G I. I A I . f ~ I P Select as the new undetermined constantsirapnica imna Iysi|s o r tontro I Dystems the amplitude A and the phase angle 4

A e- cos (w0t-,) (8)WALTER R. EVANS The constants are determined from the

ASSOCIATE AIEE initial conditions that the output iszero and its rate of change is zero at time

Synopsis: The purpose of this paper is to Consider the input to be a unit step zero. The complete solution for outputdemonstrate some graphical methods for and note that the steady state value of then becomes:finding the transient response of a controlsystem. A simple position follow-up system out0put will also be unity. Assume that / \is considered for convenience although the the output transient can be represented C1-,cos ,e 'ft cosmethod is applicable in the same form for by exponential terms, so that 00(t) =higher order systems or those in which only Aest is substituted into the differential in which tan ,==n (9)empirical frequency data is known. The . t (nbasic procedure is to find the roots of the equation, and the common factor Aetdifferential equation which correspond to the cancelled. The following numerical values areexponential trahsient terms which dominate * selected for conveniencethe response. Doctor Profos5 of Switzerland 0 =1+-(1+ TAIs)s (3) K =2/seconds Tm = 1 second (10)points out that the plot of the function which Kdescribes the system from error to output isa function of a complex variable of which Note that s appears at each point where Note that if TM were equal to one-frequency is the imaginary part and damping d/dt had occurred before. This equation tenth second, the value of 6i/0 given inis the real part. The Nyquist plot is thus in s is an algebraic one, and any value of equation 4 would be the same if s andone line of a conformal map with the root s which satisfies it represents an expo- K were both made ten times larger.of the equation being the value of the vari-able which makes the function equal to -1. nential term which can exist in the tran- Thus, the results of these problems can beAny line of plot can be calculated for sys- sient. shifted into any range of values withtems with known functions with essentially Anticipating the fact that s will replace which the reader may be normally accus-the same ease as the Nyquist plot by use of d/dt when an exponential solution is tomed. Substituting the foregoingsome graphical tricks. The amplitude of assumed the system itself can be more gany transient term is determined from the aplot once the root is known by use of a conveniently represented by the block 00(t)=1_1.07e-0.5tcos (1.32t-21l) (11)theorem of operational calculus. The de- diagram of Figure 2. The function in avelopment possibilities of the subject seem block represents the ratio of its output to Graphical Plot to Locate Real Rootsto be very great as suggested by several its input. The relationship betweentopics not yet investigated. Oi and 0, can now be set up directly. The consideration of an additional

Review of Fundamentals 0f 0,,+e e 1 delay in the control system raises the-= -=1+-=1+± (1+Tms)s (4) degree of the equation from second to

QUADRATIC SYSTEM will first be 00 00 00 K third. In this case, consider the delayanalyzed in order to emphasize For this case of a quadratic equation to be the time constant of the inductive

the important concepts in finding any the roots can be found directly by com- build-up of current in the field of thetransient response. Consider the posi- pleting the square and solving for s. generator supplying the motor. Setting-tion follow-up system shown in Figure 1. up the ratio Oi/Q,9The differential equation relating the 1 |/K (1 2 0,e 1

outputtotheerroris 2TM fTm\2TM -a=l+-=l+-(1+TGs)(l+TMs)s (12)00 00 K

d / d -Tn Jwn (5)K= lTMd--O(d (1) .The previous values of K=2,TM= 1\ , - 00(t The oscilllatory case is taken because it will be kept and Ta taken as one-fourth:

is typical of the response of a fast control second.K is the output speed corresponding to a system. The transient solution is the __________________unit error. TM is the time constant of sum of two exponentials, one for each Paper 48-85, recommended by the AWEE basicmotor acceleration, other delays are root. sciences committee and the joint subcommittee on

isth. servomechanisms and approved by the ALEEneglctedBu inpt tStteoutput plus technical program committee for presentation at:nheglected. But¢njn)+Se-n input (6) the AIEE winter general meeting, Pittsburgh, Pa.,.January 26-30, 1948. Manuscript submitted

But this can be converted to a cosine August 11, 1947; made available for printing.09^(t) =090(t) +E(t) = fucto usn*hrlto December 29, 1947.r 1 X ~d \dlfnto sngterlt WALTE:R R. EVANS, assistant professor, depart-.l+ft1+Mdt dt-it (2) C@=O5tjsna () menlt of electrical engineering, Washington Unli-

1948, VOLUME 67 Evans-raphical Analysis of C7ontrol Systems 547'

INPUT AMPLIFIER GENERATOR MOTOR OUTPUT LOAD -

Ve P)oLJ

Figure 1. Position follow-up systemoutput/ input funlctioIn is shown in the Figure 2. Simplified block diagram

block diagram of Figure 4.

Complex roots such as those which The determination of the roots of sucharose in the quadratic system always a system is generally agreed to be a of gain K=3 contracts the curve inoccur as a conjugate pair; therefore this tedious job by present methods and is toward -1 point so that one wouldthird degree equation must have either infrequently done. The frequency re- expect that the system was closer toone or three real roots. In finding the sponse method instead has been developed becoming unstable, or having less damp-real value of s which makes the function to a fine point as indicated by the many ing. A quantitative value of damping,of s of equation 12 equal to zero, one of recent articles on the subject.2 In this however, is now primarily a matter ofthe reasons for the simplicity of finding method the feed-back loop is broken and experience based on systems with knownroots of a known function will become a sinusoidal signal is impressed. The vector plots and known transient per-apparent. The factors of t/0, are al- output is determined in magnitude and formance. It is precisely at this pointready known therefore, make Gi/6o - o phase angle as a function of frequency. that the key idea of Profos becomes ef-by making e/O = -1i. In guessing s This information can be obtained by any fective.to be a negative real quantityv-u, each one of several calculating methods or byfactor ofe/0a is a real quantity and can be direct laboratory tests. A convenient -Determination of Principle Rootsplotted against -a as shown in Figure 3. way to show the results is to plot the From Vector PlotThe rangeofa-i from 0 to -1gures vector ratio of error/output. The plot

a product of factors which is negative, for the foregoing system is shown in The key idea of P. Profos' is to con-

but the magnitude is a fraction so that a Figure 5 with the frequency identified sider the vector plot to be the base lineroot eannot exist in this region. The by numbers in parenthesis. from which the complex roots of the mainregion beyond -a= -4 certainly For a stable system, the locus of e/0O damped sinusoidal term can be deter-contains a root. A guess of -o= -5 must swing outside the -1 point, physi- mined. Recall from the simple quad-gives a result of e/S0 = - 21/2. A second cally meaning that the output/error ratic system that these roots make theguess of -a= - 41/2 suggests itself to ratio is a fraction at the frequency such E/0O ratio equal -1. These roots couldmake e/O = -1 because the factor that the output is in phase with the error. be determined if a plot of the function1+ TGS is then cut in half whereas the The vector ratio of 6i/10, is greater than of s could be made as a function of bothother factors are nearly constant. The the e/0, ratio by 1 aind is therefore a - a. and w.new result is e/0O= - 63/64 with further vector to the curve with tail at the -1 One recognizes a somewhat similarcorrection probably not justified by point. Increasing the gain to K= 6 situation in plotting an electrostaticaccuracy of the data. Note the calcula- contracts the curve as showin bringing field in which the complex variable in-tion of e/0, is very rapid by maintaining it inside the -l point indicating an volves flux and voltage. The patternsthe identity of the factors, and that unstable system. An intermediate value for lines of constant voltage and lines ofguesses of values of roots are immedi- +fately suggested.

Divide by the factor s+41/2 to reduce K ___/the third degree polynomial to a quad-ratic. Solve for the complex roots of c -the quadratic as before to find s= -0.25 _ G

-jl.30. ApplIy initial conditions todetermine the constants to find that the -l _ J_total response is given by I00(t) = 1 -0.08e- 4'i'-1.036e-°025tX

cos (1.30t-270) (13) - 2

The main effect of considering the osecond delay is to cut down the damping l / / ||lrate of the oscillation.3 e

Vector Plot of Error-Output Ratio 1 1,For Sinusoidal Signals _

A standard method of counteractingl lllthe effect of a time delay is to insert in the ____

the derivative of error. The form of its locate real roots s=-

548 Evans-Graphical Antalysis of Control Systems AIEE TRANSACTIONiS

LEAD CIRCUIT BASIC CONTROL Figure 4. Block dia- and the magnitude of 2 as 6 decibels

gram of system. in Figure 7.(i+TcS) ___ Kac_L___ Lead circuit added The case of a quadratic term in the

- t[l+(Tc/oc)S]c (I+TGS)(;+TMS) S to amplifier. Gen- e/@O ratio frequently arises. Such a61oA ierator and motor term can be broken into two factors

delays considefeddelay considered having conjugate complex roots asshown below.

S2+asb=b S- (-a+jg) [S-flux is known to be a grid of curvilinear The ratio of output to input for any one (-a-ji)1 (18)

squares. If this flux plotting property time delay is a vector quantity specifiedcan be justified for e/00, the root values by amplitude and phase angle. The These conjugate roots are thus thepivotof - , and Cwn can be determined by output to input ratio for several time pointsfortheprotractorinmakingmeasketching. The justification is that e/0, delays in series is then the product of urements to the s point.is a function of s which has a particular their amplitudes and a sum of their The procedure is now simply one of

derivative with respect to s for any value phase angles. tabulating decibels and angles for each

of s in the region of interest. Thus in (1+jw T1)(1+jwT2) =A e1AJ2 term, adding decibels for the total

Figure 6 the change of e/0, along the A A2e(q1+02) (14) decibels, and adding angles for the totalcurve for an increment Ajw means that angle of the E160 ratio. The over-allthe derivative must be located 90 degrees If one keeps track of amplitudes log- amplitude factor for the product can now

clockwise from this increase. If the arithmecally, the product can be taken be established by checking the special

change in variable at this point were by adding logarithms. case of s is equal to 0. The value of

- Aa instead, the change in the e/0 e/0Ofor s= o, eliminating K and s factorsfunction would be opposite in direction logAA2= logA+ logA2 (15) is 1 as shown in the block diagram. The

to the derivative. The magnitude of The logarithmic scale commonly used sum of the decibels reading with pro-

the change of the function would be the is the decibel scale defined by the equa- tractor swung to the origin actually ob-thecageofthfuncionwuldbtheiste decibel scaledtained is due to all of the factors accumu-same for equal small changes -Aa or tion below.Ajw. Thus for fairly large, but equal decibel 20 log10 A lated in setting up the l/T+s vectors.

changes in - o and c,, a set of curvilinear (1) This decibel value should be subtractedchanges in -a andw, a set of curvilinear~ ~ ~ ~ ~ ~~~1 from all sums obtained for other values ofsquares will be formed as indicated by the The new problem introduced is to ined fo otheves of

dotted lines. express any one term l+Ts as an ampli- s. Division by the gain K IS achieved by

Note that this complex plot can be tude and phase angle. Several schemes subtracting the decibels for that K from

based on an original vector plot ob- are of course possible, but the following the net decibels previously obtained.

tained from laboratory data as well as is believed to be the most convenient. The E/0 vector is then known as an angle

from one which is calculated. This is Consider the term to be factored as and decibels of magmtude and so can be

valid however, only for systems which shown below with the T factors saved plotted using the same protractor.are linear in the test range and the for later consideration. It is convenient to make the first cal-

results are applicable only to this linear /1 the uniform changes injvu. Thofs a cur-rag.This restriction iS necessary be- 1+ Ts = TI-' (17) teuiomcagsi w hsacrrange. .

T vilinear square pattern should be formedcause the justification of the curvilinear

with any mistake in calculating one pointsquare pattern is that the derivative of The term s can be located as a vector

e/0, is dependent only on the nature of from the origin. The complete vector the grid set up by the other points. Thethe signal, not the amplitude. is one with tail at the - l/T point and plot for the system of Figure 4 is shown inOne can sketch such a grid with more head at the s point. This vector can Fi 6 f T = 1/2 and K 4 The

confidence after having calculated a few be measured by a protractor pivoted atsystems with known functional form. the - l/T point with scale in line withThis can be done conveniently by some the s point. The amplitude is desiredgraphical tricks, as shown in the next in decibels so the scale is so marked. Figure 6. Complex plot for system. Con-

section. This scale can be checked by noting the sfma n iernd0.95+j2.25magnitude of 1 is marked as 0 decibel

Calculation of Complex Plotd ( eoA

First consider the facts about present e

calculation methods as a function of fre- (h') /quency alone which simplify that task.

Figure 5. Vector plot oflIJ /lerror/output raltio with dsiu e/O / |w=3_ ( o( 5l

soidalsina impressed /e

-2 , 1 - {/XK=2 '- W6-

1948, VOLUME 67 Evans-raphical Analysis of Control Systems 549

value of s which makes e/60=-1 is can be found by any one of several ess of successive approximations. StartS=--n`Jwn= -0.95 i=j2.25. The am- methods but from the slope of the func- with the pair of complex roots suggestedplitude of the transient having this root tion they are -1.45 and + 1.66, respec- by the plot built from one side and divideas well as its initial phase angle, needs to tively. The complete solution including the function by these factors. The result-be determined before the transient can the steady state value of I becomes: ant plot will usually give good indicationbe plotted. Fortunately this is possible t of the other pair of roots since over-without need for finding the rest of the

0 e(t) = 1- 1.367e0 2co-s-(2125- 170) + ) lapping is eliminated. The first pair ofroots and substituting initial conditions roots may now be determined moreby means of a theorem of operational accurately by dividing the original plotcalculus. Procedure For Complicated Systems by the factors corresponding to the

second pair of roots.Amplitudes From Operational A solution of a fourth degree equation For systems in which the damping is

Calculus is nothing new, but this method can near critical, the behavior of the plotreadily be applied to higher order equa- near the origin must be understood. A

The concept of s thus far presented has tions. Thus additional time delays or good example to work out is for a simplebeen simply that of being a complex num- stabilization circuits will distort the fre- quadratic with roots of -1 1J1. Theber used in the exponential solution of the quency locus but the conformal map can first "square" is shaped like a triangledifferential equation. This concept is still be sketched. with base from 0 to -1, and curved sidessufficient for explaining the process of The quadratic roots found from the crossing at - 1/2+jl. The missing cornerfinding the roots to the equation. The original plot represent the transient term will be found to be at the midpoint of thesymbol s, however, has a more potent which will frequently dominate the com- 0 to -1 line.significance as it is used in operational plete transient response. If a complete Protractor measurements could bemethods such as the Laplace Transform.3 solution is desired, however, these known eliminated by substituting potentiometersThose familiar with these methods of factors can be divided out of the Oi/0O at the pivot point which turn in resistancecourse realize that this is a long study in ratio. Thus, for any value of frequency, proportional to angle. The decibel read-itself, but essentially only one fact need the angle and decibels readings for the ings could similarly be replaced withbe used. The amplitude of any transient protractors pivoted at the conjugate root potentiometers which are turned as a

term is given in terms of its root by points, with the scale at the frequency logarithmetic function of the distance topoint, should be subtracted from those of the s point. Connecting each set of po-

A,- 1 ] original vector to obtain those of the tentiometers in series will then give to-A1 d( (19) new vector. Note that in general the tal angle and total decibels as resistances.

ds6t s=SI portion of the original vector plot for These resistances could then be used as

higher frequency values will now be con- inputs to instrument servos which couldBut the derivative of the function can tracted into the -1 region. Thus, one actuate meters or locate the position of a

be determined directly from the complex would have to have frequency data in the pen.plot. It is vector with an angle of 84 region of the higher order roots in order todegrees as shown in Figure 6. The find them. Usually the corresponding Development Possibilitiesmagnitude is the change in E/0, divided transient term will be found to damp outby the change in s. An average of sev- rapidly so that they would be needed What are the possibilities for new typeseral measurements in the region of the -1 only to study the initial break away of of laboratory tests to give empirical datapoint gives the value of 0.60. the output. It is probable that other other than just frequency response? TheThe two transient terms involving the methods would be more applicable to problem would seem to be primarily one

pair of complex roots can be converted studying this region. of achieving a steady state condition longinto a single term as in the simple quadra- The construction of a vector plot for a enough to get a reading and not have thetic system. The conversion is made very multiple loop system can be readily car- natural transient of the system present.rapid by noting that the amplitude terms ried out on a completely vector basis. An exponential build-up might work inare conjugates of each other. In taking The multiplication process has already that starting from zero the transientthe sum therefore their real parts add and been shown, and the addition process is should not appear and the only readingtheir imaginary parts cancel. The re- simply the familiar completion of a necessary is the ratio of output to input,sult can thus be written as twice the real parallelogram. Many time saving tricks which should be constant during thepart of one of them. are possible however by shifting or rotat- build-up. An exponentially increasing

re- Tnte3wntl 2e-nt COs (Wot-a-j) ing an entire plot with respect to its pre- sine wave would correspond to a complex2R J = vious position. The predominance of value of s, though the reading here wouldae)abei3 ab feed-back signals ovQr feed-forward signals have to include phase angle as well as

(20) makes the inverse plot more convenient, magnitude.in which for starting at the output one can build The similarities of the conformal map-

a}=s-f+n up a diagram back to the input step by ping properties of static fields and thesestep. functions of s seem to offer the most in-

and Occasionally systems will have two teresting possibilities. Conceivably the

d leAl1 pairs of quadratic roots in about the same known system information could be set upbe'13""jr1-J I . frequency range. The Nyquist plot as boundary conditions of an electrosta-then circles the -1 point with the result tic field so that the pattern of equipoten-

The other two roots to this fourth that a plot built up from one side over- tial lines and lines of flux would corre-degree system are found to be real roots laps a plot built up from the other side. spond to the system plot. The systemof -2.58 and -16.6. The derivatives This situation can be handled by a proc- plots of the type described in this paper

550 Evans-raphical Analysis of Control Systems AIEE TRANSACTIONS

Figure 7. Vector is increased. In this situation the rootsdetermination of of the transient equation lie at the inter-

/i / 1 /T±s section of the locus for a real part of e/-0 = -1 and the locus for zero imaginarypart. These roots therefore move downthe saddle from the horn and the rear

v //\ until they meet in the center for the criti-/5> z \J cal damping case. Further increase in

gain) or raising of the saddle, causes theroots to appear as a conjugate pair mov-ing down each side toward the stirrups.

T / 30°-I ) 0 TWill a static field be capable of mapping/ 60° / -2 -I ° such a pattern? If so, what boundary

60< tude correspond to constant real part and conditions will be necessary? This seem-lines of maximum slope correspond to ingly never-ending chain of questionslines of constant imaginary part. On serves to keep the subject interesting,

however, have several values of s which this basis a quadratic function in the re- since one does not know what usefulmakes the E/90 equal to -1 where as gion of the roots has the shape of a horse's method might be encountered in theonly one voltage could exist at the -1 saddle. Imagine a view of the saddle processof findingtheanswers.point for the field. Another system from above with the origin at the horn ofplot is possible however in which the the saddle and the -a axis running Refroles of E/0o and s are reversed. Any straight back to the rear. One trace of encespoint on the plane would correspond to a zero imaginary part is along this axis and 1. A NEW METHOD FOR THE TREATMENT OFvalue of s and E/0, is plotted as loci of its the other intersects it at right angles at REGULATION PROBLEMS, P. Profos. Sulzer 7'ech-constant real part and constant imagi- the center of the saddle. Picture now the nical Review (New York, N. V.) number 2, 1945nary part. saddle to be immersed in water and E. S. Smith, McGraw-Hill Company, New York,A physical picture of the plot can be gradually lifted out. The water level N. Y., 1944.

gained by considering a surveying con- lines on the saddle give the locus for a real 3. TRANSIENTS iN LINEAR SYSTEMS (book),M. F. Gardner, J. L. Barnes. John Wiley sod Sons

tour map in which lines of constant alti- part of E/0 -1 as the gain of the servo New York, N. Y., 1942.

No Discussion

1948, VOLUME 67 Evans-raphical Analysis of Control Systems 551