Fis once again obtained by solving eqn. 27, although now

R' = [0,...,0,-ant...,-al]T (44)

Lemmas 1 and 2 again hold, and, as C(z~x) - 1,

detQ-z'1 F) = A(z-x)D(z-x)-B(z-x)G(z-x)

= T{z~x) (45)

where

F = P+QF (46)

As the parameter estimates converge, the control obtainedby calculating D(z~x) and G(z~x) from eqn. 12 convergesto that obtained by an offline calculation of eqn. 8. Thusthe calculation of F by means of eqn. 45, i.e. det(/ — z~xF)= T(z~x), gives values for the parameters of D(z~x) andG(z~x) which converge to those obtained via det (/ — z~x F)

Therefore the control obtained using u{t) — Fx'{f) canconverge to that obtained from u(t) = Fx(f), as the modelestimates converge.

Both F and F are calculated without directly employingany parameters of the C(z~x) polynomial. As the parametersof A(z~x) and B{z~x) do not converge to those of A(z~x)and B(z~x), unless Ci = c2 = 0, so Fdoes not converge to Fas the estimates converge. The asymptotic values of F canonly be calculated by consideration of the asymptotic valuesof the model estimates ax, b\ etc. This apparent anomaly iseradicated by x'(t) not converging to x(t), but the factorFx'(t) can converge, as has been shown, to Fx (r), and hencethe control input can converge to that calculated offline if themodel parametersax, bx, cx etc. were known.

5 Conclusions

With D(z~x) and G(z~x) defined as in eqn. 5, their parameterscannot be calculated explicitly by simple consideration of

eqn. 8 or eqn. 12, as there exist nx + 1 unknowns and onlyri\ equations. In the polynomial-type tuner proposed inReference 2, nd and ng are both reduced in order by one,leaving ri\ — 1 unknowns and nx — 1 equations. Hence, withthis restriction enforced, only one solution is possible foreach parameter of D(z~x) and G(z~x) at each iteration. Thishas the effect of reducing the maximum value of nt by one,from expr. 17. The reduction is carried out by effectivelyspecifying dn. as equal to zero. Then, by the nature of theproblem,#„ is therefore zero.

As mentioned in Section 4, the state-space formulationemployed here has the effect of making g0 = 0 in theequivalent polynomial-type tuner. This reduces the number ofunknowns in eqn. 8 to nx, again leading to a unique solutionfor all D(z'x) and G(z~x) at each iteration. By the samereasoning, however,g0 may be set to any value, not necessarilyzero, and the self-tuning property will still hold, each particu-lar choice leading to its own distinctly different controlleraction.

6 Acknowledgments

The author wishes to acknowledge the help of Prof. J.H.Westcott during the course of this work, and the financialassistance of the UK Science & Engineering Research Council.

7 References

1 PRAGER, D.L., and WELLSTEAD, P.E.: 'Multivariable pole-assignment self-tuning regulators', IEE Proc. D, Control Theory& Appl., 1981,128, (1), pp. 9-18

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

3 ASTROM, K.J., and WITTENMARK, B.: 'Self-tuning controllersbased on pole-zero placement', IEE Proc. D, Control Theory &Appl., 1980, 127, (3), pp. 120-130

4 ALLIDINA, A.Y., and HUGHES, F.M.: 'Generalised self-tuning•controller with pole assignment', ibid., 1980, 127, (1), pp. 13-18

5 WARWICK, K.: 'Self-tuning regulators - a state space approach',Int. J. Control, 1981, 33, pp. 839-858

IEE South Midland Electronics & ControlSection: Chairman's AddressR. G. Johnson, B.Sc. (Eng), Ph.D., C.Eng., M.I.E.E.

FEEDBACK CONTROL SYSTEMS

Indexing term: Feedback

Introduction

Although the term 'feedback' was only introduced in thiscentury, the principles of closed-loop control were being usedsome 2000 years ago. Many examplesfl] may be tracedthrough the centuries to the industrial revolution, when, in the18th century, relatively sophisticated windmill controls weredeveloped. In 1722 Meikle introduced a method of changingthe angle of the windmill sails; then in 1745 Lee's fantail wasused to rotate the sails into the wind, and in 1787 Mead used

Summary 1863D of address delivered at the University of Aston inBirmingham, 28th September 1981Dr. Johnson is with the Department of Electrical & ElectronicEngineering, University of Aston in Birmingham, 19 Coleshill Street,Birmingham B4 7PB, England

a pendulum governor for speed control. Watt adapted themethod of speed control used on windmills for speed regula-tion of steam engines, and in 1788 the flyball governor markedthe beginning of the development of control systems theory.The early Watt systems worked well until manufacturing andlubrication techniques improved. The consequence of theseimprovements was a reduction of friction in the speed controlmechanism, which caused the engines to hunt. The problem ofhunting was quite serious and led to theoretical investigationby a number of engineers and scientists. In 1868 Maxwellfounded the theory of feedback systems by showing thatanalysis could be approximated by linear differential equationsin the vicinity of an equilibrium point.

The feedback principle was not fully exploited by electrical

100 IEE PROC, Vol. 129, Pt. D, No. 3, MA Y 1982

engineers until Black showed that feedback amplifiers could beused to advantage in telephone systems to reduce distortion.

System analysis

When system complexity increased, engineers, scientists andmathematicians were encouraged to search for alternativemethods of system representation and analysis, other than interms of a set of differential equations. This led to trans-form methods and analysis of systems based on the use offunctions of a complex variable. The frequency-responsetechnique of Nyquist provided a means of basing system de-sign on practical measurements. Design methods were re-quired for improving accuracy and at the same time ensuringstability. The inverse Nyquist diagram enabled design for ad-equate performance to be achieved with feedback compensa-tors, and the Bode diagram achieved the same objective withcascade compensators.

With present-day computing facilities, both digital andanalogue computers can be used as an aid to design by relatingthe transient response in the time domain to the frequency-response loci. It is particularly useful to show how compensa-tion parameters affect performance criteria.

Nonlinear systems

The majority of practical systems are nonlinear by virtue ofthe functional relationships which exist between the inputsand the outputs of a process, or by operational nonlinearitesinvolving multiplication, division or raising to a power of oneor more variables. Such nonlinearites are frequently minimisedby feedback, whereas static accuracy is related to the accur-acies of the various measuring transducers. In many practicalsystems, nonlinearites are introduced in order to improvesystem performance or achieve an economic advantage. A non-linear system, properly designed to perform a given function,will usually outperform and cost less than a linear system de-signed to meet the same specifications. For example, optimalcontrol, for a minimum time-response specification, can beachieved by maximum energy input until the system outputvariables are within, say, 10% of desired values, at which stagemaximum damping can be applied.

As an example of how nonlinearites are incorporated inten-tionally into a system, a scheme for a convertor-fed reversingmill is shown in Fig. 1. The system incorporates speed control,armature current control and field current control. A non-linear element is used to prevent field weakening until thearmature voltage has reached a sufficiently high level, and toprevent field strengthening when kinetic energy of the motor

armature current control

and load needs to be absorbed during reversing. Other non-linear elements are used to achieve maximum forcing of timeconstants, and hence reduce loss of production during revers-ing. These practical considerations usually enable the variouscontrol loops to be designed and stabilised separately, and thecomplete system can then be checked by simulation. Thisassumes that an adequate mathematical model of the plant hasbeen identified. Fig. 2 shows the level of nonlinear interactionwhen a small DC motor is accelerated with constant field

0.2 u 1.6

Fig. 2 Comparison of predicted and measured transient responsefor a suddenly applied step of armature voltage, i = 0.3A, t = 0

predicted• ,o, X measureda Direct axis flux (scale: 1 division = 1 mWb)b Speed (scale: 1 division = 50 rad/s)c Armature current (scale: 1 division = 10A)

voltage and a peak armature current of six times full-load cur-rent. Extensive tests were needed to identify the relationshipbetween the variables before the motor speed and armaturecurrent could be predicted accurately.

Minimisation of interaction between control loops can beachieved by control action using multivariable design methods,or in some cases by feedforward techniques if the effect of achange in input is known. This method is used in strip-millcontrol when a change in strip thickness is demanded at thefirst stand. New inputs demanding a change in strip thicknessand speed can then be fed forward to other stands. Theaccurate simulation of this type of system is only possible,using modern computer facilities, if the nonlinear mathema-tical model is known. A more usual approach is to use an

speedreference

reversing

firingcircuits

firingcircuts

fieldreference

zero current

speed control

\rFig. 1 Convertor-fed reversing mill scheme

IEE PROC, Vol. 129, Pt. D, No. 3, MA Y 1982 101

approximate model for design and rely on online tuning toachieve desired performance.

Digital control

Stochastic processes, in which disturbances influence systemparameters in a random way, can constrain control actionthrough lack of information. Digital control techniques nowavailable enable very complex systems to be controlled andperformance to be optimised. There are, however, limitationsdepending on the speed of operation of the process. Complexprocesses which normally have time constants of the orderof hours can be controlled using digital computers to store in-formation about a large number of variables and issue controlinstructions. An essential feature of online control of suchsystems is to recognise that dynamic behaviour at a giveninstant in time is influenced by immediate past behaviour.Analysis originally conceived by Poincare and developed inthe 1960s by space exploration has become known as thestate-space method. An «th-order differential equation can betreated as n first-order state equations, and system control ispossible if all the state variables are measurable. Where systemstates are not measurable, they have to be estimated using

statistical models, and then, if operating time permits evalua-tion of complex algorithms, satisfactory real-time control canbe achieved.

In cases where the process is very fast, a digital controllerwill not be fast enough to achieve real-time control at everyinstant of time. This may not necessarily be a disadvantage,since executive control can enable an overall control strategyto be attained.

Conclusion

In this brief review of feedback control concepts, it has onlybeen possible to pick out a number of theoretical develop-ments that have influenced the design of better systems. Thereare many other important theories that are essential to theunderstanding of this fundamental science when applied at amore advanced level. In addition to their use in engineeringsystems, the principles can be used to describe biological,economic and social-science systems.

Reference

1 MAYR, O.: The origins of feedback control' (MIT, 1970)

CorrespondencePRACTICAL PROBLEMS IN THE USE OFCORRELATION ANALYSIS TO IDENTIFYNONLINEAR SYSTEMS

Indexing term: Nonlinear systems

Abstract: Practical limitations on a recently proposed technique,for the identification of nonlinear systems using correlationanalysis are explained. An analysis is presented which shows thatit is not, in general, possible to identify discrete-time equivalentsof the continuous linear subsystems in a system consisting of alinear system in cascade with a static nonlinear element followedby another linear system.

Introduction

The development of techniques for the identification ofsystems containing single-valued nonlinearities has been thesubject of much research effort. The techniques available maybe broadly classified [1] into those which identify thecomponents of a functional series representation of the systemand those which identify the components of various blockswhose interconnection determines the system structure. Theselatter techniques involve less computational effort than thosebased on a functional series representation and are particularlyattractive when they yield information about the systemstructure, in addition to providing data sufficient to identifythe various components.

The use of correlation analysis in both the structuredetection and the identification of the linear and nonlinearsubsystems in a class of nonlinear systems has been the subjectof a number of recent papers by Billings and Fakhouri [2, 3,4 ] . By employing the separability property of certain randomprocesses, these researchers have developed an identificationalgorithm which provides information regarding the systemstructure. When applied to the general model illustrated inFig. 1, the algorithm yields data sufficient for a par-ameterisation of the identified linear subsystem impulseresponses and the nonlinear characteristic.

The present study will show that while the algorithm ofBillings and Fakhouri [2] will permit the identification of asystem of the form shown in Fig. 1, this is only true if theidentification is performed in the same time domain as that on

n(t)

x(t)h, (t)

q(t)N(O

y(t) z ( t )

Fig. 1 Class of nonlinear systems

which the system operates; i.e. the algorithm functions if asystem with continuous linear subsystems is identified incontinuous time, or a system with discrete-time subsystems isidentified in discrete time. It will not, however, in generalyield the results claimed by Billings and Fakhouri, i.e. discrete-time equivalents to the continuous linear subsystems when acontinuous-time system is sampled and the identificationperformed in discrete time. Since this latter case is the one ofmost practical significance, a serious limitation of thealgorithm is therefore exposed.

Review of theory

If the input signal x{t) to the system illustrated in Fig. 1 is awhite nonzero mean Gaussian noise process, then, employingthe separability property of Gaussian processes and assumingthe linear subsystems are bounded-input bounded-outputstable, it can be shown that

J —

and

h\ (j)h2 (a - T)drO

(1)

(2)

where <j)xz' is the crosscorrelation between x(t) and z'(t) and(f>xxz' is the crosscorrelation between x2 (?) and z'(t) with

z'(t) = z{t)-z{t) (3)

102 IEEPROC, Vol. 129, Pt. D, No. 3, MA Y1982