Generalized Self Tuning Controller with Self Searching Pole Shift

Copyright © IFAC 10th Trienn ial World Congress, Munich, FRG, 1987

GENERALISED SELF TUNING CONTROLLER WITH SELF SEARCHING POLE SHIFT A. Chandra, O. P. Malik and G. S. Hope Department of E lectrical Engineering, The University of Calgary, Calgary T2N l N4, Canada

Abstract. A strategy for the design of a self -tuning controller that mlnlmlZes a cost function incorporating system input, output, and set-point vari ations, and adapts it in such a way that the closed loop poles are shifted radially towards the origin of the z-domain is described in this paper. It eliminates the necessity of choos i ng the closed loop pole 1ocat ions and st ill it possesses the qual i ty of robustness of 'pole-assignment' and ease of reference signal tracking of the 'selftuning controller' . KetOrds. Self tuning control, pole-shifting adaptive control, self- searching po e-shlft. INTRODUCTION Self-tuning controller developed by Clarke and Gawthrop (1975) provides one of the most successful and flexible approaches to selftuning control. The aim of it is to minimize the variance of a generalised cost function whi ch incorporates input, output, and reference control signal. It provides an elegant way of reference signal tracking, and the choice of cost function influences closed loop response and the control effort demanded. A difficulty with this scheme is that of determining the cost funct ion whi ch wi 11 provi de the most satisfactory overall control scheme performance.

very close to the origin. Under transient conditi ons thi s factor cannot be taken so small because of the practical limits on control. Therefore, the value of this factor is chosen such that it achieves the best compromise between small and large system disturbances., By utilising a self-searching pole shift factor [Cheng et al •• 1985] these difficulties can be overcome. Instead of choosi ng a fi xed val ue of a, appropriate value to match system operating conditions is computed recursively every sampling interval.

The solution to this problem was provided by the 'Generalised self-tuning controller with pole assignment' [Allidina and Hughes, 1980]. They demonstrated the link between classical control strategy of pole-assignment [Well stead et al., 1979] and the suboptimal strategy of the selftuning controller [Clarke and Gawthrop, 1975], and proposed a generalised controller which possesses the major advantages of both.

SYSTEM MODEL The system model is assumed to be of the form: A(q-1)y(t)

=

q-k B(q-1)u(t) + ~(t) (1)

where: the polyno~als A and B in the backward shift operator q- are defined as 1 -1 -na A( q-) aO + a1q + + an q a -nb B(q-1) b + b q-1 + + b q o 1 nb k represents the system t ime delay in sample i nterva 1s. Vari ab 1es y and u are the system output and input, with ~(t) an uncorrelated random sequence of zero mean which disturbs the system.

In the pole assignment strategy, the closed loop poles are placed at prespecified locations. The amount of control effort requi red is to some extent proportional to the distance of the proposed closed loop locations of the poles from thei r open loop 1ocat ions. The poor choi ce of closed loop locat i ons may resu l t in large control effort and it may make the system unstable. This can easily happen when 'a priori' decision on the system transfer funct i on cannot be made.

PROPOSED CONTROLLER

It is proposed to modify the pole-assignment strategy of the 'generalised self-tuning controller' [Allidina and Hughes, 1980] to a pole-shifting control strategy proposed by Ghosh et al., (1984). In the pole shifting strategy the closed loop poles are shifted radially towards the origin of the z-domain, and a stable controller is achieved.

The controllers suggested by Clarke and Gawthrop (1975), All i dina and Hughes (1980), and We 11 stead et al. (1979) are bri efly revi ewed in the appendix to provide the necessary background to the controller proposed in this paper and described below. Generalised Self Tuning Controller with Pole

Selection of a suitable value for the pole-shift factor, a , as proposed by Ghosh et a1. (1984), depends on the operating conditions. Under dynamic conditions, pole-shift i ng factor can be close to zero, i.e. the poles can be shifted

Shlft In the proposed controller the pole-assignment strategy of the 'General i sed self-tuning 91

92

A. Chandra. O . P. Malik and G. S. Hope

controller with pole-assignment' [Allidina and Hughes, 1980J is first modified to pole-shift technique. Based on the pole-assignment strategy, pole-shift control, while retaining the basic advantages, eliminates the requirement of specifying the locations of the closed loop poles. In eqn. (24), instead of specifyi ng the polynomial T, the closed loop poles may be shi fted radi ally towards the ori gi n of the zdomain [Ghosh et al., 1984J such that T becomes: T = A(aq - 1) = 1 + aa q -1 + ••• + (a n )a-qn ••• 1

n

(2)

with a, pole shifting factor, close to, but less than one. Here A parameters of the system are By solving eqn. (13), A and F not known. parameters can be found in terms of P parameters, and then eqn. (24) can be solved for P and Q parameters. Generalised Self Tuning Controller with Self Searchl ng Pol e Shl ft

2)

Estimate control parameters, a, by a standard recursive least squares algorithm [Strejc, 1980J, where

a

[h 0' h l' ••• , hn ' go' gl' ••• , gn ' eO' el' • •• , en J H G E

h, g, and e are the coefficients of the general polynomials H(q-1), G(q-1), and E(q-1) of orders nH' nG' and nE' respectively. Compute control from:

3)

u(t)

=

-[goY(t) + gl y (t-1) + ••• + eOw(t) +

e w(t-1) + ••• + ••• + h u(t-1) + ••• J/hO 1 1 4) Find coefficients of A and F polynomials in terms of P parameters using eqn. (13). Putting parameters A in eqn. (2), T can be found in terms of P parameters and a(t). Solve for P, 0 from:

5)

Applying self-searching pole shifting technique [Cheng et al., 1985], the algorithm for calculating a(t) can be formulated as follows:

6)

PH - q-k GQ = haFT . a. a~ au Fl nd aa' "f""' and aa as shown in eqns. (4), (5), and \~).

Rewriting eqn. (11)

7)

Find 6u as in eqn. (7).

= Py(t)

8)

Form a modification factor, 6a, as shown in eqn. (6).

9)

Calculate a(t)

~(t)

+

Qu(t-k) - Rw(t-k)

Reference fo 11 owi ng is achi eved by sett i ng R P. Jhen,.(t) can be written as ~

=

(3)

ZX

PO' P1' ••• and qo, q1' ••• are functions of a, and X = [{y(t)-w(t-k) y(t-1)-w(i--k-1)} ••• u(t-k) u(t-k-1) ••• J a. az From eqn. (3), aa (4) = aa' X From eqns.

a.

(

1) and (11), ail

Also, let the control u . (U(U ml n max Pole-shift modifi cat ion given by 6a

= -K

where :

1;~1-1 6u

=

be

hO constrained

10) Repeat from (1) for the incremented value of The algorithm may be started with coefficient Po of polynomial P set equal to unity and the rest of the coefficients of P and Q polynomials set to arbitrary values. Generally, R is equal to P. Initial value of a(t) can be taken close to, but less than, one.

(5)

A NUMERICAL EXAMPLE

as:

A nonmlnlmum phase system is considered having the system description factor, 6a, is

then [1_1.7q-1 + 0.72 q-2]y(t) u(t) + ~(t)

= q-1(a.2

~max

- u u - umln . au a.

u ) 0 u <0

(7)

go gl eo e 1) The algorithm is started with a

=

(h o h1

ainitial = 0.8 P1 initial = 0.0 qo initial = 0.0 ainitial = (0.0001 0.0001 keeping Po

a(t-1) + 6a

(9)

0.0001 0.00d1

ALGORITHM The algorithm can now be summarised as follows: Form a generalised output function:

= Py(t)

+ Qu(t-k) - Rw(t-k)

0.0001 0.0001)

= 1.0 constant.

Since, system time delay, k

~(t)

0.3 q-1)

In this study ~(t) is a random gaussian noise sequence with zero mean and 0.05 rms value.

The variable pole-shift factor, a(t), is given by

1)

+

(6)

6u

au writ i ng (8) aa ~aa ~ term can be obtained from eqns. (4) and (g1. K is some positive constant to avoid excessive variations in a(t).

a(t)

a(t-1) + 6a

t.

Z = [PO P1 ••• qo q1 ••• J

where:

=

fFrom O· the e d f·lnltlon . . but aO = 1; hence Po = fO = 1.0.

0f

= 1, polynomial F =

F (eqn. 13) , Po

=

aD f 0;

Solving eqn. in step (5) of the algorithm for P1 and qo gives

93

Generalized Self Tuning Controlle r with Self Searching Pole Shift 2 PI = (gohoa

gohl - hI - hoagO}/(h O - ~ - hOa)

The response of the system to step changes in reference input (Fig. I) is shown in Fig. 2 and the controller output is shown in ~g. 3. The output during the first 200 samples in Fig. 2 is in response to the presence of noise only. The same is the case for the samples 500 to 750. It can be seen that the response to the noise s i gnal is quite stable. Although the algorithm was started with poor initial values, it can be seen from Figs. 4 and 5 that Pl and qo converge qui ck 1y. It is shown in Fig. 6 that although during the steady state conditions the pole shift factor, a(t}, remains zero forcing the closed loop poles near the orig i n of the unit circle in z-domain, during the transient period it increases to some value 1ess than one, and thus the closed loop pol es move away from the origin of the unit circle for a short duration. It also changes the values of Pl and qo' modifying the weightage of u and y in tne cost function for a short period of time.

PAS-I03, pp. 1983-1989. Strejc, V., (19flO), 'Least square parameter estimation'. Automatica, Vol. 16, pp. 535550. Well stead, P.L, Prager, D., and Zanker, P., (1979), 'Pole assignment self tuning regulator'. lEE Proceedings, Vol. 126, No. 8, pp. 781-787. APPENDIX Bri ef out 1 i nes of the cont ro 11 ers proposed by Clarke and Gawthrop (1975), Allidina and Hughes (19flO) , and Well stead et al. (1979) are gi ven here. Self Tuning Controller In the self tuning controller proposed by Clarke and Gawthrop (1975), a cost function minimised by the control law for self tuning is considered to be of the form:

(l0) where: t(t + k} = Py(t + k} + Qu(t} - Rw(t}

(11)

CONCLUSIONS The generalised self-tuning controller with pole assignment has been modified to a pole shifting algorithm. There is no need to specify the closed loop pole locations and still the proposed controller possesses the quality of robustness of pole assignment and ease of reference signal tracking of self-tuning controller. From the example, the controller is shown to have good asymptotic as well as transient behaviour. A number of studies performed verify the generally good behaviour of the algorithm. The algorithm is now being tested as an excitation controller for a synchronous generator and the i nit i al studi es show encouraging results.

P, Q and R are polynomials in q-l and w is the control set point From eqns. (I) and (11) the following expression can be obtained: t(t + k} =

+ O) u(t} - Rw(t} +

~ ~(t

+ k) (12)

The last term in eqn. (12) can be split into two sequences as follows:

~ ~(t where:

+ k) =

F~(t

F~(t + k} v~lues,

A

It is quite feasible to implement the proposed controller on a microcomputer in real-time if the controlled system is identified by a low order model. For system identification by a higher order model, the size of the equations for the coefficients of the polynomials P and Q very large and the computational becomes requirements will make it difficult to implement in real-time the self-searching pole-shift on a microcomputer. In that case, the alternatives available are: multi microcomputer implementation with or without dual-rate sampling, or constant pole-shift factor.

(~B

+ k} +

*

~(t)

represents future disturbance

~(t)

relates to present and of disturbance sequence 1 F and G are polynominals in qF is of order (k-l)

past

v~lues

and Hence

P

7i=F+q

-k G

(13)

A

To minimize the cost function Gawthrop, 1975J the control law is

[Clarke

Hu(t} + Gy(t} + Ew(t} = 0 where:

H = BF + Q

E = -R

and (14) (15)

Pole Assignment Controller REFERENCES Allidina, A.Y. and Hughes , F.M., (19flO) , 'Generalized self-tuning controller with pole assignment'. lEE Proceedings, Vol. 127, Part D, No. 1, pp. 13-18. Cheng, Shi-jie, Chow, Y.S., Malik, O.P . , and Hope, G.S., (1985). An adaptive synchronous machine stabil i zer'. IEEE/PES 1985 Joint Power Generation Conference, Paper No. 85 JPGC 601-0. Clarke, D.W., and Gawthrop, P.J., (1975), 'Sel f tuning controller'. lEE Proceedings, Vol. 122, No. 9, pp. 929-934. Ghosh, A., Ledwich, G. , Malik, O.P., and Hope, G.S., (1984), 'Power system stabilizer based on adaptive control techniques'. IEEE Trans. on Power Apparatus and Systems, Vol.

Assumi ng the system to be of the form descri bed by eqn. (1), a control signal can be applied such ,that Hu (t) + Gy (t) = 0 where:

(16)

Hand G are polynomials in q-l.

Incorporating the control law of eqn. (16) into system eqn. (1) the following closed loop description is obtained H ~(t) (17) HA + q-k BG With pole-assi gnment [Well stead et al., 1979J the closed loop performance should be such that y(t} =

94

A. Chandra, O . P. Malik and G. S. Hope H

y(t) = f

~(t)

(l8) , .S

where T is prespecified and defines the location of the closed loop poles. For eqn. (18) to be satisfied, the following relationship must hold .....,

HA + q-k BG = T

For eqn. (19) to have a solution, the order of the regulator polynomials and the number of closed loop poles nt must be nG = na - 1 nH nb + k - 1 nt ( na + nb + k - 1

1.5

:::I

(19)

a. c: <11 U

0 .5

c: <11 '<11 "<11

~ -0 .5

-ltO----------~2.~O----------~S~OO~--------~,.O

Generalised Self Tuning Controller with Pole Assignment

Figure 1.

The self tuning controller of Clarke and Gawthrop (1975) was extended to pole-assignment by Allidina and Hughes (1980). The self tuning controller gives the closed loop description BR H y(t) = BP + AQ w(t-k) + BP + AO ~(t) (20) P and 0 are chosen such that BP + AO

=

... ~ a.

1 ,5

i5

1

.....

(21)

T

where T is a prespecified polynomial in q-1. For eqn . (21) to have a solution, the orders of polynomials P, 0, and T must be np

na - I, na

-o.S

-11~O----------~'~'~O----------~S~O.O----------~' 0

nb - I, nt ( na + nb - 1 (22) Equation (21) cannot be solved directly because parameters A and B are unknown. Hence, to obtai n appropri ate P and 0 of the performance cost function, A and B are eliminated as foll ows: =

=

Multiplying eqn. (21) throughout by F gives BPF + AOF

=

+

Figure 2.

number of samples System Output with Change in Reference

..... :::I

a.

..... :::I

FT

o

'<11

Substituting for AF from eqn. (13) BPF

number of samples Change in reference input.

o -

OP - q-k GO = FT

.....'-c:

0-

u

Further use of eqn. (15) reduces this to PH - q-k GO

=

FT

(23) Figure 3.

As shown by Allidina and Hughes (1980), to i dent ify pa rameters H, G, and E, eO can be fixed, and hence eqn. (23) is changed to PH - q- k GO

=

hOFT

number of samples Controller output u(t)

1.25

(24)

If to = 1 and PO = 1 eqn. (24) can be sol ved for polynomials P and O.

0 . '11

o eT

0 .5

"-

o

c:

0 . 25

o

.....

'" ;-0.2.

.~

>

-O"'!"O-----------:2~.~O----------~s:"!Oc:-O---------"""'"'.0 number of samples Figure 4. Variation of qO

Ge ne rali zed Self Tuning Controlle r with Self Sea rching Pole Shift

, .25

0 . 75

.....

0..

.....

...

o

c::

O . 2~

o

....,

~

'"

.~

l-

ra: -0 . 25

~---------'r----------v----'r-------------

>

-o·'t.-------L.-.;1.t;;.-------;±:__------"..'• Figure 5.

number of samples Variation of PI

0 . 25

~

0.2

~

....." 0

0 . 15

c:: o

....,

.~

'"

• •1

.~

I-

'">

0.05

·J·~-----n~um~rr.:'-r--o-f-s-amJLp-l-e-s~·~··:-------~,·· ' Figure 6,

Variation of pole shift factor, a(t)

95