Conditional Feedback Systems-ANew Approach to Feedback Control

or by any motive power other than steamwhich did not involve combustion in themotive units themselves. This act re­quired that the change of motive powerbe made on or before July 1, 1908, andprovided that a penalty of $500 per daybe exacted on and after that date forfailure to comply with its terms.

Prior to this, the Westinghouse Electricand Manufacturing Company had de­veloped the high-tension single-phase sys­tem. The New York Central plans calledfor a relatively short electrified zone andthey decided to adopt the 650-volt d-ethird-rail system. The New Haven Rail­road planned on an initial electrification toStamford and New Haven with possiblefuture extensions eastward and decidedto adopt the high-voltage single-phasesystem. These three companies were alsopartners in a pioneering rectifier motivepower application in 1913, 1914, and 1915.

A Pennsylvania Railroad combinationbaggage and passenger car (car no. 4692)was equipped with four type-30B 225-hp600-volt traction motors and d-e controlequipment borrowed from the Long IslandRailroad and with transformers, multi­anode rectifiers, and accessory a-c controlequipment supplied by Westinghouse. Thiscar was tested on the Westinghouse testtrack at East Pittsburgh and then placed

G. LANGNONMEMBER AlEE

Synopsis: In the classical single-loopfeedback system, feedback acts not only tomodify the influence of disturbances butalso to determine the basic character of theinput-output response. The inherentlyclose association of these two effects hasbeen a constant trial for designers andstudents since the inception of classicalfeedback, and has usually required thatdesign requirements be compromised.Basically new configurations for feedbacksystems are introduced in which theseeffects of feedback are separated. Inthe new systems, feedback acts solely toreduce the influence of disturbances andthus to determine the response of the systemto external loads and internal parametervariations. The character of the input­output response is independent of trans­mission around the feedback loop. Theterm "conditional feedback" has beenintroduced to distinguish the limited rolethat feedback plays in these new systemsas compared with the classical system.Conditional feedback systems permit re­quirements on input-output response andon disturbance-output response to be metindependently and offer a broad new rangeof performance characteristics for bothlinear and nonlinear systems in which thereis substantial energy storage.

in revenue service on the New York, NewHaven and Hartford Railroad. I t hauledtwo trail cars and performed 8,757 milesin revenue service on the Harlem Riverbranch and 13,588 miles on the NewCanaan branch for a total revenue serviceof 22,345 miles.

The Pennsylvania Railroad and Westing­house again pioneered rectifier car no. 4561in 1949. Here again, the PennsylvaniaRailroad supplied a combination passengerand baggage car with Long Island Railroadd-e motor and control equipment (two 559D R3 motors and associated control) whileWestinghouse supplied the transformer andrectifier equipment. This car started opera­tion July 14, 1949, and has been in con­tinuous service since then.

The Pennsylvania then purchased two2-cab rectifier locomotives which havebeen in revenue service for approximately3 years. The New Haven Railroad thenpurchased 100 new rectifier multiple-unitcars and ten rectifier locomotives.

E. W. Ames and V. F. Dowden: Table Igives the kilovolt-amperes (kva) requiredfor all auxiliaries of the rectifier car fromtest readings of amperes for each circuit.Table II is tabulated on the same basis asTable I for the stand-by kva at Ll-kv

J. M. HAMMEMBER AlEE

TH E classes of closed-loop systems in­troduced in this paper 'were evolved

after an unsophisticated review of theproblems encountered and the solutions.accepted in present-day engineering prac­tice in control and regulation. In thedesign of these new systems, the need forcompromise in performance specificationsas a result of conflicts between input­output response and disturbance-outputresponse has largely been eliminated.Such conflicts are inherent to the conven­tional feedback control system because ofthe intimate relationship existing betweenthe input-output transmission charac­teristic and the loop transmission charac­teristic. Since the latter is dictated byrequirements for loop stability and for thesuppression of disturbances, little varia­tion is allowed the former, and at that notwithout considerable compromise. Auseful analogy for the foregoing conditionis found in the intimate connection thatexists between the frequency responseand the phase characteristic of minimum

Table II. Stand-by Kva at 11-Kv Trolley for409 Motor Car and Two Trail Cars

XVI

Transformer and traction motor blower37X420= 15.6

Main transformer (805 kva) excitation kva3.4 x i, 100 = 37.4

Air-brake compressor motor (1/3 time)(36 X 347)/3 = 4.2

Car heaters* including two trail cars = 150.0M-g set l-kw output

* Car heating is maximum available.

trolley for the 409 motor car and' two trailcars. The rectifier cars accelerate at aI-mile-per-hour-per-second rate while theolder cars accelerate between the 0.5- and0.75-mile-per-hour-per-second rate.

The new cars, with a higher accelerationrate, have enabled the railroad to reducethe scheduled time for a run. The newcars offer many passenger comforts, suchas automatically controlled heating or airconditioning, fluorescent lighting, forced­air circulation, and many other comfortsnot available on the older-type equipment.Power consumption for the 409 motor carwit h two trailers is not available on thesame basis as that stated in the paper.

phase networks. These restrictions ranbe overcome by the use of networks hav­ing nonminimurn phase characteristics.In a like manner, a basic change in topol­ogy can be used to extend the pres­ent boundaries of feedback control sys­tems. 1 - 4

Classification of Control Problems

The classification of control problemsadhered to in this paper distinguishes twobasic categories defined as follows:

1. The servo class: A servo problemrequires the generation of an output signalthat bears a prescribed functional relation­ship to an input signal. This problem iscommonly met in all types of signal trans­mission. A servo system is applied to thesolution of a servo problem:2. The regulator class: A regulator prob­lem requires the elimination or reductionof the effects on a controlled quantity ofextraneous and generally poorly defineddisturbances. A regulator system is ap­plied to the solution of a regulator problem.

Paper 55-202, recommended by the AlEE FeedbackControl Systems Committee and approved by theAlEE Committee on Technical Operations forpresentation at the AlEE Winter General Meeting,New York, N. Y., January 31-February 4, 1955.Manuscript submitted October 13, 1954; madeavailable for printing December 3, 1954.

G. LANG is with the Ferranti Electric Limited,Toronto, Ont., Canada, and J. M. HAM is with theUniversity of Toronto, Toronto, Ont., Canada.

The authors wish to acknowledge the encourage­ment of A. Porter and the permission and assistancegiven by Ferranti Electric Limited in the publica­tion of this paper..

152 Lang, Ham-Conditional Feedback Systems JULY 1955

(4)

Then equation 3 becomes

or

Two distinct control problems havebeen identified: the servo or signal trans­mission problem and the regulator or dis­turbance suppression problem. The per­formance requirements for signal trans­mission and disturbance suppression inpractical applications are usually distinct,but the signal and disturbance behaviorcharacteristics of classical feedback sys­tems have been shown to be inherentlyinterdependent. Hence, the design re­quirements for input-output response anddisturbance-output response have oftenhad to be compromised. Since feedbackin many control problems is essentialsolely to effect a suitable reduction in theinfluence of disturbances, the questionarises whether there are systems in whichthe use of feedback can be so restricted.Systems possessing this property arecalled conditional feedback systems. Abasic configuration for a linear condi­tional feedback system is shown in Fig. 2.In Fig. 2, G1 is the Laplace transfer func­tion describing the main transducer, ris the input signal, u is a disturbance,and c is the output. For the systemshown, the transform C of the output c is

AGl(1+~G2)R (3)C 1 U+----~-

1+GtG2H 1+G1GiH

In equation 3, let B be defined by theequality

BG2-=G1HG2A

Configuration and Basic Propertiesof Conditional Feedback Systems

performance will usually be obtainedwhen all variations susceptible to open­loop compensation are so treated. Closed­loop regulation is required in nearly allmachine- and process-control problems.Another important role for closed-loopcontrol is to modify or synthesize trans­fer functions which, in many cases, wouldbe most intractable to synthesis in asimple manner by open-loop teehniques.!

Fig. 1 (left). A representa­tive feedback system

The importance of knowing the disturb­ance problems associated with a par­ticular system design cannot be tooheavily stressed. If there are no externaldisturbances and available system com­ponents are linear and not subject toparameter variations, an open-loop sys­tem is ideally suited to most servo prob­lems provided a suitable open-loop trans­fer function can be obtained.

Open-loop systems have the particularadvantage that real time delays are oftenunimportant. It is the character of thedelayed response that is usually of majorinterest. Further, in such systems, ade­quate control of the influence of certainnonlinearities and load disturbances canbe gained by providing stable compen­sating nonlinear elements and an outputwith a suitably low driving-point imped­ance. The influence of disturbancesentering the system between input andoutput can often be reduced by isolatingthe system from known sources of dis­turbance, e.g., transmission lines maybe transposed to reduce the effect of in­duced signals.

There will remain, however, manyclasses of parameter variation, non­linearity, and internal disturbance thatare not amenable to treatment by suchtechniques. for these a closed-loop reg­ulating system is required. A closedloop will modify the influence of thosesystem variations for which a closed loopis not necessary, but improved system

Disturbances

Fig. 2 (right). Configurationof a conditional feedback

system

concentrating gain in G2 and G1, whereasdisturbances at the input to H (con­sidered as an output-sensing device) areof a particularly troublesome nature andcannot be reduced without lowering thegain or bandwidth of H. This latterview points out the necessity for care inthe choice of components for outputsensing.

For a linear system, a disturbance atnode 2 can be considered as a disturb­ance of different amplitude and fre­quency distribution entering at node 3.Herein all external disturbances will berepresented in terms of an equivalent loaddisturbance u.

4

The response tenus on the right of equa­tion 1 express the principle of superposi­tion for the signals Un and r. It followsfrom equation 1 that

F1=F

FF,,=­

G2

(2)

Conventional Feedback ControlSystems

From equation 2 it is clear that disturb­ances at nodes 2 and 3 are quelled by

5

The dual role played by the conven­tional feedback control system is easilyexposed by considering the Laplacetransform C of the output signal c that re­sults when an additive signal Un havingthetransform U" is applied successively tothenodes n= 1,2,3,4, and 5, in the repre­sentative feedback system of Fig. 1. Foran input signal r with the transform Rand successive additive signals Un, thesuccessive outputs Cn have the transforms

C1 G1G2U1 + G1G2R =F1U1+FR1+HG1G2 1+HG1G2

C2 G1 U2 + GtG2R F2 U2+FR1+HG1G2 1+HG1G2

CUa G1G 2R

8= + FaU3+ FR (1)1+HG1G2 1+HGtG2

C -GIG2U4H G1G2R.= + F4 U.+ FR1+HGtG2 1+HGtG2

C -GtG2U5 G1G2R

5 1+HGtG2+ 1+HGtG2 =F6U5+FR

Feedback control systems are applied toboth the aforementioned problems be­cause they can act simultaneously as aservo and as a regulator. This propertyhas been found to be of the utmost valuein the solution of the many problems inwhich both servo and regulator require­ments exist. Purely servo problems donot necessarily require feedback for theirsolution, nor do purely regulator prob­lems, e.g., the glow-discharge tube is usedto provide voltage regulation withoutfeedback.

Lang, Ham-Conditional Feedback Systems 153

r c Fig. 3 (left). The condi­tional configurationequivalent to the clas-

sical system

Fig. 4 (right). Equiva­lence of conditional sys­tem to a classical system

with a prefilter

Equation 6 shows that when B is de­fined by equation 4 the input-output re­sponse of the system is unaffected by thefeedback loop and can be of a basicallydifferent character from the disturbance­output response in equation 7, the formof which is completely determined by thefeedback loop. The significance of theseequations is immense for they show thata conditional feedback configuration per­mits design requirements on input-out­put response and on disturbance-outputresponse to be considered independently.This fact implies that a broad new rangeof performance characteristics is availablefrom conditional feedback systems.

Before giving a design procedure andillustrative design for the conditionalsystem of Fig. 2, some discussion of itsinternal behavior is in order. If thetransfer function B satisfies equation 4,the feedback signal b in the absence ofdisturbances is exactly equal to the signalv produced at the output of B by theinput signal r. Hence the signal e at theoutput of the lower comparator is identi­cally zero and there is no feedback. Thetransfer ratio C/R for the signal trans­mission is then simply the product of thetransfer functions in the direct path frominput to output; see upper branch of theblock diagram in Fig. 2.

These facts point to a special signifi­cance for the system component describedby the transfer function B. The functionof B is most readily suggested by con­sidering the system when H = 1. In thiscase, the feedback signal b= c. Hencewhen B is defined by equation 4, the out­put signal from B, namely v, is equal tothe output signal c. Clearly then Brepresents the desired input-output trans­fer function and the signal v representsthe desired output response. For thisreason, the system component described

from vvhich

( ~) =AGIR U=O

and

(5)

(6)

(7)

by the transfer function B in Fig. 2 willbe called a "reference model. " The refer­ence model is an undisturbed representa­tion of the transfer characteristics of themain transducer as defined in equation 4,and will usually be realized with simple R~

L, and Celements. When H is other thanunity, v is the desired output signal asmodified by H. H commonly describesthe output-sensing device.

An interesting comparison between theclassical feedback system of Fig. 1 and theconditional system of Fig. 2 is obtainedby developing a conditional configurationthat is equivalent to the classical system.This system is shown in Fig. 3. Fromequation 3 it follows that, if A = B = 1/2,the system in Fig. 3 behaves as the classi­cal system of Fig. 1. Clearly the classicalsystem calls for an ideal reference model.Since ideal behavior is not to be expectedfrom practical equipment, it is not sur­prising that the use of an ideal referencemodel entails special system performancelimitations.

W. K. Linvill pointed out to the authorsthat the conditional feedback system ofFig. 2 is equivalent to a classical systemwith a prefilter, as shown in Fig. 4. InFig. 4, the response C to Rand U is thatgiven by equation 3. When the transferfunction B in Fig. 4 is defined "byequation4, the prefilter has the particular transferfunction A (1+G1G2H), the responsera tio C/ R = A Gl , and there is no signalin the feedback path; hence, the com­bination of this particular prefilter andthe classical system has the properties ofa conditional system. It should be ob­served that both of the foregoing equiva­lences are valid only when the systemcomponents are linear.

Since feedback in a conditional feed­back system is used solely to reduce theinfluence of disturbances, it is importantto examine carefully the action of distur­bances on this new configuration. In aconditional feedback system such as thatof Fig. 2, there is no loop feedback if thereare no disturbances. The error signal E

obtained by comparing the desired signalv with the fed back signal b serves as ameasure for the effect of disturbances.In developing conditional systems, it hasbeen observed that in the comparison ofv with b either a subtraction or a ratio

operation may he used. If a subtractivecomparison is used, an additive correc­tion is made at the input to GI, as inFig. 2. If a ratio comparison is used,the signal input to Gi is corrected bymultiplying the signal output from A.namely m, by some function of the signalE from the ratio comparator.

The action of two types of disturbanceswill be examined. First consider thesteady-state response of the system inFig. 2 to a constant additive disturbanceu. The output signal c will be in errorby a constant and the error signal E 'willbe constant whether or not there is aninput signal ~'. Next consider the systemof Fig. 2 with the subtractive comparatorreplaced by a ratio comparator and theadditive corrector replaced by a multi­plicative corrector. Let there be a con­stant multiplicative disturbance such asmay be caused by a decrease in the staticgain factor of Gl . The output signal cwill be in error by a constant factor andthe error signal E will be constant. How­ever, if additive and multiplicative dis­turbances occur together the error signalE at the output of both a subtractive anda ratio comparator will contain frequencycomponents of the input signal. Theseobservations suggest that the nature ofdisturbances has an important bearing onthe design of conditional feedback sys­terns; this fact is discussed later in thepaper.

The response of the linear conditionalfeedback system in Fig. 2 to an additivedisturbance of general form is describedby equation 7. The form of this responseis controlled by an appropriate choice forthe transfer function G2• All of conven­tional feedback theory and design tech­nique is relevant to making this selection.

It is itnportant to observe that disturb­ances influencing the system component,described by A in Fig. 1, without affectingthe reference model B, are compensatedby feedback through G2 to the output ofA. Parameter variations in the com­ponents of a conditional feedback systemsuch as that shown in Fig. 2 have essen­tially the same influence on the output asthe corresponding variations in the classi­cal feedback system of Fig. 1. Hencemost of the literature on parameter sensi­tives is relevant. To illustrate this fact

154 Lang, Ham-Conditional Feedback Systems JULY 1955

Design Procedure and IllustrativeExample

consider a variation dGl in the transferfunction of the main transducer. Thecorresponding variation in the transferfunction of the output signal is d.C. Forthe classical system of Fig. 1

(16)

(15)

G2 can be realized with gain, a lead net­work, and a lag network. The param­eters K 2, ad, Td , ai, and TI are to beselected.

Suppose that a static gain factor

In equation 15, G2 is to be designed toobtain the best response ratio. The clas­sical cut-and-try procedure is followed,The basic form for G2 is taken to be

T=O.Ol second (14)

If the time delay T, of the main trans­ducer is subject to parameter variation,the value used in equation 14for designingthe reference model is some mean value.In practice B will be realized with minia­ture elements having tolerances on theircharacteristics established by the allowa­ble tolerances in the response ratio C/R.

To complete the design of the condi­tional feedback system, a loop-compensat­ing transfer function G2 has to be designedto realize the desired disturbance-outputresponse, as defined by equation 7. Sinceit is desired to compare the performanceof the conditional system with that ofthe corresponding classical system, theselection of G2 will be based on the follow­ing considerations. The classical systemin Fig. 1 corresponds to the conditionalsystem in Fig. 2 when the function Gl,G2, and H are the same. The disturb­ance-output response of these systems isidentical and is described by equation 7.Since the input-output response of theconditional system is independent of G2,

the classical system will compare mostfavorably with the conditional system ifthe form of G2 is selected to obtain thebest input-output response from the clas­sical system.

The response ratio C/ R for the classicalsystem in Fig. 1 is

C G1G 2-=--_. H=lR 1+HG1G2'

C E- T 1S

B=-=--· T1=O.2,R l+Ts'

compensating elements which do nottransmit at zero frequency must beplaced in A.

Now an output-sensing device may beselected and its transfer function H deter­mined. To simplify the details of thisdesign example, H is considered to beunity. Design of the undisturbed refer­ence model is now carried out. Fromequation 4 and the condition H = 1, it fol­lows that the transfer function B for thismodel is

(12)

(11)

(13)

To=T=O.Ol,

1Ko=--=10r.«,

A = KoTos1+'];0$

From equations 9 and 11, the values ofKoand Toare given as

C E- T 1S----R l+Ts

EXAMPLE

It is desired to construct a positionalservomechanism using a main transducerdescribed by the transfer function

The transfer function A is readilyrealized with gain and a simple resistance­capacitance network.

It is important to note that it is notnecessary to effect the complete compen­sation of the main transducer with ele­ments placed in the box marked A. Thesynthesis of the desired input-output re­sponse transform may include tandemcompensation of G, placed in the upperbranch of the feedback loop, classicalfeedback around GI, and the like. How­ever, if the feedback loop is. required tohave nonzero transmission at zero fre­quency to fulfill its regulating function,

From equation 11 it follows that asuita ble form for A is

G1 corresponds to a time-delayed inte­gration. Pure time delays in the loop ofa classical feedback system impose a defi­nite limit on the bandwidth that can berealized in input-output response. Suchis not the case when a conditional feed­back system is employed. The config­uration of the conditional system isshown in Fig. 2. The corresponding clas­sical system is shown in Fig. 1. The stepsin designing the conditional system arenow given.

From equation 6 it follows that thetransfer function A must be synthesizedto make the product AGl represent thedesired form of the input-output response.Suppose that the desired response to aunit step has the form

c= ( 1- E- (t~Til); T1 =0.2,

T=O.Ol second (10)

From equation 10 the input-outputresponse transform is

be the same in both systems so that adirect comparison of performance charac­teristics is justified.

K 1E- T 18

G1 = - ,. K 1=10, T1=O.2 second (9)s

(8)

Forthe conditional system, the variationde has exactly the same form,

The foregoing discussion of a linearconditional feedback system has shownthat this new configuration permits theindependent control of the input-outputresponse and of the disturbance-outputresponse. The disturbance-output re­sponse can always be made as good asthat of the corresponding classical systemwhile the input-output response can haveforms hitherto unrealizable. A designprocedure for conditional feedback sys­tems based on the foregoing developmentisdescribed in the following.

These steps in design form a direct se­quence in which there is no need for com­promise among the steps. To illustratethe design procedure for conditional feed­back systems a sam ple design will begiven for a conditional system and acorresponding classical system that em­ploys the same main transducer. Thefeedback loop transmission function is to

The special properties of conditionalfeedback systems as developed in the fore­going lead to a design procedure that isunusually direct and simple. A sug­gested procedure is outlined in the fol­lowing. The particular significance ofthe steps will be illustrated in a sampledesign.

1. From a careful study of the systemproblem determine the required input­output response characteristics and therequired disturbance-output response char­acteristics; special care should be given indetermining the basic nature of dis­turbances.

2. Select a main transducer and associatewith it compensating elements to realizethe desired input-output response. Thisstep involves the choice of the transferfunctions G1 and A in Fig. 2.3. Select a suitable output-sensing deviceand determine its transfer function H.4. Design an undisturbed reference modeldescribed by the transfer fttnction B =AG1H.5. Design a loop-compensating networkdescribed by the transfer function G2 torealize the desired disturbance-output re­sponse to the extent that the fundamentalrequirement of loop stability allows.

de

JULY 1955 Lang, Ham-Conditional Feedback Systems 155

1'2

H '"/ \ - ..../' ,1'0

·9 I \ /I \ /

·8 I " /,/

u '7 I '-,I - conditional

·6.. I --- classical&"5 I (calculated by numerical..~'4 I integration)

·3,,

·2 I·1

00 '1 '2 '3 '4 '5 '6 ·7 ·8 ·9 1,0 1,1 1'2

t(sec.)

r

r

r

Fig. 7.

G

c

A simplified multiplicative system

C

O~----o..&..:::""'="'=-=~;""';""":::""::::'-~----I"--t--_

Fig. 5. Step responses of the conditional and classical design

DISTURBANCES

The casual observation that an errorcould be properly measured as a ratiorather than as a difference has led to arather novel embodiment of the condi­tional topology and to some interestingconclusions concerning the nature of dis­turbances. I t has been remarked pre­viously that a steady-state multiplicativedisturbance results in a constant errorsignal when the error signal is measuredas the ratio of the undisturbed signal tothe disturbed signal in a conditional sys­tem. In a like manner, a steady-stateadditive disturbance will result in a con­stant error signal when the error is meas­ured as a difference. Hence, it is usefulto classify arbitrary disturbances as beingdominantly multiplicative or dominantlyadditive in character accordingly as the

Extension of the Principle toNonlinear Systems

In a classical system, the effort to realizemaximum bandwidth in the input-outputresponse almost always leads to under­damped oscillatory characteristics. Thisfact is particularly true when pure timedelays are present in the feedback loop.

The foregoing example is suggestive ofthe extended range of performance charac­teristic available to feedback system de­signers* 'when conditional configurationsare used. It i,s also important to notethat the input-output bandwidth may bereduced while the disturbance outputbandwidth is maintained.

Apart from having basic significanceinthe domain of linear systems, conditionalfeedback configurations have certainspecial merits for systems containingnonlinearities and, indeed, a new class ofnonlinear servomechanism has beenevolved. The extension of the conceptof conditional feedback to nonlinear sys­tems is discussed in the following.

(19)

,, i', 1\\ I \ ,

'"" \ I\/

~

\:\:

\\

\\~

10 100

/..... _....

G1G2 plane. The open-loop response ofthe classical system is thus defined as

100E-0.28 0.25s+ 1 2.5s+ 1G1G2 = - - -

s 0.025s+ 1 100s+ 1

I/

The input-output response of the condi­tional system as defined by equation 11 isnow to be compared with the input­output response of the corresponding clas­sical system as defined by equations 9,19, and 15. In Fig. 5, the responses ofthe two systems to a unit step are com­pared and in Fig. 6 the magnitude andphase characteristics of the responseratios C/R are compared.

In evaluating the response comparisonsin Figs. 5 and 6, it is important to re­member that the disturbance-output re­sponses of the two systems are identical.The conditional system clearly has muchsuperior input-output response charac­teristics. It is important again to em­pha size that there is freedom of choicefor the character of the input-output re­sponse in a conditional feedback system.

(17)

Condit ionalClassical

·1-15 .01..------t-----_____1~---_____1~----t__--~

o in radiansper second

200

100

500

400

-10

Iiiin db.- 5

in degrees300

+5

KK 3 = - = 10 (18)KI

In equation 16, the gain factor aa of thelead network is arbitrarily assigned thevalue 0.1. From a polar plot of G1, thevalue Ta=0.025 is selected for the timeconstant of the lead network. On theassumption that the lag network acts as again factor at at frequencies in the neigh­borhood of the critical point in the G1G2

plane, at is assigned a value at = 1/40,which results in a response peak of about1.3 in ratio C/R. The time constant T1.of the lag network is assigned a valueTt=2.5 seconds, which results in a phaseshift through the lag network of lessthan 2 degrees at frequencies in theneighborhood of the critical point in the

or

K =lim SHG1G2 = 1008-+0

is required in the feedback loop. Fromequations 9 and 17 it follows that

K=K1Ks

Fig. 6. Frequency responses of the conditional and classical designs* The systems described here are the subject ofpatent action by Ferranti Electric Limited.

156 Lang, lIam-Conditional Feedback Systems JULY 1955

r

r

r

B

G

Fig. 8 (left). Ac multiplicative

.....-~.....- conditional servo

Fig. 9 (right).The incrementalresponse of asimple multipli-

cative servo

0'8

0"

c (t)0- calculated pointsx - points measured in

experimental servo.

o 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30t (sec)

(26)T

Tn+1=n+l

The exponent n in a multiplicativeservo is equivalent to the loop gain of aclassical feedback system employing asubtractive comparison. This result aswell as that given in equation 21 providesa simple example of the use of multiplica­tive feedback' to alter and calibrate thetransmission characteristics of a trans­ducer.

The nonlinear behavior illustrated inthe foregoing is produced intentionally byemploying essentially nonlinear systemcomponents. However, almost all prac­tical control systems contain nonlineari­ties that are natural to the componentsand generally undesired. The concept ofconditional feedback alters the signifi­cance of many of these residual non­linearities.

An equation of the same form governsthe response of the system to a multi­plicative disturbance such as a sudden de­crease in the gain factor K of G. Equa­tion 25 shows that response in the varia­ble cn+1 has the time constant

NATURAL NONLINEARITIES IN ADDITIVE

CONDITIONAL FEEDBACK SYSTEMS

Such nonlinearities as torque satura­tion in motors, backlash in gears, stictionon shafts, clipping in amplifiers, and non­linear gain in synchros cause the designergreat trouble. Most of these non­linearities contribute to the instabilityof a feedback loop and especially to theinstability of the classical feedback sys­tem. The reason for this condition may

Let rand c initially have the steadyvalue r= c= 1. The incremental responsein c to an additive step in r is shownin Fig. 9 for various values of the ex­ponent n. 6,7 For comparison, the stepresponse of G is shown. From the systemof Fig. 7 it is readily shown that the dif­ferential equation defining the response cis

(23)

The question now arises as to the be­havior of the system during the transientinterval. At the present time, it may besaid that, for all signals and disturbanceshaving frequency components that areconfined to a frequency range over whichG is essentially a constant, the resultantsignal c can be represented as the productof two functions: the output c caused bythe disturbance if r is maintained atunity, and the output c caused by thesignal alone. All questions pertaining tothe stability of the multiplicative loopcannot be answered at this time. Thefollowing section contains an example oftypical modes of behavior for simplemultiplicative feedback systems.

RESPONSE OF A SIMPLE MULTIPLICATIVE

SERVO

Consider the servo system shown inFig. 7. In the light of the equivalenceshown in Fig. 3, the configuration in Fig.7 is the equivalent of a classical systememploying a ratio comparator. This sys­tem has interesting modes of behavior.Suppose G has the form

KG= l+D; T=24 seconds, K=l (24)

system. Further, let the low frequencygain of B and of G be unity. In the ab­sence of a disturbance, the following rela­tions hold provided rand c are positive

Since the over-all system is linear in theabsence of disturbances, the Laplacetransform may be used to describe theinput-ouput relationship.

Assume now that a disturbance occurschanging the low frequency gain of G tok. In the ensuing steady state, vic willhave a constant value because signalvariations in v are duplicated by signalvariations in c. Since vic is a constant,the system again has a linear input-outputresponse relationship, and

n+l_ /- 'lJ 1C=R vkG and -=-~­C n+lyk

V vn-=1, rn=r, C=GR, and V=BR (22)c c

(20)

(21)

r r n +1

c=r, -=1, -n-=rc C

Hence, multiplicative feedback has ef­fectively altered the gain to a valueft+JYk. This behavior is analogous tothe reduction of an additive disturbancein the conventional feedback loop by afactor 1/(1+k), where k is the low-fre­quency loop gain.

When the system is used as a servo, itis necessary to introduce a referencemodel B. Such a system is shown inFig.8, where B(s) = G(s) as in the additive

ratio measured error or the differencemeasured error has the lower bandwidth.A significant difference in bandwidth inthese measurements, when interpreted interms of basic bandwidth restrictions onthe feedback loop, will suggest the adop­tion of a particular type of conditionalsystem for dealing with a disturbance.

In view of these observations, condi­tional feedback systems have been classi­fiedas being either multiplicative or addi­tive in character. Multiplicative condi­tional feedback systems are believed torepresent an entirely new class of systems.

A HEURISTIC DESCRIPTION OF THE

MULTIPLICATIVE SYSTEM

A simplified multiplicative conditionalsystem is shown in Fig. 7. Three com­puting devices have been introduced inthis configuration: a divider, a power­raising device, and a multiplier. Con­sider the system as a regulator when r isa positive constant and the low frequencygain of G is unity. In the absence of adisturbance, the following signal relationsexist

If the low-frequency gain of G changesfrom unity to k, where k represents a newsteady-state gain such as would resultfrom a change in load, a new systemsteady state will arise in which the newsignal relations are

n+l_ /- r 1c=r v k, and - = n+l_ I-e vk

JULY 1955 Lang, Ham-Conditional Feedback Systems 157

be expressed in an interesting mannerwith the help of Fig. 3. Fig. 3 showsthat the classical feedback system isequivalent to a conditional configurationin which the reference model is ideal.Hence the signal E in the feedback loopcontains components caused by all of thenonlinearities in the loop, the influence ofeach of which feedback is endeavoringto suppress. The classical feedback sys­tem does not permit any form of non­linearity or any form of imperfect linearityto be fully tolerated; it strives for theideal and commonly becomes unstable inso doing.

However, it is clear that in many prac­tical applications certain residual non­linearities in the relation of output tothe input are not undesirable. I t is anunfortunate limitation of the classicalfeedback system that such admissiblenonlinearities contribute to the instabilityof the feedback loop. The conditionalfeedback system, on the other hand, pro­vides a direct means for accepting tolera­ble nonlinearities in the input-output re­sponse and at the same time largely pre­vents these nonlinearities from influencingthe stability of the feedback loop. Thisremarkable behavior is realized by build­ing into the reference model, B in Fig. 2,the tolerable nonlinearities in the com­ponents A, G1, and ·H. The desired re­sponse v at the output of B then containscomponents caused by these nonlineari­ties. These components of v cancel thecomponents of the fed-back signal bcaused by the nonlinearities in A, G1, andH. Hence the stability of the feedbackloop, in so far as it is excited by the inputsignal alone, is unaffected. The condi­tional configuration does not remove theinfluence of any nonlinearities on the sta­bility of the feedback loop when excitedby an additive output disturbance.Hence some care must be exercised in ex­ploiting the aforementioned input-outputcharacteristics.

An excellent example of the use of aconditional feedback system to overcomea serious instability in a classical systemis the following. It is well known that aclassical system that is conditionallystable for small input signals may beseriously unstable for large signals whichcause an element such as a motor totorque saturate. The instability is causedby a reduction in effective loop gain. Ifthe torque-saturating characteristic isincluded in the reference model of a con­ditional system, this instability cannotbe excited by input signals.

The basic significance of the foregoingis that, for purposes of input-output re­sponse, the conditional feedback system

permits the regulating action of the feed­back loop to act in a manner best suitedto. the particular application. Perhapsno examples of this fact are more illumi­nating than those that spring from theproblem of the human operator in a feed­back system.

ApPLICATION OF CONDITIONAL

CONFIGURATIONS TO THE HUMAN

OPERATOR PROBLEM

Whether it be in an automobile, an aero­plane, a steel mill, or an economic system,the importance of system response charac­teristics in the presence of human opera­tors is profound. 8 It is recognized that thehuman operator is described by neither awell-defined nor a linear transmissionfunction. Certain basic characteristicscan, however, be distinguished. There iswhat approximates to a pure time delay inmotor response to a sudden, say, visualstimulus. There is a blocking actionagainst stimuli received in too rapid suc­cession. The pattern of response changesas experience in a given environment is ac­cumulated. Conditional feedback sys­tems provide the means for exploiting thebasic characteristics of the particular hu­man operator by achieving optimuminput-output response with adequate loopstability. The design example given issuggestive of the improvement that can berealized in the presence of pure time de­lays. By providing a linear or nonlinearrepresentation of the human operator inthe reference model, the operator is calledupon to behave as this model rather thanto behave ideally as in the classical system.If the reference model B in Fig. 2 is con­sidered to have a variable structure, it isapparent that the conditional feedbackconfiguration provides an interesting tech­nique for the determination of approxi­mate describing functions for the humanoperator. The determination is carriedout by adjusting the structure of B untilthe loop signal E falls to some rms value.

Conclusions

This paper has introduced the conceptand elaborated the basic theoretical andpractical implications of conditional feed­back. Systems are said to be conditionalfeedback systems when feedback in thesesystems is used solely in a regulating roleto reduce the influence of disturbances ofall types. It, is distinctive of linearadditive conditional feedback systemsthat the forms of the input-output re­sponses and of the disturbance-output re­sponses are independent. This funda­mental fact permits the realization of a

broad new range of response characteris­tics, especially in the presence of suchclassically difficult factors as pure timedelay. The design procedure for suchsystems is unusually direct and simple.

A classification of system disturbancesas being basically additive or multiplica­tive in character has led to the conceptand practical implementation of multi­plicative feedback systems employingratio comparisons and multiplicative cor­rections. A new range of responsecharacteristics is available from thesesystems.

Conditional feedback systems providemeans for accepting certain componentnonlinearities in the relation of systemoutput to system input and, at the sametime, for removing the influence of thesenonlinearities on the stability of the feed­back loop as excited by the input signal.This fact alters the significance for the de­signer of many residual nonlinearities andpermits certain modes of instability tobe suppressed. Conditional feedbacksystems, except through the new class ofmultiplicative systems, make no newcontribution to the regulator problembut they provide a basis for improvementfor all classes of servo problems and notleast for the class involving human opera­tors.

References

1. THEORY OF SERVOMECHANISMS, H. L. Hazen.Journal, Franklin Institute, Philadelphia, Pa., vol.218, Sept. 1934, pp. 279-330.

2. PRINCIPLES OF SERVOMECHANISMS (book), G.S. Brown. D. P. Campbell. John Wiley and Sons,Ine., New York, N. Y., 1948.

3. NETWORK ANALYSIS AND FEEDBACK AMPLIFIERDESIGN (book), H. W. Bode. D. Van NostrandCompany, Tnc., New York, N. Y., 1945.

4. SAMPLED-DATA CONTROL SYSTEMS STUDIEDTHROUGH COMPARISON OF SAMPLING WITH AMPLI­TUDE MODULATION, William K. Linvill. AlEETransactions, vol. 70, pt. II, 1951, pp. 1779-88.

5. SYNTHESIS OF TRANSFER FUNCTIONS BYACTIVE RC NETWORKS WITH A FEEDBACK Loop,D. B. Armstrong, F. M. Reza. Transactions,Institute of Radio Engineers, New York, N. Y.vol. CT-1, June 1954.

6. AN ANALYTICAL STUDY OF A MULTIPLICATIVEFEEDBACK SYSTEM, r, E. S. Stevens. Thesis,University of Toronto, Toronto, Ont., Canada,1954.

7. A PRACTICAL STUDY OF A MULTIPLICATIVEFEEDBACK SYSTEM, R. J. Kavanagh. Ibid.8. THE HUMAN OPERATOR OF CONTROL MECH­ANISMS, W. E. Hick, s. A. V. Bates. MonographNo. 17.204, United Kingdom Ministry of Supply,London, England, May, 1950.----.----

Discussion

George C. Newton, Jr. (MassachusettsInstitute of Technology, Cambridge, Mass.):The authors are to be commended for thisinteresting approach to the problem ofsuppressing disturbances without imposing

158 Lang, Ham-Conditional Feedback Systems JULY 1955

c

c

I I

~ - - - - - - - - - - - - - - - - - - - - - - - - - - - __I

r ..

Fig. 12. The conditional feedback systemdrawn in the configuration of a conventional

system

H. Tyler Marcy (International BusinessMachines Corporation, Endicott, N. Y.):It is very appropriate that the authors

output response is independent of thestability of the feedback loop. In Fig. 4,the authors have drawn a classical systemwith a prefilter which has the same transferfunction as the conditional system of Fig. 2.However, in Fig. 4, there is a signal in thefeedback path through G2, the effect ofwhich must be exactly cancelled by theprefilter.

It is possible to draw a conventionalfeedback system without a prefilter whichis linearly equivalent to a conditionalsystem. Fig. 12 shows such a system whichis topologically equivalent to Fig. 2. TheLaplace transform of the output in Fig. 12is given by equation 3 with respect to bothsignal and disturbance. Furthermore, ifthe model is defined by equation 4, thenequations 5, 6, and 7 apply in this casealso. Thus, the input-output response isunaffected by any adjustment of G2 whichmay be made to alter the character of thedisturbance-output response.

There are, however, at least two funda­mental differences between this conven­tional configuration and a conditional sys­tem. In Fig. 12, there is always signal inthe feedback path so that system stabilitydepends upon the linearity of the com­ponents; and the realizability of the transferfunctions in the feedback loop is subject tomore severe restrictions than in the condi­tional case. Thus, in special cases, linearequivalents can be formulated, but thecontribution of conditional servos is in thesimple treatment of nonlinear systems andin the new wide range of feedback configura­tions which are realizable. Models are al­ready in use" for process control problemsbut the new approach of the authors is touse the model in such a way as to eliminatefeedback signals from the error-detecting ele­ment. In the conditional system, the feed­back loop is employed only to reduce the ef­fect of deviations from model behavior andother disturbances.

(32)

Fig. 10 (Iefl).Parallel feedback

configuration

Fig. 11 (right).Single-loopequivalent to par­aUel feedback

configuration

c

HIH2=H+-Gc

the behavior of the single loop will beidentical with the behavior of the parallelfeedback configuration. Thus, it shouldnot be concluded that single-loop feedbacksystems are unable to achieve independentadjustment of the disturbance-suppressioncharacteristic and the input-output trans­mission characteristic. However, in prac­tice, realization of the required transferfunctions is usually simpler if a parallelfeedback configuration or the configurationsuggested by the authors is used.

The authors' application of their condi­tional principle to multiplicative systemsappears to be novel. In connection withprocess control problems there may beconsiderable merit in the use of conditionalmultiplicative feedback configurations.However, a number of questions are leftunanswered. For example, in the case ofcomplex G functions, is not the stabilitysensitive to the output signal level sincethe loop gain for incremental signals de­pends on this quantity? Also, are the ad­vantages of the multiplicative systemsufficient to justify the use of multiplier,divider, and power-raising devices whichare not as easily realized as summing andoperational amplifying devices? I lookforward to the future publications onthe multiplicative systems promised by theauthors.

H. C. Ratz (Ferranti Electric Limited,Toronto, Ont., Canada): Consideration ofthe conditional feedback approach to controlproblems described by the authors leadsnaturally to discussion of its relationshipto conventional feedback systems. Therelationship of conditional systems to clas­sical systems is topologically equivalentto the relationship of bridge circuits toseries-parallel circuits. In the conditionalsystem, when the model is chosen for bal­ance, as given by equation 4, there is nosignal in the feedback path through G2and, hence, the choice of signal input-

to achieve the desired input-output transferfunction.

I t has been shown that a parallel feed­back configuration can be used to suppressdisturbances without constraining the input­output transfer functions. Theoretically,it is perfectly possible to realize identicalsystem behavior using a single-loop equiva­lent to the parallel feedback configuration.Fig. 11 shows such a single loop. If, inthis figure, the transmission of the feedbackelements is set in accordance with theequation

(30)

(31)

(29)

(27)

Gc=A(1+G1G2H )

H1=[G2-A(I+GIG2H)JH

C GcGIR+U1+G1HI +G1GcH

This equation compares with equation 5.The behavior of this parallel feedbackconfiguration is identical with the authors'system providing

GcG1AG1 (28)

I+G1H1+G1GcH

Thus, in principle, the authors' systemcan be realized without need for operationson the input signal. In actual practice,the auxiliary feedback in the case of apositional servomechanism might comefrom a tachometer attached to the outputshaft and HI would be adjusted to limit theeffect of disturbances. There is no needto use equation 31 since G2 is arbitrarywithin wide limits. Gc is then adjusted

These equations may be solved for Gcand HI, assuming that A, GI, G2, and Hare known. The solutions are

constraints on the input-output transfercharacteristic. This is an important prob­lem and can be approached in a number ofdifferent ways. The authors' approach isnot an unreasonable one in those instanceswhere the input signal is available. Fre­quently, however, only the difference be­tween the .input signal and the primaryfeedback signal is available with any degreeof precision; e.g., a radar tracking systemand a position follow-up using synchrosfor error-sensing. I t may be concludedfrom this paper that the classical single­loop feedback system is unable to do suchas can be done with a conditional feedbacksystem configuration. Thus, one may feelat a loss when meeting a problem in whichthe input signal is not available for filteringprior to summing with the primary feed­back signal. I t is one purpose of thisdiscussion to show that conventional con­figurations may be used to achieve resultsidentical with those achieved by theauthors. In these configurations thereis no need for operating on the input signal.

Consider first the configuration shown inFig. 10. In this figure, an auxiliary feed­back path is used to suppress the effectsof disturbances, After the auxiliary feed­back transfer function HI has been adjusted,the other adjustable transfer functions(usually Gconly) can be adjusted to achievethe desired input-output transmission char­acteristic. To observe this consider theequation for the output signal

JULY 1955 Lang, Ham-Conditional Feedback Systems 159

should present this interesting paper at thistime. The concept of a control systemmaking use of information contained in theinput signal to decrease dynamic errorscaused by changes in the signal may be akey technique in many of the more compre­hensive process control problems which arereceiving much attention today.

There is pertinent literature applying tothis subject which makes the word "new"in the paper title subject to question. Theconcepts of feed-forward have been dis­cussed previously.t-vs It has been widelyknown that, should it be possible to operateon the input signal, dynamic errors in thecontrol system can be reduced. An im­portant consideration is that feed-forwardtechniques of error reduction do not changethe degree of stability or the natural fre­quencies of the control system, i.e., theroots of a characteristic equation remain thesame. Harris! also considered feed-forwardas a means for compensating nonlinear fac­tors such as dry friction.

Perhaps the dominant limitation with theuse of feed-forward is the nature of the inputsignal. It must first be possible to measureit directly and then to operate on the signalin a predictive sense. In the presence ofnoise, this can become impractical. Wehave, however, learned much about statis­tical treatment of this sort of problem,since feed-forward was widely discussedsome years ago. Regardless of where thedisturbance to a control system is applied,predictive knowledge of that disturbancecan be employed to decrease dynamic con­trol errors resulting from the d!sturbance.

REFERENCES

1. THE ANALYSIS AND DESIGN OF SERVOMECH­ANISM, H. Harris. Defense Research Council,Washington, D. C., Sect. D-2, 1942.

2. LINEAR SERVO THEORY, R. E. Graham. BellSystem Technical Journal, New York, N. Y.,vol. 25, Oct. 1946, pp. 616-51.

3. PARALLEL CIRCUITS IN SERVOMECHANISMS,H. Tyler Marcy. AIEE Transactions (ElectricalEngineering), vol. 65, Aug.-Sept. 1946, pp.521-29.

Rufus Oldenburger (Woodward GovernorCompany, Rockford, Ill.): This valuablecontribution to the science of automaticcontrol emphasizes the importance of de­signing for both servo and regulator perform­ance in many automatic control problems.In the design of speed governors we studythe response to speed-setting changes andload rejections, as well as other disturb­ances. We have found that the design ofour governors on the basis of characteristicroots so as to give fast closed-loop responseto sudden load changes has also given goodresponse to speed-setting changes.

Unfortunately, our most troublesome dis­turbance is what I like to call the "noise"in the speed signal. This signal is put outby the speed-measuring element. This ele­ment corresponds to the authors' feedbackelement H. As the authors point out, un­fortunately this disturbance cannot be re­duced without compromising H. The noisecan come from gear drive irregularities,prime-mover shaft runout, or other causes.It is a part of the signal one does not wishto respond to, and severely limits the mathe­matical operations that can be performedon the speed measurement in the computerpart of the con trol.

160

Prof. James Reswick! introduced anauxiliary feedback loop in which the trans­fer function A G1 of the forward part is takento be the transfer function of the extrafeedback path, except for a constant of pro­portionality. If H = 1 the authors alsointroduce an element with transfer functionAG1 into an extra loop, but in a differentmanner by placing it in a forward branch.

The inclusion by the authors of a discus­sion of nonlinear components is most timely.The American Society of Mechanical En­gineers will devote its April 1956 Instru­ments and Regulators Division conferenceto nonlinear control. The Russians spe­cialize in this area, as well as in the auto­matic control field in general; they have ajournal devoted entirely to the science ofautomatic control. The Macmillan Com­pany will soon publish an American Societyof Mechanical Engineers book entitled"Frequency Response," which I am editing.This book is to contain leading contribu­tions from all over the world on the variousphases of this subject.

REFERENCE

1. DISTURBANCE RESPONSE FEEDBACK-A NEWCONTROL CONCEPT, J. Reswick. Transactions,American Society of Mechanical Engineers, NewYork, N. Y., 1955 (55-1 RD2).

H. P. Birmingham (Naval Research Labora­tories, Washington, D. C.): This importantpaper is of particular interest to the humanengineer concerned with the man as anelement in a man-machine control system.A problem under attack is the matter ofhuman performance in pursuit tracking andin compensatory tracking. In compensa­tory tracking, the human operator is pre­sented only the error (difference betweeninput and output) and manipulates hiscontrol on the basis of this information. Inthe pursuit case, the operator sees the inputand his output, or terms proportionalthereto, on the same display and is requiredto keep the difference (error) at a minimum.It has been shown that under some condi­tions, the human is able to maintain asmaller average error in the pursuit case.It is expected that the authors' analysiscan be used to show how the man can turnin this superior performance where thecourse as well as the error is shown to him,by acting analogously to a conditionalsystem.

G. Lang and J. M. Ham: We greatlyappreciate the interest shown by the dis­cussers. Dr. Newton's observation thatinput signals are not always available withprecision is well made but we consider theoccasions when input signals cannot bemeasured with useful accuracy to be rare.

Dr. Newton proceeds to show that in alinear world a conventional feedback con­figuration, such as shown in Figs. 10 and 11,can be made to have transfer functions iden­tical with those of the basic conditional con­figuration in Fig. 2. This fact is implied bythe prefilter equivalent shown in Fig. 4. AsMr. Ratz has shown in Fig. 12, there aremany such linear equivalents. However,we wish to emphasize that our conditionalconfiguration leads to direct realizationtechniques not readily implemented forgeneral linear equivalents. For example,

Lang, Ham-Conditional Feedback Systems

Dr. Newton's equations 30 and 31 showhow to pick conventional compensatingfunctions Gc and HI to achieve equivalencewith a conditional configuration. Giventhe conditional configuration and the free­dom to pick A and G2 independently, equa­tions 30 and 31 are readily used to define Gcand HI, but we wonder how Dr. Newtonwould pick Gc and HI to give independentinput-output and disturbance-output re­sponses without prior reference to a condi­tional configuration.

In the literature on feedback control sys­tems we are not aware of ariy collected treat­ment on the problem of selecting com­pensating transfer functions to achieve in­dependent conditions on input-output anddisturbance-output responses, particularly

.when nonminimum phase components arepresent. A practical reason for the diffi­culty in achieving such conditions is readilydiscerned from equations 30 and 31. Theseequations show that any changes in Gc or inHI affect both C/ Rand C/ U so that thecut-and-try procedure is commonly resortedto. In this connection, our sample designis exemplifying the directness with which aconditional system design can be realizedeven under the difficult condition of a puretime delay in the main actuator.

A block diagram for Fig. 10 showing howto realize Gc and Hi as specified in equations30 and 31 is given in Fig. 13. If the systemis synthesized as shown, it is clear that manymore elements are required than for theequivalent conditional system. While it istrue that in a linear world Gc and HI can berealized with gain and general passive filters,it is not clear how such filters are to besynthesized in practice.

Dr. Newton's remarks about the equiva­lence of Fig. 11 to the conditional configura­tion are quite correct but are subject to thequalifications outlined in the foregoing. Itshould be observed that all of the equiva­lences discussed by Dr. Newton depend fortheir validity on linearity. In this connec­tion the remarks of Mr. Ratz are particu­lady relevant.

With regard to multiplicative systems,Dr. Newton's observation that for complexG functions the loop stability depends onthe output signal level is quite correct. Tothe question of the justification for usingmultiplier, divider, and power-raising de­vices in these systems, it may be remarkedthat multiplicative systems appear to haveother than physical worth, namely as modelelements for economic systems where per­unit changes are significant. Further, ageneralization of the concept of signal com­parison leads to insight into the nature ofdisturbances as suggested in the section en­titled "Extension of the Principles to Non­linear Systems'."

Mr. Ratz's comments may be regarded

c

Fig. 13. A synthesis For Fig. 10

JULY 1955

as an amplification of the paper and of theremarks of Dr. Newton, and deserve carefulreading. Fig. 12 represents the conven­tional single-loop feedback system equiva­lentto the conditional configuration. Againit must be noted that, if this system is syn­thesized as shown in Fig. 12, more elementsare required than for the equivalent condi­tional system.

Mr. Marcy emphasizes the relationshipof the conditional feedback systems intro­duced by us to so-called feed-forward sys­tems to which considerable attention hascertainly been given. Perhaps the workofJ. R. Moore! most closely resembles ours.Tothe extent that any linear system may be

regarded as a combination of feed-forwardand feedback elements, the conditionaltopology introduced by us can be so inter­preted. However, the concept and topologyof conditional feedback systems employinga reference model has not been describedpreviously in the literature, as far as weknow. As Mr. Marcy suggests, the conceptof conditional feedback may be particularlyilluminating for problems concerning tran­sient process control.

The remarks of R. Oldenburger are in­teresting. We have heard of ProfessorReswick's recent work and are looking for­ward to seeing his paper.

We appreciate the interest of Dr. Bir-

mingham in the application of conditionalfeedback systems to the human operatorproblem. Dr. Birmingham rightly dis­tinguishes the tasks of compensatory andpursuit tracking. These correspond to theregulator and servo definitions as used inthe paper. We believe that the condi­tional topology is significant for the en­gineering of man-machine systems. A re­search program on this topic is in progressat the University of Toronto.

REFERENCE

1. COMBINATION OPEN-CYCLE CLOSED-CYCLE SYS­TEMS, J. R. Moore. Proceedings, Institute of RadioEngineers, New York, N. Y., vol. 39, Nov. 1951,pp. 1421-32.

Design of Control Systems forBandwidth

creasing nus output with increasing band­width.

PERFORMANCE INDEX

The purpose of any control system isto constrain its output to match a de­sired output (ideal value) within an ac­ceptable tolerance when acted upon byan input (command) and disturbances.The measure used to assess the agree­ment between the desired and actualoutputs is usually called a performanceindex. For the design method of thispaper the input and desired output arearbitrary transient signals. The integralsquare of the error between the desiredoutput and the actual output is used asthe performance index. In the processof adjusting the control system to mini­mize its bandwidth, only those weightingfunctions are used which make the inte­gral-square error equal to or less than aspecified value.

SOLUTION FOR WEIGHTING FUNCTION

In the section entitled "A VariationalApproach" a general solution is obtainedfor the weighting function which the con­trol system should have in order to possessminimum bandwidth for a specified inte­gral-square error. This solution is ob­tained by variational methods. The datanecessary to obtain a particular solutionare as follows:

In connection with the bandwidth test:

BANDWIDTHADJUSTMENT

wb

CONTROL SYSTEM ......-__~

W(w)

Minimum

Fig. 1 Scheme for defining of system bandwidth

a filter and its rms value measured. Therms value of the filtered output is corre­lated with the bandwidth of the controlsystem by comparison with the filteredoutput of a standard system of any pre­scribed form but having adjustable band..width. The bandwidth of the standardsystem is adjusted to produce an nnsvalue of its filtered output equal to thatof the control system. Both the' stand­ard system and the control system aredriven from a common noise source duringthe bandwidth test. Fig. 1 is a blockdiagram showing this scheme fer relatingcontrol system bandwidth to its filteredoutput during a bandwidth test. Bydefinition, the bandwidth of the controlsystem is equal to the bandwidth of thestandard system when the rms values ofthe filtered outputs are equal. In orderthat minimizing the filtered output of thecontrol system shall be equivalent tominimizing its bandwidth, the standardsystem together with the noise source andfilter must produce a monotonically in-

GEORGE C. NEWTON, JR.MEMBER AlEE

GEORGE C. NEWTON, JR., is with MassachusettsInstitute of Technology, Cambridge, Mass.

This research was sponsored in part by the Servo­mechanisms Laboratory, Department of ElectricalEngineering, Massachusetts Institute of Tech­nology, in connection with a project subcontractedby Lincoln Laboratory, Massachusetts Instituteof Technology, and supported by the Departmentof the Army, the Department of the Navy, andthe Department of the Air Force under Air ForceContract No. AF19(122)-458.

The author wishes to acknowledge the constructivesuggestions made by his colleagues in the Servo­mechanisms Laboratory, and especially thosemade by Dr. L. A. Gould, Dr. J. C. Simons,Prof. L. S. Bryant, and G. T. Coate.

Paper 55-194, recommended by the AlEE Feed­back Control Systems Committee and approvedby the AlEE Committee on Technical Operationsfor presentation at the AlEE Winter GeneralMeeting, New York, N. Y., January 31-February4, 1955. Manuscript submitted September 24,1954; made available for printing December 21,1954.

BANDWIDTH TEST

Minimum bandwidth is obtained byadjusting the system weighting functionto produce minimum output noise underspecially contrived circumstances termedthe' 'bandwidth test." In this paper thebandwidth test is a conceptual scheme fordefining bandwidth. During the band­width test a stochastic noise signal is usedfor the control system input. The out­putof the control system is passed through

Design Specifications

Synopsis: This paper presents a methodof designing feedback control systemswhich minimizes bandwidth for a specifiedtransient error. Reduction of bandwidthis desired in order to attenuate noise,simplify the compensation, and ease therequirements on components operating athigh power levels.

JULY 1955 Newton-Design of Control Systems for Minimum Bandwidth 161