Control System Synthesis by Root O (s) = l+K,GA (s)Ka (s)The problem of finding the roots of theLocus Method differential equation here appears in the

form of finding values of s which makethe denominator zero. After these values

WALTER R. EVANS are determined by the root locus method,MEMBER AIEE the denominator can be expressed in

factored form. The zeros of the functionSynopsis: The root locus method deter- The opening section in this paper, Oo/Oi can be seen from equation 1 to be themines all of the roots of the differential Background Theory, outlines the over-all zeros of G;,(s) and the poles of Ga(s).equation of a control system by a graphical The function can now be expressed asplot which readily permits synthesis for pattern of analysis. The following sec- h idesired transient response or frequency tion on Root Locus Plot points out theresponse. The base points for this plot great usefulness of knowing factors of the Go (1-s/qi)(1-s/q2)On the complex plane are the zeros and poles open loop transfer function in finding the is (1- s/rl) (1- s/r2) (2)of the open loop transfer function, which are roots.readily available. The locus of roots is a rootsplot of the values of s which make this The graphical nature of the method Tep constan Kpeand theeo t fotransfer function equal to -1 as loop gain is requires that specific examples be depend upon the specific system but forincreased from zero to infinity. The plot used to demonstrate the method itself control systems y is often zero and Kc iscan be established in approximate form by . . often 1inspection and the significant parts of the ulder the topLis: Single Loop Example The full power of the Laplace Trans-locus calculated accurately and quickly by Multiple Loop System, and Correctiveuse of a simple device. For multiple loop Networks. The topic Correlation with form2 or an equivalent method now cansystems, one solves the innermost loop Other Methods suggests methods by be used. The transient response of thefirst, which then permits the next loop to be which experience in frequency methods output for a unit step input, for example,solved by another root locus plot. The . is given by equation 3resultant plot gives a complete picture of can be extended to this method. Thethe system, which is particularly valuable topic Other Applications includes the nfor unusual systems or those which have classic problem of solving an nth degree Go(t)=1- A etit (3)wide variations in parameters. polynomial. Finally, the section on

Graphical Calculations describes the key The amplitude Ai is given by equationHE root locus method is the result of features of a plastic device called a 4an effort to determine the roots of the "Spirule", which permits calculations to A-be made from direct measurement on the A =- -Vdifferential equation of a control system plot srz

bv using the concepts now associated Plot. The closed loop frequency response, onwith frequency response methods.' The the other hand, can be obtained by sub-roots are desired, of course, because they Backgroundstituting s=j into equation 2. For-describe the natural response of the sys- tunately, the calculation in finding Ai ortem. The simplifying feature of the The over-all pattern of analysis can be 6o/60(jw) involves the same problem ofcontrol system problem is that the open outlined before explaining the technique multiplying vectors that arises in makingloop transfer function is known as a of sketching a root locus plot. Thus a root locus plot, and can be calculatedproduct of terms. Each term, such as consider the general single loop system quickly from the resultant root locus1/(1+Ts), can be easily treated in the shown in Figurel. plot.same manner as an admittance such as Note that each transfer function is of1/ (R+jx). It is treated as a vector in the the form KG(s) in which K is a static gainsense used by electrical engineers in constant and G(s) is a function of the Paper 50-il, recommended by theAIEE Feedback-sligaccircuits. The phase shift and complex number. In general, G(s) has Control Systems pCommittecomtpeand a proed by th

attenuation of a signal of the form es1 both numerator and denominator known tion at the AIEE Winter Genlerai Meeting, Newbeing transmitted is represented by in factored form. The values of s which York, N. Y., January 30-February 3, 1950. Manu-1/(1+Ts) in which 8 in general is a com- make the function zero or infinite can able for printing November 22, 1949.plex number. The key idea in the root therefore be seen by inspection and are WALTER R. EVANS is with North American Avia-locus method is that the values of 8 called zeros and poles respectively. The tion, Inc., Downey, Calif.whicb make transfer function around the closed loop transfer function can be ex- The assitance givhentbyhis fellow woPrkers, K. R.loop equal to -1 are roots of the differ- pressed directly from Figure 1 as given in Jackson and R. M. Osborn, in the preparation of thispaper. In particular, Mr. Osborn contributed theential equation of the system. equation 1 circuit analysis example.

66 Evrans-Control System Synthesis AJEE TRANSACTIONS

ei 8r 80 Figure 1 (left). General block diagram iw'~~~~~~~~. IU (S

Figure 3 (right). Single loop root locusjWc

LOCUS OF S FOR

I L eS~.8 00+ 01 + 0. -1lo -_*00

Root Locus Plot

The open loop transfer function is so that the angles in turn can be visual-typically of the form given in equation- 5. ized. For any specific problem, however,K,MG(s)KgG6(s) many special parts of the locus are es-

K(lTs032+ 032 tablished by inspection as illustrated ins(1+ Tis)[(S+0)2+ 32] (5) examples in later sections. Surprisingly

few trial positions of the s point need beThe parameters such as T, are constant assumed to permit the complete locus to a value of s just above the real axis. The

for a given problem, whereas s assumes be sketched. decrease in Oo from 180 degrees can bemany values; therefore, it is convenient After the locus has been determined, made equal to the sum of 41 and O2 if theto convert equation 5 to the form of one considers the second condition for a reciprocal of the length from the trialequation 6. root, that is, that the magnitude of point to the origin is equal to the sum of

KMG,(s)KgGp(s) be unity. In general, the reciprocals of lengths from the trialK,,G,(s)K#Gp(s) one selects a particular value of s along point to - 1/T1 and -1/T2. If a dampingK(l/T2+s)T2 [032+cW32] the locus, estimates the lengths of the ratio of 0.5 for the complex roots is de-

s(1/T1+s)T1[(s+ 03+jco3) (s+ U3-JCO3I vectors, and calculates the static gain sired, the roots ri and r2 are fixed by the(6) K,,K = 1/G,(s)Gf(s). After acquiring intersection with the locus of radial lines

The poles and zeros of the function are some experience, one usually can select at 60 degrees with respect to negativeplotted and a general value of s is as- the desired position of a dominant root to real axis.sumed as shown in Figure 2. determine the allowable loop gain. The In calculating K for 8= ri, it is con-Note that poles are represented as dots, position of roots along other parts of the venient to consider a term (1+ Tls) as a

and zeros as crosses. All of the complex locus usually can be determined with less ratio of lengths from the pole - 1/Ti toterms involved in equation 6 are repre- than two trials each. the s point and from a to the origin re-sented by vectors with heads at the gen- An interesting fact to note from equa- spectively. After making gain K= 1/-eral point s and tails at the zeros or poles. tion 6 is that for very low gain, the roots [G(s)],$=, a good first trial for finding r8The angle of each vector is measured with are very close to the poles in order that is to assume that it is near -1/T2 andrespect to a line parallel to the positive corresponding vectors be very small. For solve for (1/T2+s). After the roots arereal axis. The magnitude of each vector very high gain, the roots approach in- determined to the desired accuracy, theis simply its length on the plot. finity or terminate on a zero. over-all transfer function can be expressed

In seeking to find the values of s which as given in equation 8.make the open loop function equal to -1, Single Loop Example 1the value -1 is considered as a vector /whose angle is 180 degrees =rn 360 de- Consider a single loop system such as (i- 1 1 (8)grees, where n is an integer, and whose shown in Figure 1 in which the transfer r1 r2/ r1magnitude is unity. Then one can con- functions are given in equation 7. The procedure in handling a multiplesider first the problem of finding the locus K loop system now can be explained.of values for which the angle condition KMG$(s) = K+ G(s) = 1(7)alone is satisfied. In general, one pictures (1 + Tis) (1+ T2s)s' Multiple Loop Systemthe exploratory s point at various posi- The poles of the open loop function aretions on the plane, and imagines the lines at 0, - 1/Ti and -1/T2 as represented by Consider a multiple loop system infrom the poles and zeros to be constructed dots in Figure 3. which the single loop system just solved

The locus along the real axis is deter- is the forward path of another loop, asmined by inspection because all of the shown in Figure 4.

_S + b3 - -w3- --< angles are either 0 degrees or 180 degrees. 0,/0i is given in factored form by equa--

Io < mum build-up rate, overshoot, naturalr < > frequency of oscillation, and the damping

rate as effective clues in solving thisproblem.

Other ApplicationsFigure 4 (above). Multiple loop block diagram i Many systems require a set of simul-

\-l/ ~~~taneous equations to describe them andFigure 5 (right). Multiple loop root locus tare said to be multicoupled. The corre-sponding block diagrams have several in-

angle at which the locus emerges from ri puts to each loop so that the root locuscan be found by considering a value of s method cannot be applied immediately.close to the point ri, and solving for One should first lay out the diagram sothe angle of the vector ( s-ri). that the main line of action of the signalsAssume that the static loop gain de- forms the main loop with incidental

sired is higher than that allowed by the coupling effects appearing as feedbacksgiven system. The first modification and feed forwards. One then proceeds tosuggested by the plot is to move the ri These examples serve to indicate the isolate loops by replacing a signal whichand r2 points farther to the left by obtain- reasoning process in synthesizing a con- comes from within a loop by an equivalenting greater damping in the inner loop. If trol system by root locus method. An signal at the output, replacing a signalthese points are moved far to the left, the engineer draws upon all of his experience, entering a loop by an equivalent signal atloci from these points terminate in the however, in seeking to improve a given the input. One can and should keep thenegative real axis and the loci from the system; therefore, it is well to indicate physical picture of the equivalent systemorigin curve back and cross the jw axis. the correlation between this method and in mind as these manipulations are car-Moving the -1/T point closer to the other methods. ried out.origin would then be effective in permit- The techniques of the root locus methodting still higher loop gain. The next as- Correlation with Other Methods can be used effectively in analyzing elec-pect of synthesis involves adding correc- tric circuits. As a simple example, con-tive networks. The valuable concepts of frequency re- sider the lead-lag network of Figure

sponse methodsi are in a sense merely ex- 7(A).Corrective Networks tended by the root locus system. Thus It can be shown that the transfer func-

a transfer function with s having a com- tion of this network is as given in equa-Consider a somewhat unusual system plex value rather than just a pure imagi- tion 9

which arises in instrument servos whose nary value corresponds to a damped sinu- V0 (1+RiCis)(1R2C2s)R3open loop transfer function is identified soid being transmitted rather than an un- - (1+R2C2S)R3+by the poles pi and P2 in Figure 6(A). As damped one. The frequency and gain VE (1+RiCis)(1+RiCiS)Ri+loop gain is increased from zero, the roots for which the Nyquist plot passes through Rl [1+ (R2+Ra) C2s]which start from pi and P2 move directly the -1 point are exactly the same values (9)toward the unstable half plane. These for which the root locus crosses the jw, The denominator can be factored alge-roots could be made to move away from axis. Many other correlations appear in braically by multiplying out and findingthe jc& axis if 180 degrees phase shift were solving a single problem by both methods. the zeros of the resulting quadratic. Asadded. A simple network to add is three The results of root locus analysis can be an alternative, it will be noted that thelag networks in series, each having a time easily converted to frequency response zeros of the denominator must satisfyconstant T such that 60 degrees phase data. Thus one merely assumes values of equation 10shift is introduced at p,. The resultant s along the jco axis, estimates the phaselocus plot is shown in Figure 6(B). angles and vector lengths to the zeros andThe gain now is limited only by the re- poles, and calculates the sum of the owl

quirement that the new pair of roots do angles for total phase shift and the prod- Pi pinot cross the jcw axis. A value of gain is uct of lengths for attenuation. The in-selected to obtain critical damping of verse problem of determining zeros andthese roots and the corresponding posi- poles from experimental data is the more

I

tions of all the roots are shown in Figures difficult one. Many techniques are al-6(A) and 6(B) as small circles. ready available, however, such as drawing I

Actually, greater damping could be asymptotes to the logarithmic attenua- o 0r-oachieved for roots which originate at Pi tion curve. For unusual cases, particu- /\and P2 if a phase shifting bridge were used larly those in which resonant peaks are vrather than the 3-lag networks. Its involved, the conformal mapping tech-transferfunction is (3-Ts)/(1+Ts) and is nique originated by Dr. Profos of Swit-of the "nonminimum phase" type of cir- zerland is recommended.3cuit. The transient response is described by p

Since these types of correction are the poles of- the transfer function. The2somewhat unusual, it is perhaps well to inverse problem in this-case is to locate the A Bpoint out that the analysis has been yeni- poles from an experimental transient re- Figure 6. (A) Basic system. (B) Correctedfled by actual test and application. sponse. One might use dead time, maxi- system

68 Evans-Control System Synthesis AIFE TRANSACTIONS

Cs ~~~~~~~~Figure 7 (left). (A)Circuit diagram. (B)

Root locusFisure 8 (right).

A

10R,0, 5c. (A*R.c2 I order term. Solve for the roots of the Several procedures are possible, but the

first loop which corresponds to the quan- over-all purpose is to successively rotatetities in brackets above and proceed as the arm with respect to the disk through

(1/R1CI+a)(1/R2C2+s)R3(R1C1)(R2c2) before for the multiple loop system. If each of the angles of interest. Thus for[1/(R2+Rs)C2+aj(R2+RP)C2RI the roots close to the origin are of most adding phase angles, the disk is held

=-1 (10) interest, substitute J= l/x first and solve fixed while the arm is rotated from a polefor root values of x. Other combinations to the horizontal, whereas the two move

Tesentdvectorsdingthis exre are rep- are, of course, possible because a single together in getting aligned on the nextrscentedinaccordingBto Thetorootlus root locus basically determines the factors pole. For multiplying lengths, the disk isscheme in Figure 7(B). The two roots of the sum of two terms, held fixed while the arm is rotated fromdotsare thereby broundedas. shown bycth two The root locus method is thus an an- the position where a pole is on the straightdotsuand the cross.ma Ther exaccateylatis alytical tool which can be applied to line to the position where the pole is oncould be estphicat medhor s. other problems than control system syn- the logarithmic curve. Rotations aremiedlocusbyfgraph modis. simply.in- thesis for which it was developed. But in made in the opposite directions for zerosThe locus of roots now is simply in-

tervals along the negative real axis be- attacking a new problem one would prob- than they are for poles.tween the open loop zeros and poles as ably do well to try first to develop ashown in Figure 7 (B). The exact location method of analysis which is natural for Conclusionsof the roots along these intervals is deter- that problem rather than seek to apply

any existing methods. The definite opinion of engineers usingmined in the usual way. Note that the * this method is that its prime advantageconstant in equation 10 is of the form GR'Cot in which R' is the effective value of Graphical Calculations is the complete picture of a system whichR'Ci, in whichR'istheeffective value ofthe root locus plot presents. ChangingR2 and Ra in parallel. The root locus plot is first established an open loop parameter merely shifts a

In more complicated networks, the ad- in approximate form by inspection. Any pointandmodifiesthelocus. Bymeansofvantages of the root locus concept over sinfcn fombinpcinAn potadmdfeshelu.Byensfvantagebraic metherod tlocusbec reatoeri significant point on the locus then can be the root locus method, all of the zeros andpalgbrticularhadvantagerisein greatainin ats checked by using the techniques indicated poles of the over-all function can be de-particular advantage oS t retalning at all in this section. Note that only two cal- termined.betwmee thearover-all netwk parelametensh culations are involved, adding angles and Any linear system is completely de-betwenth ove-allnetwrk cirmeter multiplying lengths. Fortunately, all of fined by this determnination, and its re-and the parameters of individual circuit these angles and lengths can be measured sponse to any particular input functionelements,

at the a point. Thus angles previously can be determined readily by standardIn the classical problem of finding*Inots, the classifereal problen fivending pictured at the zeros and poles also ap- mathematical or graphical methods.roots, the differential equation is given inthe form of a sum of terms of successively pear at the s point but between a hori- Refhigher order. This can be converted tothe form shown in equation 10 zeros and poles. A piece of transparent i. PRINCIPLES OF SERVOMECHANISMS (book),paper or plastic pivoted at the s point can G. S. Brown, D. P. Campbell. John Wiley and

s'+as +bs'2+ M be rotated successively through each of Sons, New York, N. Y., 1948.2. TRANsIrNTs IN LINBAR SYSTBMS (book).-[(s+a)s+b]s+ . .. +m (11) these angles to obtain their sum. M.F. Gardner,J.L.Barnes. JohnWileyandSons,

The reader can duplicate the "spirule" New York, N. Y., 1942.This corresponds to a block diagram with two pieces of transparent paper, one 3. GRAPHIcAL ANALYSIS OF CONTROL SYSTBMS,W. R. Evans. AIEE Transactions, volume 67,

with another loop closed for each higher for the disk and the other for the arm. 1948, pages 547-51.

No Discussion

1950, VOLUME 69 Evans-Control System Synthesis 69