THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING A THEORETICAL STUDY OF CONDITIONALLY STABLE SYSTEMS Loui.s F.. Kazda A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering, Syracuse University 1962 January, 1962 IP-549

PREFACE The author is indebted to his wife for her patience during the preparation of this dissertation. In addition, particular thanks goes to Professor Norman Balabanian for the interest he has shown in the work performed, for his helpful suggestions relative to the theroetical development of the dissertation, and for his helpful criticism while reviewing the manuscript. I

TABLE OF CONTENTS Preface Chapter I INTRODUCTION 1.1 Historical Background 1.2 Research of Herr and Gerst 1.3 Research of Travers 1.4 The Missing Link 1.5 Scope of the Research Chapter II CONTROL SYSTEM CONCEPTS 2.1 Transfer Function of a Linear System 2.2 Stability of Open-Loop Systems 2.3 Stability of Closed-Loop Systems 2.4 Linear System Block Diagrams Transformations 2.5 The Prototype Configuration 2.6 Root-Locus Methods 2.7 Gain-Loci and Phase-Loci 2.8 Method for Determining the Roots of an n-th Degree Polynomial in s. Chapter III SECOND ORDER CONDITIONALLY STABLE CONTROL SYSTEMS 3.1 Background 3.2 A Comparison of the Two Commonly Encountered Definitions of Conditionally Stable Systems 3.3 Conditionally Stable Second-Order Systems 3.4 Mathematical Derivations for the RootLocus of Second-Order Feedback Control Systems 3.5 Computer Study of Second-Order Systems page 1 1 4 9 13 14 15 15 16 19 21 23 27 31 35 43 43 46 47 53 56 II

TABLE OF CONTENTS (continued) Page Chapter IV HIGHER-ORDER CONDITIONALLY STABLE SYSTEMS 67 4.1 Background 67 4.2 Minimum or Non-minimum Phase ClosedLoop Systems 7 4.3 Extension of the Root-Locus Techniques 72 4.4 Crossing or Intersection of Root-Locus Branches 76 4.5 Conditionally Stable n-th Order Systems 82 4.6 A Comparison of Defining Characteristics Between Conditionally Stable and Absolutely Stable Systems 92 4.7 Relation Between Saddle Points and Root Locus Plots 101 4.8 Practical Examples of Conditionally Stable Systems 1L12 Chapter V CONCLUSIONS 130 Bibliography 133 Vita 136 III

LAISJ-T.-b OF FIGURES 1.1 Bode Attenuation Diagrams of A Minimum 5 Bandwidth System~ 1.2 Nyquist Diagram of a Minimum Bandwidth 10 Systemo 1.3 Root Locus Sketch of a Minimum Bandwidth 11 System.o 2.1 Block Diagram Representation of Linear 19 Control System. 2 2 Linear Block Diagram Transformations 24 2.3 The Prototype Configurationo 25 2,4 Reduction of a Complex System Utlizing 26 Linear Block Transformationso 2.5 Gailn and Phase Loci of a Second-Order 34 Control System. 2c6 Simulation of an n-th Order Polynomial 38 as a Control System. 2.7 Solution of an n-th Order Polynomial by 41 Root Locus Techniques.:3 1 The Prototype CProtonfigurationo 44 3.2 Root-Locus Plots Representing Two Types 48 of Conditionally Stable Systems. 3.3 Root Locus Plots o a C on itionai l1y S t able 50 System with Two Complex Right-Half Plane Poles.

LIST OF FIGURES lcontinued.~~ ~ ~aFigIure h _ f r - jPage 3.4 Root-Locus Plot of a Second-Order System 51 with a Negative Gain Constanto 3.5 The Limiting Case of a Conditionally 52 Stable System. 3.6 Example of Root-Locus Construction Utiliz- 57 ing Only a Compass and a Ruler. 3.7 Table of Equations for Constructing the 58 Root-Locus of Second-Order Systems when the Roots are Complex. 3.8 Computer Diagram Used for the Study of 60 Conditionally Stable Systems with LeftHalf Plane Closed Loop Zeros. 3.9 Computer Diagram Used for Studying Con- 61 ditionally Stable Systems Containing Right Half Plane Closed Loop Zeroso 3.10 Root-Locus Plot of the System Studied on 63 an Analog Computer. 3.11 Time Response Curves for the System of Figure 3o10 for gain, k = 1. 65 3.12 Time Response Curves for the System of Figure 3.10 for gain, k = 3, 65 3.13 Time Response Curves for the Systems of Figure 3o10 for gain, k = 10. 66 3.14 Time Response Curves for the System of Figure 3.10 for gain, k = 50. 66 V

LIST OF FIGURES (continued) Figure page 4.1 s-Plane Plots Show How the Root-Locus 69 Changes as a Function of the Location of z1. 4.2 Third Order System with Complex Zeros. 71 4.3 Showing the Multiple Mapping Property in 73 the s and KG(s) Planes. 4.4 An Example Showing the Multiple Mapping 74 Property of a Higher-Order System. 4.5 An Example Showing the Multiple Mapping 75 Property Utilizing Phase-Angle Loci. 4.6 An Example of Multiple Mapping Utilizing 77 Phase Angle Loci for a Complex System. 4.7 An Example of a System Possessing Three 79 Breakaway Points. 4.8 System of Figure 4.7 with Complex Poles 80 Moved Further Into the Left-Half Plane. 4.9 System of Figure 4.7 with Complex Poles 81 Moved Toward the Right-Half Plane. 4.10 Typical Type A Conditionally Stable Systems 86 4.11 Typical Type B Conditionally Stable Systems 89 4.12 Typical Type C Conditionally Stable Systems 93 4.13 A Graph of B vs. A for Two Values of ~ 100 4.14 Transient Response Characteristics of Two 102 Typical Systems 4.15 "Relief" Map of the Function KG(s) = 104 s VI

LIST OF FIGURES (continued) Figure Page K 4.16 "Relief" Map of the Function KG(s) s(s+ ) 106 4.17 "Relief" Map of the Control System Defined 108 by equation 4.7-1. 4.18 Root Locus Plot of the System Defined by 110 equation 4.7-1. 4.19 "Relief" Map of a Control System with Four 111 Open-Loop Poles. 4.20 Root-Locus Plot of Uncompensated System. 114 4.21 Root-Locus Plot of Compensated System. 114 4.22 Block Diagram of a Two-Loop Control System. 116 4.23 Nyquist Diagram for System of Figure 4.22 118 with "p" open. 4.24 Root-Locus Plot of System of Figure 4.22 119 with "p" open. 4.25 Nyquist Diagram of D(jw)/ E (jwO). 121 4.26a,b Bode Diagram of D(jw)/ Ei(jc). 122 4.26c Nichols Diagram of D(jco)/ E1 (jW) 123 4.27 Root-Locus Plot of the Inner Loop Function 124 Dl(s)/ El(s)o C 4.28 Nyquist Diagram of (jku) for System of 126 E equation 4.8-9. 4.29a Root Locus Plot for System of equation 128 4.8-9 with K 3.55. 4.29b Root Locus Plot for System of equation 129 4.8-9 with K4 35.5. VII

CHAPTER I INTRODUCTION 1 o1 Historical Background In the years prior to World War II the design of feedback control systems was an art that was based on a few isolated facts tempered, of course, by the designer~s experienceo This resulted in much "cut and try" engineering, with varying degrees of success, During World War II certain groups, notably those at the Radiation Laboratory and Bell Laboratories were called upon to develop certain servomechanisms to meet some of the military requirements. As a result of this concentrated effort, the design of feedback control systems advanced from an art to that of a science, based principally on the works of Bode P2 3 54 6 7 Ny Draper, Hall, Harris Hazen5, MacColl and Nyquist 1o "Network Analysis and Feedback Amplifier Design" (book), Do Van Nostrand, New York, NoY,, 1945, pages 168-169,. 451-476 2o "Design Factors Controlling the Dynamic Performance of Instruments", Co S Draper and AoPo Bentley, Transactions ASME, Vol. 62, July 1940, pp. 421-432. 3. "The Analysis and Synthesis of Linear Servomechanisms", AoCo Hall', Technology Press, Massachusetts Institute of Technology, May, 1943 4o "Frequency Response of Automatic Control Systems'", Herbert Harris, Electrical Engineering, AugustSeptember 1946, pp. 539-546o 5 "Theory of Servomechanisms", H, Lo Hazen, Journal of Franklin Institute, Volo 218, 1934, ppo 279-331o 60 "Design and Test of a High-Performance Servomechanism", Ho.L Hazen, Journal of Franklin Institute, Volo 218 November 1934, pp, 543-580.

2 7 With the exception of the work of MacColl the theory developed was a linear theory, that is one based on the solution of linear differential equationso In the past decade the linear theories which were developed in the early 440s have been refined and some new methods devised~ A considerable amount of effort has also been expended in the areas of nonlinear control system analysis and in the study of sampled-data systems A review of current literature reveals however0 that an area still exists which has been by-passed by other investigators, namely, a cmprehensive stuy of a class of linear systems commonly referred to as conditionally stable s st.ems Although individual cases of this class of systems have been treated since Ho Nyquist f8irst recognized that such systems do exist0 this general class of systems has never received any organiszed attention0 The word conditionally stable was probably first used 1d 8 )y HoWo Bode' when he described yqaist"s work in connection with feedback amplifier designo Although not formally defined by either Nyqust8 or Bode personally0 they recognized that some feedback amplifiers exist which are open-loop unstable, but which can be stablized by closing the final feedback loopo 7 "Fundamental Theory of Servomechanisms" (book) LoAo MacColl D"o Van Nostrandt New York0 NoYo 1945o 8o "Regeneration Theory" Ho Nyquist, Bell System Technical Journal0 Volo 11, January 1932, pp 126-147

3 A Conditionally Stable ystem as defined by Brown and 9 Campbell is "a sytem which is unstable for a articular gain, but stable for both larger and smaller gain values". A review of literature reveals Herr and Gerst and Travers1 have studied systems which they also defined as conditionally stableo A closer inspection of the systems studied by them would reveal that their definition of conditional stability does'not agree with the one pro9 posed by Brown and Campbello. The systems considered by these latter authors were open-loop unstable, Inspection of the root1locus plots for these systems would reveal that they satisfy the following definition: aa system that is stable for a particular value of gain, but is unstable for both higher and lower gain values'0 A comparison of these two definitions shows- that they are essentially opposite in their definition of the useful regions of operationo In an attempt to clarify this situation, in the work that follows, both types of systems will be discussedo The author's interest in conditionally stable systems stems from the paper by Travers who stated without proof that using conditionally stable control systems the following advantages could be achieved over the conventional absolutely stable system: 9 "Principles of Servomechanisms" (book",. so B row: D P. CapbelT, 1948,^ p. 172, Joh Wil & o ns Inco 10o "The Analysis and an Optimum Synthesis of Linear Servomechanisms8", Do Herr, 1o Gerst; AIEE Transactions, Volo 69, 1947, ppo 959-70o 11o "A Note on the Design of Conditionally Stable Feedback Systems", Paul Travers, AIEE Transactions, Volo 70,, 1951,

4 1. These systems sometime permit larger gain compared to an absolutely stable system of -the same bandwidth. 2. These systems sometime permit a smaller bandwidth for the same loop gain. The reasons for minimizing the bandwidth are: 1. TO siimiZ the transmission of noise by the system. 2. To require the lowest frequency response characteristic for the physical elements of the system, 1.2 Research of H!err and Gerst1.ierr and Gerst published an article summarizing their study of a class of servomechanisms having an open-loop frequency response function of the following form: (1 + j KG(,ja) 1= - 1.2-1 r I. ) t (JoW) (1 + j -) C2 w'he'e r 2 and t 1, while r + t - q 2 or 3. The par amsieers r, and t are variables which control the high, medium, axnd low-frequency attenuations rates, anrd w. and CD are the corner frequencies associated with thIe logarithmic representation of the magnitude of the open-loop'function (see Figure 1.1). The problem which they treated was: Given A,, and the h ucy at tmaxuation rate, detemie hat the high-frequency attentuation rate, determine that

5 -ATTENUATION RATE -20 r, db/oct. I \ 20 1ogA I cuo A - THE USEFUL GAIN.A.'0.3, L~ -ATTENUATION RATE -20(r -q),db/oct. IogC ATTENUATION RATE -20(r-q+t), db/oct. logc OPEN-LOOP FREQUENCY RESPONSE I I I I I I EFFECTIVE BANDWID1 I I I I I I Mm I I I I logO CLOSED-LOOP FREQUENCY RESPONSE Figure 1.1 Bode Attenuation Diagrams of a Minimum Bandwidth System

6 attenuation-rate ro, which minimizes m, the bandwidth ratio. a The authors were trying to minimize the effective bandwidth of the closed-loop functions, since this type of system would: a) minimize the amount of energy required to operate the system, b) minimize extraneous noise present in the system, and c) utilize all the bandwidth possible in a given situation~ The approach they used to solve the above problem C was: a) to create a normalized -(s) function having the E following form: A(jw) -21 X ) Ajr ) Xr (j) L1 + b x(jw) where X J a = \K, and b K 1.2-3 CD1 U)2 b) To form a new function C(X,a,b) - 2 1 g A L 1.2-4 where C is the - itercept of the M contours in the K(s) plane x M in this case is the magnitude of the system response functiono The authors chose to deal with

7 C rather than M because the analysis could be more readily carried outo c) To determine where M>1 is a maximum and C(X,ab) is a minimum in the equation g(X,a) C(X,a,b) = f(X,a,b) 1o2-5 This was done by partially differentiating this equation with respect to X, yielding jg(Xa) C(X.a,b) = X- f(X,a,b) 1o2-6 whose solution, when substituted back into equation lo2-5 yields g(X,a) C(Xm,ab) f(X a,b) l2-7 d) Now since C(X,ab) a- 1, assign M = M m Mm,isigx (design parameter) and form a second equation C-g(Xa) f(X,a,b) 1.2-8 e) Partially differentiating equation 102-8 with respect to X yiel.ds

8 C g g(X,a) = f(X.a,b) 1.2-9 X - and upon substitution this value of X back into equation 1.2-8 above yields C g(Xm,a) f(Xt,a,b) 10 210 f) Now equations.1.,2-9and 102-10 define b as a function of a o Following their method b is now maximized by differentrating equation l2-10 partially with respect to a o c dj (X, a) = f(Xm ab) 1L21l g) Equation 1o2-9, o2-10, andlo2-11 form a simultaneous system of equations which can be solved for X, aabo m Solutions are restricted to positive values of X,a,b, m for which b<a, for which b is actually maximum, and of course which corresponds to a stable system, One of the systems treated by the authors was A K + t ".1 X 1 l + aX)! A (j-,) X l(x+ bx') (1 + j - ) U2

9 In this case g(Xa) =(1 + a ) 4X8 226 4 2 f(X, a,b) b x + 2b2X6+ 4 (1 + 2b2 4ab) 2X2 The simultaneous set of equations which must be solved is 48 26 42 2~ _2 bX 2X X (1 + X 2b2 4ab) - 2X - C(l + a X m m m m m 4b4X6 + 6b2X + 2X2 (1 + 2b2 4ab) 2 - a2C m m m and ac = -4bX2 m They Nyquist diagram for this class of systems is shown in Figure 1o2, while the corresponding root-locus will be found sketched in Figure 13 1o3 Research of Travers Utilizing the same form of an open-loop function as proposed by Herr and Gerst4 Travers chose KrohnIs criterion as his starting point in developing a set of criteria for minimum bandwidth systems. Noting that Krohn's criterion * See reference 12 on following page

10 A - PLANE x=O k (I + ) ='(jW)= (I + j-T f O o~~~l Figure 1.2 Nyquist Diagram of a Minimum Bandwidth System

11 / / s - PLANE / j / / / / T3 ORDER POLE DOUBLE ORDER POLE —x- OPEN LOOP POLE e- OPEN LOOP ZERO Figure 1.3 Root Locus Sketch of a Minimum Bandwidth System

12 will be satisfied using this open-loop function if q = r 1, and in addition, making use of the following equations max KG(j jm) 1, in 1o3-1 Nm 1 max Arg KG(jc%) = Sin a 1-3-2 J; jArg KG(J m) + T( = ( - )2 =0 1.3-3 01 + 0 (02 + 0 Travers established the following relationships: CD2 _ 4(r-l)t 234.. 1 (Cos ) M max 1'cos - M max CD 2t, 1.3-6 ^-'- -- = -- 1.36 Ca. -1 1 ~- ra 120 "Theory of Servomechanism", (book), James, Nichols, Phillips, McGraw-Hill Book Co,, New York, NoYo, 1947, po 182, Krohn's criterion states that in order to utilize the maximum phase margin of a system, the openloop transfer function should be tangent to the M-contours at the point of maximum phase margin, providing the curvature of the open-loop function does not exceed the curvature of the M-contourso This curvature requirement is satisfied by making q =r-1 in equation 1 2-1.

13 Inspection of equations L,35 and Lo36 will reveal that mr is not dependent on t, the order of the pole away 51 U2 from the origin, and (- is not dependent on r, the order of the pole at the origin, as it was in the case of the work of Herr and Gersto The use of Krohn's criterion with this class of loop functions permitted the separation of the high-and lowfrequency attenuation rates about a c The effective bandwidth has been made independent of "'to "t"1 is used only to determine 2 m lo4 The Missing Link A review of literature has therefore shown that a definite gap does exist in the analysis and synthesis of linear systems which are conditionally stableo While in practice such systems do exist, their design and stabilization have been treated as a more complex problem requiring an extension of the design techniques normally usedo A class of conditionally stable systems have been discussed by Herr and Gerst, and Travers o Also brief mention about such systems will be found in the books by Bode,* James, Nichols, Phillips,** and to a lesser degree in the 13 book by Nixonl3 In each case the authors have discussed specific exampleso However, such questions as: * Reference 1, ppo 162-163 ** Reference 12, ppo 177-192 13o "Principles of Automatic Controls" (book), FoEo Nixon Prentice-Hall, N.Yo, 1953, pp. 114-121o

14 1o What are necessary and sufficient conditions for an unstable open-loop system to be closedloop stable? 2c Are all conditionally stable systems open-loop unstable? 3o Are there any advantages in making an openloop system unstable? 4o Given a prescribed unstable open-loop system, how can it be modified to give satisfactory closed-loop response? were left unanswered. An attempt to anwer these questions has led to the subject matter presented in this dissertation. lo5 Scope of the Research The objectives of this dissertation are therefore as follows: 1o To study the advantages and disadvantages of systems that are open-loop unstableo 2. To establish the design conditions which permit a feedback control system that is open-loop unstable to become stable on closing the final loop

15 CHAPTER II CONTROL SYSTEM CONCEPTS 2 1 Transfer Function of a Linear System A complex control system will in general consist of a number of electrical, mechanical, hydraulic and pneumatic elements interconnected in a given physical complex. The physical complex must in turn be broken down into a number of small sections in such a way that each section may be considered as acting independently. Under these conditions, a number of differential equations can be written relating the various sections. In addition, if these equations are linear or can be linearized and still describe the operation of the system in a satisfactory manner, they can be transformed from the real time domain into the complex s-domain using the Laplace transformation. If the initial conditions are now equal to zero, one or more transfer functions G (s), i = 1, 2, 3, o will be obtained relating the various components of this system. The procedure for obtaining the transfer function of a physical element is found discussed in such books as 14 15 Gardner and Barnes14 or White and Woodson 5 14, "Transients in Linear Systems", (book) Mo F. Gardner and Jo L Barnes, Vol. 1, 1942, Chapter Vo 150 "Electromechanical Energy Conversion, " (book) Do Co White and Ho H. Woodson, 1959, Chapter II.

16 In the work that follows, it is assumed that the problem of reducing a physical complex to its equivalent linear transfer function form has been solvedo 202 Stability of Open-Loop Systems Attention will now be focused on a typical transfer function, Gi (s) obtained as a result of applying the procedure discussed in the last sectiono It is a rational function defined as follows: X (S /a0 + als +- - - s X2(s0(s) s+s X G.(s) Zb 1 m n2-1 X1 () _ _P(s) b0 + b s + - b s which can be represented in block diagram form as shown in Figure 2.la o In representing a system in block diagram form, it is implied that he e initial conditaions operator* is assumed to be zero. In addition, if the transfer function block G1(s) is to be connected in X cascade with a second transfer function G2(s) = 3 (s), X2 (see Figure 2,lb), the following assumptions are implied, namely 1) The function X2(s) is not altered as a result of connecting G2(s) to it. In other words it assumes that X (s) operates into an open circuit or into a system G2(s) having a much higher impedance levelo * See reference 14, page 132

17 2) The impedance level of G (s) is such that it causes no change in Xl(s), the excitation function In order for G1 (s) to be physically realizable as a voltage ratio, n ~ m, while to be physically realizE able as a transfer impedance, i.eo 1 (- n = m-1 I Now equation 202-1 may be written as (s+zlU(s+z2>V (s+z)Ws X2(s)-=X(s) F(s) = X(s) _. — v. w(s+p2) (s+p) (+ where in general z, z20 o o P, may be real or complex, and u, v, w, w', v', u' may be of any order as long as they are finiteo If z, or pj is complex, however, its complex conjugate z.* or pj* must also be present0 The following question may be raised regarding the characteristics of G1(s) o "What must be the restrictions on G (s) to insure a stable response from G1(s) which will be independent of the input function x (t)?" The following approach is used to answer this question: In order to determine the stability of the function G (s), it is necessary to give the input which is described.by x1(t) an infinitesimal perturbation and note what happens to the output as a function of time0 The easiest type of perturbation to give the input is a unit impulseo When l (t) represents a delta function input occurring at

18 t = 0, X (s) = 1, and equation2.2-3 with this input function may be written in expanded form as follows: x2 (s) = u' K, iI, ^ (S+Pi i=l vj K. w Kw. + x -~ —— J —...T e-K + -j + 000 +''' Z:,(s+P2), (s+p j=l 9 =l 2.2-3 Taking inverse transforms of both sides, equation 2.2-3 becomes: u-l1 K tu-2 x (t) = K1 I+ 1-2I +. I 2 ~L (u + (u-2) j v-2 v-1 K 22t t+ V -(v-i) + 2 + *- P2t +21 (v-.)' (v 2) " +h 2.2-4 lr w-K t2t _+ L1 (w-1)! + (Wv-2) 1 +... e Pn +.... Koo 5(t) From equation 2.2-4 the following restrictions regarding the location of the poles of F(s) in the s-plane can be deduced: lo P1, P2 ~ ~ Pi cannot be negative real or cannot have a negative real part if complex, for x2 (t) to remain bounded for all values of to 2. pi P2 ~. pj cannot have pure imaginary values, or the system will be continuously oscillatory. 3, A pole or order 2 or more at the origin must be excluded since these lead to functions of the type K t where r>l

19 2.3 Stability of Closed-Loop Systems Since it is possible to talk about either the stability C of the closed-loop function (s) or the stability of the C open-loop function - (s), the definition of open-loop E stability will now be extended to include a closed-loop system of the type shown in Figure 21lc o It consists of a device known as the error detector which senses the differences between the input and output. Although in an electrical circuit this is usually the difference between two voltages, in the general case, it represents the difference between the input, or the regulated variable0 and the output, or controlled variable. x(s)~ G(s) -x(s) G1(S)~~~~~~~~~ (:s.) (a) (b) R(s).s) (c) Figiu re 2, 1 B3ock Diagcranm oRepJr'esen-tti: jo n of Linear Contrcol System

20 The closed-loop system, which is represented by the block diagram shown in Figure 201c is defined by the following equations: E(s) = KG(S) 2.3-1 and C s)-.KGes).... CS = 14KG(S) 2.3-2 R 1 + KG(s) Equation 2.3-2 can be written in expanded form as followsg dCo + c s + c - - c s~ _0 1 q _ 2.3-3 n j=l J (s+a) (s+a) - (s+a m i (S+f31) (s+32) - -- (S+5) i=l C where q ~>p for -(s) to be physically realizableo A comR parison of equation 2.2-1 with 2.3-3 reveals the closedloop function to be identical in form to the open-loop function, and therefore all the restrictions placed on the open-loop functi@ob (s), to guarantee its stability, also apply to the closed-loop function, -(s). R In the majority of feedback control systems the designer usually possesses a stable open-loop function, and he is called upon to add passive compensation so as to achieve satisfactory closed-loop performance. In the study of control systems a designer may be confronted ~however, with an open-loop system that contains a right half-plane pole and it is therefore open-loop unstableo The following questions must then be answered: 1. Is it ever possible to stabilize the closed-loop system if the open-loop system is of this type?

21 2o What restrictions must be placed on this type C C of (s) function in order that the -(s) function E R be stable? The answer to question 1 will be "yes" if it can be shown that at least one case is possible in which KG(s) is unC stable but the closed-loop function -(s) is stable As an example, consider the second-order system having the following form: a +as E(s) KG(s) - - 2 2 3-4 Ii: b b b+s o 2 where a a. b b2 are all positive real numberso Now o, 1, o, 2 in this case the poles of KG(s) lie on the jwJ axis and therefore the open-loop function is continuously oscillatoryo Equation 203-5 represents the closed-loop function of equation 2.3-4 a + als C(s) = KG(s) o 0 2 2o35 R 1 -X- KIG(s) (a + b ) + a s + b s Inspection of equation 2o3-5 reveals that the poles lie in C the left-half plane, and therefore -(s) will be stableo Question 1 can therefore be answered affirmativelyo The answer to question 2 is not so apparent since all possible types of open-loop functions must be consideredo It is therefore discussed in detail in the work that follows 2 4 Linear System Block Diagram Transformations The block diagram of a physical system consists of the

22 interconnection of a number of linear blocks of the type described in Section 2o1. If the system is complex, its reduction to a single operational expression can be time consuming. Graybeal has shown that by using eight prototype equivalences given in Figure 2.2, it is possible to change any multiple-loop system containing a number of interconnected linear blocks into a whole host equivalent systems. In order to show that these transformations are equivalent, consider as an examples the case shown in Figure 2.2.g o Starting with the lefthand configuration shown, the following equations can be written: E1 = x x2 2.4-l x= G31x 2,4-2 E 1 =3 32 2.4-3 Substituting equations 2.4-1 and 2.4-2 into equation 2.4-3 equations2.4-5 and 2.4-6 are obtained. Ei = X1 - G313 x3G32 2.4-4 i = - x3 (G31 G32) 24-5 Ce = XI - xi 2o4-6 x3 = x3(G31 G32) 2.47 Inspection of equations 2.4-5,2o4-6, 2.4-7 reveals that they are indeed the defining equations of the right-hand diagram of Figure 2 2g o 16o Graybeal, T.D., "Block Diagram Network Transformation", Trans. AIEE, Vol. 70, ppo 985-990, 1951.

x 12e G23 G,2 G,23 — (a) _ x,+i 6G,3 x3 XI Gl G3 (e) (b) (f) ~ x2 _ X (c) X, E (9) ~ (d) (h) Figure 2.2 Linear Block Diagram Transformations N' W>

2!41 2 5 The Prototype ConfDiguration In Section 2.4 the subject of linear equivalences in a feedback control system was introduced. In this section these ideas will be utilized to show that any linear block or group of blocks can be left unaltered when applying these transformations to a random configuration while reducing the remainder of the system to some prescribed configurationo Since, in general, when the transfer functions are combined, a multiple-loop system will result which will also have a random configuration, it is necessary to approach the problem in an organized fashion, and pick a specific configuration which will serve as a prototype for the work of this dissertation. After some consideration, the prototype shown in Figure 2.3 was considered most promisingo It consists of two differentials 1 and 2, interconnected by transfer functions G12(s), G20(s), G02(s), G01(s)o In this configuration, the inner loop (enclosed by the broken lines) represents a closed-loop system whose transfer function in general takes on any form, stable or unstable, representable by a rational function p(s)/q(s). The function G12 (s) and G01(s) represent the remaining portions of any multipleloop system. In order to see that the configuration of Figure 203 is completely general, let us consider the system shown in Figure 204a which contains 5 differentials and 6 transfer functions. The inner loop enclosed by the broken lines represents the unstable inner loop in this caseo This corresponds to the unstable inner loop in Figure 2~3, which is also shown within the broken lines,

25 r —-- m- - -- I 2 I I1<> I Figure 2.3 The Prototype Configuration The problem at hand is to take the configuration of Figure 2.4a and reduce it to the prototype of Figure using the equivalent linear transformations shown in Figure 2.3 o First apply transformation "e" of Figure 2,2 and in this way interchange differentials 1 and 5. Next apply transformation "b" of Figure 2.2 and combine G34, G43 o The resulting configuration is shown in Figure 2.4b e Next move G12 on the other side of differential 2 by application of "d" in Figure 2.2 and interchange differentials 2 and 5 by applying Figure 2.2e. The resulting configuration is shown in Figure 2.4c. Combine G12, G45 and G34 3 —- t — 4and its associated differential 5 with the 1 + G34G43

26 (a) R( (S ) C( 6 ) * ) s) R( s R( s R(8s (d) Figure 2.4 Reduction of a Complex System Utilizing Linear Block Transformations

27 aid of Figure 2.2b. The resulting configuration is shown in Figure 2 4d. Inspection of Figure 2,4d reveals the box in broken lines to have been unaltered in this process and therefore it represents the unstable inner loop of Figure 2 a. The transformations of Figure 22... can be applied in any order to a given linear system to reduce it to the prototype of Figure 23.,,. In this way the specific loop under consideration can be separated from the remainder of the system and its effects on the rest of the system can be analyzedo 2,6 Root-Locus Methods Since the root-locus method of analysis has been used as a major tool in this dessertation, a summary of the wellknown properties which govern the behavior of root loci will be discussedo For a more detailed treatment of these aspects of the subject, the reader is referred to a number 17, 18, 19, 19, 21 of excellent papers17 on the subjecto Some 17. "Graphical Analysis of Control Systems", Wo Ro Evans, Trans. AIEE, Vol 67, 1948, ppo 547-557 18. "Servomechanism Analysis", G. J Thaler and Ro G Brown, (book), McGraw-Hill Book Coo, Inc, New York, 1953 Chapter 14. 19. "Control System Dynamics", W R, Evans, (book)Chapto 7 and 80 McGraw-Hill Book Company, Inco, New York, 1954. 20. "Control System Synthesis", John Go Truxal, (book), McGraw-Hill Book Company, Inc., 1955, Chapter 4, pp. 224-277. 21. "Feedback Control System" - JoC. Gille, MoJo Pelegrin, P. Decauline, (book), McGraw-Hill Book Company, Inc., ppo 235-255.

28 of the more recent developments and some of the less known properties have been treated in greater detail. a) Definition of the Root-Locus The open-loop transfer function of a linear feedback control system having m zeros and n poles can be described by the following operational equation: m K 7 (s + z,) (s) = KG(s) =..1 2.6-l Er (s+pj) j=l where zi and pj represent the zeros and poles of the openloop function respectively The corresponding unity feedback closed-loop function for this system is described by m Cs)_ KG(s) K 1T (s + z i) KG (s). (S) = 1 206-2 R 1 4l KG(s) rn m - (s +pj) _ ~KK (s + z) The dynamic behavior of the system is determined, C in part, by the location of the poles of C(s), which R are the roots of the characteristic equationo In terms of the open-loop poles and zeros the roots of the characteristic equation are given by the zeros of n m +z T- (s + pj) + K (s + z ) = 0 2.63 j=l i=l In equation 2.6-3 the plus and minus signs indicate degenerate and regenerate feedback, respectively. Equation 2.6-3 when refactored will have the following form:

29 n 17 (s + j) = 0, for n > m 2o6-4 j=l which puts the roots of the characteristic equation, Bj, in evidence. A comparison of equations 2.6-3 and 2Q,64 reveals that the closed-loop poles are, in general, functions of the open-loop poles, open-loop zeros, and loop gain. In symbolic notation, fj may be expressed as Q(K, p1 p2.. p 2z z o o. z ), 2.6-5 fj I( P, 2P, 1, 2, m which puts into evidence all the factors controlling the location of the roots of the characteristic equation, The root-locus is defined as the locus of i. when K is chosen as a parametric variable with p. and z. fixed. Inspection of equation 2.6-5 reveals that it is possible to choose any of the remaining quantities as parametric variables. These have been considered more recently by 22, 23 some authors. Their results will be discussed later in this chapter. Inspection of equation 2.6-5 reveals that the root locus can also be defined as the zeros of the equation 1 + KG(s) = 0, or where 22. "Some Mathematical Properties of Root Loci for Control System Design", F.M. Reza, Transactions AIEE, 1956 DBaic Science Paper 56-125. 23. "Synthesis of Feedback Control Systems by Gain-Contour and Root-Contour", VoCo Yeh, Transactions AIEE, Vol.(App & Ind) pp. 85-95, May:.1956.

30 -1 degenerative feedback KG(s) = +1 regenerative feedback Equation 2.6-6 when written in polar form becomes 2.6-6 e ( ~ 2Nw) degenerative feedback KG(s) = e f2Nr) 2~6-7 Ks= ee(0 2N) regenerative feedback Referring to the above definition of the root locus, it is apparent that determining the root locus of a system results in determining where the fw 2N7T Arg KG(s) -= + 2 N and where KG(s) = 1. In the above equations degenerative feedback' 2.6-8 regenerative feedback N = 0, 1, 2,.. In constructing the root-locus one therefore maps either the positive or negative real axis (degenerate or regenerate feedback) in the KG(s)-plane into "slits" in the s-plane. b) A summary of Well Known Root-Locus Properties In constructing the root-locus the following well known properties are utilized: 1. A branch of the root locus starts from every open-loop pole where K = 0 and terminates at an open-loop zero where K = oo.

31 2. The root locus of a real system is always symmetrical about the real axis. 3. The closed-loop system will become unstable when a branch of the root locus enters the right-half planeo 4o For a realizable transfer function, G(s), there are as many branches of the root-locus as the number of poles of G(s)o If the numerator and denominator of G(s) are of degree m and n, respectively, there are n-m zeros of G(s) located at infinity, hence n-m branches of the locus terminate at infinity. 5, The asymptotic center is the intersection point of all linear asymptotes. For a real system this point is on the real axis: n m _ Pj. -. Arg. 180. 360 N 2.6-9 i; ArgS' = 0 n-m n-m where pj and z. are poles and zeros of KG(s), respectivelyo 6. At a junction point of the branches, the tangents to the locus are equally spaced over 2-r radians. 2.7 Gain-Loci and Phase-Loci The definition of the root-locus as discussed in section 2.6 is in reality a special case of the more general gain-loci

32 22 23 24 25 and phase-loci2 The form of KG(s) as given by equation 2.7-1 namely, m _c ~K il (s + zi) (s) = KG(s) = 2.7-1 E n jl (S + pj) where z. and pj represent the zeros and poles of the openloop function, respectively, equation 2.7-1 can be written in the following form, which puts in evidence the amplitude and phase relations of this function: K r. ej KG(s) = X ef _ iII 2.7-2 m n i' L C3- jlj 2.7-3a / n m /.j7l j While in section 2.6 y was either zero or some multiple of rr, in the case of phase loci of is allowed to take on all values. Under these conditions a new group of loci will 22. See reference 22 section 2.6 23. See reference 23 section 2.6 24. "The Study of Transients in Linear Feedback Systems by Conformal Mapping and the Root-Locus Method" Victor C. M. Yeh, Trans. ASME, April 1954, ppo 349-361o 25. "Synthesis of Feedback Control Systems by Phase-Angle Loci" Yaohan Chu, Trans. AIEE, Vol. 71, Part II, 1952 pp. 330-339.

33 be obtained with / as a parameter. Corresponndingly, by letting /6 take on values other than unity, the value it has in the case of the root-locus, a group of gain-loci will be obtained for each value of/,O. The concept of gain and phase loci will become readily understandable by considering the following example of a second-order feedback control system containing two lefthalf plane poles a,,'. gain K, and defined by the following open-loop transfer function: E(s) = s + Ce K e r e 2.7-4 j/ (s ~C)(s+L B.).je or f = e1- E2 2.7-5 and y. K = -. 2.7-6 r1 rr2 In order to determine the phase-angle loci, the parameter, J, has been assigned the following values: f = 0 + - (N = 0, 1, 2,...) in the KG(s) plane,which will be found plotted in Figure 2o5a. The corresponding "slits" produced in the s-plane by allowing K to vary along a prescribed phase-angle locus are shown in Figure 2.5b Inspection of Figure 2.5a reveals that lines of constant will be concentric

34 "q'*! s-PLANE 7 4 _ 2t I i-/' \ - 444 -5 7. -34r 2 Figure 2.5 Gain and Phase Loci of a Second-Order Control System

35 circles about the origin of the KG(s)-plane. The corresponding "slits in the s-plane are shown as the dashed lines in Figure 2.5. Inspection of equation 2.7-6 reveals that for prescribed pole:-locations of a, and., varying in KG(s)-plane corresponds to varying K in the s-plane. Thus, in reality the value of / and K are related by a constant. Since the system gain, K, is usually a design parameter, it is usually the one that is plotted. The ideas presented here will be further discussed in section 4.7. 2.8 Method for Determining the Roots of an Nth Degree Polynomial in s. A new and little publicized method of solving for the roots of a characteristic equation was recently found by the author.* The technique, which utilizes the basic concepts of the root-locus technique, is in no way restricted by the order of the system. The rules governing the behavior and construction of the root-locus also applies in this case. Given: n n-1 n-2 F(s) = s + a +a2s + —- an 2.8-1 which is an nth order polynomial in the complex variable s, and one whose n roots are desired to graphical accuracy. The procedure used to determine the roots can be * The original method, which was attributed to Walter Evans of Autonetics Division of North American, has not been published to the best of the author's knowledge.

36 described as follows: 1) The polynomial 2.8-1 is factored by first removing a and then factoring an s out of the remaining terms. The following form of equation 2.c-/ results: S t 1 - n-1 l-2 F(s) = s + a 2. 3-2 2) Now if within the brackets of 2. 2-2 the term a-.an-l is removed, the resulting terms will again contain a common factor s, which can again be removed. Eqcuation 2.-1 now tak.es on the following form: n- 2 n1-3 -.~...(s) ss (s +- a ~ s- ) + 1 + a 2.8-3 3) T'he above procedu:re is repeated until all the coefficients except a1 are removed. The final form of F(s) is F(s) = s [ — { s(s+a) +4 a2 + t3} - n-1 a 2.8-4 4) Starting with the inner-most factor namely, s(s+a), K1 the root-locus for the function F1 (s) s(sa- ) is constructed using root locus techniques. Since 2 what is really wanted are the factors of s +- als+a. attention is focused on locating on the root-locus of Fl(s) those points where K1 = a2. At this point s(s + a ) = K a, or, in other words 2 at these points s + +s + a2 0. Thus z and z2 which are the zeros of s - a1 s + a2, have been found.

37 5) Knowing the location of these two roots permits one to move on to the next step of the procedure. This consists in making a root locus plot of K2 2 S() s 2+ z (S + *2) which is the next grouping of factors in the next innermost parenthes is. After the root-locus is constructed, attention is again focused on determining where on the F (s) root-locus K9- = a3. This will yield three roots, which in factored form become (s + Z1) (s + z2) (s + 3) 6) The procedure used in 4) and 5) above is repeated until all the roots of F(s) have been determined and F(s) = (s + A1) (s + A2)(s + A3)... ( An) 2.8-5 The above procedure can be viewed as determining the root-locus of an n loop linear feedback control system. If K B1 (s) s(s+al) is consideredwhich is the inner-most function in the inner-most loop, then finding the roots of the polynomial F(s) corresponds to solving for each root-locus of this n-1 loop control system, which in this case possesses a prescribed configuration. This configuration which is shown in Figure 2.6 consists of a single pole on the real axis and a pole at the origin of the inner-most function and

38 a pole at the origin in each of the n-2 external loops. Inspection of Figure 2.6 reveals the gain constants to be ratios of the coefficients of the polynomial. In each case once the angle condition of the root-locus is determined, the closed loop roots are determined as a result of the feedback properties and the prescribed gain constants. Figure 2.6 The proceeds 1. 2. Simulation of an n-th Order Polynomial as a Control Syst:em method will now be illustrated by an example and as follows: Given F(s) = s + 12s + 56s + 88s + 68 Factor F(s) as follows: F(s) = [s (s + 12) + 56 + 88] + 68 3. Starting with the inner-most factor, namely s(s + 12) determine the locus of roots of the function F1(s) s(s + 12) using root-locus techniques. The resulting locus is shown in

39 Figure 2. 7a 4. Now determine where Is(s + 12) = K = 56. This represents the location of the roots of the polynomial within the braces, which in this case is found graphically to be (s + 6 + j 4.43)(s + 6 - j 4.43) 5. Utilizing the newly found roots at s= -6 - j 4.43 and s = -6 + j 4.43, determine the open-loop roots of the next grouping of terms, namely, the closedloop roots of K2 F2(s) s(s + 6 + j 4.43) (s + 6 - j 4.43) The locus for this case is shown in Figure 2. 7b. 6. Next determine the location on the root-locus of (5) above where s(s + 6 + j 4.43)(s + 6 - j 4.43) = K2 = 88. As a result of applying these amplitude conditions, the poles of F3(s) are found to be (s + 3.08)(s + 4.46 + j 2.95)(s + 4.46 - j 2.95). The above process is again repeated by now writing F3(s) as follows K3 F3() s(s + 3.08)(s + 4.46 + j 2.95) (s + 4.46 - j 2.95) and first determining the argument conditions on F3 (s), and, then determining where the amplitude conditions are satisfied. From the root-locus of

40 Figure 2.7c the roots of F(s) are found graphically to be F(s) = (s +.95 + j 1,.) (s +.95 - j 1.1) (s + 4.98 + j 3.05) (s + 4.98 - j 3.05) which compares very fJavorably with the exact values (known in this case) as F(s) = (s + 1 + j)(s + 1 - j)(s + 5 + j3)(s + 5 - j3) While the above description of the method leads to solutions of graphical accuracy, in the case of higher order systems there may be considerable error utilizing this method. A review of the root locus plots will r eveal that if an error is made in finding the locus in the case of the inner-most functions, these er-rors will.'. cml- t Thi.s p~roi.L elm can:)e cir:cumnven'ite. to a certain d:eg'.ree -y 2 solvincJ r: te "t oo "s ":t "id: iJLe: ):iomi. ~-. + 12s + 56'S analyticalLy using th-e cuad:iratic forzmula, an-1d th::;,l:tiiize this information in determining the roots of the higher order polynomials. In a similarE mane: it is possib)le to extend -the anaalytical approach to the real axis pole from the graphical plot and to utilize -this in the following manner: From root-locus plot (see Figure 2.7b) the real axis root was estimated to be at s = -3.. TaIking this root and dividing it out of the trinomial gives

41 jw 4 / / s - PLANE 4 i j( A if -12 -8 -4 -8 A s +6+j4.43)( +6-4).88 (s +6+j4.43)(s +6- j4.43):.88 X -OPEN-LOOP POLES A - CLOSED-LOOP POLES (b) s(s-12) =-56 X -OPEN -LOOP POLES A - CLOSED-LOOP POLES (a) ihk, ij 8 - PLANE / / (c) Figure 2.7 Solution of an n-th Order Polynomial by Root Locus Techniques

42 s -- 8.9s + 28.41 3 2 s + 3.1 s + 12s + 56s + K 3 2 s + 3.1s 8.9s2 + 56 s 8.9s2 + 27.59s 28.41s + K 28.41s + 88.071 Since remainder must be zero K = 88.071. However to satisfy the next outer bracket, inspection of F(s) reveals this constant should be 88. It is apparent therefore, that an error does exist and the root at s = -3.1 must be modified if greater accuracy is desired. This can be done with the aid of a calculator to any desired degree of accuracy. Once this has been accomplished, the remaining factor being a quadratic can be solved using the quadratic formula. Since for a third order system the approximate location of the real axis root can always be found and the argument condition of root locus position is precisely known, the above procedure can always be carried out. Thus the roots of the cubic can be determined to any desired degree of accuracy. These roots now serve as an accurate starting point by which to locate the roots of the remaining expression. In the case when a fourth degree polynomial has real axis roots, the above technique can be extended to the fourth order equation. If the roots (as determined by the root-locus) are complex, the above technique cannot be carried out since the precise location of the root locus curves is not accurately known.

43 CHAPTER III SECOND ORDER CONDITIONALLY STABLE CONTROL SYSTEMS 3.1 Background An idealized feedback control system represented by the following closed loop transfer function: 2 d s + d s + d (s) = 21 0 3.1-1 2 e2s + els + eo has been considered as the logical starting point for this investigation. While second-order systems seldom exist in practice, they lend themselves readily to analysis. If d2, d1, d are allowed to assume positive, negative, and zero values, it is apparent that all second order systems that are closed loop stable are being considered. In addition, if in Figure 2.3 G12 G = 1, 12 ol the configuration of Figure 2.3 reduces to that of Figure 3.1. The defining equations then become c C(s) = KG(s) CR() KG(s) 3.1-2 R(s) -I + KG(s)2 and E(s) - 1 R 1 + KG(s)

44 where KG(s) may in this case contain its own feedback loopo This, of course, is necessary to generate unstable C -(s) functions. E r- ------------. c(s) R(S) + EE(s) G ) +yy +1 20(S) -'pen+ _ 1 20I Loop |I~ umler. ~under C(s) E1(S) consider1I I - ation Figure 3.1 The Prototype Configuration NOW. let ER 1 + KG(s _ R + KG(s) - 2 2 e2s + els + eo 3.1-3 then C -(s) = KG(s) = E (e2 b2) s2 +(e + (e-b) ( b ) 2,.. b2s +bls + bo 3.1-4

45 and C() = KG(s) R = 1 + KG(s) (e2 b2)s + (el- bl)s + (e- b ) 2 e2s + els + e 2 1 o 3.1-5 If d 2 2 -b d = e- b, andd e - b then equation 3.1-5 is equivalent to equation 3.1-1. Inspection of equation 3.1-4 reveals the following information: C C 1, The zeros of -(s) and R(s) can be in the right or E R left half plane depending upon the relative magnitudes of b b b2 ande e e, e o, 1, 2 o, 1, 2 2o Any missing powers of s in the denominator polyC nomial of -(s) must be present in the numerator B for a system to be stable. 3. In addition consider the special case when K(s - z1) (s + z2) E S (s - 1) (s + p2) 3 where z z21 p P2, are real and positive. Under these conditions -) K(s - Z)(s + Z2) E (s - p)(s + P2 + K(s - z (s + z) K(s - z1) ( + z2) (l+K)S + + K.z) (p+ K Z) s -+ KZ (1+K)S2 + P2 KZ)-(P1+ KZ1) L S E 1P2+ K 1 2J 3.1-8

46 Solving for the closed-loop poles of equation 3.1-8 reveals that the closed-loop function contains one right-half plane pole and therefore yields an unstable closed-loop system. 3.2 A Comparison of the Two Commonly Encountered Definitions of Conditionally Stable Systems Conditionally stable systems have been treated in detail by few authors. Review of the literature reveals that in many cases authors have specifically excluded this class of feedback control systems from their discussion. As stated in Section 1.1 Brown and Campbell called a conditionally stable system a system which is unstable for a particular gain, but stable for both larger and smaller gain values. While the definition used by other authors is a system that is stable for a particular value of gain, but is unstable for both higher and lower gain values. To get a clear insight into these definitions the rootlocus of a typical system from each catagory is compared, In Figure 3,2a is shown the root-locus for a system satisfying Brown and Campbell's definition. Inspection of the open loop function which characterizes this rootlocus reveals that it contains only left half plane poles and at least two finite left half plane zeros. As the gain is increased, it is apparent that a pair of closed loop poles move into the right half plane for a prescribed range of gain, but with further increase in gain the root

47 locus and therefore the closed loop poles return to the left half plane. On the other hand, the system whose root-locus is shown in Figure 3.2b possesses a pole of order 2 or more at the origin, or one or more right half plane open loop poles. In this case the closed loop system is unstable for small gain since the locus lies in the right half plane, but becomes stable when the closed loop poles move into the left half plane. Depending upon the order of the system, as the gain is increased one or more branches of the locus return to the right half plane, thus indicating unstable closed loop behavior. Both types of systems are ones which require care in their design and are therefore avoided by many control system designers. 3.3 Conditionally Stable Second-Order Systems In section 3.2 two different types of conditionally stable systems were discussed. Inspection of Figures 3.2a and 3.2b reveals that neither of these systems is secondorder in nature, and therefore it is necessary to modify the above definitions in order to include all possible types of conditionally stable systems. Conditionally stable systems as used in this dissertation is that set of linear feedback control systems, whose internal zeros all lie in the left half plane, whose rootlocus contains one or more branches that cross the jw-axis more than once, and/or whose root-locus contains one or more right half plane open-loop, oles. This result would be cbtained for the system represented by Figure 3.2b if two additional

48 -I+ j/ =4.39 -j: 1.64 _ s - PLANE // // /7 -0 -2 1 -I-j5 -10 I ) NOTE: NOT TO SCALE C (s) E K(s2+2s +26) s(s +)(s+2) -C (s)= E K( s+ 2s +2) s (s- I)(s + )(s+2)(s +10) Figure 3.2 Root-Locus Plots Representing Two Types of Conditionally Stable Systems

49 finite left half plane zeros were added to the system. In this investigation all mathematically feasible secondC order ~(s) functions were investigated using principally the root-locus method of analysis. As a result of this analysis it was found that conditionally stable secondorder systems may arise when the open-loop transfer function contains one of the following: 1) One or two right half plane poles, one or two left half plane zeros. 2) A negative gain constant in the error channel; one or two finite left half plane zeros, two poles located anywhere. 3) A pole of order two at the origin. This represents the.limiting case of a conditionally stable system, since in this case for zero gain constant the root locus approaches the imaginary axis. As an example of 1) above, consider the open loop function KG(s)(s+ + ) +K(s+. 1 5 + jo866)(S +5 j.866)33 KG(S) 4- 2 (s 07+ j.7)(s - 7 j 7) 3. (s - 1.4s+.98) This function is characterized by two right half plane poles and two left half plane zeros. The root-locus for this function is shown in Figure 3.3. Inspection of the root-locus reveals that for small gains the closed-loop function contains two complex right-half plane poles and therefore the system is unstable. As the system gain is

50 increased from zero to infinity, the closed-loop poles move from the right half plane into the left half planeo In this case for K=lo4 the closed-loop system will be continuously oscillatory, while for K> 1 l4 the closedloop system will be stableo In this case as the gain of KG(s) increases from zero to infinity, the poles of the closed loop function always remain complexo ci iC s-plane -o5 + j.866 > S.7 + j.7 R -.5 - j.866 _.7 - j.7 Figure 3 3 Root-Locus Plot of a Conditionally Stable System.with Two Complex Right-Half Plane Poles As an example of 2) above, consider the open loop function, _ K(s29 4 lo4s 498) _ -K+o = +,7 Ls s.o-o CRa ff -q e - - -I -" l 3-2 &p"3 % 1S> p w~ s2+ + (S + s + 1) (s+.5+j 0866) (s+o 5-j 866) - ~ O

51 This open loop function is characterized by two left half plane poles, two left half plane zeros and a negative gain constant. The root-locus for this function is shown in Figure 3.4. Inspection of the locus reveals that for small system gain the closed loop function contains two left half plane poles which yield a stable oscillatory system. As the system gain is increased, the closed loop poles move into the right half plane. Still larger system gain, will yield a closed loop function having two real right half plane poles. As the gain is made still greater the closed loop roots move again into the left half plane. As the gain tends to infinity, the closed loop roots become complex and lie in the left half plane. In this case since the poles and zeros of KG(s) lie in the left half plane, the open loop function is stable although the closed loop function may be stable or unstable depending upon the system gain. More specifically the system will be unstable for 1.01 > K>.715 jW) -.5 + j.866 - -.7+ j.7 \s-plane -7 -j.7 -.5 - j.866 Figure 3.4 Root-Locus Plot of a Second-Order System with a Negative Gain Constant

52 As an example of the third class of functions, consider the open loop function 2,(.) _ ~K(s + s +K(s 5 +j.866)(s +.5 - o866 33 s2 s s The open loop function is characterized by a pole of order two at the origin, and 2 internal left half plane zeros~ The root-locus for this function is shown in Figure 3,5 o Inspection of this locus reveals that for positive gain this system cannot be rightly referred to as a conditionally stable system, since as the system gain is changed from zero to infinity, the poles of the closed loop function always remain in the left half plane o iowever# it does represent the limiting case of a conditionally stable system, since with the addition of another pole it can be shown (see Section 4o5) that it is possible for the locus of this system to move into the right half plane before finally terminating at the zeros that lie in the left half planeo -.5 + j.866 s-plane -.5 - j.866 Figure 3.5 The Limiting Case of a Conditionally I. C+FA hl <a Ca+v af.m

53 3 o4 Mathematical Derivations for the Root Locus of Second Order Feedback Control Systems In the construction of the root locus for real values C of -(s) the locus must lie on the real axiso The construction therefore imposes no real problem, When the roots of C -(s) are complex, in the usual case, the construction of the root locus is a time consuming procedure which must be done with care, if acceptable results are to be obtainedo In order to facilitate the plotting of the locus and to add mathematical credence to the locus plots, the equations C that govern the location of the roots of R(s) when they are complex have been determinedo As an example of this technique, consider the second order transfer function having the following form: (a + zl) ( + 2) KG(s) = T 304-1 which is the form of the second example on page 51 o In this case z + j P + j, and z = Z *, P2 p1* but A * of course, is realo The problem at hand is to find the location of all the complex roots of i(s), as the gain > is varied Now C KG (S) since -(s) = KG what is really desired are the R + KG(s) zeros of 1 + KG(s) for all values of gain; o Thus 1 + KG(s) 0 304-2 or * note J is used in this derivation in place of K = as previously definedo

54 (s + z) (s + z2) KG(s) -I - +p) R ( + pi1) (s + p2) 3 4-3 Utilizing the complex components of KG(s), and at the same time solving for, the following equation is obtained o9 1 | _ q;)4+ i ( ( + W1)] ^ = " T+q7)+ J (W + U2 )] [(Tc +,1) + j(W - lj r(9+V) + j (w - w1:(+T J@2t 3 4-4 Now since.- must be real, its imaginary part must be equal to zero. In other words or,)+ nfpW^Dj1a ~ n[+)+ j (-KDW2) After expanding and collecting te: in the following form: 2 2 2 22 2 " + 2 " + C2 - c)1 L j ~~~~~~~~~, |^(9+ )+ j (CO-W ) ] r(T+T)+ j(<o-32 3~ ~ z ~-1 2 3.4-5 rms this can be written 2 2 2 2 2 ~T2. l+ 32 " c 1l I. 0 —,- 3 4-6 +

55 Equation 3.4-6 will be recognized as the equation of a circle with center at 2 2 2 2,2 -1 +'2 1 c ~c - 2(. - T 304 7 ~c' 0 ~^ T -2(7 2 1) and radius 2 2 2 2 +\^c- 3o4-8 C The root locus when the roots of -(s) are complex can R thus be found using equations 3,4-7 and 3.4-8 o When the closed loop roots are real, they will fall on the real axis, and their root locus can readily be constructed. The above results will now be applied to a specific example. Let KG(s) +- j + c.7.-.j.7).(s +.7.7)4-9 KG(s) = ( 5s + o5 + j 866)(s +.5 - j,866) By application of equation 3,4-7 the center of this circle is found to be 25 -.49 + 75 - 49 410 xc =~ c 2(.5 - +7) By application of equation 304-8 the radius of the locus is found to be

56 = \1 49. *70 + (~05)2 = 1.02 Using a compass and a straight edge the complete locus for this function can readily be constructed (see Figure 3 6 )o The accuracy obtained by this method exceeds that obtained by the usual "cut and try" technique The author developed equations which would allow rapid construction of the root-locus for all secondorder systems. These will be found listed in Figure 3.7. Since this unpublished work was carried out, Yeh2 published a similar listo However, his list also included equations for the root-locus of some third and fourth order systems as wello The emphasis of Yeh's research was to develop the form of the root-locus equations and little mention was made as to its application0 3 5 Computer Study of Second-Order systems In order to verify the theoretical results on the second-order system, a computer study was made of a number of second-order systems that were open loop unstable. Although it would be desirable to test the open loop as 24. See reference 24, Section 2.7

57 jo s - PLANE Figure 3.6 Example of Root-Locus Construction Utilizing Only a Compass and a Ruler

58 Loop Functions defined in Open Loop Functions a, nd intept for copei mo m,0l-4r. ^- 1iA^ i. tord por __otd du, a 1d interce - - pt for cplex loci_ 2 + * 1 ( 0 - bo) (a + + Jc p(- + g -J)Iv bo.... -, oK I_ ________o_______________ __J___(a__ locstraight vertical linesi 2 2 + (1 - b) + (+ +1*Ja.)(s +1-JC ) - ~ c - 0. + - b______S 180 circle with center at origin (02- 2) 2 + e1+e (a++ )(a+r-Jo 1 + Ia a + q-i' 3^-.y^.^^ g^,,^, Oa~ I. ) " 2,,: M.M 2< 1 [ bJ,,2 as1 o f -cZ 4 a 2 + * ~ + (bo+ eJ ( + /1+ Ji) (a + re'-JL.) | o Q - _ - locus straight vertical lineo e' 2 + (e1 + b) + *o ( + q+JWo) (0 + 0'J -0 *-o) -I (sb)e.^, (<f^)( ~ <ro) o) | t |o | ^. 0. - 0. - f/_ + <+ o b _ L c -' c' (e2 + -2)2 e - b * + ( + ) + +Jo ))(a + Q-Ja ) O - 1 6 (e -b e+ (e b) ( + ( +'- J)Z 0 2. 0, / + 2 - 2 b2s -1 b l' 2c'~ /~_' _0 1c e2 2 + (e - b )s+ (e - b ) ( + +JaL3) (s + qj-JOmL) 2 s h 1%) 2. - -r W. w -20.,. -/%=+r0 /- = % 2 1 ~ K~ s+2 ) c o c (e2 b2)2 e) +e + (e+ e ) (ej + b ( l +JL) (a + qL-JmL) _-2 -+.2 15 - (b2sZ + b 0 XK(o+o).10 ).-O 9 (e - b2)s2 + (a +q b1+J) + -bo |j+) |2 ~ 2 S_ 2 no 1 [ — b2s+ 18+ -- + -- - -0- (b +| +ji.)X ( + I -j) |80 2 | S *o 1 b2 1 o __ ]LIM C~ [ e + (1+ b 1)a + (eo + ).(a + 2+J.,) (a + -JTT) 2 _ 2 LO -b^ a -^ b^ 0 + To - -. %' - O. f0 2 -. 2 2) (a + T.+JL) (a + q'-Jo1) 2 2 (e2 + b 2) + (e1 + b)" + *o (a + W +ja) (s + -n3) +.I- -X(a + %o) C 2V ao- 0c b2so - bLS ~' 2 < - (%+ 2%)'c (e + b2)82 + (1 b + (e- b(a -( +T+J -)j)) +1 8- 0r) 13' 0 0 X X. K.a+%(- b" - "-. no. 13 2( 1 a +Jmx (0 +.' ( -b2 - b) 82 + -(81 e.+ )(8+1bo) * 2 - 14.3 b (eX(s) + (eV r~2+-^~ JcO) (a + qJW) 18g o 2 2 (02 2)2 - ) e - h K(5 + 1+ ) (s + ~1j) ~ _ b2s2 + bs1 - b) oK(+ + ) 1(a 1 3) 1SO ( +, *- no 1 [2+t 1 2 + - r+2 ) 0 Se *_no. 7 (02 + h2) + (01+ b1PO + eso (a +V1+jG)) (5 +Q*1J(1 00 S(n, -W.,1 2 1 o -K0s 0 18 -b2' ~) 3 +1 b] "+a o 21 ^~2^b^ 1 0-K(s.+2)(a _a (+l*3) fM 3or1. 41 (e2 - b 2)s2 + (el- h1)a + e%+ bh (a + "l+Ja) (a + bT'-ja~t) d 19 b-= — +-r -— a-.. 0*10- (m q Sm 0. 1ih.3 ] b 8 - b 0. + T'2) (s'3)ol (2 2) + (ei+ b2)a + (e+ 0 ( +J) (a + b(- b oa) ( de na+jh]l)(h +e —jatn)1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 f0am a o 1 3. 20 3 - al a. + -K(a + 2)(.a +Tr)l (e2 + b 2)82 + ( e0 - bh) + (e bo (a +W +Jow) (a + ql-Jcu) Se a no. 1 21 b2+ bL b o -X(a + + )(m + 3)a no. 13 (e2 + 2).2 + (0 + b 1)a + ) (a + b) 8 T+Jid) (a +, -ja)b, b2 l be t no. 22 h a2 —-- b. UK( a +1)0 S a a eaanor L 0 23e 2 +2 s + h - (b + b) ( + a ) ( +Tja) (e2+ b2)02+ (el+ bl)a + (e06+ b) (a' +'-1+jw1) (a + d1-J ). an oo bL, b2 wLIX be eStamer noo 13 "( + a w (a L)(e + a l-Jaal) (2 2 + 2 - b) + + jo (a + (I+J - ) + o Sm as no. O e2-2+ ( ea b) + h) e+ e - h0) (a + ql+jl.i) (a + "I-Jw1L) 22 0 ~Sam as no. 7O 12 0 Figure 3.7 Equations for constructing the Root-Locus of Socond-Ocrdr Sytem When the loots are Complex.

59 well as the closed loop character of these systems, due to their unstable open loop character, the computer study was limited to the closed loop functions onlyo The following computer "road map" was utilized for the simulation of these systems which have the form 2 d s + d s + d S) =d2s +dlS+do 3.5-1 R(s) 2 35 s + els + e Equation 3.5-1 may be rewritten in the following fashion: C (s) - 1 (d2s + ds + d) 352 s+ els + e + Letting C (s) = 2 3.5-3 s + els + e0 equation 3,5-2 can be put in the following form: 2 C(s) = (d2s + d + d) Cx(s) 3,5-4 Now since the setup that is used to solve for C (s) 2 X will also contain s C (s) and s C (s) terms, the right x x hand side of equation 3o5-4 may be solved to obtain C(s) The "road map" for the simulation of this equation is shown in Figure 308 o The advantage of this "road map" over others being that it is possible for the constants e1 e d dl d to take on any positive valueo In this 1e o, 2 1, o0 way it is possible to simulate either real or complex poles and zeros lying in the left-half plane0

60 1 1 land S:-=- I cx' - 1 - Figure 3.8 Computer Diagram Used for the Study of Conditionally stable Systems with Left-Half Plane Closed Loop Zeros C In addition, if d2 d d take on negative values, -(s) 2. 1. 0 R will contain one or more right half plane zeros It is also possible by a modification of Figure 308 to simulate these non-minitnum phase functions. A road map to simulate non-mm in imuphase functions is shown in Figure 3.9

C 61 All resistance have a value of 1 meg. ohm. All condensers have a value of 1 microfarad Figure 3~9 Computer Diagram Used for Studying Conditionally Stable Systems Containing Right Half Plane Closed Loop Zeros In Table I are listed the necessary modifications that must be made in the "road map" of Figure 3~9 to simulate C any prescribed second-order non-minium phase -(s) TABL~E I oR TABLE I, Closed Loop Functions Modifications Made in Figure. a) (s) = 2 d2s -dls do 2 s + e s + e 1 o remove amplifier 5

62 Closed Loop Functions (con't) Modifications Made in Figure... (con_'t) C d2s2+ dls do b) (s) = d) (s = 2 —) —-- - s + e s + e 1 o 2 -ds + dis + d C 2 1 o e) j(s) = 2......... 2 s +e s + e~ 1 0 remove amplifiers 5 and 6 remove amplifiers 6 and 7 remove amplifiers 5, 6, and 7 remove amplifier 7 2 C -d2s - ds- d f) C(a) s= - ----- remove amplifiers R 2 s + e1s + eo 5 and 7 As an example of the simulation of a second-order system, a computer study was made of a system defined by the following open loop equation: E(s) = KG(s) = 2 4 E 2 s - 3s + 5 The closed loop function for this system becomes K ((s+4) R 2+. s s + s(K-3) + (5 + 4K) A root locus plot of this system is shown in Figure 3 10o

63 s-plane - j 4.17 -42 -5 1.5 - j 1.65 \ (t L-j 2.83 - -j 4.17 -3.5 - j 5 7 x - open-loop pole * - open-loop zero A - closed-loop pole Figure 3.10 Root-Locus Plot of the System Studied on an Analog Computer Inspection of the root locus reveals that for K1l, the closed loop system is unstable. In Figure 3.11 are recordings of the system response to a step input, It is apparent that this system is unstable for this gain. Now if the gain is increased until K=3, inspection of the root locus reveals that the closed loop system possesses

64 two poles on the jwo axis. The system will be therefore continuously oscillatory; wu = 4.12 rad/sec. being the undamped natural frequency. The computer recordings in Figure 3.12 for this value of gain clearly indicate the stable oscillatory nature of this system. For K greater than 3 and less than 22 the closed loop poles will be complex and lie in the left half plane. Inspection of the recordings of Figure 3.13 indicate the damped oscillatory nature of the system for K = 10; the damping constant for the system in this case being ) =.52. For still larger gains the closed loop roots become two real left half plane roots. In Figure 3.14 will be found the recordings for K = 50.

65 SANBORN,.......................................................................' zit'"ii............'.:"...................' _ _;............. _:,::..::i::.:::::::.:::::::.::::i ii:::::::::::i::-::::!:::! -: -:/. *: - -.,,..',".. *:::.:::: i::::::::::::.:::::::1: i::-::i:::: i::::::::~ii i{::::i {:.:::-:-:::!!:/.: i::::,!?:l::i L::.::::::::::::-:-D.:{_......... _ ~~~~~~~~~~~~............. __'....... ~::=:=.' _:"__-'=....=.=_,_..I.1,.,._._:.'_.!,!!........, _t...!..., -:..';..,..., ~' __ _!...,I. _._..._...., _".''.''." -_.._' -"' __".," *''' -:".._......'....,:: —._ _"'.."'''''..'_..__.... _...,. _., -~. _______ _" ~ _" ~. ~ _ ~ _" ~ _ ~ _" ~ _." ~ __ ~." ~ __"~. ~.. _.'.' _-. ".".' _ _'.' _ ~' ~ _ _ _ __~-.,,., _"."'._..~' __... __." __.' _._.._.'._''...'~. _" _" __. __"''.'._ _ _' _";._ ='.`.___ ____ __.'._' __.. _' __._ _ —_.- - = -—.- -" " T" -t - - _""" _ —-.',.-;.:-~;:";-~ _- _~ _~ ___~ _~.__............ _. _.......__ ""."....... rv-ez -~- ___- _-..._._,. ___.........~......._____.........-..-......__ _.............. _ *." "" "': S"tt _ _-; _ --- t --- -t —-— ~-. — _ _-,-~ _ ~~..~~ _ - _~ ~~~. * — t-ri- ~ ~~, — _ - r -~- _- - ~ —- ~. _ ~- ~~ ~- _ _ f. -- _ __ ____X[ fi4fizFfi_.- _~-; -- 1; -.~-._~ _*.:_ I — ~ —, _ _ |. _ - ___ ~1 1 t_ _ _i5- m -'__ _ — _- _ t — _~ J_ _ =i r t r _ _ __'- -. —-. —. - t ir-.... 4-t ~ ~ ~ ~ ~ ~ ~ ~ ~ -~ TZ — ~ ~ ~ ~ ~ ~ - -HE ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - _H Mi - 4i _~~~~~~~j ++ r' i I I i r___+.z_.__ _ __ -;;.,_,__.=1_,-;,:4S''$r..x.. _~~~~~~~'_; I j; _. i_ I _.._. i ____-. - t, i,___. i tI Ii t _ _. iI/ I:.:-. tr. tt _._+,,_..__tr.. - — ~~~ +~ 1t-lt ~- j0 -t-:': - tFz. _._ I -...._.. ——.'t4__..._._ i I I E _ T - I~e-t-t_~,.._.t_._ 4_ i t +r, -.-.i <~...tt i&;t' - 1v {7 + ^;"t.v. _"~4tr!.t4+ ~-t'_. _;+rm. e tt +-tt'; -'~-. —-g __x- t- -ei. i 1 i ffi g...................... H4!~~~~~- -_ ~ W i bf Figure 3.11 Time Response Curves for the System of Figure 3.10 with K = 1 IX:~~~~~~~~~~~~C - - - - - - - - - ---------- ~ ~1 — ~ 1 l~! t 1: ri; 7i~~~~~~~~~~~~~~~~~~qA:'" fti: "'('"": ~~~~~~LC= A11.ii ii-i!,. i~~~77-H M-+~- ~j' RM Figure 3.12 Time Response Curves for the System of Figure 3.10 with K = 3

66 It't Jo-,~~~~~~~~~~.,...'...,.-....,-,..... -,..-.__>__X.. ~~t?`'" -t'-::::'::::f —::::I I I................................. 14i- ~ l ~ ~ ~~- I ~ ~I~ ~ I ~ ~ I ~ ~I~~~'~ — 1~~'' ~'~'-~'~~- ~~ - ~ 1"" 1- "~ 4 -...... 4444-~~~~.,.I. AH i;~~~i~tuttt~+ H~tetttll-~eet~ti~t~4J It 1 Figure 3.13 Time Response Curves for the System of Figure 3.10 with K = 10 ISANBORN iRecota g &U~nq petA.......... || | IIf || | + -- I |0 11 | | I | | II 1 | ||I 11:illlmii 1111 lllli~ii~imillltllliXll'If: I;I;:;;::::::illl Ago::,;i ^ p:ym iii'.,','..........m:^:;;;! ti ^ Figure 3.14 Time Response Curves for the System of Figure 3.10 with K = 50

67 CHAPTER IV HIGHER ORDER SYSTEMS 4 1 Background In chapter III the root locus method of analysis was used to analyze second order control systems that are conditionally stable. The ideas developed in that chapter can be extended directly to higher order systems. However, a more general approach is desirable, since the complexity and the number of possible system' combinations increases rapidly as the order of the system increases. Attention in this chapter is therefore directed toward the more general properties of the root locus as applied to conditionally stable systems. These include the following topics: 1. The behavior of the root locus for large values of s. 2. The sufficient conditions which will insure that the locus terminate in the left half plane, 3. The possible pole-zero configurations of KG(s), 4. Minimum or non-minimum phase closed-loop systemso 4,2 Minimum or Non-minimum Phase Closed-Loop Systems Up to the present time no specific mention has been made as to what characteristics the closed-loop function must possess. In this section the characteristics of these

68 systems are discussed and will serve as a justification for the material in the following sections. Consider the open-loop function C KK(s + Z1) - - (s + z1) (s + z) E(s) = KG(s) = (s + ) p - (s +p) -.(s ) 4a2-1 and the closed-loop function K(s + z) (s + z,) - (s + zm) c(s) =.....-......_..... 4 2-2 R) (s + -- (s + (s )+ K(s + z.) - (s + z) 4 which in this form focuses attention on the relation between the poles and zeros of the open and closed-loop functionso A comparison of equations 4,2-1 -and 4.2-2 reveals that the zeros of the open-loop function are also the zeros of the closed-loop function..while the poles of the closedloop function are dependent upon the zeros and the poles of the open-loop function plus the loop gain, Ko In order to investigate the effect a zero has at various locations in the s-plane, two examples will be studied, Example of a third-order system with one real zero Consider first the simple third-order system shown in Figure 4ola and let a finite zero be added on the real axis in the extreme left half plane. The behavior of the root-locus will now be investigated as this zero, z,, moves along the real axis. As z,- p3 the system becomes more like a second-order system with vertical asymptotes located

'z o0 uoT;eoIo eqa;o uo^Toun e s se sueqo snooq-ooI- aq4 MOH BuTMO4S s9oTd aueTd-s T'V aezn6Ta (U) (a).0 --.. A. ____ ____.o I 0'2 MDr D'd'| d i' d'd Xd (P)I (O) mr__ > -f 1-! —I'd *z'd Td p'd m~~~~~~~~~~~~~~~~~~~~~~,~ -i,, I I (q) (D).0.0 4d I I mi mf 69

70 at 0 (see Figure 4olb )o The case when z is located so as to cancel p3 is shown in Figure 40lc As z, is allowed to move toward the origin0 successively different root-loci are obtained which, of course, lead to a different set of closed-loop poles. When z, cancels P2 a second-order system results with vertical asymptotes at boy o When z, cancels the pole at the origin, another second-order system results with a vertical asymptotes at Now let z, enter the right half planeo Inspection of the root locus (see Figure 4olf ) reveals that no value of positive real gain exists that would lead to a stable closedloop system, since one of the branches of the root-locus always lies in the right half plane. Exam le of a third-order system with complex zeros As a second example consider a simple third order system, but now let a pair of complex zeros be introduced far out in the left half planeo Under these conditions0 the system will behave similar to a third-order systemo However0 the addition of the complex zeros will force the complex branches back into the left half plane for some large gain0 and therefore adding these zeros has made it a conditionally stable system of a type described by Brown and Campbell, namely, a system that is stable for small gains but which becomes unstable for larger gains and then becomes stable again as the gain tends to infinityo In Figure 4~2 is shown how the root-locus changes with changes in the location of the zeros. Attention is

so.zto xa.Cduto3 qqTm uraSOAs o p' tn6. Iq) D 3NVd- s (o) m _- _ - - -31 0 O: MO'3*? l k.~ 3-.1V3S 01 ION:31ON IZ 3NVld -- -- At,_ —-~ ~. o 00 *, _ __ _ _ _ ~ ~ ~'0000 O IL

72 directed toward Figure 4.2b where the complex zeros are allowed to move into the right half planeo Under these conditions the system will become unstable for some gain. Truly, this is an undesirable condition, because the closedloop performance has not been basically improved by the addition of the complex zeroso As a result of th abe above observations the author has restricted the work that follows to systems whose finite zeros all lie in the left half plane, or in other words, to closed-loop minimum phase systems, 4.3 Extension of the Root-Locus Techniques In section 2.6 some of the well-known properties of the root-locus method of analysis were presentedo However, in the application of the root-locus method of analysis to n-th order systems the author has found the multiplicity relationship between the s and KG(s) functions not clearly defined. Consider, for example, the simple open-loop function KG(s) 4o3-1 3 s and let KG(s) = U + j V = o e J and s =- + j = r e Substituting these quantities into equation 4o3-1 and then solving for r and & yields

73 / - J( and & S r --....... —~ The root-locus in the KG(s) plane (see Figure 4.3a ) is investigated by letting 0 < /< oo and =5 Tv- 2NT, while N, which acts as a parameter, takes on all integer values. For each new value of N, KG(s) will always map the negative real axis. However, a study of the rootlocus in the s-plane (see Figure 4.3b ) will reveal that iv I jio / / / / KG(s )-plane s-plane / / /. —-W a (a) (b) Figure 4.3 Showing the Multiple Mapping Property in the s and KG(s) Planes

74 N=0 = 3 N= 1 8 =- N - 2 a =7Nr N 3 7 = - - 3 3 Thus, we arrive at the conclusion that associated with each N there exists a certain curve or branch of the rootlocus in the s-plane. In this case, for N ~ 3 no new branches of the root-locus are obtainedo In general, however, the number of branches that occur depend upon the number of poles and zeros in the KG(s) function. The multiple mapping of the negative real axis in the KG(s) plane results in "slits" being mapped in the s-plane, An example of a more complex root-locus diagram is shown in Figures 4.4a and 44b, The different values of N being distinctly labelled. JV J KG(s)-plane s-plane N = 1, c = 37 _ N = 2, cp =.. N = 2, cp =5 ____, (a) (b) Figure 4.4 An Example Showing the Multiple Mapping Property of a Higher-Order System

75 If. in place of letting 9< take on fixed values in eauation 4.3-2 it is allowed to be a variable so that a sector in the KG(s) plane is mapped, there will be a corresponding sector in the s-plane which is defined by e = -. These are shown in Figures 4.5a and 4.5b, In a more complex case a sector in the KG(s) plane will map into an odd shaped region in the s-plane. jW - PLANE K6(s) -PLANE ENCLOSED SECTOR (a) (b) Figure 4.5 An Example Showing the Multiple Mapping Property Utilizing Phase-Angle Loci

76 As an example of a somewhat more complex plot, consider the open-loop function K(s + z1) (s + z ) ( + (s + z) (s + z4) KG(s) 1 4.3-3 KG() s(s + p1) (s + p2) (s + p3) (s + p4) where Iz41> z3 > IZ2l > lzl > IP31 > IP2 > IP1l A plot of the root-locus for this system is shown in solid lines in Figure 4.6, while the cross-hatched area represents the region associated with a sector in the KG(s) plane (in this case around / = t 2Nr). As the number of poles and zeros of KG(s) increase and as their locations change the sectors in the s-plane may take on odd shapes, However, for a fixed zero-pole configuration each section will always start at a pole and terminate at a zero of KG(s)o 4o4 Crossin or Intersection of Root-Locus Branches In the study of n-th order systems sooner or later the following question must be answered: "Is it possible for the branches of the root-locus of a given system to cross or intersect?" In section 4.3 it was shown that once the zero-pole configuration of KG(s) is given, a definite angular condition exists over the entire surface of the s-plane, Any fixed angular value in the KG(s)-plane will map "slits" in the s-plane, which start at each pole of KG(s) and terminate

77 (^/) / — -7r CONTOUR ^S^^^g 3 CONTOUR NOTE: SIMILAR SECTORS IN 3rd QUADRANT NOT SHOWN Figure 4.6 An Example of Multiple Mapping Utilizing Phase Angle Loci for a Complex System

at each zero of the system as the value of is increased from zero to infinity/ Now, in order for two distinct branches of a rootlocus to cross it is necessary for this crossing point in the s-plane to possess two different argument valueso The only place such a condition can exist for a rational function KG(s) is at its poles and zeros, Therefore it is impossible for two branches of the root-locus which possess different argument values to intersect anywhere in the S-plane excepta the poles and zeros of the K() funtion o The other type of intersection can be brought about by the intersection of two branches of the root-locus having the same argument valueo Inspection of the system shown in Figure 4,7 will reveal it to be such a system, It represents the one of a family of zero-pole configurations obtainable from this type (fourth-order) of system, that yields a unique angular condition everywhere on the four branches of the locus, which in this case is = -o A slight movement of the poles p, P,*, to the right will result in the zero-pole configuration shown in Figure 4 8, while a slight movement of the poles, p, p,*, to the left gives the zero-pole configuration shown in Figure 4 9,

79 -2+j4 -\I \ \ \ \ ~\ F~~~~~~ / / / / / e / / / / s-plane. A A2 k CG(s) ~ s(s+4)(s2+4s+20) / / / / / / / / / N — - N / -2-j4 Figure 4.7 An Example of a System Possessing Three Breakaway Points A more detailed investigation into the characteristics of systems which possess intersecting root-locus branches will reveal, that it is a characteristic of even the simplest second-order system. These systems which contain two closed

80 -3+j4 jW / I i / / \ I / / y ~ ~// s- PLANE / \I / _I // I-3-j4 \ Figure 4.8 System of Figure 4.7 with Complex Plane Poles Moved Further Into the Left-Half Plane / I / \ / I \ / \ / I / -3-j4 Figure 4.8 System of Figure 4.7 with Complex Plane Poles Moved Further Into the Left-Half Plane

81 -2+j4 s -PLANE Ir/4 W-4/ I \ /I / / I \ 2 —j4 Figure 4.9 System of Figure 4.7 with Complex Plane Poles Moved Toward the Right Half Plane

82 loop poles that lie on the negative real axis for small loop gains, but which become two'-critically damped poles for some larger gain, and then two complex poles for still larger gains. This in effect represents the same type of argument condition as the example shown in Figure 4.7 In the case of the second-order system, however, there exists only one point where the closed-loop system has two identical roots, while in the case of the fourthorder system for small gains (i.e. K =,8) the closedv loop system possesses two real roots, while for K = 1o25 v the system possesses two pair of complex conjugate roots at s = -2 + j 2.45. In section 4.7, it will be shown that the points in the system where two roots occur are in reality saddle points or points of stagnation of the KG(s) functiono 4o5 Conditionally Stable n-th Order Systems In Chapter 3 the subject of conditionally stable secondorder systems was introduced and a number of second-order systems were studied~ One of the topics considered at that time was that of stability. In attempting to extend the ideas presented to a general class of n-th order systems, it is necessary to return to the n-th order openloop function K(s + z1) ( + z2) - - (s + z) - ( + m) KG(s) = (s +.p ),(s +p ) - - KG(S) (s + p) (S + P) -- (S + Pj) -- (S + p 26

83 which was discussed in detail in section 206 In order to define a stability criterion for a general class of systems, it has been found necessary to classify the general n-th order system and to obtain an answer to the following question: "Given a rational openloop function, KG(s), which satisfies the definition of a conditionally stable function, what set of sufficient conditions can be placed on KG(s) which will insure C stability of the closed-loop function, C-(s)?" As a R result of this research three classes of open-loop functions were found which will always lead to a stable closedloop function for some real gain, Ko These will be discussed in turn: Class A Conditionally Stable system is defined as one possessing the following characteristics: 1) In equation 26-1 (n - m) = 1, or in other words, the open-loop function contains a single order zero at infinity, 2) KG(s) contains (n - 1) left half plane zeros, and 3) KG(s) contains one or more right half plane poles, or, a pole of order 2 or more at the originB Class B Conditionally Stable system is defined as one possessing the following characteristics: 1) In equation 26-1 (n-im).- 2, or in other words, the open-loop function contains a double order zero at infinity 2) KG(s) contains (n - 2) left half plane zeros,

84 3) KG(s) will contain one or more right half plane poles, or, a pole of order 2 or more at the origin. Class C Conditionally Stable Systemsare defined as ones possessing the following characteristics: 1) In equation 2.6-1 (n-m) j 3 or in other words, the open-loop function contains a zero of order three or more at infinity, 2) KG(s) contains (n - 3) or less left hand plane zeros) 3) KG(s) will contain one or more right half plane poles, or a pole of two or more at the origin~ The conditions that must exist for each class of systems to be stable will now be discussed, Consider first systems that satisfy the conditions defined under class A. The open-loop function will be of the form KG(s = K(s + zl) - - - (s + Z ) 451 KG(s) == _ ____ 1_ ______, ____ n-1 4o5-1 (s + p) - - (s + Pn) where all the z. lie in the left half plane, while p can fall in either the right or left half plane, To satisfy the conditionally stable requirement, i it is necessary that at least some pj be in the right half plane, or, as a limiting case a pole of order 2 exist at the origin. In the above class of functions, however, all the open-loop poles may lie in the right half plane.

85 Now since all the (n - 1) internal zeros lie in the left half plane, there will be (n - 1) branches of the root locus which terminate at these zeros~ The remaining external zero terminates at infinity along the negative real axis, and therefore some gain can always be found which will place all the closed-loop poles in the left half plane, Stability Criterion for a Class A Conditionally Stable System The stability requirement of the class A type system can now be stated: A system meeting the requirements given under Class A above can alwa ys be stabilized for some finite real gain, Ko Examples of root-locus plots for Class A conditionally stable systems are shown in Figure 4in~.. Inspection of the root-locus and the corresponding open-loop function, will reveal that a number of distinct zero-pole configurations have been included. Consider next systems that satisfy the conditions defined under class Bo The open-loop function will be of the form K (s+z,) ------ (s+ KG(s).: - n-2 4(5n2 (S+Pn) where all the z, lie in the left half plane, while pj can fall in either the right or left half plane, However, to satisfy the conditionally stable requirement, item nO 3)

86 L ij c0 a, i jw i j( i C0 -VP Figure 4.10 Typical Type A Conditionally Stable Systems

87 listed under this class of systems must be addedo Now since all the (n - 2) internal zeros lie in the left half plane, there will be (n - 2) branches of the root-locus which will terminate at these zeros as the loop gain tends to infinity, For these branches of the root-locus, therefore, some gain can be found which will place (n - 2) of the closed-loop poles in the left half plane. It is now necessary to investigate what happens to the remaining 2 branches of the locus which tend to infinity along the asymptotes, which were defined by equation 2.6-9 in section 206 and which is repeated here for convenience, n m 1i- + 7 2Nr 0 n m n For the case of (n - m) = 2, these asymptotes have an argument condition of - - + N7r, which makes the remaining branches of the root-locus tend to infinity in the jc direction. Now, if in addition, the pole - zero configuration is such as to make 0 < Q, these asymptotes will lie in the left half plane, and so will the remaining 2 closed-loop roots, Stability Criterion for a Class B Conditionally Stable System The stability requirement of the class B type of system can now be stated A system meetin the requirements given under Class B, above, can always be stabilized

88 for some finite real gain, K, if in addition to these requirements, a restriction is place on the zero-pole configuration so as to make lie in the left half plane. Examples of root-locus plots for Class B conditionally stable systems are shown in Figure 4.11 o They were chosen because of their distinct zero-pole character. Consider now the systems which satisfy the conditions defined under Class Co They represent a number of distinct types as (n - m) takes on various values greater than three In order to investigate the behavior of these systems, attention will be focused on the specific class defined by (n - m)= 3*. Under these conditions the openloop function will be of the form: k(s + 1) - - - - + Zn3) KG(s) =(S + p (s + 4. —3 (s + P1) n (S + Pn) where again all the zi lie in the left half plane, while pj can fall in either the right or left half planeo However, to satisfy the conditionally stable requirements, item no0 3) listed under this class of systems must be added0 Now in this case there are (n - 3) internal zeros which lie in the left half plane, and correspondingly there are (n - 3) branches of the root-locus which terminate at these zeros as the loop gain tends to infinityo For these branches of the root-locus it is therefore apparent that some gain can be found which will place

89 I I I / 1 - I/ 0-OPEN-LOOP ZERO / X-OPEN-LOOP POLE I ju I Figure 4.11 Typical Type B Conditionally Stable Systems

90 (n - 3) of the closed-loop poles in the left half planeQ It is now necessary to investigate what happens to the remaining 3 branches of the root-locus, which tend to infinity along the asymptotes as defined for this class of functions by n m n n-3 0-T~~~ _4 ~ Z~~pi. -4.5-4 n - m 3 and Arg n-T 2 N7 - + 2Nvr 4 55 Arg s- 4o5-5 n ~ m 3 Inspection of equation 4.5-5 reveals that the asymptote of the remaining branches of the root-locus tend to infinity along lines which make an angle of and - radians with respect to the positive real axiso Thus, it is apparent that for this class of systems, if the gain is made large enough the system will always become unstableo Intuitively at first sight, it appears that a system possessing one or more right half plane open-loop poles and having two of its asymptotes terminating in the right half plane will always be unstable However, a closer investigation into various zero-pole configurations will reveal that it is possible to have a zero-pole configuration which will lead to a system that satisfies the requirement of being conditionally stable, namely, of being unstable for small gains and then becoming stable

91 as the gain of the system is increased. However, for this class of system, if the gain is made sufficiently large these systems will always become unstableo It is apparent that the asymptote requirement, at least by itself, does not produce sufficient conditions for stability in the case of Class C systemso The above discussion on Class C (n - m = 3) type of systems can in general be extended to systems in which (n- m>3) The Arg s' will of course be different for each change in n - m For example, for (n - m = 4) the Arg s' =,.- 3 radians. This change in the Arg s' requirement will not basically alter the ideas developed for Class (n - m = 3) type of systems, and with slight modification they can be extended to these higher order systemso As (n - m) becomes larger, the possible range of K or the choice of a suitable zero-pole configuration will probably be severely restricted. To date no general criterion has been found which can be applied to Class C type systems However, certain statements can be made regarding this class of systems which will intutively help the designer when confronted with such a system0 They are: 1) Since in this type of system it is the branches of the root-locus which are farthest in the right half plane that must be controlled, the one way possible to "bend" these root-locus branches back into the left half plane is to cause a portion of

92 the Arg s' asymptote to lie in the left half plane. 2) The addition of one or more left half plane zeros near the origin will aid in causing the locus to move into the left half plane. 3) Routh's Stability criterion can then be applied to determine whether the root-locus has really crossed the jw axis. Numerous conditionally stable systems which are of the (n - m = 2) or (n - m = 3) type can be found in practice. However, no practical applications have been found where (n - m > 3). In Figure 4.12 will be found a number of typical zero-pole configurations for the Class C (n - m = 3) type of conditionally stable system. Inspection will reveaL that they are all for systems possessing few poles and zeros. For more complex systems, a wide variety of configurations is possible. One such example is discussed in detail in Section 4.8. 4.6 A Comparison of Defining Characteristics Between Conditionally Stable and Absolutely Stable Systems In previous sections attention was directed toward defining what is meant by a conditionally stable system and toward discussing the stability aspects of such a system. In this section attention will be directed toward answering such questions as:

93 s- PLANE s - PLANE / / / / / /. LIMITING CASE OF CONDITIONAL S ABILITY ORDER 2 ORDER -ORDER 3 (a) (b) NOTE: NOT TO SCALE / s - PLANE s-PLANE /f/ / / \ /\ \/ // / -- \ / \\ \ \\ \ ~~\. (c) (d) Figure 4.12 Typical Type C Conditionally stable Systems

94 1) How do the closed-loop frequency responses of these systems compare for the same magnitude of K? v 2) What frequency response characteristics do the closed-loop systems possess if the open-loop amplitude characteristics are identical? 3) How does the closed-loop time response of a conditionally stable system compare with that of an absolutely stable system? Since in general the answers to these questions depend upon the zero-pole configuration of the functions involved, it was necessary to define the relative zeropole position of the functions utilized in the investigation, After some deliberation, it was decided to utilize the closed-loop zero-pole configuration first proposed by Guillemin6 and later exploited by Truxal. However, both authors restricted their discussion to systems possessing left half plane poles. In the work that follows, all types of systems will be considered. As mentioned by Guillemin, his choice of a closed-loop function leads to open-loop systems which have negative real axis poles and zeros, and can therefore be readily synthesized as R-C networks, providing of course, that the "plant" possesses real axis poles and zeros. The closed-loop function utilized in his study was 26. J. G. Truxal "Servomechanism Synthesis through Pole-Zero Configurations", MIT Research Laboratory of Electronics Tech. Report 162, August, 1950 27. J. G. Truxal, "Automatic Feedback Control System Synthesis", (book) McGraw-Hill Book Company, Inc., 1955, Chapter 5.

95 2 P W C1 (s + Z1) R() =. 2 4.6-1 (s + 2 ny Sn + S + )(S + ) which contains two control poles and a real axis pole and zero. The values of z1, pi, Pt and Y are chosen in such a manner so as to meet the closed-loop requirements of frequency response, time response, specified K, and etc., or some combination of these quantities. One of the reasons for choosing this function in this study is that an explicit solution of the open-loop function can be obtained directly. Carrying out this procedure leads, after some manipulation,to: 2 p z lC (s + 1) E(s) = Z.. 1 4.6-2 2 ~2 2 PiWn s s + (2 OWn Pl )S+ p)s + + u nP ) L z1 1 The roots of the quadratic term with the brackets are: - 2 nn 1 + 1' 2 2 2 co + 2 () Pl - n n 1 4.6-3

96 Thus, given the closed-loop poles and zeros, it is possible to substitute these values directly into equation 4.6-3 and obtain the location of the open-loop poles, which in general are not located at the origin. Inspection of equation 4.6-3 reveals that depending upon the relative magnitudes of z1, P, and n) this closed-loop function can possess the following types of open-loop poles: ~ 2 - 2 nl 1) For the case when a + 2 nPl --- > 0,?n J nn 1 z s and s2 are two left half plane open loop poles; s3 is located at the origin. 2 2 PlWn 1 2) When [n + 2 t n - z1 z < 0' C -(s) will contain one right and one left half E half plane pole; s3 is located at origin. 2 n 3) When W + 2 p l -n = 0, ~n n' z2 C E(s) will possess a double order pole at the E origin and one left half plane pole. Further investigation into the conditions which lead to the open-loop pole configuration of 3) above, will reveal that it is directly related to Truxal's* equation which relates the K of a unity feedback control system V to the zero-pole configuration of the closed-loop function. As derived by Truxal, this is * See reference No. 27, page 284, equation 5.21

97 n m 1 >!r - r 1 1_ sC~~ 1 _ 7~ 1 4.6-4 K = p z. v j J ij= For the system defined by equation 4.6-1 this reduces to 1 + - - - 4.6-5 K 03 pl Zl v n Now for K t=c, it is apparent that v + 1- - =0 4.6-6 n P1 1 or 2 2n- 2 n P 0. 4.6-7 n +rWnPlz The right side of equation 4.6-5 is the constant term in the quadratic factor of equation 4.6-3. Thus, condition 3) above corresponds to the case of infinite K, which is v in agreement with the other definition, namely, 2 C Pl ~n K = lim s (s) = lim s n (s+ z) v E z 1 s -;O. s,.0 1's? (s + s. ) = oo for s-o0 It is also apparent from 2) above that certain values of Z 1, P', and 0n, correspond to certain closedloop requirements. These requirements can be met only

98 if a right half plane open-loop pole is utilized. By dividing the expression in 2) above by pi and 2 (which of course assumes neither to be zero) the following expression is obtained: 31-4 2 _ 1 <0o 4.6-8 P1 On C 1 Thus, if the sum of the reciprocal of the closed-loop poles is less than the reciprocal of the left half plane zero, the open-loop function will contain a right half plane pole. In addition, it should be pointed out that whenever a closed-loop zero-pole configuration is of this type, it will always lead to a system which contains a negative K, (a characteristic of systems possessing an v odd number of right half plane open-loop poles). The relative position of the poles can be better observed if in the expression 4.6-8 the substitution P1 PI A = and B _- is made. The following expression w z results: 1 + 2A - B <0. 4.6-9 In Figure 4.13 will be found a plot of this expression for A and B greater than zero, and with as a parameter. Thus for a specified Y when B falls in the shaded area, a right half plane open-loop pole results. In order to gain further insight into this subject, a comparison is made between two systems whose closedloop functions are of the form given by equation 4.6-1, but whose zero-pole configurations are such that one of

5 4 B= P/Z, 3 2 I 4 0~ A - F /wn Figure 4.13 A Graph of the B vs. A for Two Values of.

100 the systems possesses a right half plane open-loop pole, while the other system possesses all left half plane openloop poles. In addition, a further constraint is placed on the open-loop transfer functions, namely, that they possess the same magnitude of velocity constant, K. v The two systems considered possess the following transfer functions: System I. _ and (s) Is 75 (s + 5/3) (s + 5)(s2 + 5s + 25) 75 (s + 5/3) s(s + 12.07)(s - 2.07) 4.6-10 4.6-11 System II - II and II 75 (s + 5/3) -_8. (s + 1.55)(s2 + 12.59s + 80.5)' 4.6-12 75(s + 5/3) s(s + 12.07)(s + 2.07) 4.6-13 Pi Pi In system I above - - 1 and = 3, while in system II n 1 Pi Pi -.173 and - =.928. It is apparent from equation 0) Z n 1 4.6-11 that system I contains an open-loop right half

LCI plane pole, while a study of equation 4.6-13 reveals that system II does not. Taking the inverse transforms of equations 4.6-10 and 4.6-12 gives equations 4.6-14 and 4.6-15, respectively. -5t c(t) = 1 + 2e + 4.6-14 3.16 e 25tSin(4.34t - 79.2~) and -l.55t -6.3ts c(t) = 1-.0915 e +155 1.29e 3tin(6.4t - 135.40) 4.6-15 A comparison of the time responses of these systems will be found plotted in Figure 4.14. Inspection of these curves reveal that the time response of system I possesses a high degree of overshoot. This seems to be a characteristic of systems possessing right half plane open-loop poles. On the other hand however, if the criterion of performance is to be the rise-time, it is apparent that the rise-time of system I is considerable less than that of system II. It is left as a future investigation to make a detailed comparison regarding the defining characteristics. 4.7 Relation Between Saddle Points and Root-Locus Plots In previous parts of this dissertation various aspects of root-loci and phase-angle loci were discussed, In this section it will be shown that the breakaway point on a root-locus plot is in reality a saddle point of the KG(s) function.

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10 As discussed in Chapter 2, the root-locus plot in the s-plane graphically represents where the Arg KG(s)= r + 2N7r radians. It was shown in section 2.8 that the root-locus is a special case of the more general phase-angle loci where the Arg KG(s) = + 2Nr radians, in which 9 takes on all positive real values. In place of having Arg KG(s) as a parameter it is possible to consider IKG(s)I as a parameter. In the KG(s) - plane this corresponds to a series of concentric circles about the origin. In the s-plane the corresponding contours will take on some odd shape depending upon the zero-pole configuration of the system. In Figure 2.5a will be found amplitude and phaseangle plots of KG(s) in the KG(s)-plane, while in Figure 2.5b will be found the corresponding plots in the s-plane for a simple second-order system containing two real poles. Additional information can be obtained however, if a three dimensional plot is made which involves the magnitude of KG(s) as an axis perpendicular to the CC and jw axes of the s-plane. The plot IKG(jo) I vs. cu is a special case in this multi-dimensional plot which can be obtained by passing a plane through the |KG(s)I and jw axes, and then letting s = jw be the variable. If however IKG(s)I is plotted as s is allowed to take on all complex values, a three-dimensional plot is obtained which looks like a "relief" map. On such a map, constant amplitude lines of KG(s) would appear as level contours of constant elevation. In Figure 4.15 is shown a three dimensional plot of the

104 IKG(s)I * 8 - PLANE Figure 4.15 "Relief" Map of the Function KG(s) = -

105 function |KG(s) s= s. Inspection of this plot shows a pronounced characteristic, namely, it looks like a "pole", which in this case is located at the origin. Common usage of the term "pole" for a root which lies in the denominator of a function, F(s) dates back to other fields of applied mathematics where this type of function is often found plotted. However, to date, in the area of control systems little work has been published which utilizes the multi-dimensional contours in the analysis or synthesis of feedback control systems. In this section attention is directed toward relating the breakaway points on a root-locus to saddle points as defined in potential theory. Consider now a simple second-order system of the type depicted by the phase-angle and amplitude loci of Figure 2.81a and 2.81b. The root-locus plot which is a special case of the more general phase-angle loci is given in the Figure by Arg KG(s) =- = -T. If s is allowed to take on all complex values and | KG(s) | vs. s is plotted in a three-dimensional fashion as described above, Figure 4.16 will result. Inspection of this Figure reveals that the "relief" map for this function contains two "poles" which occur at each of the singular points of KG(s). In addition, it is apparent that a trough or minimum point exists at point, s m along the real axis between these two poles. Since there are points removed from the real axis for which

IKG(s)| / 0 K ON Figure 4.16 "Relief" Map of the Function KG(s) = s(s + a) s (s + a)

107 |KG(s)) < IKG(sm)L in reality this trough looks rather like a "mountain pass" or has the familar shape of a "saddle". Thus, point s is referred to as a saddle point of KG(s). Since s is a minimum point for s = T, it is possible to determine the minimum value by setting KG(s)' = 0. It should be noted that since Arg KG(s) is not plotted in this case, it is only a coincidence that this point occurs along the root-locus curve. In this particular case it is part of the negative real axis that lies between the poles. Sometimes in finding KG(s)' = 0, it is possible to obtain more than one value for s. While inspection of m a root-locus plot may reveal there is only one possible breakaway point. The following questions then arise: 1) Which, if any, of the values of s is correct? m 2) What do the other values obtained as a result of setting KG(s)' = 0 represent? In an attempt to answer these questions a third example will be considered, namely, a second-order system that contains a left half plane open-loop zero. In Figure 4.17 is shown the "relief map" for the system defined by the following open-loop function C K(s + 2) 4.7-1 (s) = KG(s) (s + 1) E s(s + 1) A study of Figure 4.17 reveals that the addition of a real left half plane zero has altered the characteristics of the "relief map" of the simple second-order system of

108 Figure 4.17 "Relief" Map of the Control System defined by equation 4.7-1.

109 Figure 4.16. Setting KG(s)' = 0 in this case yields two values of s, namely, s = -.59 and s = -3.41. From m m m Figure 4.17 it is seen that these values define where a minimum and a maximum occur in KG(s) as s is allowed to vary along the negative real axis. A study of the rootlocus for this system, which is shown in Figure 4.18, shows that the minimum or saddle point of KG(s) occurs at the one breakaway point of the root-locus, while the maximum point occurs at the other breakaway point where the complex root-locus returns to the real axis. As a final case, the system whose root-locus is given by Figure 4 7 will be investigated. As it will be recalled, this is the system which contains intersecting root-locus branches. In Figure 4.19 will be found the "relief map" for this system. A study of the figure reveals that this system contains three saddle points, one on the negative real axis mid-way between the real axis poles and two others which occur at the complex values of s, namely at s = -2 + j 2.5. In addition, as mentioned previously, m the root-locus does not in general indicate the bottom of the valley between two poles. The characteristics relating the breakaway point of a root-locus plot to the saddle point commonly used in potential theory will now be summarized: 1) Setting KG(s)' = 0 is a convenient method for locating the breakaway points on the root-locus of a low-order transfer function.

i10 S-PLANE Figure 4.18 Root-Locus Plot of the System defined by equation 4.7-1

il IKG(S)I f Figure 4.19 "Relief" Map of a Control System with Four Open-Loop Poles

112 2) Since not all values of sm determined in this fashion are breakaway ~points, it is necessary to utilize the Arg conditions of 1 + KG(s) =0 to determine which values of sm apply. 3) For higher order systems the determination of s requires the solution of a high order polynomial in s. This leaves some question as to the desirability of utilizing this approach if extreme accuracy is not needed. 4.8 Practical Examples of Conditionall Stable Systems As mentioned in Section 3.2 conditionally stable systems can occur in two ways, namely, either as the result of the designer being "given a plant" that contains one or more right half plane open-loop poles and which is therefore open-loop unstable, or as the result of his changing an inner loop of a complex system in order to achieve some desired performance that couldn't be obtained by using an absolutely stable inner loop. Under the former conditions the closed-loop system will always be conditionally stable, and care in design must be exercised if the closed loop function is to be a stable one. One of the prevalent examples of the former type can be found in the missile industry at the present time, where the dynamic behavior of the missile in the vertical plane in inherently unstable in uncontrolled flight, and suitable control must always be devised. A typical example of such an open-loop function is the following:

113 KG(s) - K (s +.025) -4.8-1 (s +.473s + 9.43) (s -.063) It is apparent that without feedback the system is unstable, since it possesses a right half plane pole. From the root-locus plot of the uncompensated unity feedback system shown in Figure 4.20, it is also apparent that the closed-loop system will also be unstable for K< 23.8. In addition since the complex closed-loop poles possess very low damping the system will be highly oscillatory even for K>23.8. Inspection of equation 4.8-1 and/or the root-locus in Figure 4.20 reveals it to be an n - m = 2 system possessing one right half plane pole and it is therefore a class B conditionally stable system. Utilizing the ideas developed in previous sections, but principally the results of Section 4.5, it is found that with a series compensating network of the form k — it is possible to modify the characteristics of this system and thus improve its performance by moving the complex poles away from the jw axis to the =.5 line, thereby allowing an increase in K and allowing the v system to have a larger bandwidth. From the root-locus of modified system which is shown in Figure 4.21, it is apparent that the addition of the series compensation network has forced the asymptote line from T. = -.18 to T = -13.65. In the normal case this would permit a range of loop gains to be utilized, unless very specific bandwidth requirements are specified.

114 s - PLANE I I I. I I P I I I G) iw - wC -PLANE wv —. I I 1 I.- -30 I / - o =Q185 \ OX s -0.236+ 3.06 -13.65s - 0.236 - 3.06 Root-Locus Plot of Compensated System Figure 4.20 Root-Locus Plot of Uncompensated System Figure 4.21

115 As an example of the advantages of utilizing a conditionally stable system to achieve a prescribed performance, consider the double-loop system shown in block diagram form in Figure 4.22. The physical form of this system may consist of the following possible types of components: K = an amplifier 4.8-2 K2.-... = a field controlled motor 4.8-3 s(s + 1) sK = an electric tachometer 4.8-4 s K 2-.... =2 an active filter 4.8-5 (s+1) (1 +.25s) The performance of this system will be studied under various operating conditions. First consider the system, if it were operated without the tachometer feedback loop. This is equivalent to opening the tachometer loop at point, p, (see Figure 4.22). The resulting system is now a singleloop second-order system, whose performance is defined by the following open-loop and closed-loop functions: C K1 2 (s) = 4.8-6 E s(s + 1) C K1 K2 R(s) = 2.. 4.8-7 + S + K K

R (s Figure 4.22 Block Diagram of a Two-Loop Control System

117 In order to study the system's performance two methods of analysis will be applied, namely, the Nyquist diagram and the root-locus technique. The results obtained in this fashion are shown in Figures 4,23 and 4.24. From Figure 4.23 the following information is obtained: for -1 an M = 1.3, K = 1.37 sec and cu = 0.9. From the rootp v r locus plot in Figure 4.24, utilizing the same K, the v closed loop poles are found to be located at s = - - + jl.05 ~~~~~~~~1 2 and s = - - jl.05. The addition of the normal passive compensation tLe-cwork in cascade with the open-loop function of equation 4.8-6 will result in only a mild increase in the system K and rc for the same M. Attempts to V rp improve materially the K or wr using only a single-loop system are hopeless. If the feedback loop is now closed at point, p, an entirely different approach is being made to the problem. The system now contains two closed loops (labelled 1 and 2 in Figure 4.22). It is now possible to vary the over-all system performance over a wide range by adjusting the gain, K4, (see equation 4.8-8) in loop 2. In order to demonstrate the type of performance that may be obtained, the over-all performance will be evaluated for two specific values of K. Consider now the two-loop system shown in Figure 4.22 with point, p, closed. The equations governing the behavior of the system under these conditions are: 3 K K Kt s D 2 2 t - (s) = or 1 (s + 1)' (1 +.25 s) 3 K s D (s) = -.4 - 4.8-8 E1 (s + 1)' (s + 4)

118 iv (jI) - PLANE p= 1.3 jl.O.5 U -i.5 -ji.o -jl.5 -j2.0 Figure 4.24 Nyquist Diagram for System of Figure 4.22 with p open

119 j(w RELATIVE DAMPING j1.o.75 S-PLANE j.5 j.25:5.75 A CLOSED-LOOP POLES X OPEN-LOOP POLES 1i0 Figure 4.23 Root-Locus Plot of System of Figure 4.22 with p open

120 and C K (s + )2 (s + 4)...... _(s) = 4 4.8-9 E 4 1C~~ 3 2 s Is + (7 + K4) s + 15 s + 13 s + 4 The behavior of the system is now investigated by first studying the behavior of the inner loop using a Nyquist diagram, Bode diagrams, and the root-locus plot. These plots are shown in Figures 4.25, 4.26a, 4.26b, 4.26c, and 4.27. The three methods of analysis are carried along more or less in a parallel fashion as in the case of the single loop, primarily for comparison. Inspection of Figure 4.25 which contains the Nyquist diagram, reveals that for values of gain K4> 12 the system will encircle the -1 + jO point in the - (jO) E, plane two times as w is varied from - o to + M o Applying Nyquist's criterion in its most general form, namely, iz = N-P. to this diagram indicates that the inner loop system contains two right half plane closedloop poles. Figure 4.27 shows that this closed loop system contains two real left half plane poles and two complex poles which move from the left half plane to the right half plane as the system gain, K4, is increased from 0 to Xo. For the value of K = 12, two branches of the root-locus lie on the jwX axis. This corresponds to the D - (ju) locus passing through the -1 + jO point in the Nyquist diagram.

,jv j.6 -(jw) — PLANE E I I:=3.0 j.2 U -.6 -.4 -.2.4.8 Figure 4.25 Nyquist Diagram of D (jo) i 1

122 I o? Co N 8Q 0 - -20 - -40 -60 -80 - -100 - -120 / -140 -.01.10 1.0 10 log w Figure 4.26a Log Amplitude Response of K (jO) vs. Log K5E1 100 Z3 IU. o z Q 0.10 1.0 log W Figure 4.26b Argument of D vs. Log 51~ (m

(D 4. 0 0 I E- 0 (0 0o 0 o. 4 too 9 0 I-h t" I-h (D (D I 2 21-II ---— \ — 1. I — -- -- -} i _ _ _,_ i _/lI%[_ _ j _ ~ _I_ _ —_ _ _ _ _-' /Irr l^ ~ ^ Z ^ ^ " L _ ^._'I -4 l\ [! -, I-: / I-! - -./:, i ~ iXt\I. I I - -.. -'" < *""' *T' - K -i^S y ^ ^ ^ ^ ^' - ^ ^ - ^ C S ^ L -48 * -. -,I ^^^^^g^ ^ -- _ _.,o -2 - - ". -- -' *6 - ~[,l, -...1..l l- - - - < _-3E2k~ -^'~p^3I l/ -38I -40 Al.mat a=ort o' s' 40 r d- - - - -S' -2 -- - - - -- - - -- III I I yw lw PHAE ANGLE OF JQ() E

jw S - PLANE =-.155 + j.65 S, =+. +j.3 S4 =-147.2 S =20.5 S2=+.- j. S2=+.I- j.3.155-. 65 H N) 4.>1 DF Figure 4.27 Root-Locus Plot of the Inner Loop Function - (s) 1

125 The performance of the over-all system will now be studied for two specific values of inner-loop gain, namely, K = 3.55 and K = 35.5. For K = 3.55 equation 4.8-9 in factored form becomes: 1)2 C ) K(s +....(s + 4) 4.... E s(s + 20.5) (s +.155 + j.65) (s +.155 - j.65) while for K = 35.5 equation 4.8-9 takes on the following form C(5) _ K(s + l)2 (s + 4) 4 s-l E s(s + 147.2) (s -.1 + j.3)(s -.1-j.3) A study of equation 4.8-10 and 4.8-11 reveals that changing the gain by 20 db has resulted in the following modifications: 1) Moving the complex poles of -(s) from the left half E plane to the right half plane. 2) Moving the real axis pole much farther into the left half plane. In Figure 4.28 will be found the Nyquist diagrams corresponding to the two values of gain, K4. It is apparent from an inspection of the diagrams that the overall openloop behavior is completely different in the two cases. This is due to the pair of right half plane poles in the open-loop function in equation 4.8-11 which causes a C large amount of phase shift in the -(jw) function. E

900 C Figure 4.28 Nyquist Diagram of C(jo).for System of equation 4.8-9 E H O0\

127 The closed-loop behavior of the over-all system will not be investigated using the Nyquist diagram, and the associated M circles. The closed-loop frequency response for these two values of K4 gain setting are shown in Figure 4.28 for K adjusted to give an M = 1.3 in each P case. The velocity constant for this system when adjust-l ed in this fashion is K = 150 sec, for K4 = 3.55 and 4 -1 v K = 10 sec for K4 = 35.5. Thus, by adding an unstable v 4 inner loop, the K of the system has been increased by v more than sixty fold. From Figure 4.28 it is apparent that the resonant frequency of the two systems is also widely different. In Figures 4.29a and 4.29b will be found the rootlocus diagram for the system when operating under the above conditions. Inspection of the diagrams indicates the presence of the right half plane open-loop poles cause the complex branches of the root-locus to move rapidly far into the left half plane as the gain, K, is increased. A constant damping line, f =.5 for the two cases indicate that much larger values of K1 can be used before the system's complex closed-loop poles cross this line for the case of K4 35.5 than for K = 3.55. This yields, therefore, a large velocity constant and a correspondingly smaller steady-state velocity error. In addition, moving the closed-loop complex poles farther into the left-half plane results in a system having a large bandwidth.

128 S- PLANE t N.5 LINE -— 0.155.+ j0.65 -20.5 -4 -2 -I -Q s - PLANE 5 =.5 LINE A LOCATION OF CLOSED POLES MIRROR IMAGE OF TOP BRANCH Figure 4.29a Root Locus Plot for System of Equation 4.8-9 with K = 3.55 4

129 S-PLANE ENLARGED VIEW OF.I+jO.3 Figure 4.29b Root Locus Plot for System of equation 4.8-9 with (4 = 35.5

130 CHAPTER V CONCLUS IONS The major objective of this dissertation was to establish a better understanding regarding the characteristics of conditionally stable systems. Although these systems are found discussed in technical articles and texts on feedback control systems, no organized treatment of the subject was to be found. The subject matter was introduced with a brief discussion of control system terminology and methodology. This was followed by a discussion (see Section 3.2) of the meaning of a conditionally stable system, where it was shown that the word conditionally stable has been used to define two types of systems possessing widely different characteristics. In this work the definition of a conditionally stable system is one which becomes unstable as the loop gain is decreased from some maximum value to zero. Utilizing this definition allowsit to include systems that possess both of the above characteristics. Also to be found in this chapter are the results of an extensive study of conditionally stable second-order systems. The insight in studying these systems was then utilized in the study of n-th order systems, which culminated in a set of sufficient conditions for conditionally stable n-th order systems. More specifically, it was shown that

131 if the closed-loop function is constrained to be a minimum phase function, the open-loop function could be catagorized into three types, A, B, C. Sufficient conditions were found for types A and B systems which would guarantee stable operation of these systems for some specified loop gain, K. In addition, in Section 2.8 it will be found a description of a novel method for solving for the roots of an n-th degree polynomial utilizing a modified rootlocus technique, which to the best of the author's knowledge has not been published. While in Section 4.4 will be found a discussion regarding the intersection of rootlocus branches, where it was shown that it was impossible for root-locus branches not possessing the same argument condition to cross except at a pole or zero of the openloop transfer function. Section 4.7 discusses the relation between saddle points of potential theory and the breakaway points of the root-locus. Three dimensional contour plots, indicate that breakaway points of a root-locus may C be either maxima or minima of the -(s) function. It was ~C ~~E shown that setting -(s)' = 0 is a convenient method for locating the breakaway points on the root-locus of a loworder transfer function. Since not all the values of s determined in this fashion are breakaway points, the approximate shape of the root-locus must be known beforehand. For higher order systems the determination of s requires the solution of a high order polynomial in So

132 This leaves some question as to the desirability of utilizing this approach if extreme accuracy is not needed. Although much of the emphasis in the control system theory at the present time is directed toward nonlinear systems, there are numerous topics in the realm of linear control theory which have not been adaquately treated. These include 1) An adequate method of describing the requirements about a given physical system, so that the synthesis problem becomes unique, 2) Extending the ideas of the root-locus so that it can be more readily adapted to higher order systems, 3) To utilize more of the complex variable theory which is probable available and known to the mathematicans in order to enhance our understanding of systems.

133 BI BLOGRAPHY 1. Bode, H.W., "Network Analysis and Feedback Amplifier Design", (book), D. Van Nostrand Book Co., Inc., New York, N.Y., 1945, pp. 168-169, 451-476. 2. Draper, C.S., Bentley, A.P., "Design Factors Controlling the Dynamic Performance of Instruments", Transactions ASME, Vol. 62, pp. 421-432, July 1940. 3. Hall, A.C., "The Analysis and Synthesis of Linear Servomechanisms", Technology Press, Massachusetts Institute of Technology, May, 1943. 4. Harris, Herbert, "Frequency Response of Automatic Control Systems", Electrical Engineering, AugustSeptember 1946, pp. 539-546. 5. Hazen, H. L., "Theory of Servomechanisms", Journal of Franklin Institute, Vol. 218, 1934, pp. 279-331. 6. Hazen, H. L., "Design and Test of a High-Performance Servomechanism", Journal of Franklin Institute, Vol. 218, November 1934, pp. 543-580. 7. MacColl, L.A., "Fundamental Theory of Servomechanisms" (book) D. Van Nostrand Book Co,, Inc., New York, N.Y., 1945. 8. Nyquist, H., "Regeneration Theory", Bell System Technical Journal, Vol. 11, January, 1932, pp. 126-147. 9. Brown, G.S., Campbell, D.P., "Principles of Servomechanisms", (book), John Wiley & Sons Book Co., 1948, p. 172.

134 10. Herr, D., Gerst, I., "The Analysis and an Optimum Synthesis of Linear Servomechanisms", AIEE Transactions, Vol. 66, 1947, pp. 959-70. 11. Travers, Paul, "A Note on the Design of Conditionally Stable Feedback Systems", AIEE Transactions, Vol. 70, 1951, pp. 626-630. 12. James, H.M., Nichols, N.B., Phillips, R.S., "Theory of Servomechanism", (book), McGraw-Hill Book Co., Inc., New York, N.Y., 1947, p. 182. 13. Nixon, F.E., "Principles of Automatic Controls", (book) Prentice-Hall Book Co., N.Y., 1953, pp. 114-121. 14. Gardner, M.F., Barnes, J.L., "Transients in Linear Systems," (book), Vol. 1, Chapter V, 1942. 15. White, D.C., Woodson, H.H., "Electromechanical Energy Conversion", (book), 1959, Chapter II. 16. Graybeal, T.D., "Block Diagram Network Transformation", Transactions AIEE, Vol,70, pp. 985-990, 1951. 17. Evans, W.R., "Graphical Analysis of Control Systems", Transactions AIEE, Vol. 67, 1948, pp. 547-557. 18. Thaler, G.J., and Brown, R.G., "servomechanism Analysis", (book), McGraw-Hill Book Co., Inc., New York, 1953, Chapter 14. 19. Evans, W.R., "Control System Dynamics", (book) Chapter 7 and 8, McGraw-Hill Book Co., Inc., New York, 1954. 20. Truxal, J. G., "Control System Synthesis", (book) McGraw-Hill Book Co., Inc., 1955, Chapter 4, pp. 224-277.

135 21. Gille, J.C., Pelegrin, M.J., Decauline, P., "Feedback Control System", (book), McGraw-Hill Book Co., Inc., 1959, pp. 235-255. 22. Reza, F.M., "Some Mathematical Properties of RootLoci for Control System Design", Transactions AIEE, 1956, Basic Science Paper 56-125. 23. Yeh, V.C., "Synthesis of Feedback Control Systems by Gain-Contour and Root-Contour", Transactions AIEE, (App. and Ind.), May 1956, pp. 85-95. 24. Yeh, V.C., "The Study of Transients in Linear Feedback Systems by Conformal Mapping and the Root-Locus Method", Transactions ASME, April 1954, pp. 349-361. 25. Chu, Yaohan, "Synthesis of Feedback Control Systems by Phase-Angle Loci", Transactions AIEE, Vol. 71, Part II, 1952, pp. 330-339. 26. Truxal, J. G., "Servomechanism Synthesis through Pole-Zero Configuration", MIT Research Laboratory of Electronics Tech. Report 162, August 1950. 27. Truxal, J.G., "Automatic Feedback Control System Synthesis", (book) McGraw-Hill Book Co., Inc., 1955, Chapter 5.

136 VITA Name: Date and Place of Birth: Louis F. Kazda September 21, 1916. Dayton, Ohio Elementary School Junior High School Senior High School: College: Degree: Date: Graduate Work: College: Graduate Appointment: Degree: Date: College: Graduate Appointment: College: Degree: G. A. Lange School, Dayton, Ohio Lincoln Junior High, Dayton, Ohio Stivers High School Dayton, Ohio University of Cincinnati, Cincinnati, Ohio Electrical Engineer June, 1940 Electrical Engineering University of Cincinnati, Cincinnati, Ohio September, 1941, Teaching Fellow Master of Science in Engineering June, 1943 University of Michigan Ann Arbor, Michigan September, 1947, Instructor. Syracuse University, Syracuse, New York. Doctor of Philosophy, January, 1962.