TH E classes of closed-loop systems introduced. Conditional Feedback Systems-A New Approach to Feedback Control. J. M.

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1 or by any motive power other than steam whih did not involve ombustion in the motive units themselves. This at required that the hange of motive power be made on or before July 1, 1908, and provided that a penalty of $500 per day be exated on and after that date for failure to omply with its terms. Prior to this, the Westinghouse Eletri and Manufaturing Company had developed the high-tension single-phase system. The New York Central plans alled for a relatively short eletrified zone and they deided to adopt the 650-volt d-e third-rail system. The New Haven Railroad planned on an initial eletrifiation to Stamford and New Haven with possible future extensions eastward and deided to adopt the high-voltage single-phase system. These three ompanies were also partners in a pioneering retifier motive power appliation in 1913, 1914, and A Pennsylvania Railroad ombination baggage and passenger ar (ar no. 4692) was equipped with four type-30b 225-hp 600-volt tration motors and d-e ontrol equipment borrowed from the Long Island Railroad and with transformers, multianode retifiers, and aessory a- ontrol equipment supplied by Westinghouse. This ar was tested on the Westinghouse test trak at East Pittsburgh and then plaed in revenue servie on the New York, New Haven and Hartford Railroad. It hauled two trail ars and performed 8,757 miles in revenue servie on the Harlem River branh and 13,588 miles on the New Canaan branh for a total revenue servie of 22,345 miles. The Pennsylvania Railroad and Westinghouse again pioneered retifier ar no in Here again, the Pennsylvania Railroad supplied a ombination passenger and baggage ar with Long Island Railroad d-e motor and ontrol equipment (two 559 D R3 motors and assoiated ontrol) while Westinghouse supplied the transformer and retifier equipment. This ar started operation July 14, 1949, and has been in ontinuous servie sine then. The Pennsylvania then purhased two 2-ab retifier loomotives whih have been in revenue servie for approximately 3 years. The New Haven Railroad then purhased 100 new retifier multiple-unit ars and ten retifier loomotives. E. W. Ames and V. F. Dowden: Table I gives the kilovolt-amperes (kva) required for all auxiliaries of the retifier ar from test readings of amperes for eah iruit. Table II is tabulated on the same basis as Table I for the stand-by kva at Ll-kv Table II. Stand-by Kva at 11-Kv Trolley for 409 Motor Car and Two Trail Cars XVI Transformer and tration motor blower 37X420= 15.6 Main transformer (805 kva) exitation kva 3.4 x i,100 = 37.4 Air-brake ompressor motor (1/3 time) (36 X 347)/3 = 4.2 Car heaters* inluding two trail ars = M-g set l-kw output * Car heating is maximum available. trolley for the 409 motor ar and'two trail ars. The retifier ars aelerate at a I-mile-per-hour-per-seond rate while the older ars aelerate between the 0.5- and 0.75-mile-per-hour-per-seond rate. The new ars, with a higher aeleration rate, have enabled the railroad to redue the sheduled time for a run. The new ars offer many passenger omforts, suh as automatially ontrolled heating or air onditioning, fluoresent lighting, foredair irulation, and many other omforts not available on the older-type equipment. Power onsumption for the 409 motor ar wit h two trailers is not available on the same basis as that stated in the paper. Conditional Feedbak Systems-A New Approah to Feedbak Control G. LANG NONMEMBER AlEE Synopsis: In the lassial single-loop feedbak system, feedbak ats not only to modify the influene of disturbanes but also to determine the basi harater of the input-output response. The inherently lose assoiation of these two effets has been a onstant trial for designers and students sine the ineption of lassial feedbak, and has usually required that design requirements be ompromised. Basially new onfigurations for feedbak systems are introdued in whih these effets of feedbak are separated. In the new systems, feedbak ats solely to redue the influene of disturbanes and thus to determine the response of the system to external loads and internal parameter variations. The harater of the inputoutput response is independent of transmission around the feedbak loop. The term "onditional feedbak" has been introdued to distinguish the limited role that feedbak plays in these new systems as ompared with the lassial system. Conditional feedbak systems permit requirements on input-output response and on disturbane-output response to be met independently and offer a broad new range of performane harateristis for both linear and nonlinear systems in whih there is substantial energy storage. J. M. HAM MEMBER AlEE TH E lasses of losed-loop systems introdued in this paper 'were evolved after an unsophistiated review of the problems enountered and the solutions.aepted in present-day engineering pratie in ontrol and regulation. In the design of these new systems, the need for ompromise in performane speifiations as a result of onflits between inputoutput response and disturbane-output response has largely been eliminated. Suh onflits are inherent to the onventional feedbak ontrol system beause of the intimate relationship existing between the input-output transmission harateristi and the loop transmission harateristi. Sine the latter is ditated by requirements for loop stability and for the suppression of disturbanes, little variation is allowed the former, and at that not without onsiderable ompromise. A useful analogy for the foregoing ondition is found in the intimate onnetion that exists between the frequeny response and the phase harateristi of minimum phase networks. These restritions ran be overome by the use of networks having nonminimurn phase harateristis. In a like manner, a basi hange in topology an be used to extend the present boundaries of feedbak ontrol systems. 1-4 Classifiation of Control Problems The lassifiation of ontrol problems adhered to in this paper distinguishes two basi ategories defined as follows: 1. The servo lass: A servo problem requires the generation of an output signal that bears a presribed funtional relationship to an input signal. This problem is ommonly met in all types of signal transmission. A servo system is applied to the solution of a servo problem: 2. The regulator lass: A regulator problem requires the elimination or redution of the effets on a ontrolled quantity of extraneous and generally poorly defined disturbanes. A regulator system is applied to the solution of a regulator problem. Paper , reommended by the AlEE Feedbak Control Systems Committee and approved by the AlEE Committee on Tehnial Operations for presentation at the AlEE Winter General Meeting, New York, N. Y., January 31-February 4, Manusript submitted Otober 13, 1954; made available for printing Deember 3, G. LANG is with the Ferranti Eletri Limited, Toronto, Ont., Canada, and J. M. HAM is with the University of Toronto, Toronto, Ont., Canada. The authors wish to aknowledge the enouragement of A. Porter and the permission and assistane given by Ferranti Eletri Limited in the publiation of this paper Lang, Ham-Conditional Feedbak Systems JULY 1955

2 Fig. 1 (left). A representative feedbak system 5 4 Feedbak ontrol systems are applied to both the aforementioned problems beause they an at simultaneously as a servo and as a regulator. This property has been found to be of the utmost value in the solution of the many problems in whih both servo and regulator requirements exist. Purely servo problems do not neessarily require feedbak for their solution, nor do purely regulator problems, e.g., the glow-disharge tube is used to provide voltage regulation without feedbak. Conventional Feedbak Control Systems The dual role played by the onventional feedbak ontrol system is easily exposed by onsidering the Laplae transform C of the output signal that results when an additive signal Un having thetransform U"is applied suessively to thenodes n= 1,2,3,4, and 5, in the representative feedbak system of Fig. 1. For an input signal r with the transform R and suessive additive signals Un, the suessive outputs Cn have the transforms C 1 G 1G2U1 + G 1G2R =F1U1+FR The response tenus on the right of equation 1 express the priniple of superposition for the signals Un and r. It follows from equation 1 that F1=F F F,,= G 2 1+HG1G2 1+HG1G2 C 2 G 1 U 2 + G tg2r F 2 U 2 + FR 1+HG1G2 1+HG1G2 G 1G2R Ua C8= + 1+HG 1G2 1+HGtG2 FaU3+ FR (1) -GIG2U4H G1G2R C.= + 1+HGtG2 1+HGtG2 F4U.+ FR C -GtG2U5 G1G2R 5 1+HG tg2+ 1+HG tg2 =F 6U5+FR (2) From equation 2 it is lear that disturbanes at nodes 2 and 3 are quelled by Fig. 2 (right). Configuration of a onditional feedbak system onentrating gain in G 2 and G 1, whereas disturbanes at the input to H (onsidered as an output-sensing devie) are of a partiularly troublesome nature and annot be redued without lowering the gain or bandwidth of H. This latter view points out the neessity for are in the hoie of omponents for output sensing. For a linear system, a disturbane at node 2 an be onsidered as a disturbane of different amplitude and frequeny distribution entering at node 3. Herein all external disturbanes will be represented in terms of an equivalent load disturbane u. Disturbanes The importane of knowing the disturbane problems assoiated with a partiular system design annot be too heavily stressed. If there are no external disturbanes and available system omponents are linear and not subjet to parameter variations, an open-loop system is ideally suited to most servo problems provided a suitable open-loop transfer funtion an be obtained. Open-loop systems have the partiular advantage that real time delays are often unimportant. It is the harater of the delayed response that is usually of major interest. Further, in suh systems, adequate ontrol of the influene of ertain nonlinearities and load disturbanes an be gained by providing stable ompensating nonlinear elements and an output with a suitably low driving-point impedane. The influene of disturbanes entering the system between input and output an often be redued by isolating the system from known soures of disturbane, e.g., transmission lines may be transposed to redue the effet of indued signals. There will remain, however, many lasses of parameter variation, nonlinearity, and internal disturbane that are not amenable to treatment by suh tehniques. for these a losed-loop regulating system is required. A losed loop will modify the influene of those system variations for whih a losed loop is not neessary, but improved system performane will usually be obtained when all variations suseptible to openloop ompensation are so treated. Closedloop regulation is required in nearly all mahine- and proess-ontrol problems. Another important role for losed-loop ontrol is to modify or synthesize transfer funtions whih, in many ases, would be most intratable to synthesis in a simple manner by open-loop teehniques.! Configuration and Basi Properties of Conditional Feedbak Systems Two distint ontrol problems have been identified: the servo or signal transmission problem and the regulator or disturbane suppression problem. The performane requirements for signal transmission and disturbane suppression in pratial appliations are usually distint, but the signal and disturbane behavior harateristis of lassial feedbak systems have been shown to be inherently interdependent. Hene, the design requirements for input-output response and disturbane-output response have often had to be ompromised. Sine feedbak in many ontrol problems is essential solely to effet a suitable redution in the influene of disturbanes, the question arises whether there are systems in whih the use of feedbak an be so restrited. Systems possessing this property are alled onditional feedbak systems. A basi onfiguration for a linear onditional feedbak system is shown in Fig. 2. In Fig. 2, G 1 is the Laplae transfer funtion desribing the main transduer, r is the input signal, u is a disturbane, and is the output. For the system shown, the transform C of the output is or Then equation 3 beomes AGl(1+~G2)R (3) C 1 U+----~- 1+GtG2H 1+G1GiH In equation 3, let B be defined by the equality B G 2-=G1HG2 A (4) Lang, Ham-Conditional Feedbak Systems 153

3 r Fig. 3 (left). The onditional onfiguration equivalent to the lassial system Fig. 4 (right). Equivalene of onditional system to a lassial system with a prefilter from vvhih ( ~) =AGI R U=O and (5) (6) (7) Equation 6 shows that when B is defined by equation 4 the input-output response of the system is unaffeted by the feedbak loop and an be of a basially different harater from the disturbaneoutput response in equation 7, the form of whih is ompletely determined by the feedbak loop. The signifiane of these equations is immense for they show that a onditional feedbak onfiguration permits design requirements on input-output response and on disturbane-output response to be onsidered independently. This fat implies that a broad new range of performane harateristis is available from onditional feedbak systems. Before giving a design proedure and illustrative design for the onditional system of Fig. 2, some disussion of its internal behavior is in order. If the transfer funtion B satisfies equation 4, the feedbak signal b in the absene of disturbanes is exatly equal to the signal v produed at the output of B by the input signal r. Hene the signal e at the output of the lower omparator is identially zero and there is no feedbak. The transfer ratio C/R for the signal transmission is then simply the produt of the transfer funtions in the diret path from input to output; see upper branh of the blok diagram in Fig. 2. These fats point to a speial signifiane for the system omponent desribed by the transfer funtion B. The funtion of B is most readily suggested by onsidering the system when H = 1. In this ase, the feedbak signal b=. Hene when B is defined by equation 4, the output signal from B, namely v, is equal to the output signal. Clearly then B represents the desired input-output transfer funtion and the signal v represents the desired output response. For this reason, the system omponent desribed by the transfer funtion B in Fig. 2 will be alled a "referene model." The referene model is an undisturbed representation of the transfer harateristis of the main transduer as defined in equation 4, and will usually be realized with simple R~ L, and Celements. When H is other than unity, v is the desired output signal as modified by H. H ommonly desribes the output-sensing devie. An interesting omparison between the lassial feedbak system of Fig. 1 and the onditional system of Fig. 2 is obtained by developing a onditional onfiguration that is equivalent to the lassial system. This system is shown in Fig. 3. From equation 3 it follows that, if A = B = 1/2, the system in Fig. 3 behaves as the lassial system of Fig. 1. Clearly the lassial system alls for an ideal referene model. Sine ideal behavior is not to be expeted from pratial equipment, it is not surprising that the use of an ideal referene model entails speial system performane limitations. W. K. Linvill pointed out to the authors that the onditional feedbak system of Fig. 2 is equivalent to a lassial system with a prefilter, as shown in Fig. 4. In Fig. 4, the response C to Rand U is that given by equation 3. When the transfer funtion B in Fig. 4 is defined "byequation 4, the prefilter has the partiular transfer funtion A (1+G 1G2H), the response ra tio C/ R = A G l, and there is no signal in the feedbak path; hene, the ombination of this partiular prefilter and the lassial system has the properties of a onditional system. It should be observed that both of the foregoing equivalenes are valid only when the system omponents are linear. Sine feedbak in a onditional feedbak system is used solely to redue the influene of disturbanes, it is important to examine arefully the ation of disturbanes on this new onfiguration. In a onditional feedbak system suh as that of Fig. 2, there is no loop feedbak if there are no disturbanes. The error signal E obtained by omparing the desired signal v with the fed bak signal b serves as a measure for the effet of disturbanes. In developing onditional systems, it has been observed that in the omparison of v with b either a subtration or a ratio operation may he used. If a subtrative omparison is used, an additive orretion is made at the input to GI, as in Fig. 2. If a ratio omparison is used, the signal input to Gi is orreted by multiplying the signal output from A. namely m, by some funtion of the signal E from the ratio omparator. The ation of two types of disturbanes will be examined. First onsider the steady-state response of the system in Fig. 2 to a onstant additive disturbane u. The output signal will be in error by a onstant and the error signal E 'will be onstant whether or not there is an input signal ~'. Next onsider the system of Fig. 2 with the subtrative omparator replaed by a ratio omparator and the additive orretor replaed by a multipliative orretor. Let there be a onstant multipliative disturbane suh as may be aused by a derease in the stati gain fator of G l. The output signal will be in error by a onstant fator and the error signal E will be onstant. However, if additive and multipliative disturbanes our together the error signal E at the output of both a subtrative and a ratio omparator will ontain frequeny omponents of the input signal. These observations suggest that the nature of disturbanes has an important bearing on the design of onditional feedbak systerns; this fat is disussed later in the paper. The response of the linear onditional feedbak system in Fig. 2 to an additive disturbane of general form is desribed by equation 7. The form of this response is ontrolled by an appropriate hoie for the transfer funtion G 2 All of onventional feedbak theory and design tehnique is relevant to making this seletion. It is itnportant to observe that disturbanes influening the system omponent, desribed by A in Fig. 1, without affeting the referene model B, are ompensated by feedbak through G 2 to the output of A. Parameter variations in the omponents of a onditional feedbak system suh as that shown in Fig. 2 have essentially the same influene on the output as the orresponding variations in the lassial feedbak system of Fig. 1. Hene most of the literature on parameter sensitives is relevant. To illustrate this fat 154 Lang, Ham-Conditional Feedbak Systems JULY 1955

4 onsider a variation dg l in the transfer funtion of the main transduer. The orresponding variation in the transfer funtion of the output signal is d.c. For the lassial system of Fig. 1 de Design Proedure and Illustrative Example (8) Forthe onditional system, the variation de has exatly the same form, The foregoing disussion of a linear onditional feedbak system has shown that this new onfiguration permits the independent ontrol of the input-output response and of the disturbane-output response. The disturbane-output response an always be made as good as that of the orresponding lassial system while the input-output response an have forms hitherto unrealizable. A design proedure for onditional feedbak systems based on the foregoing development isdesribed in the following. The speial properties of onditional feedbak systems as developed in the foregoing lead to a design proedure that is unusually diret and simple. A suggested proedure is outlined in the following. The partiular signifiane of the steps will be illustrated in a sample design. 1. From a areful study of the system problem determine the required inputoutput response harateristis and the required disturbane-output response harateristis; speial are should be given in determining the basi nature of disturbanes. 2. Selet a main transduer and assoiate with it ompensating elements to realize the desired input-output response. This step involves the hoie of the transfer funtions G 1 and A in Fig Selet a suitable output-sensing devie and determine its transfer funtion H. 4. Design an undisturbed referene model desribed by the transfer fttntion B = AG1H. 5. Design a loop-ompensating network desribed by the transfer funtion G 2 to realize the desired disturbane-output response to the extent that the fundamental requirement of loop stability allows. These steps in design form a diret sequene in whih there is no need for ompromise among the steps. To illustrate the design proedure for onditional feedbak systems a sam ple design will be given for a onditional system and a orresponding lassial system that employs the same main transduer. The feedbak loop transmission funtion is to be the same in both systems so that a diret omparison of performane harateristis is justified. EXAMPLE It is desired to onstrut a positional servomehanism using a main transduer desribed by the transfer funtion K 1 E- T 18 G 1 = -,. K 1=10, T 1=O.2 seond (9) s G 1 orresponds to a time-delayed integration. Pure time delays in the loop of a lassial feedbak system impose a definite limit on the bandwidth that an be realized in input-output response. Suh is not the ase when a onditional feedbak system is employed. The onfiguration of the onditional system is shown in Fig. 2. The orresponding lassial system is shown in Fig. 1. The steps in designing the onditional system are now given. From equation 6 it follows that the transfer funtion A must be synthesized to make the produt AG l represent the desired form of the input-output response. Suppose that the desired response to a unit step has the form = ( 1-E- (t~til); T 1 =0.2, C E- T 1S ---- R l+ts A = KoTos 1+'];0$ To=T=O.Ol, 1 Ko=--=10 r.«, T=O.Ol seond (10) From equation 10 the input-output response transform is (11) From equation 11 it follows that a suita ble form for A is (12) From equations 9 and 11, the values of K o and Toare given as (13) The transfer funtion A is readily realized with gain and a simple resistaneapaitane network. It is important to note that it is not neessary to effet the omplete ompensation of the main transduer with elements plaed in the box marked A. The synthesis of the desired input-output response transform may inlude tandem ompensation of G, plaed in the upper branh of the feedbak loop, lassial feedbak around GI, and the like. However, if the feedbak loop is. required to have nonzero transmission at zero frequeny to fulfill its regulating funtion, ompensating elements whih do not transmit at zero frequeny must be plaed in A. Now an output-sensing devie may be seleted and its transfer funtion H determined. To simplify the details of this design example, H is onsidered to be unity. Design of the undisturbed referene model is now arried out. From equation 4 and the ondition H = 1, it follows that the transfer funtion B for this model is C E- T 1S B=-=-- T 1=O.2, R l+ts' T=O.Ol seond (14) If the time delay T, of the main transduer is subjet to parameter variation, the value used in equation 14 for designing the referene model is some mean value. In pratie B will be realized with miniature elements having toleranes on their harateristis established by the allowable toleranes in the response ratio C/R. To omplete the design of the onditional feedbak system, a loop-ompensating transfer funtion G 2 has to be designed to realize the desired disturbane-output response, as defined by equation 7. Sine it is desired to ompare the performane of the onditional system with that of the orresponding lassial system, the seletion of G 2 will be based on the following onsiderations. The lassial system in Fig. 1 orresponds to the onditional system in Fig. 2 when the funtion Gl, G 2, and H are the same. The disturbane-output response of these systems is idential and is desribed by equation 7. Sine the input-output response of the onditional system is independent of G 2, the lassial system will ompare most favorably with the onditional system if the form of G 2 is seleted to obtain the best input-output response from the lassial system. The response ratio C/ R for the lassial system in Fig. 1 is C -=--_. G 1G 2 H=l R 1+HG 1G2' (15) In equation 15, G 2 is to be designed to obtain the best response ratio. The lassial ut-and-try proedure is followed, The basi form for G 2 is taken to be (16) G 2 an be realized with gain, a lead network, and a lag network. The parameters K 2, ad, T d, ai, and T I are to be seleted. Suppose that a stati gain fator JULY 1955 Lang, Ham-Conditional Feedbak Systems 155

5 1'2 H 1'0 / '"\ /'-..., 9 I \ / I \ / 8 / I ", / u '7 I '-, I - onditional.. 6 I --- lassial &"5 I (alulated by numerial..~'4 I integration) 3,, Fig. 5. I 0 '1 '2 '3 '4 '5 ' ,0 1,1 1'2 t(se.) K = lim SHG1G2 = Step responses of the onditional and lassial design is required in the feedbak loop. equations 9 and 17 it follows that K=K1Ks or +5 Iii in db.- 5 I / O~----o..&..:::""'="'=-=~;""';""":::""::::'-~----I"--t--_ / in degrees 300 Condit ional Classial (17)... _... ~ \: \: \ \ \ \,, i', 1\ \ I \, '"" \ I \/ ~ r r r Fig. 7. G 1G2 plane. The open-loop response of the lassial system is thus defined as 100E s+1 2.5s+1 G 1G2 = s 0.025s+1 100s t ~--- 1~----t --~ o in radians per seond From K K 3 = - = 10 (18) K I In equation 16, the gain fator aa of the lead network is arbitrarily assigned the value 0.1. From a polar plot of G 1, the value T a=0.025 is seleted for the time onstant of the lead network. On the assumption that the lag network ats as a gain fator at at frequenies in the neighborhood of the ritial point in the G 1G2 plane, at is assigned a value at = 1/40, whih results in a response peak of about 1.3 in ratio C/R. The time onstant T1. of the lag network is assigned a value T t=2.5 seonds, whih results in a phase shift through the lag network of less than 2 degrees at frequenies in the neighborhood of the ritial point in the (19) Theinput-outputresponse of the onditional system as defined by equation 11 is now to be ompared with the inputoutput response of the orresponding lassial system as defined by equations 9, 19, and 15. In Fig. 5, the responses of the two systems to a unit step are ompared and in Fig. 6 the magnitude and phase harateristis of the response ratios C/R are ompared. In evaluating the response omparisons in Figs. 5 and 6, it is important to remember that the disturbane-output responses of the two systems are idential. The onditional system learly has muh superior input-output response harateristis. It is important again to empha size that there is freedom of hoie for the harater of the input-output response in a onditional feedbak system. G A simplified multipliative system In a lassial system, the effort to realize maximum bandwidth in the input-output response almost always leads to underdamped osillatory harateristis. This fat is partiularly true when pure time delays are present in the feedbak loop. The foregoing example is suggestive of the extendedrange of performaneharateristi available to feedbak system designers* 'when onditional onfigurations are used. It i,s also important to note that the input-output bandwidth may be redued while the disturbane output bandwidth is maintained. Apart from having basi signifianein the domain of linear systems, onditional feedbak onfigurations have ertain speial merits for systems ontaining nonlinearities and, indeed, a new lass of nonlinear servomehanism has been evolved. The extension of the onept of onditional feedbak to nonlinear systems is disussed in the following. Extension of the Priniple to Nonlinear Systems DISTURBANCES The asual observation that an error ould be properly measured as a ratio rather than as a differene has led to a rather novel embodiment of the onditional topology and to some interesting onlusions onerning the nature of disturbanes. It has been remarked previously that a steady-state multipliative disturbane results in a onstant error signal when the error signal is measured as the ratio of the undisturbed signal to the disturbed signal in a onditional system. In a like manner, a steady-state additive disturbane will result in a onstant error signal when the error is measured as a differene. Hene, it is useful to lassify arbitrary disturbanes as being dominantly multipliative or dominantly additive in harater aordingly as the C Fig. 6. Frequeny responses of the onditional and lassial designs * The systems desribed here are the subjet of patent ation by Ferranti Eletri Limited. 156 Lang, liam-conditional Feedbak Systems JULY 1955

6 r r G Fig. 8 (left). A multipliative...-~...- onditional servo 0'8 r B Fig. 9 (right). The inremental response of a simple multipliative servo 0" (t) 0- alulated points x - points measured in experimental servo. o t (se) ratio measured error or the differene measured error has the lower bandwidth. A signifiant differene in bandwidth in these measurements, when interpreted in terms of basi bandwidth restritions on the feedbak loop, will suggest the adoption of a partiular type of onditional system for dealing with a disturbane. In view of these observations, onditional feedbak systems have been lassifiedas being either multipliative or additive in harater. Multipliative onditional feedbak systems are believed to represent an entirely new lass of systems. A HEURISTIC DESCRIPTION OF THE MULTIPLICATIVE SYSTEM A simplified multipliative onditional system is shown in Fig. 7. Three omputing devies have been introdued in this onfiguration: a divider, a powerraising devie, and a multiplier. Consider the system as a regulator when r is a positive onstant and the low frequeny gain of G is unity. In the absene of a disturbane, the following signal relations exist r r n + 1 =r, -=1, -n-=r C (20) If the low-frequeny gain of G hanges from unity to k, where k represents a new steady-state gain suh as would result from a hange in load, a new system steady state will arise in whih the new signal relations are n+l_ /- r 1 =r v k, and - = n+l_ I- e vk (21) Hene, multipliative feedbak has effetively altered the gain to a value ft+jyk. This behavior is analogous to the redution of an additive disturbane in the onventional feedbak loop by a fator 1/(1+k), where k is the low-frequeny loop gain. When the system is used as a servo, it is neessary to introdue a referene model B. Suh a system is shown in Fig.8, where B(s) = G(s) as in the additive system. Further, let the low frequeny gain of B and of G be unity. In the absene of a disturbane, the following relations hold provided rand are positive V v n -=1, rn=r, C=GR, and V=BR (22) Sine the over-all system is linear in the absene of disturbanes, the Laplae transform may be used to desribe the input-ouput relationship. Assume now that a disturbane ours hanging the low frequeny gain of G to k. In the ensuing steady state, vi will have a onstant value beause signal variations in v are dupliated by signal variations in. Sine vi is a onstant, the system again has a linear input-output response relationship, and n+l_ /- 'lj 1 C=R vkg and -=-~ C (23) The question now arises as to the behavior of the system during the transient interval. At the present time, it may be said that, for all signals and disturbanes having frequeny omponents that are onfined to a frequeny range over whih G is essentially a onstant, the resultant signal an be represented as the produt of two funtions: the output aused by the disturbane if r is maintained at unity, and the output aused by the signal alone. All questions pertaining to the stability of the multipliative loop annot be answered at this time. The following setion ontains an example of typial modes of behavior for simple multipliative feedbak systems. RESPONSE OF A SIMPLE MULTIPLICATIVE SERVO n+lyk Consider the servo system shown in Fig. 7. In the light of the equivalene shown in Fig. 3, the onfiguration in Fig. 7 is the equivalent of a lassial system employing a ratio omparator. This system has interesting modes of behavior. Suppose G has the form K G= l+d; T=24 seonds, K=l (24) Let rand initially have the steady value r= = 1. The inremental response in to an additive step in r is shown in Fig. 9 for various values of the exponent n. 6,7 For omparison, the step response of G is shown. From the system of Fig. 7 it is readily shown that the differential equation defining the response is An equation of the same form governs the response of the system to a multipliative disturbane suh as a sudden derease in the gain fator K of G. Equation 25 shows that response in the variable n + 1 has the time onstant T T n + 1 =n+l (26) The exponent n in a multipliative servo is equivalent to the loop gain of a lassial feedbak system employing a subtrative omparison. This result as well as that given in equation 21 provides a simple example of the use of multipliative feedbak' to alter and alibrate the transmission harateristis of a transduer. The nonlinear behavior illustrated in the foregoing is produed intentionally by employing essentially nonlinear system omponents. However, almost all pratial ontrol systems ontain nonlinearities that are natural to the omponents and generally undesired. The onept of onditional feedbak alters the signifiane of many of these residual nonlinearities. NATURAL NONLINEARITIES IN CONDITIONAL FEEDBACK SYSTEMS ADDITIVE Suh nonlinearities as torque saturation in motors, baklash in gears, stition on shafts, lipping in amplifiers, and nonlinear gain in synhros ause the designer great trouble. Most of these nonlinearities ontribute to the instability of a feedbak loop and espeially to the instability of the lassial feedbak system. The reason for this ondition may JULY 1955 Lang, Ham-Conditional Feedbak Systems 157

7 be expressed in an interesting manner with the help of Fig. 3. Fig. 3 shows that the lassial feedbak system is equivalent to a onditional onfiguration in whih the referene model is ideal. Hene the signal E in the feedbak loop ontains omponents aused by all of the nonlinearities in the loop, the influene of eah of whih feedbak is endeavoring to suppress. The lassial feedbak system does not permit any form of nonlinearity or anyform of imperfetlinearity to be fully tolerated; it strives for the ideal and ommonly beomes unstable in so doing. However, it is lear that in many pratial appliations ertain residual nonlinearities in the relation of output to the input are not undesirable. It is an unfortunate limitation of the lassial feedbak system that suh admissible nonlinearities ontribute to the instability of the feedbak loop. The onditional feedbak system, on the other hand, provides a diret means for aepting tolerable nonlinearities in the input-output response and at the same time largely prevents these nonlinearities from influening the stability of the feedbak loop. This remarkable behavior is realized by building into the referene model, B in Fig. 2, the tolerable nonlinearities in the omponents A, G 1, and H. The desired response v at the output of B then ontains omponents aused by these nonlinearities. These omponents of v anel the omponents of the fed-bak signal b aused by the nonlinearities in A, G 1, and H. Hene the stability of the feedbak loop, in so far as it is exited by the input signal alone, is unaffeted. The onditional onfiguration does not remove the influene of any nonlinearities on the stability of the feedbak loop when exited by an additive output disturbane. Hene some are must be exerised in exploiting the aforementioned input-output harateristis. An exellent example of the use of a onditional feedbak system to overome a serious instability in a lassial system is the following. It is well known that a lassial system that is onditionally stable for small input signals may be seriously unstable for large signals whih ause an element suh as a motor to torque saturate. The instability is aused by a redution in effetive loop gain. If the torque-saturating harateristi is inluded in the referene model of a onditional system, this instability annot be exited by input signals. The basi signifiane of the foregoing is that, for purposes of input-output response, the onditional feedbak system permits the regulating ation of the feedbak loop to at in a manner best suited to. the partiular appliation. Perhaps no examples of this fat are more illuminating than those that spring from the problem of the human operator in a feedbak system. ApPLICATION OF CONDITIONAL CONFIGURATIONS TO THE HUMAN OPERATOR PROBLEM Whetherit be in an automobile, an aeroplane, a steel mill, or an eonomi system, the importane of system response harateristis in the presene of human operators is profound. 8 It is reognized that the human operator is desribed by neither a well-defined nor a linear transmission funtion. Certain basi harateristis an, however, be distinguished. There is what approximates to a pure time delay in motor response to a sudden, say, visual stimulus. There is a bloking ation against stimuli reeived in too rapid suession. The pattern of response hanges as experiene in a given environment is aumulated. Conditional feedbak systems provide the means for exploiting the basi harateristis of the partiular human operator by ahieving optimum input-output response with adequate loop stability. The design example given is suggestive of the improvement that an be realized in the presene of pure time delays. By providing a linear or nonlinear representation of the human operator in the referene model, the operator is alled upon to behave as this model rather than to behave ideally as in the lassial system. If the referene model B in Fig. 2 is onsidered to have a variable struture, it is apparent that the onditional feedbak onfiguration provides an interesting tehnique for the determination of approximate desribing funtions for the human operator. The determination is arried out by adjusting the struture of B until the loop signal E falls to some rms value. Conlusions This paper has introdued the onept and elaborated the basi theoretial and pratial impliations of onditional feedbak. Systems are said to be onditional feedbak systems when feedbak in these systems is used solely in a regulating role to redue the influene of disturbanes of all types. It, is distintive of linear additive onditional feedbak systems that the forms of the input-output responses and of the disturbane-output responses are independent. This fundamental fat permits the realization of a broad new range of response harateristis, espeially in the presene of suh lassially diffiult fators as pure time delay. The design proedure for suh systems is unusually diret and simple. A lassifiation of system disturbanes as being basially additive or multipliative in harater has led to the onept and pratial implementation of multipliative feedbak systems employing ratio omparisons and multipliative orretions. A new range of response harateristis is available from these systems. Conditional feedbak systems provide means for aepting ertain omponent nonlinearities in the relation of system output to system input and, at the same time, for removing the influene of these nonlinearities on the stability of the feedbak loop as exited by the input signal. Thisfat alters the signifiane for the designer of many residual nonlinearities and permits ertain modes of instability to be suppressed. Conditional feedbak systems, exept through the new lass of multipliative systems, make no new ontribution to the regulator problem but they provide a basis for improvement for all lasses of servo problems and not least for the lass involving human operators. Referenes 1. THEORY OF SERVOMECHANISMS, H. L. Hazen. Journal, Franklin Institute, Philadelphia, Pa., vol. 218, Sept. 1934, pp PRINCIPLES OF SERVOMECHANISMS (book), G. S. Brown. D. P. Campbell. John Wiley and Sons, Ine., New York, N. Y., NETWORK ANALYSIS AND FEEDBACK AMPLIFIER DESIGN (book), H. W. Bode. D. Van Nostrand Company, Tn., New York, N. Y., SAMPLED-DATA CONTROL SYSTEMS STUDIED THROUGH COMPARISON OF SAMPLING WITH AMPLI TUDE MODULATION, William K. Linvill. AlEE Transations, vol. 70, pt. II, 1951, pp SYNTHESIS OF TRANSFER FUNCTIONS BY ACTIVE RC NETWORKS WITH A FEEDBACK Loop, D. B. Armstrong, F. M. Reza. Transations, Institute of Radio Engineers, New York, N. Y. vol. CT-1, June AN ANALYTICAL STUDY OF A MULTIPLICATIVE FEEDBACK SYSTEM, r, E. S. Stevens. Thesis, University of Toronto, Toronto, Ont., Canada, A PRACTICAL STUDY OF A MULTIPLICATIVE FEEDBACK SYSTEM, R. J. Kavanagh. Ibid. 8. THE HUMAN OPERATOR OF CONTROL MECH ANISMS, W. E. Hik, s. A. V. Bates. Monograph No , United Kingdom Ministry of Supply, London, England, May, Disussion George C. Newton, Jr. (Massahusetts Institute of Tehnology, Cambridge, Mass.): The authors are to be ommended for this interesting approah to the problem of suppressing disturbanes without imposing 158 Lang, Ham-Conditional Feedbak Systems JULY 1955

8 Fig. 10 (Iefl). Parallel feedbak onfiguration Fig. 11 (right). Single-loop equivalentto parauel feedbak onfiguration onstraints on the input-output transfer harateristi. This is an important problem and an be approahed in a number of different ways. The authors' approah is not an unreasonable one in those instanes where the input signal is available. Frequently, however, only the differene between the.input signal and the primary feedbak signal is available with any degree of preision; e.g., a radar traking system and a position follow-up using synhros for error-sensing. I t may be onluded from this paper that the lassial singleloop feedbak system is unable to do suh as an be done with a onditional feedbak system onfiguration. Thus, one may feel at a loss when meeting a problem in whih the input signal is not available for filtering prior to summing with the primary feedbak signal. It is one purpose of this disussion to show that onventional onfigurations may be used to ahieve results idential with those ahieved by the authors. In these onfigurations there is no need for operating on the input signal. Consider first the onfiguration shown in Fig. 10. In this figure, an auxiliary feedbak path is used to suppress the effets of disturbanes, After the auxiliary feedbak transfer funtion HI has been adjusted, the other adjustable transfer funtions (usually Gonly) an be adjusted to ahieve the desired input-output transmission harateristi. To observe this onsider the equation for the output signal C GGIR+U 1+G1HI+G1GH G=A(1+G 1G2H ) H 1=[G 2-A(I+GIG2H)JH (27) This equation ompares with equation 5. The behavior of this parallel feedbak onfiguration is idential with the authors' system providing GG1 AG 1 (28) I+G1H1+G1GH (29) These equations may be solved for G and HI, assuming that A, G I, G 2, and H are known. The solutions are (30) (31) Thus, in priniple, the authors' system an be realized without need for operations on the input signal. In atual pratie, the auxiliary feedbak in the ase of a positional servomehanism might ome from a tahometer attahed to the output shaft and HI would be adjusted to limit the effet of disturbanes. There is no need to use equation 31 sine G 2 is arbitrary within wide limits. G is then adjusted H. C. Ratz (Ferranti Eletri Limited, Toronto, Ont., Canada): Consideration of the onditional feedbak approah to ontrol problems desribed by the authors leads naturally to disussion of its relationship to onventional feedbak systems. The relationship of onditional systems to lassial systems is topologially equivalent to the relationship of bridge iruits to series-parallel iruits. In the onditional system, when the model is hosen for balane, as given by equation 4, there is no signal in the feedbak path through G 2 and, hene, the hoie of signal inputto ahieve the desired input-output transfer funtion. I t has been shown that a parallel feedbak onfiguration an be used to suppress disturbanes without onstraining the inputoutput transfer funtions. Theoretially, it is perfetly possible to realize idential system behavior using a single-loop equivalent to the parallel feedbak onfiguration. Fig. 11 shows suh a single loop. If, in this figure, the transmission of the feedbak elements is set in aordane with the equation HI H2=H+- G (32) the behavior of the single loop will be idential with the behavior of the parallel feedbak onfiguration. Thus, it should not be onluded that single-loop feedbak systems are unable to ahieve independent adjustment of the disturbane-suppression harateristi and the input-output transmission harateristi. However, in pratie, realization of the required transfer funtions is usually simpler if a parallel feedbak onfiguration or the onfiguration suggested by the authors is used. The authors' appliation of their onditional priniple to multipliative systems appears to be novel. In onnetion with proess ontrol problems there may be onsiderable merit in the use of onditional multipliative feedbak onfigurations. However, a number of questions are left unanswered. For example, in the ase of omplex G funtions, is not the stability sensitive to the output signal level sine the loop gain for inremental signals depends on this quantity? Also, are the advantages of the multipliative system suffiient to justify the use of multiplier, divider, and power-raising devies whih are not as easily realized as summing and operational amplifying devies? I look forward to the future publiations on the multipliative systems promised by the authors. output response is independent of the stability of the feedbak loop. In Fig. 4, the authors have drawn a lassial system with a prefilter whih has the same transfer funtion as the onditional system of Fig. 2. However, in Fig. 4, there is a signal in the feedbak path through G 2, the effet of whih must be exatly anelled by the prefilter. It is possible to draw a onventional feedbak system without a prefilter whih is linearly equivalent to a onditional system. Fig. 12 shows suh a system whih is topologially equivalent to Fig. 2. The Laplae transform of the output in Fig. 12 is given by equation 3 with respet to both signal and disturbane. Furthermore, if the model is defined by equation 4, then equations 5, 6, and 7 apply in this ase also. Thus, the input-output response is unaffeted by any adjustment of G 2 whih may be made to alter the harater of the disturbane-output response. There are, however, at least two fundamental differenes between this onventional onfiguration and a onditional system. In Fig. 12, there is always signal in the feedbak path so that system stability depends upon the linearity of the omponents; and the realizability of the transfer funtions in the feedbak loop is subjet to more severe restritions than in the onditional ase. Thus, in speial ases, linear equivalents an be formulated, but the ontribution of onditional servos is in the simple treatment of nonlinear systems and in the new wide range of feedbak onfigurations whih are realizable. Models are already in use" for proess ontrol problems but the new approah of the authors is to use the model in suh a way as to eliminate feedbak signals from the error-deteting element. In the onditional system, the feedbak loop is employed only to redue the effet of deviations from model behavior and other disturbanes. H. Tyler Mary (International Business Mahines Corporation, Endiott, N. Y.): It is very appropriate that the authors r.. I I ~ I Fig. 12. The onditional feedbak system drawn in the onfiguration of a onventional system JULY 1955 Lang, Ham-Conditional Feedbak Systems 159

9 should present this interesting paper at this time. The onept of a ontrol system making use of information ontained in the input signal to derease dynami errors aused by hanges in the signal may be a key tehnique in many of the more omprehensive proess ontrol problems whih are reeiving muh attention today. There is pertinent literature applying to this subjet whih makes the word "new" in the paper title subjet to question. The onepts of feed-forward have been disussed previously.t-vs It has been widely known that, should it be possible to operate on the input signal, dynami errors in the ontrol system an be redued. An important onsideration is that feed-forward tehniques of error redution do not hange the degree of stability or the natural frequenies of the ontrol system, i.e., the roots of a harateristi equation remain the same. Harris! also onsidered feed-forward as a means for ompensating nonlinear fators suh as dry frition. Perhaps the dominant limitation with the use of feed-forward is the nature of the input signal. It must first be possible to measure it diretly and then to operate on the signal in a preditive sense. In the presene of noise, this an beome impratial. We have, however, learned muh about statistial treatment of this sort of problem, sine feed-forward was widely disussed some years ago. Regardless of where the disturbane to a ontrol system is applied, preditive knowledge of that disturbane an be employed to derease dynami ontrol errors resulting from the d!sturbane. REFERENCES 1. THE ANALYSIS AND DESIGN OF SERVOMECH ANISM, H. Harris. Defense Researh Counil, Washington, D. C., Set. D-2, LINEAR SERVO THEORY, R. E. Graham. Bell System Tehnial Journal, New York, N. Y., vol. 25, Ot. 1946, pp PARALLEL CIRCUITS IN SERVOMECHANISMS, H. Tyler Mary. AIEE Transations (Eletrial Engineering), vol. 65, Aug.-Sept. 1946, pp Rufus Oldenburger (Woodward Governor Company, Rokford, Ill.): This valuable ontribution to the siene of automati ontrol emphasizes the importane of designing for both servo and regulator performane in many automati ontrol problems. In the design of speed governors we study the response to speed-setting hanges and load rejetions, as well as other disturbanes. We have found that the design of our governors on the basis of harateristi roots so as to give fast losed-loop response to sudden load hanges has also given good response to speed-setting hanges. Unfortunately, our most troublesome disturbane is what I like to all the "noise" in the speed signal. This signal is put out by the speed-measuring element. This element orresponds to the authors' feedbak element H. As the authors point out, unfortunately this disturbane annot be redued without ompromising H. The noise an ome from gear drive irregularities, prime-mover shaft runout, or other auses. It is a part of the signal one does not wish to respond to, and severely limits the mathematial operations that an be performed on the speed measurement in the omputer part of the on trol. 160 Prof. James Reswik! introdued an auxiliary feedbak loop in whih the transfer funtion A G 1 of the forward part is taken to be the transfer funtion of the extra feedbak path, exept for a onstant of proportionality. If H = 1 the authors also introdue an element with transfer funtion AG 1 into an extra loop, but in a different manner by plaing it in a forward branh. The inlusion by the authors of a disussion of nonlinear omponents is most timely. The Amerian Soiety of Mehanial Engineers will devote its April 1956 Instruments and Regulators Division onferene to nonlinear ontrol. The Russians speialize in this area, as well as in the automati ontrol field in general; they have a journal devoted entirely to the siene of automati ontrol. The Mamillan Company will soon publish an Amerian Soiety of Mehanial Engineers book entitled "Frequeny Response," whih I am editing. This book is to ontain leading ontributions from all over the world on the various phases of this subjet. REFERENCE 1. DISTURBANCE RESPONSE FEEDBACK-A NEW CONTROL CONCEPT, J. Reswik. Transations, Amerian Soiety of Mehanial Engineers, New York, N. Y., 1955 (55-1 RD2). H. P. Birmingham (Naval Researh Laboratories, Washington, D. C.): This important paper is of partiular interest to the human engineer onerned with the man as an element in a man-mahine ontrol system. A problem under attak is the matter of human performane in pursuit traking and in ompensatory traking. In ompensatory traking, the human operator is presented only the error (differene between input and output) and manipulates his ontrol on the basis of this information. In the pursuit ase, the operator sees the input and his output, or terms proportional thereto, on the same display and is required to keep the differene (error) at a minimum. It has been shown that under some onditions, the human is able to maintain a smaller average error in the pursuit ase. It is expeted that the authors' analysis an be used to show how the man an turn in this superior performane where the ourse as well as the error is shown to him, by ating analogously to a onditional system. G. Lang and J. M. Ham: We greatly appreiate the interest shown by the disussers. Dr. Newton's observation that input signals are not always available with preision is well made but we onsider the oasions when input signals annot be measured with useful auray to be rare. Dr. Newton proeeds to show that in a linear world a onventional feedbak onfiguration, suh as shown in Figs. 10 and 11, an be made to have transfer funtions idential with those of the basi onditional onfiguration in Fig. 2. This fat is implied by the prefilter equivalent shown in Fig. 4. As Mr. Ratz has shown in Fig. 12, there are many suh linear equivalents. However, we wish to emphasize that our onditional onfiguration leads to diret realization tehniques not readily implemented for general linear equivalents. For example, Lang, Ham-Conditional Feedbak Systems Dr. Newton's equations 30 and 31 show how to pik onventional ompensating funtions G and HI to ahieve equivalene with a onditional onfiguration. Given the onditional onfiguration and the freedom to pik A and G 2 independently, equations 30 and 31 are readily used to define G and HI, but we wonder how Dr. Newton would pik G and HI to give independent input-output and disturbane-output responses without prior referene to a onditional onfiguration. In the literature on feedbak ontrol systems we are not aware of ariy olleted treatment on the problem of seleting ompensating transfer funtions to ahieve independent onditions on input-output and disturbane-output responses, partiularly.when nonminimum phase omponents are present. A pratial reason for the diffiulty in ahieving suh onditions is readily diserned from equations 30 and 31. These equations show that any hanges in G or in HI affet both C/ Rand C/ U so that the ut-and-try proedure is ommonly resorted to. In this onnetion, our sample design is exemplifying the diretness with whih a onditional system design an be realized even under the diffiult ondition of a pure time delay in the main atuator. A blok diagram for Fig. 10 showing how to realize G and Hi as speified in equations 30 and 31 is given in Fig. 13. If the system is synthesized as shown, it is lear that many more elements are required than for the equivalent onditional system. While it is true that in a linear world G and HI an be realized with gain and general passive filters, it is not lear how suh filters are to be synthesized in pratie. Dr. Newton's remarks about the equivalene of Fig. 11 to the onditional onfiguration are quite orret but are subjet to the qualifiations outlined in the foregoing. It should be observed that all of the equivalenes disussed by Dr. Newton depend for their validity on linearity. In this onnetion the remarks of Mr. Ratz are partiulady relevant. With regard to multipliative systems, Dr. Newton's observation that for omplex G funtions the loop stability depends on the output signal level is quite orret. To the question of the justifiation for using multiplier, divider, and power-raising devies in these systems, it may be remarked that multipliative systems appear to have other than physial worth, namely as model elements for eonomi systems where perunit hanges are signifiant. Further, a generalization of the onept of signal omparison leads to insight into the nature of disturbanes as suggested in the setion entitled "Extension of the Priniples to Nonlinear Systems'." Mr. Ratz's omments may be regarded Fig. 13. A synthesis For Fig. 10 JULY 1955

10 as an amplifiation of the paper and of the remarks of Dr. Newton, and deserve areful reading. Fig. 12 represents the onventional single-loop feedbak system equivalentto the onditional onfiguration. Again it must be noted that, if this system is synthesized as shown in Fig. 12, more elements are required than for the equivalent onditional system. Mr. Mary emphasizes the relationship of the onditional feedbak systems introdued by us to so-alled feed-forward systems to whih onsiderable attention has ertainly been given. Perhaps the work ofj. R. Moore! most losely resembles ours. Tothe extent that any linear system may be regarded as a ombination of feed-forward and feedbak elements, the onditional topology introdued by us an be so interpreted. However, the onept and topology of onditional feedbak systems employing a referene model has not been desribed previously in the literature, as far as we know. As Mr. Mary suggests, the onept of onditional feedbak may be partiularly illuminating for problems onerning transient proess ontrol. The remarks of R. Oldenburger are interesting. We have heard of Professor Reswik's reent work and are looking forward to seeing his paper. We appreiate the interest of Dr. Birmingham in the appliation of onditional feedbak systems to the human operator problem. Dr. Birmingham rightly distinguishes the tasks of ompensatory and pursuit traking. These orrespond to the regulator and servo definitions as used in the paper. We believe that the onditional topology is signifiant for the engineering of man-mahine systems. A researh program on this topi is in progress at the University of Toronto. REFERENCE 1. COMBINATION OPEN-CYCLE CLOSED-CYCLE SYS TEMS, J. R. Moore. Proeedings, Institute of Radio Engineers, New York, N. Y., vol. 39, Nov. 1951, pp Design of Control Systems for Synopsis: This paper presents a method of designing feedbak ontrol systems whih minimizes bandwidth for a speified transient error. Redution of bandwidth is desired in order to attenuate noise, simplify the ompensation, and ease the requirements on omponents operating at high power levels. Design Speifiations BANDWIDTH TEST Minimum bandwidth is obtained by adjusting the system weighting funtion to produe minimum output noise under speially ontrived irumstanes termed the' 'bandwidth test." In this paper the bandwidth test is a oneptual sheme for defining bandwidth. During the bandwidth test a stohasti noise signal is used for the ontrol system input. The outputofthe ontrol system is passed through Paper , reommended by the AlEE Feedbak Control Systems Committee and approved by the AlEE Committee on Tehnial Operations for presentation at the AlEE Winter General Meeting, New York, N. Y., January 31-February 4, Manusript submitted September 24, 1954; made available for printing Deember 21, GEORGE C. NEWTON, JR., is with Massahusetts Institute of Tehnology, Cambridge, Mass. This researh was sponsored in part by the Servomehanisms Laboratory, Department of Eletrial Engineering, Massahusetts Institute of Tehnology, in onnetion with a projet subontrated by Linoln Laboratory, Massahusetts Institute of Tehnology, and supported by the Department of the Army, the Department of the Navy, and the Department of the Air Fore under Air Fore Contrat No. AF19(122)-458. The author wishes to aknowledge the onstrutive suggestions made by his olleagues in the Servomehanisms Laboratory, and espeially those made by Dr. L. A. Gould, Dr. J. C. Simons, Prof. L. S. Bryant, and G. T. Coate. Bandwidth GEORGE C. NEWTON, JR. MEMBER AlEE Minimum a filter and its rms value measured. The rms value of the filtered output is orrelated with the bandwidth of the ontrol system by omparison with the filtered output of a standard system of any presribed form but having adjustable band.. width. The bandwidth of the standard system is adjusted to produe an nns value of its filtered output equal to that of the ontrol system. Both the' standard system and the ontrol system are driven from a ommon noise soure during the bandwidth test. Fig. 1 is a blok diagram showing this sheme fer relating ontrol system bandwidth to its filtered output during a bandwidth test. By definition, the bandwidth of the ontrol system is equal to the bandwidth of the standard system when the rms values of the filtered outputs are equal. In order that minimizing the filtered output of the ontrol system shall be equivalent to minimizing its bandwidth, the standard system together with the noise soure and filter must produe a monotonially in- BANDWIDTH ADJUSTMENT w b reasing nus output with inreasing bandwidth. PERFORMANCE INDEX CONTROL SYSTEM...- ~ W(w) The purpose of any ontrol system is to onstrain its output to math a desired output (ideal value) within an aeptable tolerane when ated upon by an input (ommand) and disturbanes. The measure used to assess the agreement between the desired and atual outputs is usually alled a performane index. For the design method of this paper the input and desired output are arbitrary transient signals. The integral square of the error between the desired output and the atual output is used as the performane index. In the proess of adjusting the ontrol system to minimize its bandwidth, only those weighting funtions are used whih make the integral-square error equal to or less than a speified value. SOLUTION FOR WEIGHTING FUNCTION Fig. 1 Sheme for defining of system bandwidth In the setion entitled "A Variational Approah" a general solution is obtained for the weighting funtion whih the ontrol system should have in order to possess minimum bandwidth for a speified integral-square error. This solution is obtained by variational methods. The data neessary to obtain a partiular solution are as follows: In onnetion with the bandwidth test: JULY 1955 Newton-Design of Control Systems for Minimum Bandwidth 161