~

Plant-2

Plant-3

y*( k)

-;:

i;t-:i

o!l. i ·.

:.

'y(u\':

'1 3

o

DlSt( \

•

.~: •.

I'

F

......;,....-.i 100

~ . ;

;..;"

t I '\.----l

200

300

~',

5

o 21

11

[,

~

c

I

~ .i

U

.,,<0,

O' 0

Fig. 2

aI il_l '-

cn, ,. 2 I I

n ... n, ). J I

I

I

I

i

I[ 100

200

300

Simulation result for the first example (c)-estimated parameters. (d)-estimated order 300

Figure 2 (c) shows lh,· ,'stimated [",r"m"If'SrS of the non-mlnlmill mouel ilnd 2 (dl shows estlmilt''tl oroelr of the cmmon factor.

~·ig.

4 Slmulilllon re~ult fur the third example (a)plant uutput. (b)control input. (c)estimated order

In the second example. the disturbance is a sinusoid described as follows. w(k);0.2 sin(O.25k) w(k);O

(k;71 •...• 171) (otherwise)

Goodwin, G.c. and Ch an. S. W. (1983). Model Reference Adaptive Control Systems Having Purely Deterministic Disturbances. IEEE Trans. Aut. Control. AC-28. 855-858. Goodwin. G.c.. Hill. D.J. and Palaniswami. M. (1984). A Perspective on Convergence of Adaptive Control Algorithms. Automatica. 20-5. 519-531.

The conditions in adaptive system are the same as the first example except that the upper bound of the plant order is set at 5. The simulation result is given in Fig. 3 la)-Id).

Grimble. M.J. (1982). Weighted Minimum-variance Self-tuning Control. ~.L Control. 36-4, 597-609. KOlvo. H.N. (1980). A Multivariable Self-tuning Controller. Automatica. 16-4. 351-356.

The third example:

Konishi. K. and Yoshimura. T. (1990). Adaptive PolePlacement Algorithm Based on Common Factor Elimination. Proceedings .Q.!. IFAS:: Symposium on Adaptive Systems J.D. Control and Signal Processing, Glasgow, !,2~. 107-112.

In this case. only the adaptation gain is changed to r(O); 100001 compare to the second example. Figure 4 shows the simulation result.lt can be seen that the higher adaptation gain gives the faster convergence. From these simulation results. the proposed adaptive controller can fairly well estimate the parameter and order changes in the unknown plant and decouple the disturbance.

Kreisselmeier. G. K. and Smith. M. C. (1986). Stable Adaptive Regulation of Arbitrary nth-order Plants. IEEE Trans. AC-31. 299-305. Lozano- Leal. R. and Landau. I.D. (1982). Quasidirect Adaptive Control for Non-minimum Phase Systems. Trans. AS~h ser.• g. 104. 331-316.

6. CONCLUSION In this paper. we have presented the adapt ive pole placement algori thms for plants with unknown order subjected to deterministic disturbances. The proposed adaptive control algorithm could work fairly well for parameter changes in the unknown plant and some deterministic disturbance. 7. REFERENCES Elliott. H. (1982). Direct Adaptive Pole Placement with Application to Non-minimum Phase Systems. IEEE Trans. Aut. Control. AC-27-3. 720-722. Elliott. H .• Cristi R. and Das M. (1985). Global Stability of Adaptive Pole Placement Algorithms. IEEE Trans. Aut. Control. AC-30. 348-356. Fujii. S. and Mizuno. N. (l985a). A Discrete Model Reference Adaptive Control System for Plants with Unknown Deterministic Disturbances. Trans. So. Instrum. and Control Ei!&, 21-9. 914-920. Fujii. Sand Mizuno. N. (1985b). A Discrete Time Adaptive Control System Based on Pole Placement Method for Plants with Unknown Deterministic Disturbances. ibid., 21-10. 1021-1028. Giri, F.. Dugard. L.. Dion, J. M. and M'Saad. M.( 1988). Stable Pole Placement Direct Adaptive Control for Systems with Arbitrary Zeros. ~.L of Control, 47. 1143-1454. Giri. F .• Ahmed-Zaid. F. and loannou. P. A. (1992). Stable Indirect Adaptive Control of Cont inuousTime Systems with no A Priori Knowledge on the Parameters. Preprints .Q.!. .!!:~~ ~!!)posiu!!) on ~daptive Syste!!)~ J..!! Control and ~ Processing, Grenoble. France. 497-502. Goodwin. G.c. and Sin, S.W." (1981). Adaptive Control of NOlll11lJlIl11UI11 I'husc Syslellls. J.!JJ.: TrailS. I\uto. COlllrol, AC-26, ~78-~8L.

Lozano-Leal. R. and Goodwin. G. C. (1985). A Globally Convergent Adaptive Pole Placement Algorithm without a Persistency of Excitation Requirement. IEEE Trans. AC-30. 795-798. Lozano-Leal, R. and Zhao. X. H. (1992). SingularityFree Adaptive Pole Placement for 2nd Order Systems. Preprints .Q.!. IFA~ ~posium on Adaptive Systems on Control and Signal Processing. Grenoble, France. 503-508. Mirkin, L. B. (1991). Closed-Loop/Open-Loop Discrete Adaptive Control of Continuous-Time Plant with Arbitrary Zeros. Preprints of IFAS:: ~mposium on Identification and System Parameter Estimation. Budapest, Hungary, I, 322-327. Samson. G. (1982). An Adaptive LQ Controller for Non-minimum-phase Systems. ~ .L Control. 35-1, 1-28. Sebakhy. O.A. and Salama. A.I.A. (1982). Adaptive Control of Linear Plants with Step Disturbances. ~.L.Q.!. Control. 36-2. 235/247. Shinnaka. S. and Suzuki, T. (1983). Adaptive Pole Placement Applicable to Nonminimum Phase Systems of Unknown Order and Unknown Dead Time. Trans. So. Electro. Comunl. J66-A, 12. 1228-1235. Suzuki. T., Shinnaka. S. and Tanaka, M. (1983). A Design Method for Adaptive Pole-placement Systems. Trans. So. Instrum. and Control, 19-1, 2835. Xlanya, X. and Evans. R.J. (1984). Adaptive Control of Discrete-time Time-varying Systems with Unknown Deterministic Disturbances. lEE proc. Q... Control theory ~ ~ 131-3. 81-84. Youral. Y., Najlm, K. and Najim, M. (1989). Regulurl/ed I'ole I'lucemell.t Adapllve Control. l'ro(,('I'dlllgs ~ J..!::0~ 2t!!)POSlU!!! ~ !3obusl I\dapllv(' Cont rol. Newcast le, Australia. 73-77.