1965 Short Papers 87

COSCLUSIOSS

For a given rectangular region of uncertainty, the multivariable dichotomous optimum-seeking method requires fewer function eval- uations than does the previously proposed multivariable Fibonacci technique. The dichotomous method is applicable to rectangularly unimodal functions, a class which includes the strictly convex func- tions needed for successful application of the Fibonacci method.

REFERENCES [ I ] Sugie. X., An extension of Fibonaccian searching to multidimensional cases.

[2] Grolak. P.. and L. Cooper, An extension of Fibonaccian search to several I E E E Trans. on Autmnalic Control (Conespottdence). vol AC-9 Jan 1964. p 105.

[3] Kieier. J.. ,%uential minimax search for a maximum, Proc. Am. M u l k Soc.. variables, Comm. Assoc. Comp. Mach.. vol 6 Oct 1963. pp 639-41.

[I] Wilde. D. J.. Optzmum Seeking Methods. Englewood Cliffs, N. J.: Prentice-Hall. vol.1. 1953, pp 502-6.

[SI Wild& D. J., Optimization by the method of contour tangents. Am. Insl. C k m . 1964 P. 33.

Engrs. J., vol 9, Mar 1963, pp 18G90.

Approximate Parameter Invariance with Nonlinear Feedback

J. AI. HAM, SESIOR NENBER, IEEE, AXD h.1. 4. HASSLAN

Abstract-The principle of invariance, first discussed by Kule- bakin [ Z ] for systems subject to external disturbances, is extended to the problem of building systems that are insensitive to plant- parameter variations. Through the use of nonlinear signal com- parators and plant models, several classes of systems that are rela- tively insensitive to large parameter variations have been devised. These feedback configurations implicitly identify major parameter changes and modulate the plant drive to compensate for these changes. The control of nonminimum phase plants and systems with noisy output transducers may be improved with these schemes.

INTRODCCTION

The problem of the sensitivity of system output and state to plant-parameter variation is an old one that is receiving renetved in- terest as connections between sensitivity, controllability and stabilitJ- are becoming better understood. The virtues of using high-gain feed- back to reduce parameter sensitivity have long been recognized [ 1 ]. However, high-gain feedback has perhaps too often been regarded as an effective answer to all problems, for example that of rejecting dis- turbances. 4 more general approach to the reduction of the depend- ence of system output on external disturbances is given by Kulebakin [ 2 ] . In this work, which has been called the “theory of invariance,’’ necessary conditions for the freedom of the system equations from disturbance terms are used to derive the structure of the control system.

For the problem of sensitivity to plant-parameter variation, more general approaches than high-gain feedback originated with the con- cept of adaptivity [3]. “Passive-adaptive” schemes [4] using non- linear operations to reduce sensitivity, and “active-adaptil-e’ schemes [j], where the plant is identified and this information is used to adjust the compensator, have been described. The complexity of these adaptive schemes continues to make difficult a general ap- preciation of their sensitivity and convergence properties.

This paper develops in a heuristic way a theory of invariance for parameter variation that leads to some novel nonlinear feedback configurations. The systems arrived at are intended to be relatively insensitive to large variations in plant parameters. Such parameter- insensitil-e systems should be distinguished from optimum adaptive systems [6].

Manuscript received May 25, 1964; revised October 6, 1964. Facilities for this research were provided in part by the Defence Research Board, Canada, under Grant No. D.R.B. 4003-03

Canada. J. AI. Ham is with Dept. of Electrical Engineering, University of Toronto,

Egypt. C.A.R. M . .a. Hassan is with the Atomic Energy Research Establishment. Anshas,

Consider a linear plant P as shown in Fig. 1 described by a dif- ferential equation of the form,

L b ) = K(tjm (1)

where L( ) is a linear operator of the form

and K( t ) is a time-varying gain factor. Suppose the inverse of L( ), namely L-’( ) is realizable. Then the system in Fig. 1 when excited from rest will haxre the input-output equation,

L(y) = K d r (2)

where & is a desired value for K(t) . In Fig. 1, Kd is a pure-gain operator. Note that (2) is independent of K ( t ) , the varying plant gain. The scheme of Fig. 1 of course suffers (as do some of the schemes of [2 ] ) from the need for in\-erse operators. However, the basic con- figuration of nonlinear arithmetic operators and the “model” (& i n Fig. 1) will be retained in practical systems now to be discussed.

APPROXIblATE PaRANETEK INVARIANCE

We shall now devise feedback configurations that will provide approximate parameter invariance (invariance after a transient period) but which are free from the need for in\-erse operators. The systems will contain only plant models and arithmetic operators. In the heuristic analysis to follow, it will be assumed that each variable parameter is piecewise constant. Initial conditions at the beginning of a segment are neglected. These assumptions will permit us to use Laplace transform analysis to develop system configurations. Simu- lation data that establish the usefulness of the configurations will be given.

T o begin let us consider the first-order plant described by the scalar equation,

- = dy - a(t)y f K ( t ) m at

where a( t ) and k ( t ) are unknown functions of time. If k ( t ) only varies, the plant will be called a “variable-gain” plant, while if a ( t ) only varies, the name ‘‘mo\ing-pole’’ w i l l be used. The control configura- tions for these two types of plants will be different.

VAFSABLE GAIT I N A FIRST-OKDER PLAST

The plant equation is,

Figure 2 show the system configuration for approximate parameter invariance. In this system, we can write, under the assumptions stated above, in pseudo Laplace notation,

where k is the current unknown x-alue of k ( / ) , and

x=- 112 K d

s + a

Hence,

and

VI = r e = - r K d k

Using (8) in (5) \\-e have,

y=- * K d s + a

which indicates that the system (within the approximations made) is insensitive to the variation of k ( t ) .

Multiplication , , Plant

Division

Fig. I . An impractical example oi invariance.

P l a n t

Fig. 3. The variable-pole scheme.

Plant

I I Model

Fig. 2. The variable-gain scheme.

The plant equation is now,

-= - a(t)r + Km dt Ls-"

Figure 3 shows the control system configuration. Using again the assumption of piecewise constancl- for a(t) and neglecting initial con- ditions a.e can write in pseudo Laplace notation,

Fig. 4. Second-order plant with variable 81 contrullecl bb- a moving-pole scheme.

rt Plant

Model

T

+F Model Model

X, -

or

Fig. 5. The composite variable-gain, moving-pole scheme.

which indicates that the system is insensitive to the variation of a ( t ) (xvithin the approximations used).

If fixed poles and zeros are added to the simple plant and model in Fig. 3 and the above analysis is repeated, the result shows independ- ence of the one poles= --a that does shift.

A SECOSD-ORDER PLAXT KITH VARIABLE DAMPISG Consider a second-order plant described by the equation

The damping factor 7 is varying. Figure 1 shows a control configura- tion for achieving approximate parameter invariance.

rlgain assuming that ~ ( t ) has a piecewise constant fluctuation and neglecting initial conditions n-e can write with reference to Fig. 4.

1 1 n z (151

m = (r - e)

Short Pafie7-s 89 I965

and

R Y =

s* + 2 i 7 d Z S + lib2 The schemes of Figs. 2, 3 and 1 are suited to handling changes in

one parameter only. In practical situations involving changes in more than one parameter the separate schemes can be combined. The variable-gain scheme, for example, may function satisfactorily in the presence of a moving pole. Then we can build a compound scheme, as shown in Fig. 5, by using the variable-gain configuration directly on the plant and controlling the resulting configuration through the use of the moving-pole scheme. Such a system has been shown to be capable of handling large simultaneous changes in the gain and cer- tain poles of the plant.

The pseudo Laplace transform analysis of all of these systems suggests that they will yield invariance in the sense that the response of the system to an input will approximate the response of the model to the Same input. The variable-gain schemes yield exact invariance in the steady-state when the plant gain is fixed but unequal to the model gain. The following simulation results illustrate the degree to Lvhich transient and steady-state invariance is achieved.

SIMULATIOS RESULTS

The results were obtained on a general purpose analog computer. The actual system response is drawn as a solid line and where p o s sible the response of the fixed plant model to the same system input is shown as a dotted line.

The presence of dividers and multipliers in the control configura- tions presented leads to ob\-ious practical difficulties. For example in Fig. 2, if z and yare initially zero. e is indeterminate and if set initially to zero, no system excitation can occur. A problem is clearly present whenever the value of the denominator of a divider passes through zero, while that of the numerator does not, for example in the scheme of Fig. 4 when the undamped natural frequency of the model does not coincide with that of the plant. The signal I then, theoretically cm- tains singularities. I t has been found that such difficulties can be handled in practical systems by providing suitable amplitude limiting on the outputs of dividers and providing where necessary appropriate initial conditions, and thresholds of sensitivity.

Figure 6 shows the step response from rest of the moving-pole scheme of Fig. 4. In Fig. 6(aj, the damping factor of the plant is one iifth of that of the model. In Fig. 6(bj, there is an additional discrep ancy in the parameter T = l/w. Note that the plant-drive signal m ( f ) no\\- contains evidence of spikes.

Figure 7 shows the response of a variable-gain scheme controlling a nonminimum phase plant. Figure 7(a) shows the response of the system output to a step change in the input from 10 to 20 with fixed gain differences between the plant and model. Figure 7(b) shows the regulator action for step variations in plant gain. In a simple linear unity-feedback system, the maximum value of k for stability is 1.0. Under the conditions of Fig. 7(a), the response of the system to the indicated step change in input is indistinguishable from the response of the model to the same input.

COXLVSIONS The brief results given indicate that the proposed schemes for

parameter invariance have some quite useful properties. In the situa- tions shown the steady-state deviations of the system output are zero. The duration of the transient deviations are on the same scale as the settling time of the output of the plant model when excited from rest by a step input. Except for the divider 'spikes!" which may be clipped, there is little evidence that plant inputs w i l l be overdriven by the schemes. The system configurations include the use of dividers and multipliers as signal processors. They are conceptually different from high-gain linear feedback schemes and despite certain peculiarities appear particularly useful in some situations such as those involx-ing nonminimum phase plants. By examining these systems in terms of equations of state, the conditions for exact and approximate invari- ance can be given more precisely and extended to some classes of multivariable systems.

(b)

Fig. 6. Second-order moving-pole scheme.

P ( s ) = K

P s 2 + 27Ts +y Kd = 5 7d = 0.5

1

X'd T , = -- = I; r = 2.5 volt step.

Tolerance in one of the fixed parameters. Srep response irom rest. (a) Complete correspondence of fixed paramerers. Step response from rest. (bj

"""n 13 I

Fig. 7. Nonminimum-phase plant controlled by a variable-gain scheme

Pb) = K(T?s - I ) '

( T I S i Ij(Tar i 1) T?, = 2 ; T w = 1 Krl = I ; Tid = 3 ~-

(a) Step response irom 10 to 20 units, up and down. (b) Regulator action to step changes in gain.

REFERESCES [ I ] Horowitz. I. A I . , Plant Adaptive vs. Ordinary Feedback. I R E Trans. on Auto.

[21 Kulebakin 1:. S.. The theory of invariance oi regulatinz and control systems. inatic Control. vol. AC-i Jan. 1962, pp 48-56

[3] Aseltine. J . A,. A. R. Marcini and C. W. Sarture, A survey of adaptive control Proc. IFAC, 1960. vol 1. pp 106-115.

(41 Borgiorno, J . J.. Stability and convergence properties oi model reference adap- systems. I R E Trans. om Aafornatu Control. vol AC-6. Dec 1958. pp 102-108.

tive control systems. I R E Trans. on Aalmnalic Conlrol. vol BC-7, Apr 1962. nn 30-41

[5] Smith, Kelvin C., Adaptive control through sinusoidal response, I R E Trans.

[6] Bellman. Richard, Adaptine Coxfro1 Procezs: A Gxi&d Tour, Princeton. K.J.:

r_ .. ...

on Alrfolnaiic Controls. vol AC-7, Mar 1962, pp 129-139.

Princeton University Press, 1961. ck. 16.