Generalised self-tuning controller withpole assignment

A.Y. Allidina, M.Sc., A.M.I.E.E., and F.M. Hughes, M.Eng., Ph.D., C.Eng., M.I.E.E.

Indexing terms: Closed-loop systems, Control system synthesis, Poles and zeros, Tuning

Abstract: This paper presents a self-tuning controller that minimises a cost function incorporating systeminput, output and set-point variations, and adapts it in such a way that the closed-loop poles are located onprespecified locations. The link that exists between the classical control strategy of pole assignment and thesuboptimal strategy of the self-tuning controller is demonstrated and made use of to produce a generalisedcontroller which possesses the major advantages of both. The controller combines the robustness of poleassignment with the ease of reference following provided by suboptimal self tuning.

Principal symbols

A(q~1),B(q~1),C(q~l) = polynomials of order na, nb

and nc corresponding to, re-spectively, system output, con-trol input, and disturbanceinput; a0 = c0 = 1

E{'}= expectation operationk = system time delay (integer)

-P(<7~1)>(2O7~1)>-K(<7~1)=:: costing polynomials of ordernP, riq and nR acting on, re-spectively, system output, inputand set point

t = time measured in sampleinstants

y(t),u(t), w(t) = system output, input and setpoint, respectively, at time t

£(/) = uncorrelated zero-mean randomsequence

q'1 = backward shift operation:q~ku(t) = u(t-k)

E{q~l), F(q-'), G(q~l), H(q~l), T{q~l) = general poly-nomials

A polynomial A{q~l) of degree na is denoted by

A(q~l) = a0 +aiq-x + . . . + anaq-n°

The argument is dropped for convenience in the text.

1 Introduction

Self-tuning control of single-input/single-output systems hasbeen much studied of late. Optimal regulation of suchsystems, where the aim is to minimise the variance ofthe output, has been studied by Peterka1 and Astrom,2

while Clarke and Gawthorp3 have developed a suboptimalstrategy in which the aim is to minimise the variance of ageneralised cost function. Clarke's approach provides anelegant way of reference signal tracking, and can also beused by a 'cut and try' procedure to control nonminimumphase systems where optimal controllers1'2 fail.

An altogether different approach has been adopted byEdmunds12 and Wellstead et a/.4'5'6 for pole-assignmentregulators, where the controllers are chosen in such a waythat the closed-loop-system poles are placed at prespecifiedpositions depending on the required transient response and

Paper 474D, first received 30th April, and in revised form 24thSeptember 1979The authors are with the Department of Electrical Engineering andElectronics, UMIST, PO Box 88, Manchester M60 1QD

IEEPROC, Vol. 127, Pt. D, No. 1, JANUARY 1980

on maximum system response rates. This regulator is there-fore more robust and can be applied with more ease to non-minimum phase systems. However, some difficulties arepresented when tracking of reference inputs is involved.The problem has been tackled by including servo compen-sation in the loops,5 but the price paid for this is a signifi-cant increase in computational effort. An alternative andmore favoured approach has been to incorporate digitalintegrators into the loop6'8 and have the controller operateon the difference of the system output and the set point.

Pole placement has also been considered by Astrom etal.n However this work differs from that of Wellstead etal.*'s> 6 in that it focuses entirely on the servo problem; nostochastic disturbances are considered in deriving the con-troller.

The pole-assignment technique has been developed fromthe classical control viewpoint and may be considered to bea departure from the suboptimal approach of Clarke. How-ever, since a specified pole location can be uniquely trans-lated into a suboptimal performance criterion, the twoapproaches are obviously linked. In this paper, the relation-ship between the approaches is demonstrated and made useof to provide an extended suboptimal controller, incorpor-ating pole assignment, that possesses the robustness of poleassignment4' 6 and the ease of reference signal tracking ofthe self-tuning controller.3

Before deriving the extended controller, the self-tuningcontroller and pole-assignment regulators are summarised,and the success of the derived controller is demonstrated byapplication to a nonminimum phase system.

2 Self-tuning controller

A novel class of control laws was developed by Clarke andGawthrop which provided several advantages over self-tuning strategies which only minimise output variance. Byintroducing a cost function incorporating system input,output and set point variations, the facility for controlweighting and set point following is provided in addition tothe capability of dealing with nonminimum phase systems.

Single-input/single-output randomly disturbed systemsare considered which may be described by:

Ay(t) = 0)where A, B and C are polynomials in q x, with a0 = c0 = 1,and k represents the system time delay in sample instants.Variables y and u are the system output and input, with £an uncorrelated random sequence of zero mean whichdisturbs the system. The argument of the variables cor-responds to the sampling instant.

13

0143- 7054/80/010013 + 06 $01-50/0

The control law for self tuning minimises a cost functiongiven by:

where

00 + k) = Py{t + k) + Qu(t)-Rw(t)

(2)

(3)

and P, Q and R are polynomials in q l and w is the con-trol set point.

Combining eqns. 1 and 3 gives the following expression:

00 + k) = f j + Qj u(t)-Rw(t) + ™ \{f + k) (4)

The disturbance term can be re-expressed as follows:

PC G— %(t + k) = FW + k) + -S(t)

where F and G are polynomials in q'1 and F is of order(k — 1). The disturbance term has now conveniently beensplit into two sequences: F%(t + k), which relates to futuredisturbance values, and (G/A) £ (t), which relates to presentand past values of the disturbance sequence.

Consequently

PC

A

Suitable manipulation of eqns. 1,4 and 5 gives

0 0 + k) = Hu(t) + Gy(t) + Ew(t) + F%{t + k)

(5)

(6)

where

H = BF+QC E = -CR

If the applied control is such that

= 0

C*=C-l (7)

(8)

the last term in eqn. 6 vanishes,3 giving

4>(t + k) = Hu (0 + Gy if) + EW(t) + F${t + k) (9)

The control law (eqn. 8) therefore minimises the cost func-tion I.3

In self-tuning, the parameters of H, G and E can beidentified from eqn. 9 (since 0 0 + k) is known from eqn.3), using recursive least squares,7 for example, and thenthese estimated parameters can be employed in the controllaw of eqn. 8.

3 Self-tuning pole assignment regulator

This technique has been developed by Wellstead4'6 and usesa classical control approach whereby the control objectiveis to locate the closed-loop-system poles at prespecifiedlocations. Although such an approach does not aim foroptimality, by specifying pole positions appropriate accountcan be taken of engineering constraints in the form of per-missible control signal magnitudes. The major points infavour of the pole-assignment approach are that it providesa robust controller6 and can cope more readily with non-minimum phase systems than the previously outlined sub-optimal self-tuning controller where suitable adjustment3

of the control signal weighting in the performance costfunction is required.

Considering a system described by eqn. 1, a controlsignal can be applied such that:

Hu(t)+Gy(t) = 0 (10)

where H and G are polynomials in q l .When the control law of eqn. 10 is incorporated into the

system (eqn. 1) the following closed-loop description isobtained:

HC

HA+q-«BG(ID

With pole assignment the closed-loop performance shouldbe such that:4' 6

y(t) = f (12)

where T is prespecified and defines the location of theclosed-loop poles.

Inspection of eqn. 11 indicates that for eqn. 12 to besatisfied the following relationship must hold:

HA+q'hBG = TC (13)

Also, for eqn. 13 to have a solution, the order of the reg-ulator polynomials and the number of closed-loop polesnt must be:4 6

nG = na -

nH = nb + k—\

nt < na + nb

(14)

k — nc —

So if the parameters A, B and C are known, we can solveeqn. 13 to get the controller parameters H and G.

When self tuning, the system parameters A, B andC are not known, and hence eqn. 13 cannot be solveddirectly to get the controller parameters. In the self-tuningliterature4*6 two distinct forms of system models havebeen employed. It has been shown4'6 that the system ofeqn. 1 under control described by eqn. 10 may be describedmore conveniently by the following two models:

(a) Model I4 can be described by

y(t) = (15)

where e(t) is a moving-average process of order (k — 1), i.e.e{t)- V%(t), where V is a polynomial of order (k— 1).The orders of the polynomials .4! and^! are, respectively,(na — 1) and (nb + k— 1). The procedure adopted is toidentify A i and Bx from eqn. 15, and then obtain H and Gfrom the following equation:

= VT (16)

In eqn. 16 there are (na + nb + 2k — 2) unknowns to befound.

(Z>) Model 26 uses the following description:

y(t) = B2u(t-l)+A2y{t-\) + l(t) (17)

where the orders of the polynomials A2 andi?2 are> respect-

ively, (na — 1) and (nb + k— 1). This model is not aspecial case of model 1 with k = 1, but is valid6 for any k.

14 IEEPROC, Vol. 127, Pt. D, No. 1, JANUARY 1980

A2 and B2 may be identified from eqn. 17, and H and Gfound from

= T (18)

Eqn. 18 has (na + nb + k — 1) unknowns, which is {k — 1)less unknowns than in the identity of eqn. 16. Model 2 isthus of computational advantage compared with model 1.

It has been shown4' 6 that solving eqn. 16 or 18 gives thesame controller parameters as solving eqn. 13.

4 Generalised controller with pole assignment

It is now shown how the self-tuning controller3 may beextended to achieve pole assignment.

The self-tuning controller3 described in Section 2 givesthe closed-loop description

y{t) =BR H

BP + AQ v ' BP + AQ

If Pand Q are chosen such that:

BP + AQ = T

(19)

(20)

where T is a prespecified polynomial in q l. For eqn. 20to have a solution, the orders of the polynomials P, Q andT must be

= na — l,nQ = nb — and nt<n nb

(21)It should be noted that, if the A and B polynomials havecommon factors, owing, for example, to the system orderbeing over-estimated, then eqn. 20 cannot be solved. How-ever, reasonable familiarity with system dynamic behaviourcoupled with the natural complexity of most real systemsnormally ensures that this situation does not arise.

Making use of the expression in eqn. 20, eqn. 19 may berewritten as

y{t) = ~^(t- (22)

If w(t) = 0, i.e. reference following is not required and thecontroller only regulates for random disturbances, eqn. 22reduces to

(23)

and we achieve the same closed-loop behaviour as the pole-assignment regulator4' 6 of eqn. 12.

When self tuning, the system parameters A and B areunknown and hence eqn. 20 cannot be solved directly toobtain the appropriate P and Q of the performance costfunction. However, multiplying eqn. 20 throughout by Fgives

BPF + AQF = FT

Substituting for AF from eqn. 5 results in

BPF+QPC-q~hGQ = FT

Further use of eqn. 7 reduces this to

PH-q-kGQ = FT (24)

In the self-tuning controller,3 the parameters H, G and Eare identified recursively from eqn. 9, Obviously for the

first evaluation, starting values for P, Q and R need to beassigned. Having estimated H and G, eqn. 24 may then besolved to get improved values for P and Q that will be usedin the next step. Note that the polynomial F is alsounknown.

From the definition of F (eqn. 5), copo = a0f0, butao=Co = 1;hencep0 =/0 ,and, from eqn. 24, poho =f0t0,thus giving ho = t0. When estimating parameters fromeqn. 9, one parameter needs to be fixed,2 so we could fixho = t0. However, it has been shown2 that the parameterestimated could diverge if h0 is fixed at a much smallervalue than its true value. This problem can be avoided byfixing e0, a parameter that is not system related, instead ofh0.

3 If h0 is estimated freely from eqn. 9, then the relation-ship hQ = t0 (from eqn. 24) is not necessarily satisfied.However, by introducing a constant multiplying factord into the right-hand side of eqn. 24, the following isachieved:

,-kPH-q-*GQ = dFT (25)

Consequently after estimation of h0, d is chosen to satisfythe relationship hQ = dt0, and p0 is fixed to unity. Similarly,eqn. 20 becomes:

BP + AQ = dT (26)

If o = 1 is chosen for convenience, then d = h0, giving(na + nb + k — 2) unknown parameters.

The algorithm can be summarised as follows:(i) Form: 0(0 = Py(t) + Qu{t - k) -Rw(t - k)

(ii) Estimate H, G and E from:0(0 = Huit -k) + Gy(t-k) + Ew(t -k) + e(t)

(iii) Apply control: Hu{t) + Gy(t) + Ew(t) = 0(iv) Solve for P, Q from: PH ~q~hGQ = h0FT (see

Appendix)(v) Repeat from (i) for the incremented value of t.

The algorithm may be started with coefficient p0 of poly-nominal P set equal to unity and the rest of the coefficientsof the P, Q polynomials set to arbitrary values (NB.R = P). The routine to solve for P and Q (step (iv) above)need not be continuously on line. P and Q may be tuned inperiodically.

5 Simulation

A demonstration of how the generalised controller withpole assignment works will now be given in terms of simu-lated examples.

Example 1A nonminimum phase system is considered having thesystem description

The aim is to have the closed-loop poles defined byT= 1 — 0-5#-1. Since polynomials A and B are known,eqn. 26 can be solved directly to find the required values ofP and Q to which the self-tuning algorithm should converge.

For the first-order example chosen, P and Q are simplyconstants, and values of P = p0 = 1, Q = q0 = 4 provide therequired pole location and reference following achieved bysetting R = P.

If 6 is defined by the parameter set

0 = i,h2,g0,eQ,ex)

IEEPROC, Vol. 127, Pt. D, No. 1, JANUARY 1980 15

then the parameter set dtrue which satisfies the requiredcontrol constraint that T = (1 — OSq'1) is

Qtrue = (50, 1-5, 1-2, 0-8, - 1 - 0 , 0-2)

In the first case the algorithm is started with initial valuesof/>= 1,(2= l,/? =

0initial = (2-5, 0-75, 0-6, 0 - 4 , - 1 0 , 01)

Fig. 2 shows the system output and Fig. 3 the manner inwhich Q approaches the required value of Q = 4. In this

u

12

1 r\10

86

U

2

0

-2-4

-

-

0 50 100 150 200 250

Fig. 1 Reference input sequence

Fig. 2 System output for example 1

100 150 200 250

Table 1 :

Time innumber of samples

300600900

1200

Comparison of losses for cases a and

Loss

468105016342229

Increasein loss

(a)

468582584595

Loss

1570331750586821

\b

Increasein loss

(b)

1570174717411763

50 100 150 200 250 300

Fig. 3 Variation ofq0 for nonzero initial estimates

16

Fig. 4 Variation ofq0 for zero initial estimates

example, since P = p0 = 1 and R = ro= 1, the extendedalgorithm finds the required value of control-signal weight-ing to provide the specified closed-loop dynamic perform-ance. In the basic self-tuning controller of Reference 3, asuitable value of control weighting would have had to befound by a 'cut-and-try' method.

In the second case the starting values were P — 1, Q = 0,R = 1 and

6 initial = ( 0 , 0 , 0 , 0 , - 1 , 0 )

Fig. 4 shows that Q converges to its final value much moreslowly than in the previous case, and as a consequence thesystem performance is initially degraded. This is especiallytrue when following a rapidly changing reference signal.

Next, the loss (y(t) — w(t — 2))2 is compared for(a) the generalised controller with pole assignment(b) pole assignment with the controller operating on the

difference between the system output and set point asexplained in References 8 and 6.It can be seen (Table 1) that the increase in loss for case (a)is approximately one third of that for case (b). For a slowlychanging reference signal the difference between the twowould be smaller.

Example 2A second-order system which is stable and minimum phaseis now considered:

06q

= q-\l-5-0-53q~l + 0-9q-2)u(t) + (1 -

By setting T= 1 — 0-5q~l, the required P and-(? polynom-ials to which the self-tuning algorithm should converge are

P = (1-0-54?"1) and 0 = (0-2 + 0-818?"1)

The algorithm is started with p0 = 1 and px = q0 = qx = 0with R made equal to P. The system output is shown in Fig.

IEEPROC, Vol. 127, Pt. D, No. 1, JANUARY 1980

5, and Figs. 6, 7 and 8 show that p1, q0 and qx converge totheir required values.

Other systems considered exhibited similar character-istics to the example given in this Section. Even for mini-mum phase systems, if initial estimates of the parametersare set to zero (apart from e0 = — 1), then Q convergesslowly to its required value so that the performance overthe initial period will be poor. One way to improve initialperformance, other than starting with better estimates, is toplace limits on the control input amplitude.8 The use of asmaller forgetting factor when estimating the parameters ofH, G and E also improves initial convergence.

Although the generalised controller has proved success-ful in the applications considered to date it is appreciatedthat, with the time-varying parameters involved in the esti-mation procedure, convergence problems could arise, andthis aspect of the work will be the subject of future work.

150

Fig. 5 System output for example 2

200 2 50

100 200 300 400 500

Fig. 6 Variation ofpt for example 2

6 Conclusions

The self-tuning controller which minimises a performance-cost function has been extended to achieve pole assign-ment. The generalised controller combines the robustnessof a pole-assignment regulator4*6 with the ease of referencetracking of a self-tuning controller.3

In deriving the extended controller, the link whichexists between the classical control strategy adopted by

Wellstead4'6 and the suboptimal strategy of Clarke,3 isdemonstrated. It is shown how the two approaches canreadily be combined to provide a generalised controllerwhich has the advantages of both. Unlike the extendedcontroller presented here, pole-placement controllers,considered by Astrom et a/.,11 are not able to deal withstochastic disturbances, and require on-line polynomialfactorisation.

Fig. 7 Variation of qt for example 2

2 0

1-5

10

0-5

-0 5

-2

100 200 300 400 500

Fig. 8 Variation of qx for example 2

50 100 150 200 250

-U

Fig. 9 Variation of q0 for example 1 using modified identifi-cation procedure

IEEPROC, Vol. 127, Pt. D, No. 1, JANUARY 1980 17

7 Acknowledgments

The authors would like to thank Dr. P.E. Wellstead,Control Systems Centre, UMIST, for many helpful dis-cussions concerning the work, and UMIST for the financialsupport and computing facilities provided.

8 References

1 PETERKA, V.: 'Adaptive digital regulation of noise systems'.2nd Prague IFAC symposium on identification and process para-meter estimation, 1970

2 ASTROM, K.J., and WITTENMARK, B.: 'On self-tuning regu-lators', Automatica, 1973, 9, pp. 185-199

3 CLARKE, D.W., and GAWTHROP, P.J.: 'Self-tuning controller',Proc. IEE, 1975, 122, (9) pp. 929-934

4 WELLSTEAD, P.E., EDMUNDS, J.M., PRAGER, D., andZANKER, P.: 'Self-tuning pole/zero assignment regulators', Int.J. Control, 1979, 30, pp. 1-26

5 WELLSTEAD, P.E., and ZANKER, P.: 'Servo self-tuners', ibid.,1979, 30, pp. 27-36

6 WELLSTEAD, P.E., PRAGER, D., and ZANKER, P.: 'Poleassignment self-tuning regulator', Proc. IEE, 1979, 126, (8) pp.781-787

7 ASTROM, K.J., and EYKHOFF, P.: 'System identification - asurvey', Automatica, 1971,7, pp. 123-162

8 WITTENMARK, B.: 'A self-tuning regulator'. Department ofAutomatic Control, Lund Institute of Technology, report 7311,1973

9 ASTROM, K.J.: 'Introduction to stochastic control theory'(Academic Press, 1970)

10 GAWTHROP, P.J.: 'Some interpretations of the self-tuning con-troller,'Proc. IEE, 1977, 124, (10), pp. 889-894

11 ASTROM, K.J., WESTERBERG, B., and WITTENMARK, B.:'Self-tuning controllers based on pole-placement design'. Depart-ment of Automatic Control, Lund Institute of Technology,TRFT-3148, 1978

12 EDMUNDS, J.M.: 'Digital adaptive pole-shifting regulators'.Ph.D. thesis, Control Systems Centre, UMIST, 1976

9 Appendixes

9.1 A comment on the solution of the identityPH- q~kGQ = FT

Depending on k, the system time delay, and nc, the orderof the C polynomial, the number of equations obtainedfrom the identity by equating coefficients of powers of q~l

can be more than the number of unknowns. For nH = orderof//, and similarly for other polynomials, from the defini-tion of/ /and G (eqns. 5 and 7)

or

whichever is the greater,

nc),

and

«G = (na - 1) or (nc + rip - k),

whichever is the greater.

From eqn. 21

"p = («a - 1), riQ = (nb - 1)

Substituting for nP and nQ into nH and nG,

(27)

and

= (nb+k-l) or (nb +nc- 1),

whichever is the greater,

= ("a-1) or (na-l +nc-k),

whichever is the greater. (28)

The identity has (nP + nQ + k) unknowns, and (nH + nP)equations may be formed by equating powers of q~x. Bymaking use of eqns. 27 and 28 it can be seen that

if k > nc, number of equations

= number of unknowns

and

if k < nc, number of equations

> number of unknowns

In the second case, the extra number of equations, formedby equating coefficients of higher powers of q'1, are dis-carded. In practice, it may be possible to assume thatnc < k\ then, the above duplication of equations does notoccur.

The set of equations is solved by triangulating the co-efficient matrix, using successive row manipulations. Sincethe matrix is sparse, this approach provides a faster solutionthan a general matrix-inversion method.

9.2 Form of estimation

The form of estimation used (eqn. 9) means that both Pand Q affect the control-law estimation. This is because theparameters of H, G and E which are identified depend onboth P and Q. A different form of estimation as suggestedin Reference 10 estimates parameters that are independentof<2-

From eqn. 9

0(0 = Hu(t-k) + Gy(t -k) + Ew(t e{t) (29)

By making use of the definitions of// and E (eqn. 7), eqn.29 may be rewritten as

0(0 = (BF)u(t - Gy(t-k) + Cy(t -k) + e(t)(30)

where j(t — k)=Qu(t — k)— Rw(t-k). Eqn. 30 can thenbe used to estimate parameters of (BF), G and C, which areall independent of Q.

Fig. 9 shows the variation of q0 for example 1 (startingwith zero initial estimates) when the modified estimationstructure is used; this may be compared with Fig. 4. Noimprovement is obtained when this modified structure isused for example 2; this is because here P also affects theestimation.

18 IEE PROC, Vol. 127, Pt. D, No. 1, JANUARY 1980