HIGH RESOLUTION ROOT LOCUS PLOTS ------~------~~~----~~--~~--~~------Abstract Approved Redacted for Privacy --·~O~----~·~·----~(M--A~Jo R__P-Ro~F~E-Ss-O~R~)

AN .ABSTRACT OF THE THESIS OF

Edward Janes Swenson for the M. S. in Electrical Engineering (Name) (Degree) (Major)

Date thesis is presented June 30, 1966 ------~~~------Title HIGH RESOLUTION ROOT LOCUS PLOTS ------~------~~~----~~--~~--~~------Abstract approved Redacted for Privacy --·~o~----~·~·----~(M--a~jo_r__p-ro~f~e-ss-o~r~)------

A technique is presented for obtaining high resolution root locus plots and

Bode diagrams. This is done by translating the origin of coordinates to the area of interest by an appropriate change of variable. The method is applicable to hand calculations or use of a potential analog computer.

Examples include the analyses of systems that contain very highly damped, closely spaced pole zero pairs. These are actual design problems encountered in the design of high-speed tracking antenna systems and booster control systems. HIGH RESOLUTION ROOT LOCUS PLOTS

EDWARD JAMES SWENSON

A THESIS

submitted to

OREGON STATE UNIVERSITY

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

June 1967 HIGH RESOLUTION ROOT LOCUS PLOTS

EDWARD JAMES SWENSON

Litho U.S.A. APPROVED:

Redacted for Privacy Associate Professor of Electrical Engineering In Charge of Major Redacted for Privacy

Hedd/of Department of Elbctrital and Electronics Engineering (

Redacted for Privacy

Dean of Graduate Schoo I

Date thesis is presented June 30, 1966 ------~------

Typed by Loraine Riley ACKNOWLEDGMENT

The author wishes to express appreciation to Professor

Solon A. Stone for his constructive suggestions offered during the preparation of this thesis. TABLE OF CONTENTS

I. The Log s Plane 1 1.1 Introduction . . 1 1.2 The Log s Plane 2 1.3 Need for Scale Expansion 5

II. Origin Translation in the Log s Plane 7 2. 1 Basic Equat·ions 7 2.2 Origin Translation 8 2.3 Axis Location 9

Ill. Real Axis Translation 13 3. 1 Introduction 13 3. 2 Roots Near Pole or Zero 13 3.3 Angle Breakaways 17

IV. Root Locus Analysis for Closely Spaced Complex Pairs 21 4. 1 Translation Along jw Axis . . . . 21 4.2 Translation in the Plane .... 24 4.3 Bode Diagrams of Resonant Pairs . . 25

v. Compensation Synthesis in Log x Plane 36 5. 1 Introduction . . . . . 36 5.2 Damping Ratio Template 38 5. 3 Design Example . 41

VI. Origin Translation in the z Plane 53 6. 1 Introduction . . 53 6. 2 The z Plane and Log z Plane 53 6.3 Log z Plane Origin Translation 56 6.4 Cone luding Comments . . . 57 Bibliography ...... 59 Appendix A: Changing Factor Forms 61 Appendix B: Generalized Frequency Response Plots 65 FIGURES

1. 1 The s and log s planes in rectangular and polar coordinates 3 1.2 Danping ratio lines in the s and log s plane 4 2. 1a Translation along the negative real axis •• 10 2. 1 b Translation along the positive real axis •• 10 2.2 Translation along the imaginary axis 11 2.3 Translation in a complex direction 12 3.1 System block diagram · • . . . • • 13 3.2 Root locus plot of Equation 3.2 • 14 3.3 Root locus plot of Equation 3.4 15 3. 4 Root locus plot of Equation 3. 13 • 19 3.5 Real axis plot or "Siggy" plot of Equation 3.13 20 4.1 Root locus plot of Equation 4.1 • • . • • • •••••••• 22 4.2 Log x plane root locus plot of Equation 4.1 (st = +j30) • • ••• 24 4.3 Log x plane root locus plot of Equation 4.1 (st = -1 + jlOO) •• 26 4.4 Phase plot of Equation 4.1 • • • • • • • • • • • • • • 27 4.5 GainplotofEquation4.1 • . • • . . . • • . . • • • 28 4 .6a log x plane phase plot of Equation 4.1 for w positive (90° line) •• 29 4 .6b log x plane phase plot of Equation 4. 1 for w negative ( 270° line) • 30 4. 7 Phase plot of Equation 4. 1 with the log x plane information added • 31 4.8a log x plane gain plot of Equation 4.1 for w positive . . • • . . 33 4.8b Log x plane gain plot of Equation 4.1 for w negative . . . . . 34 4.9 Gain plot of Equation 4.1 with the log x plane information added • 35 5.1 Damping ratio lines in the log s plane . . . . . • . . . • • • 36 5.2a The s plane dcrnping ratio and undamped natural frequency lines plotted on the log x plane (at positive) . . . . • . . . . . 37 5.2b The s plane damping ratio and undanped natural frequency lines plotted on the log x plane (at negative) • . . . . • • • . . 38 5.3 The s plane damping ratio and undamped natural frequency lines plotted on the log x plane (Wt positive) . • . • . • . . . • 39 5.4 The s plane dcrnping ratio and undamped natural frequency lines plotted on the log x plane (at negative and Wt positive) • 40 5.5 Template for constructing damping ratio lines • • • • • •• 40 5.6 Calibrated damping ratio template ( DRT) • 41 5.7 System block diagram ...... • • 41 5.8 Root locus plot of Equation 5.2 • . . . . . • 42 5.9 Root locus plot of Equation 5.2 with lead network compensation added . . . . • . . • • • . • ...... • . • . • 43 5. 10 log x pI ane root locus plot of zero pole pairs x3 - x 13 and x5-x15with lead network z1 -P1 added (st=+j42.1)... 45 5.11 log x plane root locus plot of zero pole pair x7- x17 with lead network Z1- P1 added (st = +j85.1) ...... • . . . . . 46 5.12 log x plane root locus plot of zero pole pair x9- x19 with lead network zl- pl added (st=+201.4) ...... 47 5.13 Symmetrical twin-tee or parallel-tee network ...•..••.• 48 5.14 log x plane root locus plot of zero pole pair X9- x19 with lead network zl - pl and twin-tee network p2 - p3 and z2- z3 added . 50 5.15 Root locus plot of Equation 5.2 with lead network Z1 - P1 and twin-tee network P2 - P3 and Z2- Z3 added ••...... •. 51 5.16 Root locus plot of Equation 5.2 W,ith lead network and twin-tee compensation added . . . . . : ...... • . • . . . . 52 6. 1 The s plane, z plane and log z plane . . . • . • . . . . 54 6.2a The log z plane with s plane damping and undamped natural frequency lines (positive real axis in center) ...... • . 55 6.2b The log z plane with s plane damping and undamped natural frequency lines (negative real axis in center) . . . • . • 56 6.3 Figure 6.2a when translated to the z1 plane (z = z1 + 1) . 57 6.4 Figure 6.2b when translated to the z 1 plane (z = zl + 1) 58

TABLES

4. 1 Translating the origin to +j30 ...... 23 4.2 Translating the origin to -1 + jlOO ... . . 25 4.3 Transforming Figures 4 .6a and 4 .6b to the s plane . . 31 4.4 Transforming Figures 4.8a and 4.8b to the s plane ...... 32 5. 1 Translating the origin to +j42.1, +j85.1 and +j201.4 . 44 HIGH RESOLUTION ROOT LOCUS PLOTS

CHAPTER I

THE LOG s PLANE

1 . 1 Introduction

As automatic control systems have become increasingly complex through the years, new and more powerful techniques for designing and analyzing them have been required. Although studies in the time domain using digita I and ana log computers do provide answers to systems whose performance has been we II specified, transform methods are extremely important in the design of systems with a number of variable parameters. The transform methods have been partic ularly useful for design since they allow a wide variety of performance criteria and visualization aids to be applied. The most widely used of the visualization aids are the Nyquist Diagram, Nichols Chart, Bode Plot and Root Locus Plot.

Of the four graphical methods, the most popular by far has been the

Bode plot. Its popularity stems from its simplicity of construction and its ability to represent all of the singularities of the transfer function on one plot. Diffi culties with the Bode diagram are encountered, however, if several quadratic terms appear in the transfer function.

In recent years the root locus method has become popular because it overcomes the difficu Ities encountered with the Bode diagram in that it com pletely specifies the variation of the closed loop pole locations as the open loop 2 gain, or some other parameter, is varied. Normally, root locus plots are plotted on a linear coordinate system as originally developed by Evans (3, 4). The s plane was adopted because of its familiarity and ease of calculation. Unfortu nately, however, there are several disadvantages with the s plane root locus plot. These are:

(1) Accuracy is difficult to obtain for a system with widely separated roots.

(2) Gain versus frequency and system band width (which may be specified} are not apparent.

(3} The effect of sma II gain changes in the system performance cannot be pre dicted readily.

1 . 2 The Log s Plane

The difficulties encountered with the root locus plot can be overcome if the log s plane is used since:

(1} All the poles and zeros of the transfer function can be represented with equa I resolution and accuracy over every decade.

(2) The system band width is completely specified and can be sketched directly from the log s plane root locus plot because the root locations line up directly with the break frequencies of the standard Bode diagram.

(3} The effect of small gain changes can be observed, particularly at lower fre quencies.

Figure 1 . 1 shows the s plane and log s plane for both rectangular and 3

s PLANE LOG s PLANE 90•

270•

a:> crz c:; cr :IE + 0z cr cr..J +j "'a: +

Figure 1.1 The s and log s planes 1n rectangular and polar coordinates. polar coordinates. Note that the quadrants of the s plane appear as bands in the log s plane. The origin of the s plane occurs in the log s plane at nega tive infinity. In polar coordinates, the circular magnitude lines become straight vertical lines in the transformation while the radial angle lines become straight horizontal lines. The horizontal and vertical lines (a= constant and jw = constant) of the s plane transform into curvi I in ear Iines in the log s plane transformation. These curvilinear lines are all of the same shape and are 4

described by the same equation. For most practi co I problems four-decade logo

rithmic paper is used. Damping ratios (Figure 1 .2) are now straight horizontal

lines in the log s plane and are shown on the right-hand edge of the log s plane

graph paper.

I PLANE LOG s PLANE

360" 1.0

141 .J -0.5 (!) z 270° <1: +8.2 .._ +0.5 !:! 0 1 z +0.8 c <1: ,.,, o• a:: :: 360" 180" + 1.0 Ill 141 +0.8 ~ 0 Q, ::::) +0.5 ~ .... +g.2 0 z 90° (!) <1: -0.5 ::l

o• -1.0

+ 1.0 (: a:> -0.5 <1: -j 0 .._ z +0.2 0 (5 +0.5 <1: 1 ::l +0.8 ~ + + 1.0 Ill 0 +0.8 ~ z Q, <1: +0.5 ~ .J +g.2 0 <1: +J 141a: (: +

Figure 1 .2 Damping ratio lines 1n the sand log s plane. 5

Both the polar and rectangular graph papers are available commercially

from Electro Scientific lndustries1 Inc . 1 Portland 1 Oregon (Forms< 46 and 47 1

respectively).

Although not as popular as the s plane root locus1 there are severo I

methods proposed for log s plane computations. Devices similar to the Spirule

have been devised by Klugman (8) and Nomoto (16) for rapid hand construction of root locus plots. The rules originally developed by Evans (3) for root locus

plotting can be directly extended for log s plane root locus plotting. One of

the major difficulties encountered when applying root locus to design is the tre mendous amount of time required to draw accurate root locus plots for practical systems. These problems can be overcome for log s plane computations if a po

tential analog computer (12 1 13 1 14 1 15) is employed. This allows extremely

rapid root locus and frequency response plotting for systems with a large number of singularities.

1.3 Need for Scale Expansion

Several problems may arise when root locus plots are constructed graphi

cally or by means of a computer. Closely spaced poles and zeros 1 either near

the origin or elsewhere in the plane 1 may result in insufficient resolution. Simi

lar problems may be encountered in finding roots very close to a pole or zero or

1 Commercially available from the Spirule Company1 Whittier1 California 6 in finding breakaway points of the angle locus. These difficulties can be greatly reduced if the origin of coordinates is shifted to the troublesome area and the scale expanded. When the scale is expanded in the s plane to give sufficient detail, the other poles and zeros move off the plotting paper. This does notal low, then, the study of the effect of other poles and zeros on the locus of inter

est. In the log s plane, however1 a logarithmic expansion around the new origin is obtained when the origin is translated, yet those poles and zeros far from this area are still represented and their effects can be studied. To under stand this translation we must go back and carefully study the equations of interest and the effects the translation has on these equations. 7

CHAPTER II

ORIGIN TRANSLATION IN THE LOG s PLANE

2.1 Basic Equations

If all the exponents of an algebraic function of a complex variable are integers, the function is said to be a rational algebraic function. A rational algebraic function may be written in the form:

F : (2. 1)

This can be written as a product of factors:

no Oc F = K1s Il(s-sif where K1 : (2 .2) I bd itO

no oo F: K2s n~-~yi where K2 = (2.3) 1 s· bo j;tQ I

Each n is± some integer. K and K are real for a rational function by F, s 1 2 and the s. 1s may be complex. Note that the s. 1s are the pole or zero locations I I in the s plane--for the factor (s + 3), for example, s. =-3. The singularity at I the ongm' ' o f t h e s pane I ·IS s no .

Graphical computation or potential analog computer computation starts from equations in the form of Equation 2. 2 or 2. 3. 8

2 . 2 0 r1g1n.. Trans Iat1on . l

If F{s) is a function of a complex variable s, and if s is to be examined in the vicinity of st' we translate the origin from 0 to the point st by changing the independent variable frorn s to x, where:

S = X+ St (2 .4)

This results in a new function:

F(x) = F(s-st) (2. 5)

In a linear two-dimensional scale for s, this places the point s =st at x =0, the origin of the new x plane. The scale can be greatly increased to improve resolution around x =0. As was pointed out earlier, on a linear plot the poles and zeros farther away from this area of interest wi II move off the plotting area unless the plotting paper is increased in the same proportions as the scale.

Equation 2.2 could have been written as:

0 1 F(s) = K, (s-s0r(s-str ~ (s-sit (2 .6) itO i ;t t

where the s.rs are the pole and zero locations and (s- s 0 )n° represents a pole I or zero at the origin of the s plane. Equation 2.4 was s =x + st' so for specif ic values of s and x we have:

Si = Xi + St (2. 7)

Portions of this material have been published prior to the presentation of this thesis {17). 9

SO = XO + St (2 .8)

St : Xt + St (2. 9)

Substituting Equations 2.4 and 2.7 through 2.9 in Equation 2.6 we have:

F(x} =K,·(x-xoro (x-xtrt r;I (x-xjri (2. 10) itO i ;{ t

Noting that xt =0 and including (x - Xo) in the product n, we have:

F= K,xn' r;I (~-xri (2. 11) i=O itt

Note that the point st in the s plane is now at the origin in the x plane and is represented by the factor x nt. There are three possible cases for st: a real number (a), a pure imaginary number (jw), or a complex number (a+ jw).

Each of these specific cases will be studied in the following chapters.

2. 3 Axis Location

If the translation is to be used in conjunction with root locus plotting, then the original axes need to be located on the log x plane plot. The trans- lotion along the negative and positive real axis is shown in Figures 2.1a and

2.1b respectively. Note that for a a negative (Figure 2.1a), the original axis appears to be located on the positive real axis. The curvilinear lines emanating from the original origin located on the log x plane plot are the original jw posi tive and negative axes. To locate the original origin, simply sets= 0 in 10

Equation 2.4, and solve for the required value of x. For a translation along the

positive real axis the original origin appears to be located along the negative

real axis of the log x plane. Again, the original axes are sketched and labeled

as shown in Figure 2. 1b.

a z ~ ·:·:· ..J ""' ~ ILl a::

-j

s = X + CTt (SHOWN FOR CTt

Figure 2. la Translation along the negative real axis.

+j en >< ~· ~ >p :::::;:. .:3: tiP 't.-=3: 3 t .,... .~ ....J .:3: z :· 1:: x PLANE ~ ORIGINAL-~ ORIGIN ~ s PLANE ;,..o x=Os=a:1 t- ORIGIN I + a z ~ w;;._;:.. ..J ~ ;:ffib ~ ILl a:: ~p~ t-t-- x PLANE

s =x + CTt (SHOWN FOR CTt POSITIVE)

Figure 2.lb Translation along the positive real axis. 11

Translation along the jw positive or negative axis is by far the most prac

tical of the three possibilities. We wi II consider a translation along the jw posi

tive axis only, since the functions under consideration appear in complex conju

gate pairs. Figure 2.2 shows a translation in the x and log x planes. Again,

the origina I origin is located and the original axes are sketched. The curvi Iin-

ear lines of the curvilinear graph paper that pass through the origin are the

original positive and negative real axes. Note for an wt positive the original

ongm IS located on the jw negative axis.

+j : ·: ·: lA

:" ::·

0z cl ...J cl w a: ·: ::: :: :: . : :: 1--t-t-++~t-+-+-~H:,,). tt ;;; t + -j s = x + jwt (SHOWN FOR Wt POSITIVE)

Figure 2.2 Translation along the imaginary axis.

If the troublesome area lies away from both axes, the origin can be trans

lated to the area of interest. This translation is thus far more of academic than

practical interest since all the systems encountered to date have contained light

ly damped pole zero pairs very close to the jw axis, so that a translation along

the jw axis gives good resu Its and is much easier to perform and interpret. 12

+j len I>< -~ ~~ ::: IC:i E~ - '~ f~

[x~R~t~ : :: : r~ .t.l !Arl' iE'R:~ in'1 il : : : + f-- 0 : :: z ::

""..J tc RfGI N,IL 'NE< ;A "lVI F'F.ll . L&.l 1~IS ""It: 1 II ~PLI .~J.R~:.~- -j s = x +crt + jw (SHOWN FOR crt NEGATIVE, Wt POSITIVE)

Figure 2.3 Translation in a complex direction.

Figure 2. 3 shows the translation for a negative and jw positive. Again

note that the original axes are located by first solving for the original origin and

then darkening the curvilinear lines that intersect at that point. 13

CHAPTER Ill

REAL AXIS TRANSLATION

3. 1 Introduction

If a root occurs on the real axis and is very close to a pole or zero, the exact root location cannot be determined. In many problems, this is incense quentia I. However, when a root occurs very close to a low-frequency open loop zero that remains in the closed loop expression, the exact root location and its corresponding residue may be of vita I interest since the transient from the root can be a major contributor to the transient error. The real axis translation can a Iso be used to obtain greater accuracy for an angle locus breakaway on the rea I axis and to factor polynom ia Is more accurately using the root locus method.

3.2 Roots Near Pole or Zero

Consider the system shown in Figure 3. 1.

BIN K (s+O.I54)(s+0.726) Bour s3 (s+3.44) (s+7. 56)

Figure 3. 1 System block diagram.

The closed loop expression may be written in the familiar form: 14

Bo K (s+O.I54)(s+0.726) BIN= s3(s+3.44Hs+7.56) + K ( s+O.I54)(s+0.726) (3.1)

Setting the denominator equal to zero and rearranging yields:

K (s+O.I54)(s+0.726) (3. 2) s3 (s+3.44) (s+7. 56) =-I

Figure 3.2 is a root locus plot of Equation 3.2. Note that for both K values chosen, the root near the zero at -0. 154 cannot be resolved.

;::;:::;;;;::: :<• >•< t\?? I•?(} ??? t::;:: •::::::::::;:=::;:;:::;.

270' •=::b :::::::: 1=::::::;:::;::: ·.1:=:=. r:• ::•: ••••••••• •••••••••••••• ===•=•== ''' T•••f'! •:•}•••••••••::::~:::=:::=: :=:=:' (: -90° K=~O ...,I,. ~~~~~~~~-~ r~~~~~+o•TU.> Ls 0 >= I I < 2. (K= 25) .. -+ ]9.,154~K, 125) A • ..... 2 .,. .. ~~~--~~~~~~~L.1~15•40(K-~1 ~-+0)-+~~~~4-0•)~~~~~--~--~~ ~+'(!) =40) z I a: l: < Cl

Figure 3.2 Root locus plot of Equation 3.2.

We can resolve this root location more accurately and study its movement as K is varied if the origin is translated to the zero location -0.154. This will place 15

the zero at the origin of the new x plane. Substituting in Equation 2.4 yields:

s =x-0.154 (3. 3)

and substitution in Equation 3.1 for s gives:

-I= Kx(x+0.572) (3 .4) (x -·0.154)3 (x+3.29) (x +7.41)

Figure 3.3 is a plot of Equation 3.4 with the two roots of interest indicated.

The roots are x = 0. 0058 and x = 0. 0038.

Figure 3.3 Root locus plot of Equation 3.4.

If we now substitute these values for x in Equation 3.3 we have:

s: -0.1482 (3. 5) and 16

s= -0.1502 (3 .6)

These are the desired s plane root locations. The denominator of Equation 3.1 can now be replaced by its factored equivalent, as read from Equation 3 .5 or

3.6 and Figure 3.2. Expanding into partial fractions and taking the inverse transform will give the time domain transient response, and the effect of the transient contributed by the root near the zero at -0.154 on the total transient response can be studied.

The same approach may be used if greater accuracy is desired when fac taring polynomials using the root locus method. Consider the fifth order poly nomia 1:

s5 + lls4 + 26s3 + 25s2 + 22s + 2.8 = 0 (3. 7)

To factor Equation 3.7 by the root locus method, we first rearrange which gives:

(3 .8)

Dividing by the coefficient of the highest power and applying the quadratic formu Ia to the two quadratics gives:

25(s+O.I54)(s+o. 726) = -l(s3Hs+3.44)(s+7.56) (3.9)

Rearranging we have:

25(s+0.154) (s+ 0. 726) (3. 10) s3 (s+3.44) (s+7.56) :-I

This equation has already been solved. Referring to Figure 3.2 and Equation 3.5, we have the five roots s = 0.93 L 105; s = 0.93 L 255; s =0.1482 L 180; 1 2 3 17 s 2. 15 L180; s = 8 . 1 L180. 4 = 5

3.3 Angle Breakaways

The expanded resolution near the origin of the log s plane can sometimes be a disadvantage. Angle locus breakaways which occur very close to the origin are expanded to such a degree that the exact location of the breakaway cannot be found. Often, if the breakaway is close to the origin the angle locus cannot be resolved at all. This is the opposite of what might be expected. It would appear that we could translate the origin to the approximate breakaway location, and then, using the results of this first translation, do a second translation to further improve the accuracy. This method is used quite successfully for finding breakaway locations in the s plane (6, p. 351-360). In the log s plane, however, as the breakaway moves closer to the origin one finds that the whole plotting area appears to be a saddle point. Hence, only within limits can the real axis translation be used to obtain better resolution for the breakaway.

In many high-speed aircraft and missile problems, a right half plane zero is encountered on the positive real axis. This often leads to a breakaway that oc curs very close to the origin. The locus is often almost totally unresolvable. To overcome this difficulty, we move away from the origin to decrease the resolu tion at the breakaway. An example is again used to illustrate the procedure.

Consider the characteristic equation encountered in the analysis of a multi loop booster contro I system. 18

254(s- 5.96Hs+l2.1) +I= (3. 11) (s-0.94Hs+l.71)(s+4.72)(s+IOHs+ 15.7)

Plotting the root locus we find the angle locus breakaway between the pole at

+0.94 and the pole at -1.71 is unresolvable. To obtain better resolution for this angle locus breakaway, we move the origin to +0.94. Substituting for st

in Equation 2.4 we have:

s=x+0.94 (3. 12)

Substituting for s in Equation 3.11 gives:

254 (x- 5.02Hx+l3.04) +I= (3. 13) x(x + 2.65)(x+ 5.66)(x +10.94Hx+ 16.64)

Figure 3.4 is a plot of Equation 3.13. Note the location of the original origin and the angle locus breakaway. The breakaway in the x plane is x = -0.96. Substituting this value for x in Equation 3.12 gives:

s =-0.96 + 0.94 =0.02 (3. 14) which agrees v~ry closely with the calculated value.

The breakaway may also be solved for by plotting a real axis or "Siggy"

plot of Equation 3.13. The breakaway occurs where the plot becomes tangent to the w axis. (For a more detailed discussion of real axis plots, see Appendix B.)

Figure 3.5 is a real axis plot of Equation 3.13. Note that the breakaway loco tion agrees very closely with that found by the root locus method. 19

Figure 3.4 Root locus plot of Equation 3.13. 20

~ 60 8 7 I 5' I ~ I I ~ . ~

2 I' ! I~ i ~\ @. I 40 8 7

5 "'"'- IFI ' 3 ' "'Il""''IIII ~ 2 ~ ..;;J r-- ~.... irl ~ 0 [jQJ: I:""' 20 8 'THE ANGLE LOCUS 7 BREAKAWAY LOCATION

3 tO

0 m 2 3 ' 5 6 8 9 2 3 ' 5 6 1 a 9 2 3 ' 5 6 7 9 2 3 ' 5 6 7 8 9 0.01 0.1 rr ~lsi I 10 tOO

Figure 3.5 Real axis plot or 11 Siggy 11 plot of Equation 3.13. 21

CHAPTER IV

ROOT LOCUS ANALYSIS FOR CLOSELY SPACED COf.v\PLEX PAIRS

4.1 Translation Along jw Axis

If the poles and zeros of the open loop transfer function lie very close to

the imaginary axis 1 the origin can be translated along the jw axis to the trou blesome area. This finds a major application in high-speed aircraft and missile

control system analysis and synthesis1 where root locus plots must be constructed between lightly damped pole zero pairs. Let us again consider an example:

+K(s+20) (s+O.J±j31) (s+2±jl00) -1=~------~ (4. 1) s(s+JO) (s+40) (s+O.I±j29) (s+J±jJOO)

Equation 4.1 is typical of the type encountered when designing booster control systems or high-speed tracking systems. A root locus plot of Equation

4. 1 is desired with K being the parameter. The high-frequency pairs of poles

and zeros near the radian frequencies of 30 and 100 represent fuel sloshing1 structural resonances and other mechanical characteristics in the open loop transference of the system.

Figure 4. 1 is the root locus plot of Equation 4. 1 in the log s plane. In

the vicinity of the structural resonances1 the plot is too coarse to yield the

stability information desired. For additional information 1 we translate the origin to the area of interest st and plot the root locus of the function F(x). Let us examine Equation 4.1 between the pole at s =+j29 and the zero near x =+j31. 22

To examine this pair, we shall translate the origin to st =+j30, which is half way between the pole and zero. Substituting this value in Equation 2.4 gives:

s=x+j30 (4. 2)

Figure 4.1 Root locus plot of Equation 4.1.

On more complicated equations it is advisable to tabulate the poles and zeros of the function. To simplify identifying the poles and zeros in the log x plane, each pole and zero should have a subscript assigned. If the subscripts are carried throughout, the original poles and zeros may be identified in the x plane.

Table 4.1 shows the poles and zeros of Equation 4.1 with their subscripts and co ordinate values s. and x.. Note that the function F(x) has poles and zeros I I 23 which do not occur in complex conjugate pairs.

TABLE 4.1 TRANSLATING THE ORIGIN TO +j30.

+K(1+20) (s+O.I:tj31) (1+2:tjl00) F(1) • •-1 1(1+10) (1+40) (I+O.I:tj 29)(1+0.1:tjl00)

Si Xi si =xi- j30

II • -20 x, • -20-j30

(/) 12 • -O.I+j31 X2. -O.I+jl 0 a:: 13 =-O.J-j31 X3 = -O.I-j61 L&J N 14. -2+jl00 X4: -2+J70 s5 = -2-jiOO X5 • -2-jl30

IQ • 0 xo = -j30 ss = -10 X6 =-IO-j30

(/) 17 = -40 X7 • -40-j30 L&J ...J 0 18 =-O.I+j29 xa =-0.1-jl 0.. Sg • -O.I-j29 Xg =-O.I-j59

IJQ. -1 + jiOO x,o• -1+ j70 111 = -1-jiOO XII • -l-jl30

Figure 4.2 is a plot F(x) for this case. There is an obvious simplification that can be made because of the close spacing of the other lightly damped pole zero pairs. We may omit the pairs at -j60, -j130 and +j70, since they essen tially cancel. Note the location of the negative real axis of the original s plane. The poles and zeros on this axis are still represented, and their effects on the locus between the pole at x and the zero at x may be studied. Since 8 2 the root locus plot is symmetrical around the 180° line in the s plane, we need not examine st =-j30 as it will be the mirror image. We could make a similar change of variable to examine the pole zero pair near s =jlOO, but for maxi mum accuracy a translation in the plane wi II be used to explore this area. 24

Figure 4. 2 Log x plane root locus plot of Equation 4. 1 (st = +j30).

4.2 Translation in the Plane

If a pole zero pair lies in the plane, then the origin may be translated in the plane to afford better accuracy and resolution. The example used in the previous section (Figure 4.1) has a pole at -1 + j100 and a zero at -2 + j100.

As was mentioned, we could translate along the jw axis to s =+j 100, but for maximum accuracy, let us put the origin of a new x plane at the pole location, s = -1 + j100. Substituting in Equation 2.4 yields: t

s =X -I +jiOO (4. 3) 25

The pole and zero of Equation 4.1 for this translation are listed in Table 4.2.

Figure 4.3 is a root locus plot of F(x) for this translation. Again we may omit the closely spaced pole zero pairs near -j70, -j130 and -j200. We see now

TABLE 4.2 TRANSLATING THE ORIGIN TO -1 + jlOO.

Sj Xi Xj"Si+l-jiOO ., • -20 x, • -19-jiOO 12 • -O.I+j31 X2 • +0.9-j69 0 "'a:: 13 • -O.I-j31 X3 • +0.9-jl31 UJ N 14 • -2+jl00 X4 • -1

15 =-2-jiOO X5 • -l-j200

IQ • 0 xo • +1-jiOO

16 =-10 X6. -9-jiOO

17 • -40 X7 • -39- jiOO UJ "'...J sa • -O.I+j29 xe • + 0.9- j71 0n. sg • -O.I-j29 Xg • +0.9-jl29 s 10 • -l+jiOO x1o • 0 =x"' Ill. -1-jiOO xu•-j200 that the root locus path of the pole zero pair of interest indicates only condition- a I stabi Iity. Note the gain values located along the root locus path.

4. 3 Bode Diagrams of Resonant Pairs

The jw axis translation introduced in Section 4.1 is the most useful and commonly used of the three translation possibilities. One of the advantages of the imaginary axis translations is that it yields very high resolution Bode plots, as well as root locus plots. A chapter or a complete thesis could be devoted to the use of the imaginary axis translation for high resolution Bode plotting, par ticularly in conjunction with the design and synthesis of high attenuation filters. 26

However, this investigation will be left for future study. We shall consider here

Figure 4. 3 Log x plane root locus plot of Equation 4. 1 (st = -1 + j 100). only the method as it applies to control system problems.

Limiting st in Equation 2.4 to a positive imaginary number we have:

s = x+ jwt (4 .4)

Limiting s to the imaginary axis also limits x to the imaginary axis. This can most easily be seen by examining Figure 2.4. Limiting s to the jw axis in

Equation 4.4 gives:

(4. 5) 27

(4 .6)

Equation 4.6 tells us that to get the w value we add the constant w to each s t w read from the x plane Bode diagram. X Again consider the example used in Section 4. 1. Figure 4.1 is the root

locus diagram of Equation 4.1, and Figures 4.4 and 4.5 are the corresponding

phase and gain diagrams. As was the case with the root locus diagram, the plot

-90' -...... ~...... ""' I r--.... -...... L F ...... I I 1""1~~ 00} ...... I I i ' I ' I

ffiQj

I I 1 i I '

I' 60 i I I i 2 3 9 J 56 18 9 ) 3 ) t '8 9 2 ] 5 6 7 8 9 ' 5 '' ' ' ' 0.1 I W -lsi 10 100 1000

Figure 4.4 Phase plot of Equation 4.1.

is too coarse to give any stability information around the resonances near 30 and

100 radians. Figure 4.2 is the root locus diagram used to analyze the pole zero 28

' K!l ,&..

~ D:QM-9 ~ '7 ' ""~ .' "" I'll. J " ~ , "" \.. [JID; ~ '7 ' ' 1111... IFI "' J " ~ t® , iii •0 ...0 0 \ 9 :ill '7 ' I' , ' 0 , ~~ ITEl; ' 2

'7 .I i ~ Cr.iiJ J l'-=

) \

I \ 1o l. lr.;;, 9 ~

'7 ~ .' ~ I.:;;J J -"""' , '~ ~~ -r.::::l I0-3 i , J 4 89 ) J 4 6 7 8 9 i J 4 5 6 7 89 2 J 4 5 6 7 8 9~ 0.1 ' ' I w cis I ' 10 100 1000

Figure 4. 5 Gain plot of Equation 4. 1. 29 pair near +j30. To examine this resonant pair we proceed in exactly the same manner as if we were plotting a root locus plot. Rather than replot the pole zero diagram refer to Figure 4.2.

Note on Figure 4.2 that the original s plane positive jw axis includes all of the positive x plane imaginary axis plus that part of the negative imagi nary axis up to -j30. Hence, to retrieve all of the positive s plane imaginary axis from the x plane diagram we must plot ty,o Bode plots--one for the positive

jw and one for the negative jw axis up to the original s plane origin, which X X in this case is -j30.

i I I 4 I <3) < ) I I LF~--~-+-+~~Hi~~~~~ti~~~~--~~(5)~H+----~~~~~~~ r I ...... (6) I i ~~~

I I J :~:I(Ill I ' I ~·~---+----+-+--+-++~----+---+--+--t-t-+-+~++------1--+---+-t-+++-t-+---~-~~t-----+--+- ~+- -+

r.;a I i' L::..!lli-"H----t-+-+-H-+-HH----+-t--t--+-+-t+tt---+--+--+-+-++ttt--t--t--t-+-t--+-+i+i ':!l1 I 1 ! I ' (2) I I I I I

~(I J.;;;;l~~rrrrn---r--n~+++-t-++------+----+-+---+-++-+++-----1---+--+r 1trT

---+----+--+-+++-l-++-----+----+---+-++++-++--+----+--+-t-++-t-1!, +--+---+- -+--t-L . ' '

I I I I i ~I"H--~,~43~+~l~6~7~89H---+I--~l-+~\~o~'~"+,+---~,-4,~~~-~<>H----+-~l-.~~~-~-~~~- 0.1 1&1 =lXI 10 100 1000

Figure 4.6a Log x plane phase plot of Equation 4.1 for w positive (90° line). 30

Figures 4 .6a and 4 .6b are the phase diagrams for the positive and

negative jw axes. Note in Figure 4.6b that the plot stops at -30 radians X

! ! I_] I:]

I I !T jl E2Qj ! I ....., ) i LF (10) , J I I I' i I I EI!Q} / f<9l i j~t I

! (8) ~-Uti I I': ' I I I: §:} i

! I Ill (7) .rlI I 11 I f I i til i I I : i! I . ~ 2 J 9 2 J 56 7 8 9 } J 5 6 71:39 2 3 ' '0 7 ' ' ' 567:~ 0.1 I -lxl 10 -100

Figure 4 .6b Log x plane phase plot of Equation 4. 1 for w negative ( 270° line). where the phase shift is 90°. (This, of course, v.ould be expected since this is the original s plane origin where there is one pole which contributes 90° of phase shift.) Severa I points have been chosen on Figures 4 .6a and 4 .6b, entered

in Table 4.3 and retransformed back to the s plane. Figure 4.7 is Figure 4.4, the original phase diagram with the new phase information obtained from Figures

4 .6a angA .6b added. 31

TABLE 4.3 TRANSFORMING FIGURES 4.6a AND 4.6b TO THE s PLANE.

Ws =W, +j30

w. Ws POINT IFROM FIGURE 4.6A) (PLOTTED ON FIGURE 4.n

(I) I.OL-31 0 30.1L-310 (2) · 0.8L-285 30.8L-285 (3) 1.4L-150 31.4L- 150 (4) 10 L-145 40 L-145 (5) 50 L- 160 80 L-160 (6) 200L-175 230 L-175

w, (FROM 4.68) W5 IPLOTTED ON 4.n

(7) -O.IL-310 +29.9L-310 (8) -0.95L-240 +29.05L-240 (9) -1.2L-180 +28.8L- 180 (10) -14L-130 +16.0L- 130

~ -~.... ~ ~- I ~ ~ ~~ I L F ~5 ) 1 I I( 3) (9) I I ~) illQj r-...... II I i II ! I : (8) I~ II II I ~ . ,_ I I, 1' ~ 2 3 '0 7 9 3 56 7 8 9 2 5 6 7 8 9 2 J ' ' J ' ' 5 6 7 8 9 0.1 I W= lsi 10 100 1000

Figure 4.7 Phase plot of Equation 4.1 with the log x plane information added. (See Figures 4 .6a and 4 .6b) 32

TABLE 4.4 TRANSFORMING FIGURES 4.8a AND 4.8b TO THE s PLANE.

Ws • Wx +j30

Ws POINT (FROM FIGURE4.8Al"'· (PLOTTED ON FIGURE 4.91

(I) 0.110.341 30.1 10.341 (2) 0.810.031 3o.8lo.o31 (C1l 1.010.0181 31.010.0181 (3) 2.210.151 32.210.151 (4) 6.0,0.221 36.010.221 (5) IOOI0.031 13010.031

Wx (FROM 4.88) W5 (PLOTTED ON 4.9)

(6) -0.110.451 +29.910.451 (7) -o.8l4.ool +29.214.001 (C2l -1.ol7.49l +29,0 17.491 (8) -211.201 +28.o 11.20 I (9) -1511.001 + 15.0II.OOI

Figures 4 .8a and 4 .8b are the x plane gain plots for the positive and negative jw axes. Again, several points on Figures 4 .8a and 4 .8b are chosen and transformed back to the s plane as shown in Table 4.4. The points c and 1 c on Figures 4 .8a and 4 .8b were calculated to cross-check the depth and 2 height of the notches. Figure 4. 9 is the original gain plot with the information obtained from Figures 4.8a and 4.8b added.

The root locus diagram (Figure 4. 2) indicated that the system was un stable for K = 500. The Bode plot should give the same information since it was plotted for K = 500. From Figure 4. 7 and Table 4 .~1 we find that the phase shift is -180° at 28.8 radians per second (Point 9),. and from Figure 4.9 and Table 4.4 we see that the gain is greater than one between 15 and 29.5 radians. 33

i 1!2:!) ) A 5 0 89 2 l 4 5 6 7 8 9 3 4 5 0 7 89 0.1 r.J •lXI 10 100

Figure 4 .Sa log x plane gain plot of Equation 4. 1 for w positive.

Hence the system is unstable, since the gain is greater than one and the phase shift 180° or greater at the same corresponding frequency. 34

rn. ~ 8 7 6 5 I 4 i ~

ill, :mJ 8 7 ~r-C2

5 .- 4 (7) IF\ 3 "\ ~ VI \ J ;rl ' .. 8 J8l , 0 ~~ L (9~ [1Q-i} tm 8 7 ...... '(5) .... ~ jlllll"""' ' 4 3 tBJ

2f-----

2 ~ 3 4 5 6 7 89 3 4 5 6 7 89 2 3 4 5 6 7 89 J 4 5 6 7 8 9 0.1 ' I 'W :\xi 10 100 ' 1000

Figure 4 .Sb Log x plane gain plot of Equation 4. 1 for w negative. 35

J ~

2 ~

ITQD, \.. [-§] 8 7 ~ 5 " ' ' "' J "!""" ~

2 \..

[iQ} \.. 20 8 7 ~ c2 1 5 " ' (7) ' lj J ~ /0 I' ll 2 ~ ,, (8)-H I (9)., I 0 9 8 I I 7 ... ., I 0 ~ 5 1' I (G) u"' ,..,_ 0 IFI' J (l)t~4) Bl 2 - ~~.~

I _, ~3)~ 9 ~ 8 7 --·

5 I il ' ,, \,<_~) J <2> I ~

2 c, ~ \~ i ~9 4 8 ' I 7

5 ' ' 5 3 ''"

! 2 '~ I I I I I I i I 6 ~ 2 J 5 6 7 89 2 l 5 6 18 9 2 ] ,, 6 '8 c, ) ] s'~II :, l 8? ' ' ' 0.1 I w -lsi 10 100 1000

Figure 4.9 Gain plot of Equation 4.1 with the log x plane information added. (See Figures 4. 8a and 4. 8b) 36

CHAPTER V

COMPENSATION SYNTHESIS IN LOG x PLANE

5. 1 Introduction

Although the imaginary axis translation in the previous chapter displayed the stability of the system (i.e., was the system stable or unstable?), it did not tell the degree of stability. Since, generally, we are interested in synthesizing compensation to achieve certain predetermined specifications, additional infer- motion is needed from the translated presentation. Specifically, we need to know the damping ratio and undamped natural frequency of the closed loop roots

Figure 5.1 Damping ratio Iines in the log s plane. 37

s = x + cr1 (SHOWN FOR cr1 POSITIVE)

Figure 5.2a The s plane danping ratio and undamped natural frequency lines plotted on the log x plane (at positive). as the gain and/or compensation poles and zeros are varied.

Figure 5.1 shows the damping ratio lines of 0.2, 0.5 and 0.8 plotted in rectangular coordinates in the log s plane. Also plotted are the undamped natu ral frequencies 0.5, 1 .0 and 10. Figures 5.2a and 5.2b show how these lines are transformed when the origin is translated along the positive or negative real axis. Figure 5.3 shows these same damping and undamped natural frequency lines when the origin is translated along the positive jw axis (the one usually of interest). Figure 5.4 shows how these lines are transformed when the origin is 38

s =x + CTt (SHOWN FOR CTt NEGATIVE)

Figure 5.2b The s plane damping ratio and undamped natural frequency lines plotted on the log x plane (at negative). translated into the second quadrant ( i. e., for a negative and w positive).

Note that for all three cases the damping ratio lines approach the origi nal log s plane lines after two decades. Also, they emanate from the original origin and are the same shape. Hence a template could be constructed to draw the damping ratio lines. This template is shown in Figure 5.5.

5.2 Damping Ratio Template

In the previous chapter we pointed out that the imaginary axis translation 39

1 =x + jwt (SHOWN FOR lilt POSITI'(E)

Figure 5.3 The s plane damping ratio and undamped natural frequency lines plotted on the log x plane (wt positive). is particularly useful for the analysis of resonant systems. Often in these reso nant systems we are concerned about damping ratios far less than 0. 1, which is the minimum calibration of the log s plane graph paper. Examination of Figure

5.3 and our experience with the imaginary axis translation in the previous chap ter suggest that the template could be calibrated for damping ratios much less than 0.1. Figure 5.6 shows this calibrated damping ratio template ( DRT) used with the imaginary axis translation. It can be constructed of clear plastic, which simplifies its use. 40

lxl (SHOWN FOR ITt NEGATIVE, "'t POSITIVE)

Figure 5.4 The s plane damping ratio and undamped natural frequency lines plotted on the log x plane (at negative and wt positive).

Figure 5.5 Template for constructing damping ratio lines. 41

0 coco 0 0 0 0 0 00000 0 0 0 0 000000 0 0 0 0 Qggggg g g g 000000 0 0 0 0 (D(D ..... a'l Ut .$:!l (..W N -00000 0 0 0 Q (D(D ..... a'lUt ... (..W N

Figure 5.6 Calibrated damping ratio template ( DRT).

4 Note the calibration on the DRT is from ! = 1 .0 to ! = 10- • To use the DRT, place the calibration mark of the desired damping line at the original origin with the center line of the template parallel with the negative real axis of the log x plane.

5.3 Design Example

The use of the DRT is best illustrated by an actual design example.

BIN BouT K1G(s)

K2 A(s) H(s)

Where: G(s)= (s+0.31± j31.4)(s+0.42 ± j42.1)(s+0.27± j56.2)(s+0.85± j85.7)(s+2.01±j201.4) s(s+0.29±j7.25)(s+0.38±j38)(s+0.47 ± j47.4)(s+0.86± j86.3)(s+2.19±j219.3) H(s) =s A(s)=Compensation Network to be Introduced

Figure 5. 7 System block diagram . 42

Figure 5.7 is a block diagram of a high-speed tracking antenna system.

The closed loop equation is given by:

K1 G(s) (5. 1 ) = ------1+ K1 K2 G(s) H(s) A(s)

Setting the denominator equal to zero and rearranging for root locus analysis (for the moment let A(s) = 1): ( 5. 2)

K1K2(s+0.31±j3.4)(s+0.42±j42.1)(s+0.27±j56.2)(s+0.85±j85.7)(s+2.01±j201.4) -1=~~------~------~------(s+ 0.29 ± j7.25) (s+0.38 ± j38) (s+0.47 ± j47.4) (s+0.86± j86.3) (s+ 2.19 ±j219.3)

Figure 5.8 is a root locus plot of Equation 5.2. Note a subscript has been assigned to each of the poles and zeros.

Figure 5.8 Root locus plot of Equation 5.2. 43

We wi II first attempt to compensate the locus between the poles s and 11 s and the zeros s and s . A simple lead network for A(s) will be used as a 12 1 2 starting point. We will temporarily ignore the effect of the lead netlM:>rk on the higher frequency resonant pairs. Due to component limitations, the maximum pole location ( P ) is 20 times the zero location ( z ). Leaving this spacing 1 1 fixed, we will move the pole zero pair z -P along the 180° axis until the 1 1 desired frequency and damping ratio is achieved for the low frequency locus.

Other spacing between the zero and the pole and other compensation techniques could be studied, but this method proves quite adequate. Note in Figure 5.9 that a zero location ( z ) between -5 and -10 radians and a pole ( P ) between 1 1 -100 and -200 radians wi II stabilize the low-frequency locus.

z1 =5 p1 = 100 -j-

Q '< "'

0 z 0..

<( 0"

Figure 5.9 Root locus plot of Equation 5.2 with lead network compensation added. 44

TABLE 5.1 TRANSLATING THE ORIGIN TO +j42.1, +j85.1 AND +j201.4.

Sj Xj Xj Xj

(FIGURE 5.8) Xi • lj- j42.1 Xj • Si- j85.7 Xi 0 lj -j201.4

II • -0.31 +j31.4 X! • -0.31-j 10.7 x, • -0.31- j54.3 x, •-0.31-j 170 12. -0.31-j31.4 X2 ° -0.31-j73.5 X2 • -0.31-jl71.4 X2 ° -0.31 -j232.8 13 • -0.42+j42.1 X3 'z -0.42 X3 • -0.42-j43.6 X3 • -0.42-jl59.3

14 • -0.42-j42.1 X4 ° -0.42-j84.2 X4 • -0.42-jl27.8 X4 • -0.42-j243.5 (/) 0 15. -0.27+j56.5 X5. -0.27+jl4.4 x5. -0.27-j29.2 x5. -0.27-jl44.9 a: loJ s6 • -0.27-j56.5 xs • -0.27-j98.6 x6 • -0.27-jl42.2 x6 • -0.27-j257.9 N S7 • -0.85+j85.7 X7 • -0.85+j43.6 X7 • -0.85 X7 • -0.85-j 115.7

sa • -o.85-i85.7 xe • -0.85-jl27.8 xe • -0.85-jl71.4 x8 • -0.85-j287.1 lg • -2.01 + j201.4 Xg • -2.19+jl77.2 Xg • -2.19+jl33.6 Xg • -2.01

s,o. -2.01- j201.4 x,o• -2.19-j261.4 x, 0 • -2.19-j305.0 x10 • -2.01- j402.8

s,,. -0.29+j7.25 X11 • -0.29-j34.85 Xi! • -0.29-j78.45 X!! • -0.29-jl94.15 lt2" -0.29-j7.25 x,2. -0.29-j49.35 x,2. -0.29-j92.95 x,2. -0.29-j208.65

Sf3 ° -0.38+j38.0 Xt3" -0.38-j 4.10 x,3. -0.38-j47.7 X13 ° -0.38-jl63.4

114. -0.38-j38.0 Xt4 • -0.38-j80.10 Xt4" -0.38-il23.7 Xt4. -0.38-i239.4 (/) loJ s15 • -0.47+j47.4 x,5. -0.47+j 5.30 Xt5. -0.47-i38.3 X15• -0.47-jl54.4 -l 0 a. Sl6" -0.47-j47.4 xl6• -0.47-j89.5 Xf6 • -0.47-jl33.1 Xts• -0.47-j248.8 't7. -0.86+j86.3 x17• -0.86+j44.2 x,7. -0.86+ j 0.6 x,7. -0.86+j 115.1

5t8. -0.86-j86.3 XIB • -0.86-j128.4 X!B • -0.86-jl72 X!B" -0.86-j297.7

't9. -2.19+j219.3 X19" -2.01+jl59.3 x,g. -2.01+ jll5.7 x19 =-2.19+j 17.9

Sro" -2.19-j219.3 Xro" -2.01-j243.5 x20 = 2.01 - j287.1 X20" -2.19-j420.7

Now we must study the effect of moving the zero pole pair z - P on 1 1 the resonant pairs. For each resonant pair we will plot the locus for the maxi mumandminimum locationsof z1- p1 (i.e., z1 =-5, p1 =-100 and z = -10, P = -200). Also, let K K = K to simplify the notation. 1 1 1 2 For the first translation let us move the origin to +j42. 1. This expands the pairs s - s and s - s . The second column of Table 5.1 lists the new 3 13 5 15 pole and zero locations for G(s), and Figure 5.10 is the root locus diagram for the pairs x - x and x - x . Note we have plotted only the locus between 3 13 5 15 45 the pairs of interest, and, as was done in the previous chapter, we have omitted the pairs x - x , x - x , x - x and x - x • We see that the locus 4 14 6 16 8 18 10 20 between x - x and x - x indicates that both pairs are stable, and the 3 13 5 15 movement of z - P has very Iittle effect. 1 1 Using the calibrated DRT, we draw several damping lines and find the damping of the roots for K = 2 and K = 5 is greater than ~= 0.05.

Figure 5.10 Log x plane root locus plot of zero pole pairs x - x and 3 13 x5- x15 with lead network z1 - pl added (st =+j42.1 ).

Next we translate the origin to st +j85.7 to examine the pair s = 7 s T The third column of Table 5.1 lists the new pole zero locations, and Figure 1 5.11 shows the log x plane root locus plot. Again we see the pair is stable, 46 and the movement of z - P has virtually no effect on the locus. (For z = -5 1 1 1 and P = -100 the damping of the root is ~= 0.01 and for z = -10 and 1 1 p = -200 €= 0.008.) We have omitted the pairs x - x , x - x , 1 I 4 14 6 16 x - x and x - x since they effectively cancel. 8 18 10 20

Figure 5.11 Log x plane root locus plot of zero pole pair x - x with lead 7 17 network z1 - p1 added (st = +j85.1 ).

To examine the last pair s - s , we move the origin to +j201 .4. The 9 19 fourth column of Table 5.1 lists the new pole zero locations, and Figure 5.12 is the root locus plot. Here we find that this pair is unstable for K = 2, for either placement of lead network z - P • Note a Iso that the locus moves farther into 1 1 the right half plane as the pole is moved farther out along the negative real axis. 47

Figure 5.12 Log x plane root locus plot of zero pole pair x - x with lead 9 19 network -P added(st=+j201.4). z1 1

The pairs X - X X - X X - X X - X 1 X - X X - X 1 1 1 11 2 121 3 131 4 14 5 151 6 16 x - x ~ x - x and x - x were omitted. 7 17 8 18 10 20

The compensation of the low-frequency locus has created a high-frequen cy instability that will require additional compensation. The approach generally used to stabilize systems of this type is a rejection filter in the region of the resonance. (In this case around 200 radians.) The twin-tee and bridged-tee networks are easy to synthesize and are usually effective for this purpose.

The major problem encountered when using these notch networks is the phase lag introduced at lower frequencies. Networks have been devised which 48 minimize this effect ( 1 ), but they are difficu It to implement. Often, too, the lightly damped pole zero pairs do not remain fixed but vary with operating con ditions. Hence, several investigations must be performed to find compensation that will stabilize the system over its total range of operation, or a notch net work must be employed that adjusts its notch frequency with the pole zero move ments. A method for designing this type adaptive notch is discussed by

Greensite (5).

With the aid of the DRT we wi II show that a simple twin-tee network with its notch in the vicinity of the unstable pair s - s wi II stabilize the system. 9 19 We assume for this example that the pole zero locations of G(s) remain fixed 1 with antenna position (i.e., the system is linear and stationary).

R R

2C B. 2

Figure 5.13 Symmetrical twin-tee or parallel-tee network.

The symmetrical twin-tee network or parallel-tee network, when adjusted for source and load resistance, introduces a pair of zeros on the imaginary axis

In the actual system this assumption is quite good since the pole zero pairs migrate over a relatively small range. 49 and a pair of poles on the negative real axis. Hollister and Thaler (7) have shown that the transfer function of a loaded and null adjusted twin-tee network shown in Figure 5.13 has a transfer function of the form:

= (4a/3+a+J3).p2+(4a/3+4a+4/3+2)p +(a+13 +2) ( 5.3) Rs Where: P =RCs a=- J3= RL 1 R 1 R

Curves of versus can be drawn to give the real axis pole locations versus source and load resistance. The curves can be used to place one of the real axis poles of the transfer function, while retaining some control over the loca tion of the other pole provided either the source resistance, the load resistance or R can be varied. After choosing specific real axis pole locations and notch frequency, the transfer function may be written in the form:

= K2(s2+C&Im2) (5.4) (s+o)(s+b)

Let us fix for our analysis the real axis poles at -150 and -300 radians and adjust the zero location (notch ) around 200 radians as needed.

Figure 5.14 is the root locus of the pair x - x with the twin-tee net 9 19 work added and z - P fixed at -5 and -10. The zeros introduced on the 1 1 imaginary axis have been placed at three different locations--T , T and T . 1 2 3 The real axis pole at -150 in the s plane is P and the pole at -300 in the s 2 plane is P . The zeros on the positive and negative jw axis are z and z 3 2 3 respectively. 50

Figure 5.14 Log x plane root locus plot of zero pole pair x - x with lead 9 19 network z1 - p1 and twin-tee network p2 - p3 and z2 - z3 added.

We see that if the zeros are placed between the zero x and the x 9 19 (T ) the locus curves farther into the right half plane. If, however, the zero 1s 1 placed above x (T and T ) then the system is stabilized. This indicates that 19 2 3 the notch frequency of our filter could vary somewhat without causing serious trouble. Our design problem is now complete provided the phase shift introduced by the twin-tee has not created an instability at lower frequencies. Reexamine tion of the resonant pair with the notch filter added shows that the notch does not have serious effect upon the stability of the other resonances. The most seri ous ly affected are the pairs s - s and s - s · 3 13 4 14 51

Figure 5. 15 Root locus plot of Equation 5.2 with lead network z - P and 1 1 twin-tee network p2- p3 and z2- z3 added.

Figure 5.15 is a replot of Figure 5.10 with the twin-tee added. In com paring the two figures note that the damping has been decreased from a damping ratio of t = 0.05 to t = 0.04 for K = 2. For this plot we have chosen the notch frequency to be ws = 245.4 radians (T in Figure 5.13). 3 Figure 5. 16 shows the complete system plotted in the s plane. Note the excellent agreement between the damping ratios calculated in Figures 5.14 and

5. 15 using the DRT and damping found in Figure 5. 16. 52

Figure 5.16 Root locus plot of Equation 5.2 with lead network and twin-tee compensation added. 53

CHAPTER VI

ORIGIN TRANSLATION IN THE z PLANE

6.1 Introduction

The extensive use of digital computers and the time sharing of communi cation channels and compensators have confronted the control engineer with sys tems that operate with variables in the form of a sequence of pulses. Often the designer chooses to analyze and synthesize sample data systems using the z trans form. Fortunately, the root locus method can be used to analyze the z domain characteristic equation in the same way it is used in the s plane.

When a system containing lightly damped poles and zeros is sampled and subsequently analyzed using the z transform, the lightly damped pairs wi II now appear on the boundary of unit circle of the z plane. The same resolution problems encountered in the s plane are now encountered in the z plane (as would be expected). In order to analyze these systems a log z plane origin translation can be used. This analysis is covered by Mclane (10) in a paper un published to date and wi II therefore only be outIined here.

6. 2 The z Plane and Log z Plane

To provide rapid interpretation of system performance from closed loop poles on the z plane root locus, lines of constant s plane damping ratio and constant undamped natural frequency (radials and circles from and about the s p one, respective! ) 54 originI" of the I y ore mopped into the mes ore th z >plene. The some en mapped into the log z pIane.

(a) s PLANE (b) Z PLANE0..~ ' ~~

0 Im(Ts)

-3 100

Figure 6. 1 shows the relationship between the 1 z plene. The left half of the s p one, z plene end log · s plane maps into th e area · 'd h In the z plene. A I" lnSI e t e unit circle lne of constant d amping ratio ~ in th heart-shaped curve in th I e s plane maps into a 1m e z p ane and into . og z plene. The real and. . a straight line in the I aglnary part of s can a Iso be plotted on the Figure 6.1 since: z graph os shown in 55

(6. 1 )

IOQe (z) =o-T + jwT =sT (6.2)

Figures 6.2a and 6.2b show the log z plane with the positive real axis

located on the center of the graph paper (as it was in Figure 6.1) and the posi tive real axis located on the outer edge of the paper as in log s plane analysis.

Also plotted on Figures 6.2a and 6.2b are normalized's plane undamped natural frequencies and damping ratio lines. These are plotted on the log z plane graph paper to provide rapid interpretation of system performance from the closed loop poles on the log z plane root locus.

-6 -5 -4 -2 -I o Re(Ts) +I +2 [iiD v~ j..-o' ~ + J ·9 ~ 0 1.7rn ,!!"I""'~ f"" --. 1.1 T ..,.. ""' + i 1-111 ~ I ~ 1111.. [illj ~ ~ i ~~,. r---...... ]) ~ i'ar ...... +I ..... ~ 1 I ... ~--~--~- I I - llillj~~ LZ I ~' r>c ~ I r-- I I I .... ~ I , ~ o.r~\ ' I' ' 0 01 I I ! ~ ' 1,..11 ~,\ o'Z- ~ J 1 ,_ ~· . ._...... ~ ?Jw ~ :~~'w Q. "" 'L -I ~ ~ ll. ~ r\ 1,..."" 0.4T I I l- ~ ~ f--- g& l.. / ~ ~~ , v I ~ ~~ I ~ (\J i I' ~ ~i) o·

j i I f- Ei!Q} i I ~~ J ~ il ) 4 6 7 9 ~~ 'B 9 2 J 2 3 4 5 6 18 9 I J ' ' ' 3 ' 5 6 7 6 9 0.00 0.01 lzl 0.1 I 10

Figure 6 .2a The log z plane with s plane damping pnd undamped natura I frequency lines (positive real axis in center). 56

~~ ~, n 1.7'f .,.,.. ~ ~ ~~ ~~~ ..J II"""' """' II"" ~ ,..~ /j, C\J ~ ~~ ~~ / 0 II"""' ~ II"" "v" ~~ ~ p ~ ,. ~ ~ll ~ ~~ 1'1!! Jj_ V' """' 1111'~ T, ~ '\ LZ ~ ~ lL ~ n ..,~~ ~:;,fr:r j_ ~ 1"'-- n~ m2J ...... T !'- V"' I-'I""' I ~ ~ n J.j_ 1"-11~ I' n... I" 0.2TJL lilT I\ ~ ~~ ' \ ~ "' I"' I/ t-0.4~' r-----~ !'- ~ 00 ~ "" ~ ~ ~ ~~ !"'Il~ r----- ...... IL ,_.~ ~~ r- ~~-- ... "'I ~--~ "' "' ~ ~ ,~ ~ ~ ~...... ~~~~~~ l ll 2 3 4 ") 0 7 9 4 o:J. 3 5 6 7 8 9 2 3 ' 5 6 7 8 9 2 3 4 5 6 7 8 9 0.001 0.01 lzl 0.1 I 10

Figure 6.2b The log z plane with s plane damping and undamped natural frequency lines (negative real axis in center).

6.3 Log z Plane Origin Translation

As mentioned earlier, quite frequently in designing sample data systems a cluster of poles and zeros are encountered near the point z = 1. We therefore translate the origin to z =+1 and form a new z plane. Stated in equation 1 f~rm we have:

z = z, +I (6.3)

For the untranslated root locus, plotting the coordinate configuration shown in Figure 6.2a is preferable because of its similarities to the s plane. 57

However, if we use Equation 6.3 to translate the origin we find that an expan sion of the unstable area is achieved as shown in Figure 6.3. If the coordinate configuration shown in Figure 6.2b is used we achieve the desired expansion shown in Figure 6.4. Also plotted on Figure 6.4 are several of the s plane damping ratio lines and undamped natural frequencies. The complete template can be found in Mclane's paper ( 10) which includes damping ratios to

0.000411/T.

Figure 6.3 Figure 6.2a when translated to the z plane (z == z + 1). 1 1

6.4 Concluding Comments

The use of the z plane has proven very usef~:~l. However, the r or w 1 58 transform can also be used which would allow the s plane DRT to be applied.

The best approach and its usefulness in the design require additional investigation.

Figure 6.4 Figure 6.2b when translated to the z plane (z = z + 1 ). 1 1 59

BIBLIOGRAPHY

1. Ainsman, Erwin. Higher order notch networks Nike-Zeus. Santa M;,nica, Douglas Aircraft Company, 1964. p. 1-21. (Report no. SM-45074)

2. Axelby, G. S. Practical methods of determining feedback control loop performance. In: Proceedings of the First International Congress of the International Federation of Automatic Control, M;,scow, 1960. Vol. 1. London, 1961. p. 68-75.

3. Evans, Walter R. Contra I system synthesis by root locus method. Trans actions of the American Institute of Electrical Engineers 69:66-69. 1950.

4. . Graphical analysis of control system. Transactions of the American Institute of Electrical Engineers 67:547-551. 1948.

5. Greensite, Arthur L. A frequency sensor and roving notch filter for con trol system application. Proceedings of the National Electronics Confer ence 21:558-563. 1965.

6. Harris, L. Dale. Introduction to feedback contra I systems. New York, Wiley, 1961. 363 p.

7. Hollister, Floyd H. and George J. Thaler. Loaded and null adjusted symmetrical paralie 1-tee network. Proceedings of the National Electronics Conference 21:753-758. 1965.

8. Klugman, Dale R. Root locus technique in the log s plane for design and analysis of feedback control systems. Master's thesis. Monterey, California, United States Naval Postgraduate School, 1964. 46 numb. leaves.

9. Kusters, N. L. and W. J. M. Moore. GeneraIi zation of the frequency response method for the study of feedback contra I systems. In: Automatic and manual control: Papers contributed to a conference at Cranfield, 1951. New York, Academic Press, 1952. p. 105-122.

10. Mclane, Robert C. Sample-data system design templates for origin shift on potential-plane analog of the log z plane. Unpublished paper prepared by Honeywell Military Products Group Research Laboratory, St. Paul, Minnesota, 1960. 60

11. McRuer, Duane T. Unified analysis of linear feedback systems. Dayton, Wright-Patterson Air Force Base, July 1961. p. 21-52. (ASD Technical Report no. 61-118)

12. 1-Aorgan, M. L. A computer for algebraic functions of a complex variable. Ph. D. thesis. Pasadena, California Institute of Technology, 1954. 114 numb. leaves.

13. • A new computer for algebraic functions of a com plex variable. In: Proceedings of the First International Congress of the International Federation of Automatic Control, Moscow, 1960. Vol. 3. london, 196 1 . p . 121 -128 •

14. . Algebraic function calculations using analog pairs. Proceedings of the Institute of Radio Engineers 49:276-282. 1961.

15. 1-Aorgan, M. L. and J. C. looney. Design o(the ESIAC algebraic com puter. Institute of Radio Engineers, Transactions on Electronic Computers EC-10(3):524-529. 1961.

16. Nomoto, A. log-root-locus method for systematic study of linear feed back centro I systems. In: Proceedings of the Fifth Japan National Con ference for Applied Mechanics, Tokyo, 1955. Sec. 4. Science Council of Japan, Tokyo, 1956. p. 523-526.

17. Smith, 0. J. M. and E. J. Swenson. Enhancing the resolution of a potential plane analog by changing the variable to translate the origin. Institute of Electrical and Electronics Engineers, Transactions on Auto matic Control AC-9(3):303-307. 1964. APPENDICES 61

APPENDIX A

CHANGING FACTOR FORMS

A. 1 Introduction

When using an ES lAC®computer or when plotting graphically in the log s plane, it is advantageous to express the product of factors in the form ( 1 - s/s.) I rather than in the (s- s.) factor form. This form is more accurate for small I values of s and is therefore the most desirable form to use in the translated plane.

We have seen in Chapter Ill that in the ( s - s.) factor form, the constant I

K remains unchanged in the translation to the (x -xi) form. This is no longer 1 true when the translation is made to the ( 1 - x/x.) factor form. Another con- I stant multiplier is introduced which in general is a complex constant.

A.2 Changing Factor Forms in F(s)

Before we study changing factor forms in the x plane let us study the process in the s plane. For clarity we will rewrite the basic equation:

F(s} =K1 Q(s-siri (A. 1) itO i~ t If we multiply and divide each factor in Equation A. 1 by its corresponding singularity we have:

F(s} =K1sn° I;J (-siri ~-:)nj (A.2) j;tQ ------

Now define a new constant K as follows: 2

K2 =K1 Q(-siri (A.3) itO

Substituting Equation A.3 in Equation A.2 yields:

F(s) =K2s n0 ni ~l-Si s)nj (A.4) itO which is the desired factor form.

A.3 Changing Factor Forms in F(x)

Again for clarity consider the original function:

F= K,xn' J;I (x-xri ( A.5) i=O itt

If we multiply and divide each factor in Equation A.5 by its corresponding singularity we have:

(A.6)

We now define a new constant Kt as follows: 1

Kt, = r;I (-xiri (A.7) i=O i;tt

Substituting Equation A.7 in Equation A.6 gives:

n ~~-.!.)ni (A.S) I X· i=O I itt 63

This is the equation to use if the original equation was in the form of Equation

A.l.

If the original equation was in the form of Equation A.4, with K given, 1 we may substitute Equation A.3 in Equation A.6 to obtain:

(A. 9)

Let us define a new constant Kt as follows: 2

(A.10) !;I (-si)"i ili!O i=t

Substituting Equation A. 10 1n Equation A. 9 yields:

(A. 11)

Since the xi's do not occur in complex, conjugate pairs Kt and Kt in 1 2 general are complex. Both Kt and Kt may easily be evaluated numerically if 1 2 the poles and zeros are listed in tabular form. If the values of s. and x. are I I converted to polar coordinates, the calculation of the constants is simply a mul tip lying and dividing of magnitudes and adding and subtracting of angles. If an

ES lAC computer is used, these constants may be measured rather than ca leu Ia ted. 64

A.4 Changing Factor Forms, But Leaving K Unchanged

To leave the original K unchanged in a root locus plot, the new constant may be accounted for if it is divided out to give a new constant value for F on the left-hand side of the equation.

If the original equation is in the form of Equation A.l, with K given: 1

IJ ~-.!.)"i (A. 12) 1 x· i=O I iil!t

If the equation is in the form of Equation A.3, with K given, then: 2

(A. 13)

Since in general Kt and Kt are complex, the new Ft will usually be com 1 2 plex. This means then, the x plane angle locus plot will be plotted for some angle other than the fam iIiar 0° or 180° locus and the gain plot for some magnitude other than one. 65

APPENDIX B

GENERALIZED FREQUENCY RESPONSE PLOTS

Consider a c lased loop expression of the form

A(s) A(s) G(s) = I+A(s)B(s) = I +H(s) K(1+ -:;-)(1+-:;) ••• (1+-t) (B. 1 ) Where H(s) = (I +b~ ) • • • (I+ bsm )

Setting the denominator equal to zero and rearranging gives:

H(s)=-1 (B.2)

To avoid factoring this expression, the Bode method (or the Nyquist plot) ex amines for roots on or in the proximity of the imaginary axis. A closed loop root is located on the imaginary axis when the IH(jw)l =1.0 and LH(jw) = 180°.

A gain versus log frequency and phase angle versus log frequency plot may be used to determine the closed loop roots by evaluating H(s) in Equation

B.2 along axes other than the familiar jw axis of the standard Bode diagram.

Closed loop roots are located on any axis when IH(s)l = 1.0 and the

LH~s) = 180° at the same complex frequency.

This method may be used to find the closed loop roots on the real axis.

Kusters and Moore ( 9) first proposed this method as an extension of the Bode method, and McRuer ( 12) has extended the method to evaluate roots along any 66 line of constant damping ratio.

To solve for roots on the real axis, set s =a and plot the IH(a)l versus log a and the LH(a) versus log a. A simple graphical technique for plotting real axis plots is discussed by Axelby ( 2 ), or potential analog methods may be used.

As an example, consider the characteristic equation:

K2 (1+ 2~8 ) H(s) = ------~ ( B.3) s (1+ o.:2s) (1 + ;o)2

Figure B.1 is the root locus plot of Equation B.3. Figure B.2a is the real

360" I. O'

0. r- 910 270" ~ -90' ~~ I.0 a·o 6.0 ~~ ·o.5 0.~~!11' ,~ l• ~ ~0 ~~~'\ 0 ;::: 0.~ <( ..... 0.17{ lX43 5\ "' I• + _._ .... 2 ~0 ~ 180 +I.0 o" " I C) 50 50 z o.k r 51~ a: :1: <( f"'~~ 0 I \-., 0.6 ~1 +0.5 1.0 2.o 6.0 90 0 '~~ -2 70° 910 ~ -.. 0.5 I

0 -I. 3 4 5 6 7 8 9 2 ] 5 6 7 8 9 2 3 5 6 189 2 ] 4 5 6 7 -J•o' ' ' ' 9 0.1 I lsi 10 100 1000

Figure B. 1 Root locus plot of Equation B. 1. 67

2 q 8 8 7 I ' '• 4 I Iii :! II li ~ 2 iii !i li II I ui ~9 60 8 7 I ii ,!IIIII.. !

4' .4 1: ...~ , i 1 I ~ ~ f-THE ANGLE locus I BREAKAWAY LOCATION i: I

[]QQ} ! ! 40 8 ! I I 7 ~ I ' ' 1\

] 1\ 30

2 \

10 \ 20 9 8 7 ~ ' "'8 ' 0 4 IFI ~ 3 ~ THE ANGLE LOCUS 2 ~ BREAKAWJrTION IJ~ ,, ~..t CD-, ~ 8 ROOTS ROOTS 7 ~ '\

'4 THE ANGLE LOCUS BREAKAWAY LOCATION 3 ~

2 I I I 0.1 -ill 8 7 I

5 I II 4 l

] != 1\ :§

i•:I 2 \ ! ~ 0.01 I I 4 2 ] .'i 6 7 g 9 5 6 7 8 9 ) ) s n, 2 \1 4 5 tl '8 Q ' 2 '""' ' ' ' 0.1 I w =lsi• 10 100 1000

Figure B.2a Real axis gain plot or 11 Siggy 11 plot of Equation B. 1. 68

axis plot of the IH(cr)l with K =50, and Figure B.2b is a plot of the LH(cr). 2 Note the phase plot is simply a restatement of the real axis angle locus from the

1 root locus plot. In other words, the phase angle is either 0° or 180° •

rn I I lj li I I 82:1 I I I lj

L F li I I EiiQ1

Ellil

EM} 2 J ' ' 6 7 9 2 J 4 56 7 8 9 2 J ' ) 6 /8 9 2 J ' 56 7 8 9 0.01 0.1 CT =Is I 10 100 1000

Figure B.2b Real axis phase plot of Equation B.2.

Comparing Figure B.1 with Figure B.2a, we see that whenever the real axis plot becomes tangent to the log a axis, this corresponds to the breakaway of the angle locus. This makes sense since at that point the two real axis roots

Note that the loci on the negative real axis of the log s plane can be sketched as they are in the s plane. A locus exists to the right (as opposed to the left in the s plane) of an odd number of poles or zer~ Count any poles or zeros at the origin or on the positive real axis as being to the left. 69 become coincident. Note also the four crossings of IFl = 1 line. These are the four closed loop real axis root locations, and they correspond exactly with the roots found on the root locus diagram in Figure B. 1 •

From Figure B.1 note that for a gain of K = 1.0, there are a pair of 2 complex roots at W = 0.85, ~ = 0.5. Hence, a frequency response plot of the n

~ = 0.5 damping line with K = 1.0 should indicate this root. Figures B.3a and

B.3b show the gain and phase plots and the expected root locations.

OQQJ-, ~ 8 7

J ~

[JQJ; \.. ~ 8 7 ' 5 1" 4 ~ IFI J ~ ~~ 2 ' ~ u"' '~ 0 o:::J, ~ 8 7 I'

4 ' j J ' ~ 2j---- J ' I 0.1 2 f\2 4 5 6 7 8 9 4 5 6 7 89 J 4 5 6 7 2 J 4 56 7 89 ' 3 2 J 8 9 0.1 I 10 100 1000 w = ! = 0.5 LINE

Figure B.3a Gain plot of Equation B.1 for the ~ =0.5 contour (K =50).

To summarize, a plot of any axis or radial lin.e gives a picture of the 70 roots in the vicinity of that line, while the root locus plot shows all of the root locations for any gain. If the root locus plot is plotted in the log s plane, the correspondence between the root locus plot and the frequency response methods is obvious.

-90' ...... ~ L F ...... r- ~ I' ~~ ...,~ Bill:- - "~ ' I ~

L Ellil 1\. ~ " I ! I ' § 2 J 5 0 7 9 2 3 s 6 7 8 9 ) J 5 0 2 3 5 6 7 8 9 ' ' ' '' 9 ' 0.1 I 10 100 ' 1000 w = ! = 0.5 LINE '

Figure B.3b Phase plot of Equation B.l for the ! = 0.5 contour.