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I Bui Tien Rung FEEDBACK CONTROL SYSTEMS: DESIGN WITH REGARD TO SENSITIVITY, ' I u. S. Naval i'o->tuia<!uate Monterey. California Schocrt UNITED STATES NAVAL POSTGRADUATE SCHOOL FEEDBACK CONTROL SYSTEMS: DESIGN WITH REGARD TO SENSITIVITY by BUI TIEN RUNG, LT., VIET NAM NAVY GEORGE J. THALER, DR. ENG. RESEARCH l*9PeiCr NO. 41 JANUARY 1964 1 I FEEDBACK CONTROL SYSTEMS: DESIGN WITH REGARD TO SENSITIVITY Research Report submitted by Bui Tien Rung, / George J. Thaler, Dr. Eng Lieutenant, Viet Nam Navy . , Professor of Electrical Engineering U. S. NAVAL POSTGRADUATE SCHOOL Monterey, California January 1964 i ERRATA. SHEET FOR RESEARCH PAPER NO. la PAGE LINE INSTEAD OF PLEASE READ Iv 17 satisfied satisfies 1 If wrong using 19 1 clockwise counterclockwise 22 28 irregularities singularities 22 29 irregularities plant singularities 2k 7 OZ QZ 2k 8 OP QP 25 29 JON=JON J0N=J0M .25 29 JN=JN JN=JM 29 18 TT< dq = q<2< TK^ - ^<2ir 55 8 expect P^ expect Pp 55 10 then the 59 12 unidentical n identical 59 15 August 196 August 1965 .rj? V. S. r a.-’’ . *■■ Sv.h. j, 'VT«>nf'ei c V* - ' FEEDBACK CONTROL SYSTEMS: DESIGN WITH REGARD TO SENSITIVITY Table of Contents Chapter One : I-l : 1-2 : 1-3 : 1-4 : 1-5 : 1-6 : 1-7 : Root sensitivity to parameter changes Introduction Purpose of feedback Small parameter changes, large parameter changes System sensitivity and root sensitivity Survey of previous works and scope of this work Root sensitivity: Definition Property 1 - Relationship between S^, S^> S^ K. P Z Property 2 - Relationship between root-sensitivity of -q^ and residue at Property 3 - Sum of all S^ and S^ in a system Chapter Two : II- 1 : II- 2 : II- 3 : II- 4 : II- 5 : II- 6 : II- 7 : II- 8 : A sensitivity design method Introduction Practical aspects of problems with small parameter changes Graphical method for determining root-sensitivities: the vector diagram An example of application of the vector diagram til f ' ' Particular case of open-loop singularities of n — order A design philosophy Locus of U on the s- plane Limit of locus of U ii Table of Contents (Continued) Chapter Two ; II- 9 ; II-IO : II-ll : 11-12 : Appendix References A sensitivity design method (continued) Design techniques The three- step procedure Design for minimum root sensitivity Design for constant damping when gain varies Design for constant damping when a singularity varies Other possible sensitivity designs using the U>locus Design examples Design example No. 1; minimum sensitivity Design example No. 2: constant damping when 1 varies Design example No. 3: constant damping when p varies Analog computer simulation Single-stage or multiple-stage compensator? iii ABSTRACT The purpose of using feedback in a control system is not merely to improve its static and dynamic performance and eliminate or minimize the effect of noise, but also to eliminate or minimize the effect of unpredictable changes within the plant itself. Such changes are expressed in terms of variations in the plant's parameters, i.e., gain constant, plant poles, plant zeros. This paper starts with a definition of "root sensi- tivity", relating the changes in plant's parameters to corres- ponding changes in the system's roots. Interesting properties of root-sensitivity are shown, then applied to the derivation of a laborless graphical method for obtaining the sensitivity of a given root. Finally a compensation design method is proposed, which not only secures a desired location for the system's dominant roots on the s-plane, but also simultaneously satisfied conditions concerning the sensitivity of these dom- inant roots to the varying plant parameter (ters). Examples are solved using the proposed method, and the results verified with the analog computer. iv CHAPTER ONE Root Sensitivity to Parameter Changes 1; Introduction I s It is well established that the reasons for wrong feedback in con- trol systems can be classified as follows; 1 - To improve static and dynamic performance of the system . Feed- back can stabilize an unstable system or increase the stability of a stable system. Feedback can shape the system response in to some desired pattern. Feedback can reduce the steady-state error of a class of control systems. This aspect of feedback has been dealt with abundantly in the past» and well known re- sults may be found in the literature as well as textbooks. 2 - To minimize the effects of man”s ignorance of the plant’s en- vironment . What man cannot predetermine in the plant's en- vironment is commonly referred to as disturbance or noise. Feedback in fact reduces the effect of this ignorance on the system output to an acceptable value. This side '“of the pro- blem also has been carefully investigated and the related results well established. 3 - To minimize the effects of man's ignorance of the plant it- self . Man's Ignorance of the plant, which he wishes to con- trol, can be of various categories. Some plants cannot be readily analyzed and a mathematical model cannot be readily obtained. Such is the case for problems in biology, medicine or other natural sciences. Other plants are easier to anal- yze, but a rigid mathematical model is difficult to obtain due to the changing nature of the plant, resulting in vari- tlons in the plant's parameters. Such is the case with pro- blems in econon^, industry, or management. Such is also the case with a large number of engineering problems, of which a few examples will be given in the next section. Plant para- 1 - parameter variations result either from the .basic nature of the plant itself (chemical processes), or from environmental changes (climatic and other ambient conditions). Variations of system’s response as related to plant parameter changes are expressed by the "sensitivity" of the system. The sensitivity-re« ductlon aspect of feedback is the purpose of this report. This chapter introduces the notion of root-sensitivity and its properties. Chapter Two will make use of them in a idesign procedure. At this point, it seems necessary to classify plant parameter changes into two kinds; incremental or small parameter changes, and large para- meter changes. The technique for treating each class of problem is diff- erent and no extension from one class to the other seems to be possible nor recommendable. Examples for small parameter changes can be found in many situations: chemical processes where the speed of various chem- ical reactions changes with pressure, with ambient temperature, humidity; electronic circuitry where component values change with temperature or aging; rotating generators where small changes in the field resistance cause proportional changes in the voltage gain as well as in the time constant; pneumatic or hydraulic systems in which fluid properties change with temperature and aging; mechanical systems in which friction, spring characteristics, etc., are far from being constant. Larger parameter changes, on the other hand, are common in a number of other problems, ranging from automatic steel rolling mills where the thickness of the slab varies within wide limits, paper mills where roll diadteter starts from zero and ends up at its maximum value, to the more recent problems of missle and space technology, where the vehicles are called upon to function at extreme environmental conditions, with wild changes in mass due to the burning out of fuel. This report will be concerned with analysis and synthesis methods for problems with small parameter changes. 1-2; System sensitivity and root sensitivity; Several definitions of sensitivity have been used in the past. The 1 first one, as far as is known to the author, is by Bode , defining - 2 - sensitivity of the overall transfer function T to the gain constant K ass d K T ^ K _ 51n K d_T ain T T Horowitz took the inverse of Bode's definitions - T L T K T Defined one way or the other, S is generally known under the name of "class- K ical sensitivity" of more suggestively "system sensitivity" since it relates’ the change in system transfer function to the change in parameter K. Another kind of sensitivity is based on the location of system's dominant tp lx roots. Such a sensitivity relates the change in (i — dominant root of the closed-loop system) to the change in x; where x may be the gain constant, or an open-loop zero, or an open- loop pole of the plant. The sensitivity thus de- fined is known as "root-sensitivity". Formal definitions of root-sensitivity vary from author to author. H‘Or- 10 2 owitz and Ur defined the sensitivity of closed-loop root with respect to parameter x (where x may be gain constant, or pole, or zero) ass • A ^ X " dx Huang , on the other hand, used; 9qi i ^ ■*1 S = ox X ( 1 ) ( 2 ) 4 More recently, McRuer and Stapleford prefer different definitions for sensi- tivity to gain (K), and sensitivity to poles or zeros (x)s & K X 5q. K 9x (3) - 3 - (4) It will be shown in Section 1-5 that definitions (3) and (4) are most suit- able for the work presented here, and therefore will be adopted. 1-3; Survey of previous works and scope of this chapter; a) As far as large parameter variations are concerned, the most signi- 10 ficant work known to the author is Horowitz’s book in which an exten- sive treatment of passive-adaptation is given, concerning systems with one or more parameters varying simultaneously and independently within wide ranges. Horowitz's methods are mainly based on frequency response, and since P h + — lf_ L + 1 o (L =loop transfer function. Subscript o means original value, f means final value) , the problem is to select L ( jcc) so as to achieve tolerances T P ° on _o , despite the variations in . This is called "loop shaping" of L .^f o P. Variations in P are represented on the polar plane as an area (section 3.5, reference 10). As a consequence, the method becomes impractical for more than 2 changing parameters. Horowitz's work extends well beyond the limits of the sensitivity problem alone, but in the treatment of the latter his certainly is one of the most valuable contributions up to the present time. Along the same passive-adaptive line is the recent work of Liu, Han and Thaler^. For a second order system with tachometer feedback, the three parameters are gain K, open-loop pole p, and tachometer gain K^.. A graph- ical method is proposed to determine the optimal values for K and K^., when p changes, in order to maintain the damping-ratio^ [within a certain limit. When K changes, p and K^. are similarly determined graphically. The pro- cedure is also extended to third order systems. This is the economical way to solve the problem, using to its best the limited amount of passive- adaptation inherent to any feedback systems. On the other hand, in many cases, passive-adaptation may not be suffi- cient and one must have recourse to active adaptation, which has bean the Subject of a profuse literature. Mention must be made of the APRACS tech- - 4 - nique, for ’‘Amplitude and Phase Regulated Adaptive control sys- tems**, and the recent work of Horton and Eisner^, who propose a method whereby the system's dominant poles are maintained fixed despite changes in the plant P(s)* In order to do so, gain and phase of the controller C (s) must change in such a way as to com- pensate for similar changes in P. A test signal is injected into the system and the output measured* Amplitude and phase of such output are compared with the input test signal. Differences are used as driving force to adjust gain and phase of C(s) in order to null the affects of changes in P. b) Turning next to small parameter changes, a great deal of work has been done in the recent past concerning the analysis of the pro- blem but so far no significant effort has been spent on synthesis. 2 Ur derived interesting root-locus properties and proposed a graphical i 3 method for evaluation of S^. Huang showed. by a number of examples the usefulness of root-sensitivity in a wide variety of analysis pro- 4 bleras. McHuer and Stapleford derived interesting properties of root sensitivity and worked out various graphical and analytical methods for computing not all of which are practical. X Considering what has been done in the past, the remainder of this chapter will be devoted to a study of root-sensitivity properties, and in the next chapter, use will be made of these results to formulate a design method. In the literature mentioned above as well as in what follows, em- phasis is laid on the location of dominant system roots. One may argue on the validity of such a philosophy when applied to synthesis, since nothing guarantees that dominant roots remain dominant after the system has been compensated. In practice, however, it usually happens that if any extra root is introduced by the compensation, either it is far away enough, to be negligible, or it will be close enough to a system zero, so that its effect on the transient is thereby cancelled. In case of doubt, however, it is advisable to perform an analytical or analog- computer chhck after a solution has been obtained, in order to make sure it does satisfy the specifications. - 5 - 1 1 1 fi • 1-4; Root-sensitivity; definition In this section, it will be shown how a definition of root-sensitivity is arrived at. In the next three sections, some important properties of rbot- sensitivity are derived. Let P be the transfer function of the plant to be controlled, C that of the cascade controller, and F that of the feedback con- troller(fed back around C and P) . We define G =PC as forward transfer function, and L =GF =PCF as loop transfer function of the system. Then the system char- acteristic equation is 1 + L =0 and if is a systems root, then 1 + L(s) _ = 0 s - q. If K is the gain constant of L(s), and z., p. its open-loop zeros and poles, J J then one can write; L = L (s,k,Zj ,p^) I and take the total differential of L; dL = ds 4p. j=i ^ j=i On a root locus, L =-l =constant, ie: the total differential dL is zero for s = Let dL = 0 and s =q^^ in the above equation, this gives; Rearranging: 0 =■ ds .1 ^ ^L dK + s =-q/- 9z . 1 J dz. + s =-q 3L Sp, dPj s =-q dz . S = -q . 1 ( 5 ) But q. itself is a function of K, z . and p. i J J q. = q. (K, z, , p,) . ^i ' J J Taking the total differential of q^ , = K- ^ ^ ‘^‘^i 9K K V V vs ( 6 ) Equation (6) suggests that dq^ be written as: dK zj dZj + dp. PJ J (7) Equations (6) and (7) thus define the sensitivity of root q^ to gain K as; (8) (9) ( 10 ) Equations (8) through (10) are the same as definitions in equations (3) and (4) given earlier, used by McRuer and Stapleford. relates the change in q, with Ix i i ^ the corresponding percent change in gain K, while S . and S , relate the change ZJ PJ in q^ with the total change in z^ or p^ . There is no reason why other definitions cannot be adopted. It is just a matter of convenience. 1-5; Property 1: Relationships between S^, and Comparison of equations (5) and (7) yields: 7 But Then i / sl/bk ] K 1^9L/Bk/ s =-q^ ^ y^L/Sz. *zj *yaL/3 s / s =-q li PJ 5L/Sp, L = K 9L/5s / s =-q J (s + z ) - — — - m n (s + p^) fl ( ^ +: z\) li ( s + p.) Then (11) becomes ( 11 ) ( 12 ) (13) L s=-q, S=-q, K = K - 1 Similar derivation for/ ^ s.i. = z , J CO z ; - 3 ‘li - CO s — ‘li " (14) leads to: (15) (16) Elquations (15) and (16) show; the convenience of the definitions used. The sensitivities to all singularities are directly proportional to S^, and in- lx versely proportional to the distance between q. and the singularity in- ^ i i volved. Thus, whatever properties are found for S may be extended to S iL K. z or Sp. A particular case of equation (16) is for = 0, ie; the root- sensitivity to the pole at origin of the s-plane. 8 Mit I •4^ III iU I • 4|i mmi • » » •• t m* • ••* <4 I#* .« :| 1 I I I . # ( 17 ) gi Po ‘li is proportional to S.^ , the constant of proportionality being — ^ Po ^ ‘^i (complex quantity) . 1-6: Property 2: relationship betweeh r66t“sensitivity of -q^ and res- idue at -q^. In section 1-5 it has been shown that S^, and sf" are all proportional. K Z p In this section it will be shown that, if q^ is a single system root, then ^ -q. '’t (18) where F (’q^) is the feedback transfer function F evaluated at s = -q^^, and is the residue of the system transfer function at q^^. For unity feedback, (18) very simply becomes: 4 = (19) th '' Finally it will be shown that when q^^ is an N — order pole, (18) becomes: For unity feedback it becomes: 4 = < Qin ( 20 ) ( 21 ) The remainder, of this section, is concerned with proofs of equations (18) and (20). The overall transfer function is X (s) = -PCsJ.. C .(s ) ^ . l-(s) 1 + L (s) F(s)Li + L(s)J The residue of T(s) at q^^ is = (s + q^ T(s) (s + q^) L(s) s =-^1 F (s) [1 + L( s )]3 ^ - 9 - (22) Let the rightmost expression be denoted as R^, ie, by definition *^i <*‘»i> = ^i F (1 + L) R. =(s + q.) L Take the drivative with respect to s of both sides: If [1 + L] Rj + PR. If + F [1 + U 5^ = L + (s + q^) If At s = -q^, ie; at a point on the root locus, L = -1 and the above equation reducers to: 9,L FR i ds ’ = -1 s = -q. or; R ( -q.) ‘ = -q^ (23) Compare (14) with (23) and obtain equation (18) which is thereby proved. Turning now to the case of order root at q^^, theory of Heai/rside's partial fraction gives: ^il T (s) + ‘i2 s + q- / \2 1 (s + q^) other roots 'iN (s + q^^) N + terms from (24) where 0 = ^ gN-k >C (s + q^)L ^ik (N -k)l as”-'' F(1 + L) (25) s = -q. Again .defining the quantity inside the small bracket as R^, then repeating the operations as for equation (22) above, one obtains after repeated diff- erentiation: ^ .. (26) ‘iN F(”qi) ^ ^-N ds s = -q. - 10 The next step in the derivation of equation (20) is to obtain the til equivalent of equation (14) for the caseii of N — order root at the repeated differentiations leading to equation (26), it is found that the detivatives of L: forl<k<N-l (27) = 0 and the original definition of root sensitivity as by equation (14) becomes infinite. In order to avoid this difficulty, a more suitable definition is suggested by writing an expansion of the total differential dL to include higher order terms, than retain only the lowest order terms for each para- meter and at the following equation, counterpart of equation (7) . dq^ = 5 L(3 ). i k os = 0 s = -q. This means that for N /.s = -q from which: K (-1)” N! (29) which is the counterpart of equation (14) . Complete derivation of the above may be found on Appendix Ir-Combinlng equations (26) and (29) dir- ectly yields equation (20^ which is thereby proved. Appendix 2 shows that equations (15) and (16), which relate with i iL til and S , are still valid when q. is a N — order system root. z p 1-7: Property 3: Sum of all and in a system. Z , P c i. fl J J When -q^ is a single-order system root, the sum of the sensitivities 11 I ( 30 ) of -q^ to all open-loop zeros and poles is equal to 1. j j This is easily seen by referring to the construction of root loci. If all open-loop zeros and poles are displaced by the same amount Q , then all closed-loop roots are displaced by the same amount, i.e., if dZj = dp^ = 6 for all j, then dq^ = 6 for all i. This interesting property will be of great utility later on. til When -q^'is a N — order system rdot, setting dK = 0 in equation (28), and using the same reasoning as above, i.e., shifting all open-loop zeros and poles by 5 , one obtains dq^ = I N But closed-loop roots shift by the same amount 6 becomes ; Then the above equation Vs' Z. ^ p. J j ^ which no longer has a universal character as equation (30) since it depends on the magnitude of shift 6 . 12 - CHAPTER TWO A Sensitivity Design Method II ~ 1; Introduction In chapter one, a number of properties of the toot- sensitivity to gain, poles and zeros have been derived. In particular it was shown (equation (30)) that for any system, the sum of the sensitivities of a system root to each and every open- loop singularity, is always equal to unity. It was also shown (equation'^ 15,16) that the root-sensitivity . « to each singularity is directly proportional to the root-sensitivity to gain, and inversely proportional to the distance from the root to the singularity involved. It is now desired to apply these results to a number of design pro- blems where specifications include condition] on the sensitivity of the dominant roots. These specifications may be in the form of an upper limit for the magnitude of the sensitivity of the dominant root, or for the change of damping factor, or the change of natural frequency and bandwlth, when gain and/or singularity (ties) of the plant vary with time. This chapter will be presented in the following sequence. The practical aspect of problems with small parameter changes is discussed first. Then a graphical method to obtain root-sensitivity values is formulated and other properties of sensitivity are derived therefrom. Finally a design procedure is presented and applied to several examples. II - 2; Practical aspects of problems with small parameter changes . In section I - 1 a number of situations where small parameter changes frequently occur have been mentioned. A desirable quality of control sys- tems is undoubtedly the reliability of their response under varying oper- ating conditions, and perhaps one of the most objectionable shortcomings is the unpredictable variations in system response, variations due to the combined effects of small changes in the plant gain or time constants or both. 13 - The question then arises as to when the plant gain Is affected and when the plant time constants are, and whether they affect each other la mutually. There Is no unique answer to this question, and for each In- dividual problem, an analysis Is needed to determine, from physical sit- uations, what parameters are changed and what Is the extent of the change. A simple example may be found In the amplldyne whose transfer function; is k k. q ^ O = g ^ ^ Vs + ^ + Pi + P2^ where e Is output voltage, e the control voltage,! subscript q refers to the quadrature field, subscript c refers to the control field. One can see that If r^ changes with temperature, only p^ Is changed. If r^ changes, only p. Is changed proportionally. But If an Inductance value changes, not only the corresponding pole varies, but'k does so as well. As another example, take a mechanical system with inertia and friction; J 0 + f 6 = KE where E Is the driving error signal. e !L_ Vj ® Js^ + fs s(s + J ) In this case, a change In the friction modifies the time constant alone, while a fluctuation In the value of the Inertia causes both gain and time- constant to vary accordingly. In some Instances, even the open-loop pole at the origin of the s-plane varies. This Is the case of the above mechanical system when a shaft. Intended to be rigid, is twisted under load, or when a transmission belt, designed to be of fixed length, is elongated under tension. Then; 0 = K ^ Vj ® Js^ + fs + k (^s^ J® ■*" j ^ _ ^ Vj (s + P^)(s + Pj^) - 14 - ^ • -f • « ####• where Po+ Pi and The last equation shows that in the Ideal case, k = 0 giving p = 0, o but if some k exists ^ then p exists* As long as Inertia J is fixed, the K ^ gain constant — does not vary. If k alone varies, then both p and p- are or changed since the sum p + p. = — is constant. If both torsion k and O X. «J friction f vary, p^ may change alone, or change simultaneously with Pj^. Finally if J varies, then both gain constant and poles p , p. are changed O jL proportionally . Another similar example of parameter change may be found in mechanical systems with springs the constant of which varies in use. II - 3; Graphical method for determining root-sensitivities . Equations (15) and (16) derived in chapter one suggest that to singu- larities which are close to system root is more sensitive, and for singularities which are farther away, q^ is less sensitive until it becomes Insensitive to singularities at infinity. The above concerns the magnitude of sensitivities. But sensitivities are vector quantities, since the change of a parameter may move the roots in different directions. It is then help- ful to make use of equation (30) together with equations (13) and (16) . Refer to equation (15), where z and p indicate open-loop singularities and q^ is the system root in question. Draw a vector from q^ toward each and every pole, and away from each and every zero. The length of each vector will be inversely proportional to the distance from q^^ to the singularity concerned. Then construct the siun U of all th^se vectors, which is a vector Itself. If U is taken as unity vector in magnitude and phase, then the other vectors measure the sensitivity of q^ to each singularity respectively. Here after, the vector • II- 1 1-0 U = le will be baptized ''unity vector for root-sensitivities to poles and zeros", or more conveniently "unit-sensitivity vector". The first lengthy name 15 - ¥ ^ « m \ t I « vM I im ■ -m^ I I ■^1 §(► • l<^ •► |►»^!|■ «w»4 i« emphasizes the fact that this unity scale, applies to sensitivity poles and zeros only, i.e., S, and Sp , since equation (30) only concerns these two quantities. This unity scall does not apply to The diagram just described is from now on referred to as the “vector diagram", as compared to the ^circle diagram" to be introduced later. The vector diagram offers a quick way to measure both magnitude and phase of the vectors or for any j. Only one little detail needs be kept in mind; phase of sensitivity vectors must be measured as positive in the clockwise sense starting from the U vector. This seemingly arbitrary sign convention in fact comes from equations (15) and (16) which are the basis of the vector diagram; K K K I - "j (-Pj9 (-q^) (31) V . j The denominator is the vector from root (-q^) toward pole (”Pj) as shown on figure 2.' The phase relationship of the above equation is; S = - V . Pj ^ J (32) where the hat sign reads "phase of". The above is true no matter what conventions are applied to the measurement of the angles. Since the U- vector has been^found to be equal to le'^^^ as far as and are con- p z cerned, angl^^p will^e measured starting from U as zero phase. On the other hand, and V are measured in the conventional way, i.e., starting from the positive real axis and counting positively counter- clockwise. In equation (32), the-quantity S is not dependent on j, i.e., lx it is the same for all j's, S = constant Pj Thus, for each j; V . J ^^aning that the larger the value of , the smaller must be Sp ^^^..^ince V. is measured conventionally (positively counter-clockwise) ^ S^ must P T be measured positively clockwise. 16 - As an example, on figure la: P2 ♦ 37 1.05 = .352 sj; = + 120° P2 This means that if pole p^ moves by = .1 (to the left) while other parameters remain constant, then will move by dq^ = (.352 /1 20° ) ( .1 /0° ) = .035 /120° . To say that q^ moves by .035 /120° means that the root (-q^^) moves by .035 7-60° , since q^ and -q^ are two oppo- site quantities and move in opposite directions. Finally note that the difference in the ways the sensitivity vectors are drawn for poles and zeros (toward the former, away from the latter) comes from the different signs in the denominators of equations (16 and (15). A different sign corresponds to a rotation of 180°. Before this section is ended, another interesting feature of the vector diagiram is presented. This concerns the root-sensitivity to gain, S , which so far has not been mentioned on the vector-diagram. One recalls, ^ i i however, that at the end of section 1-5 a relation between S„ and S "was K ."'ti p given, p^ being the open-loop pole at the origin of the s-plane; Po 'll (17) Equation 17 shows that, for a particular system root under investigation, is equal to x q. the phase relationship isi K. P X 4^4 * "i O /A ^ Figure 2 shows the angles q. and S = a , the latter being measured from U. Since q^ + gives the direction of U on the s-plane, it is thus established that on the s-plane, always lies on U. Since S^ indicates K K the direction in which the root moves when K varies, i.e., the direction of the root locus, the above result can be stated as follows: 17 - I * *At any point on the root-locus, the U vector is tangent to the root- locus . ** * Again note that the direction of U-vector indicates the direction in which moves when K increases. The direction in which (-q^^) moves when K increases differs by 180°. It seems worthwhile to state once more the results obtained in this section which the reader should keep i|in mind before going on: Phases of root-sensitivities to p and z are measured positively clockwise starting from the U vector. Phase of root-sensitivity to gain K is measured in the conventional fashion, that is from the horizontal and negatively clock- wise. On the s-plane, always lies on U which is tangent to the system root locus at the point of contact. II - 4; An example of application of the vector diagram . In order to show the practical character of the vector diagram, the following numerical example is taken from reference 4, but solved by use of the vector diagram. A look at the lengthy arithmetical and graphical methods of reference 4, some exact and others approximate by nature, will convince the reader of the rapidity of the vector diagram. Results ob- tained here and those obtained in reference 4 are compared to show the relative degrees of accuracy. Given the open- loop transfer function s(s + l)(s + 5) For K = 2.07, the closed-loop roots are at locations indicated on figure 3. It is desired to calculate the sensitivity of the root at to the gain K; (other sensitivles can be readily obtained, it is merely a matter of measuring a length and a phase on the graph) . The vector diagram is constructed and U - vector is drawn on figure 3. From measurement of vectors and phases, it is found that Sp = Itff = .753 Zz5l! *: this result may be very useful in the construction of root loci, since it readily gives the orientation of the root locus at any point. 18 - • 14 I* • ••«■■•« 19 «» lij • ii •- t4* ^ I V ipa»t< L (Minus sign because clockwise from U-vecCor) Since the system root -q2 is at 0.6 4/135° , is at 0.64 /-45° . Then q 2 = (0.735 /-51 °) (0.64 7-45° = (0.481 7-96° Note that on s-plane, the U-vector also lies in the direction -96°. 2 Hence, S is a vector lying on U. Note the tangency of U to the root K. locus at point “^ 2 ' The same example is worked in 10 different ways in reference 4. Results of only 5 of the most accurate methods are reproduced here for the purpose of comparison (order of increasing accuracy) . Method 2 Value obtained for S (1) Root locus metho4^ by gain perturbation 0.356 7262° (2) Closed loop Bode Asymptotes 0.451 7270° (3) Root locus method by phase perturbation 0.457 7264° (4) Open loop Bode and plot 0.491 7266° (5) Direct calculation 0.492 7264? 47 (6) Method of this report 0.481 7264° The vector diagram method is in fact an exact method since no approx- imation of any kind was made in the derivation. The more accurately the diagram is constructed and measured, the better th$ results. In order to improve accuracy, one may choose a scale for the sensitivity vectors different from the scale used for the root-locus. til II - 5; Particular case of open-loop singularities of N — order . Looking at the construction of the vector diagram as Illustrated on 19 - figure la, intuitively one can see that in case of a double pole or double zero (or hi gher- order ) , the same construction still applies, providing two vectors be drawn toward the double pole (or away from the double zero) . This can be seen by assuming that pole -p^ moves toward and reaches pole -Pj^. Simultaneously the sensitivity vector of would move toward that of -p^ and reach the same magnitude as that of ■*p^ . Note that in such case, the length of the sensitivity vector of each pole at -p^ remains the same as before, but the actual value of the sensitivity of -pi is different, since the scale-vector U has changed. *k The same result may be obtained analytically as follows. Assume L has an order pole at p^, then L = J r.*,) ( . . . ) (s + Pi ) * Pi> (s + p^) n-1 ll = 0 + L ^ ^1 (s + p^)^n = h (s + Pj^) n = n s + Pi But s + Pi (51), S is equal to ^ if p^^ , . , ^ 9L is proportional to ^ were a single pole. Also, I' . Then the dbove result: from equation — — — — bl dL apl = n X Pj^ is order pole a Pi were a single pole is equivalent to saying that the root-sensitivity to a n — order pole is n times that to the same pole assumed single. th c The same reasoning applied to a n — order zero leads to identical con- clusion. II - 6: A design philosophy It has been shown from the vector diagram that the root-sensitivity of q with respect to each singularity is given by the vector associated with that singularity, measured with U-vector as unity scale, both in magnitude Casual reader may skip. 20 and phase. If the U-vector is changed, either in magnitude or in phase or both, then root-sensitivity changes. In particular, the larger the magnitude of U-vector, the smaller the root-sensitivity value, i.e., the more Insensitive the system root. This leads to a design philosophy whereby the U-vector would be mod- ified in order to meet particular specifications or restrictions imposed on the sensitivity of the dominant root of the system under investigation. In- tuitively one can see - and this will be shown to be true later - that in a given situation it is possible to maximize the magnitude of U-vector in order to minimize root- sensitivities; or to orient U in a certain fashion so to make the damping factor Insensitive to the variations of a certain pole, or to make it insensitive to the variations of gain- constant, etc. In a number of control problems, the system response specifications are expressed as a desired location on the s-plane for the dominant sys- tem roots. Such a location determines a damping factor ^ and a natural frequency (0^ for the system. Thus, fixing the location of the dominant roots and fixing the frequency response of the system with phase-and gain- margins and bandwlth, are two ways of expressing the same conditions. One can find, in the literature, a comprehensive treatment of the problem of compensating a given system in order to place the dominant roots where they are desired. The simplest way is to use lead or lag networks in cascade in single or multiple-stage. In fact, there exists an infinite number of solutions to the problem of forcing the root-locus of a system to go through a certain point on the s-plane. All what is needed is that the lead- or lag-network contribute the desired phase shift so to make the total phase at that point equal to 180°. Hoever, if another contralnt is placed on the system, the number of possible solutions decreases and eventually becomes unique. Such is the case of the problem in which the desired location of the dominant roots, and the steady-state accuracy of the system are to be satisfied simul- taneously. The latter condition fixes the value of gain constant when the dominant roots are at their assigned location. This problem was solved in - 21 0 7 8 detail by Ross, Warren and Thaler , also by Poliak and Thaler , and re- 9 cently Hsu proposed a graphical method. The following is concerned with two simultaneous conditions; loca- tion of dominant roots, and sensitivity of these dominant roots. The second condition on sensitivity may take a number of different forms, such as sensitivity of or sensitivity of co^, with respect to vari- ations of gain or poles or zeros. In other situations, the dominant roots may b^ restricted to moving only within a certain area. As stated earlier, the philosophy of the (design method is to modify U in mag- nitude and/or in phase. Thus the first step is to investigate what the possibilities are in modifying U using lead or lag networks; in ether words, how the U-vector changes on the s-plane. 11-7; Locus of U on the s-plane . This section, as a preliminary to the design procedure to be pre- sented next, is devoted to the determination of the geometric locus of the tip of U-vector on the s-plane. Knowledge of this locus tells the designer how and how much he can change magnitude and phase of the U- vector, and what good such change will do. Consider the pl&nt to be compensated; K(s + z, ) Q ^ ^ S(s + Pj^)(s + P2> and the desired dominant-root location -q as indicated on figure 4a. A spirule measurement shows that an additional phase of + 0 is needed at location -q. Thus a lead network with a zero at Z and a pole at P is needed. The question is; how does the tip of the U-vector move on the s-plane, when Z and P take all possible values on the negative real axis? (avoid positive real axis to avoid possible conditional instability) . Figures 4a, b and c illustrate the answer to the above question. First, ignoxe the compensator irregularities Z and P, and draw the unity vector for the irregularities p^, Pj^, P 2 , and Zj^, alone. This "uncompensated" unity vector is labeled QI on figure 4a. Now if a pole P is added, it has associated with it a sensitivity vector Qin of mag- nitude . Then the unity vector QI is augmented by the vector quan- 22 - w > rMil: II #• tity IM = Qm. Similarly, when a zero Z is added, it has associated with it a sensitivity vector Qu, directed away from Z and of magnitude r— . Then the unity vector QM is augmented by the vector quantity MU = Qu. The question is to find the geometric locus of U. Figure 4b shows that when P moves along the real axis, since QM ™ moves on a circle of radius R = , resulting from geometrical inversion of the real axis, inversion with center ^ and ratio 1. Such a circle will be referred to as the (M) circle for convenience. The locus of point M is the (M) circle, of radius R = -^r- and with I as its uppermost point. 2d Figure 4c shows that lU = 2R sin 0 = ^ sin 0 which is a constant quan- tity for each problem, and hence U move on a circle centered at I and of radius r = sin 0. The reasoning attached to figure 4c is as follows, a Draw vector IN equal and opposite to M U. Since Z is a point on the real axis, N.is on the (M) circle due to figure 4b. Then MN = 2R sin 0, and since I U = MN, I U = 2R sin 0 = constant. The circle on which point U moves will hereafter be known as the (U) circle. The (M) circle, locus of point M, of radius R= and the (U)' circle, locus of 'point U, of radius r = ■^ sin 0 are shown on figure 4a. For con- venience in terminology, the diagram just drawn will be given the name of "circle diagram", as opposed to the "vector diagram" shown on figure 1. For purpose of reference, the above result is restated below! The geometric locus of the tip of the unity vector is a circle, cen- tered at I and of radius r = — sin 0; where I is the tip of the "un- compensated" unity-vector, d the imaginary part of the dominant system root, and 0 the phase shift to be introduced by the compensation. II - 8; Limit of locus of U It does not make sense to define a geometric locus without specifying its limits. This is the purpose of this section. There must be 2 limit points on the locus of U, a right hand limit U^, and a left hand limit U^'. Point is defined by the extreme condition where Z, th^compensator *s zero, would be at the origin, and P on its left, such that PQZ = 0. Point U. is defined by the other extreme condition with - 23 - A P at -» and Z on its right, such that PQZ = 0. Any other possible case is between these two extremes. Determination of U and U. may be done by first noting the following r detail on figure 4a (or 4c); IN // QZ| IM // QP, if J is the midpoint of MN then lU is perpendicular to OJ I U J_ 0 J This can be used to obtain U when P and Z are known; draw IN// OZ, draw IM //OP; take midpoint J; draw lU J_ OJ. A quick way is to take the angle NOJ = 0, thus avoiding the trouble of obtaining the midpoint of an arc. Conversely, and this is more important, one can start from any de~ sired location of U and go back to obtain the corresponding Z and P, by doing the above construction in reverse order. This is the essence of this design method, whereby one changes the U-vector by proper compensation. When a particular location for U on its locus is chosen to satisfy some specification (next sections), obtain P and Z as follows; Draw OJ _[_ lU which cut (M) circle at J. On (M) circle, measure arc JN = JM = 0. Then the direction of IN is the direction of QZ (thus one gets Z), and the direction of IM is the directlnn of QP (thus one gets P). An example is shown on figure 6a. A desired U is given (purely as an example, for no particular reason). Perform the construction as indicated above and obtain and ot as angles of the direction of P and Z with re- A Zj spect to the vertical. This determines the compensator pole and zero as on figure 6b. Obviously a_ + a_ = 0. A Lt One may now go back to the problem of determining the limit points and of the locus of U. This is. merely' a:pair of problems similar to the one just solved. For the extreme right case (subscript r), refer to figure 5a. A measure on figure 4a shows that when Z is at the origin, QZ makes -30° with the horizontal. Draw IN at -30° from the horizontal. N is the r r extreme-right position of N. Measure N^O J^ = 0 and draw lU^ J_ OJ^. is the extreme right limit of the locus of U. For the extreme-left case (subscript A), refer to figure 5b. Here P - 24 - is at This calls for rotating angle N^I of figure 5a, until it reaches the extreme-left position Nj^I of figure 5b. Then is at I, and is at intersection of the two circles. The same construction gives U^> the left limit of the locus of U. Once the limits are found, one can see how far one can change the mag- nitude and phase of the unity vector, which is represented by QU on figure 6a. As an example, on figure 6a, the maximum magnitude that the U -vector can reach is QU^, and the rightmost direction it can have is given by'QUj^. Finally note that in the case of a lag network, the points M and N are simply interchanged and the deteirmination of limit points and is still the same, with P at origin in one limit case, and Z at infinity in the other case. Throughout the work, point M will be associated with pole P, point N will be associated with zero Z to help the reader follow the argument more easily. II - 9; Design techniques . In the above section it has been derived a method for finding P and Z, given the desired location of point U on its locus, i.e., the magnitude and phase of QU, the unity-vector. This method is re-stated below in a step-by- step form (refer figure 4a) . Step 1 - Considering Q as if it already were a point on the root locus, draw the vector diagram at Q for the vincompensated system and obtain QI, the "uncompensated" unity vector. Step 2 - Draw the circle diagram , composed of; the (M) circle, of radius R = ^nd whose uppermost point is I.; and the (U) ^ 1 circle centered at I and of radius r = ~~r~ sin 0. Fix the a limits of the locust of U on the (U) circle. Step 3 - Given QU as the desired unity-vector for the compensated system, draw OJ lU which cuts (M) circle at J. Draw angles JON = JON = 0 (or arcs JN = JN = 0) . Then Z and P are determined by drawing QZ // IN and QP // IM. In this section, it will be seen how the U-vector (i.e., QU, t\4»> the location of U on its locus) is selected to satisfy a particular condition. a) Design for minimum root-sensitivity . Since the" U-vector is the scale used to measure the individual sensi^ 25 - 4 iTt III • t ^3 <^tM ••i-i «l ^ «• r n* 1 j I tivity vectors, the larger the scale, th^csmaller the magnitude of the sensitivity measures. Thus one possibility is to design the compensation for maximum magnitude of U-vector, that is, minimum root sensitivity to open loop singularities. Figure 4a shows that maximum magnitude of QU is obtained by placing point U near the lowermost part of the (U) circle, or more exactly, on the extension of Ql. If such point is not within the locus of U , then it can|fot be a location for U , and one must select the lowest polQt which ^ on the locus. In figure 6a, this is point U^. Thus, QU^ ts the selected unity vector, and with this given, one can proceed tp the 3 step procedure outlined in the beginning of this section. With such a design, and are all minimum, for all i“s. For ex- Pf q ample, the minimum value of will be equal to the ratio of the sensitivity vector associated witn p^ divided by vector qU^, both in magnitude and phase. Note that these are minimum values of sensitivities obtained with only one filter stage. It will be seen later that by use of multiple-stage filters, results may be improved, but more often than not. Improvements are small and do not justify the extra cost. Design example No. 1 given in the next section applies the above technique. b) Design for constant damping when gain K variesCgefer to figure 4a) Another practical problem is to compensate a system in such a way that when gain K varies about its nominal value, the dynamic response of the system doesn'. change. This calls for a constant i.e., a root- locus that remains tangent to the radial line OQ at the neighborhood of Q. How can Z and P be found to obtain such a root-locus? It is now shown that this can be done by merely selecting point U so that the unity vector Qll^ goes through the origin 0 of the s-plane . In other words, choose U so that Q, 0, U be in line. (If the locus of U doesn” t permit such a choice, this means it is not possible to obtain a constant C about Q for the given system. One can then choose the best solution available, by taking the U location that is closest to a straight line with QO) The above statement can be proved very simply if one recalls 26 - ^ n$mrn ^ O I ’-Tsyt equation (17) of Chapter One o ( 17 ) The specification here is to force Sj^ to have same direction^^ QO, thus This means S must be making move on a radial line when K varies equal to the phase of QO (namely -30 on figure 4a) which is also phase of q^, or^^« But from equation (17) /\ /\ /\ *^o The phase of must be 0, this means that the sensitivity vector S ^o ^o must lie on the U-vector, or conversely, the U-vector must pass through p^ at the origin of the s-plane, qed. A faster way to prove the above is to come back to section II - 3 where it has been shown that always lies on the U vector. In order to i ^ keep C constant, must be radial, thus U-vector must be radial. Design example No. 2 of next section illustrates this part. c) Design for constant damping when a singularity varies . Again refer to figure 4a. Another practical design problem is the following: the nature of the plant is such that pole -pj^ varies more than the other singularities, (see section II - 2). It is desired to compensate the system in such a way that: 1-the dominant roots be at -q, and 2-that the sensitivity of the damping factor C with respect to r changes in p. , namely S be nullified, or at least made as small as possible. ^ i It is now shown that such problem is solvedLby simply making S ^1 equal to or that is, forcing the U-vector to a postion such that the phase of S^, be equal to the phase of q. or of -q.. The latter P X i differ by II, so the above underlined condition may be written as: ( 33 ) 27 - ^ /\ /\ where D reads s "either alone, or q^^ + U Equation (33) is the condition for C to be insensitive to p. , that is for = 0» i p. Note that, as for previous design problems, it may en that the limits of the locus of U do not permit a (33) be satisfied. of U”vector such that .c Im such case it is always possible to minimise S* ^ i^ by making S as close to II as possible o Design example NOo3 ?! illustrates this method o The following is proof of condition (33) o The proof is based on this remark (see fig® 7a» d)g for to be kept constant, the change in that is dg^^, must be on the same radial line as q^, ioOo, dfli = @ n where the sign has same meaning as i^equ^^on the natural frequency u) increases. If dq, = q, + II, co decreases. n i X n From the definition of root-sensitivity, when Pj^ changes by dpj^, robt q^ changes bys /N /\ If dq^ = q^. or, phasewise; dq^ ^Pl /X dq. = s + 1 Pi /\ Since Pj^ varies on the negative real aaeiSj = 0© n. (35) In more I detail, if the pole at -p, moves to the left, dp, > 0, and dp, = 0. If jL JL L pole moves to the right, dp,< 0, and dp, = II. Combining equations (34) JL JL and (35), the following condition for = 0 obbainsi ^ /A. ’’i which is the same as equation (33), since an addition of II to the right hand side is equivalent to the same addition to the left hand side. d) Other possible sensitivity designs using the U locus Three practical design problems have been discussed; a) design for minimum root sensitivity^ b) design for constant damping when K is per- - 28 - turbed, assd c) desig® fof comstamt damping whe® a plarat singularity is perturbed. Still other problems may be solved using the same technique. If it is desired to keep constant (constant bandwidth) when gain K or plant irregularities varys, and condition (32) becomes s >A n - ^l± 2 then dq. must be perpendicular to q. Combination of (20) and (21) yields; + n= t n that is or /\ *^1 /X ^ n = qi± 2 /\ n /\ n= q. ± n (37) which is the condition for keeping co constant when p, is perturbed. It is also possible to design the system in such a way that a part- icular plant-parameter pertubation has stabilising effect (or destabil- izing effect, if it is so wishedS) on the system response. Fig. 7b shows that if has a phase between 0 and II, i.e., 0< dq - q<II , ^ dq the root variation has stabilizing effect. Figure 7c shows that if ~ has a phase between II and 20, i.e., Il< dq - q< 2<, then the root variation has destabilizing effect. A reasoning similar to that of part (c) of this section will yield the conditions for obtaining one or the other of the above effects. II - 10s Design eaeamples . The preceding section shows that the proposed sensitivity design technique is an exact method, involving no approximation or cut-and- try. It is a reasonably quick method, all that is required as pre- liminary work is the construction of the U-locus. It is versatile, can be readily applied to various practical design problems involving small plant-parameter perturbation. - 29 In this section three design problems will be worked out in details in order to illustrate the techniques presented above, then analog computer simulations are done to check the results « a) Design example NOo It The plant to be compensated has the transfer function? K ® 0 CoTj 0 Dynamic and bandwidth requirements lead to the desired location for dominant roots at -q = -0.2 + jO.35 = 0.4 /120*^ , that is, q = 0.4 7-60*^ . All three plant poles are subject to fluctuations. It is desired to de- sign a cascade compensator satisfying the above dominant root requirment, and in addition, guaranteeing a Mnimum value of sensitivity' of q to the poles ' fluctuations . Plant singularities and desired root location are represented on figure 8. Step 1 ; Considering -q as if it already were a point on the root locus, the vector diagram is drawn and the ^'uncompensated" unity vector QI obtained (figure 8). Using a spirule, measure the phase shift 0 necessary to make root- locus pass through -q. Found 0 = -38° (lag network needed) . Step 2 ; Circle diagram (figure 9); Draw the (M) circle, of radius R = = 1.43 Draw the (U) circle, of center I and radius r = 2R Sim 0 = 2 X 1,43 X sim 38° = 1.76 The geometricc locus of U is on the (U) circle. The limits of this locus are found as explained in II - 8; For extreme right limit, filter pole P is at origin, then QP makes 30° with vertical. On circle diagram of figure 9, draw IM^ making 30P with vertical. On circle diagram of figure 9, draw IM^ making 30° with vertical. Draw angle M^O = 0 = -38°, thus get J^. Draw lU^ ,_L 0 (see explanation of section II - 8; thus get right limit-point U . ‘‘‘ IT - 30 - For extreme left limit, filter zero Z is at minus infinity, then QZ is horizontal. On circle diagram, draw IN^ horizontal (Njj coincides with I). Draw angle NjjO = 38°, thus get Draw lU^ I 0 Jjj, giving the left limit-point ^ Here is simply the Intersections of the (M) circle and the (U) circle. This is true for all problems where a lag filter is needed. (Observe on figtire 5b that for problems where a lead filter is needed, the left limit-point is diametrically opposite to the intersecting point of (M) and (U) . Step 3 : Select a location for U, and from this derive the necessary compensator. In this problem, it is desired to minimize the sensitivities, thus one must maximize the magnitude of U-vector. The maximimi length that this vector can reach is being on the extension of QI. Now from U^, find P and Z, using the construction presented in section II - 8. Draw [ lUj^. Measure arcs = '^1^1 “ = 0. Then the direction of IN^^ is the direction of QZ(thus get Z = as shown on figure 9b. The direction of IM^^ is the direction of QP (thus get P = .36 on figure 9b). The complete compensated system's pole zero configuration and corresponding U vector are shown on figure 9b. b) Design example No. 2 . It is desired to compensate the plant given in design example No. 1 in such a way that; 1 - the dominant roots be located at -q = 0. 4/120° . and 2 - when plant gain K is perturbed, dominant root may move about the desired location but the system's damping faci&or will not change. r This amounts to designing for = 0, or to say the same thing diff- K. erently, to force the system root-locus to follow a radial line in the vicinity of -q. From part b) of section II - 9, it has been determined that this can be done by forcing the U-vector to go through the origin of the s-plane. This means, for this problem, that U-vector must make 30° with the vert- ical direction; that is, it must occupy the position QU 2 indicated on - 31 - *1 ^ t figure 10a. Thus Che location of U is fixed. (Subscript 2 used for this example) The design is accomplished by performing the now familiar construction. Draw OJ 2 I this gives J 2 . Measure arcs J 2 M 2 “ “^ 2^2 ~ ^ given on figure 10b by drawing QP // IM^, and Z is given by QZ // IN 2 « The result is P = .14, Z = .39. The complete compensated system? s pole-zero configuration is drawn on figure 11 and U-vector is constructed thereon. As expected, it goes through Che origin of the s-plane. As a check, Che entire root locus is drawn on figure 11 and it does follow the radial line in the vicinity of -q. The same problem is simulated on the analog computer and the results obtained are reported later in this same chapter. c) Design example No. 3 . In Che plant given in previous examples, it is observed that Che pole at the origin, -p^ = 0, fluctuates most. Moreover, since -p^ is the closest Co Q among all plant poles, it has most effect on the location of Q (sensi- tivity relatively highest, at least in magnitude). The compensator must be designed such that the effect of the fluctuations of the pole at -p = 0, o on the damping factor he nil or minimized. From part c) of preceding section, it was found that, in order to make C insensitive to p , one must have: y\ y\ = <ii ©n 0 In the present problem, = -60 , and the above condition becomes /\ = -60° or + 120° Po Recalling Chat this phase is measured negative counter clockwise, the above condition calls for a U-vector making an angle of 60° with QO and on the left of QO. A look at figure 12, however, shows that the leftmost -position U can reach is which gives a Z at minus infinity and a P at -0.65, as showt\,on figure 12b. , This^ yields a U-vector making only 48° with QO instead of Che required^ 60°.^ This is the. best one can do to minimize , using a 32 f single stage compensacat. By use of multiple stage, this result can be improved by making the angle UQO exactly 60°, as will be seen later in section II - 12. Another remark may be made on figure 12b. According to equation (37) ^ n if some pole p. is so located that S = q. + o' > then variations in p. J p - 2 J J / \ n 0 do not affect to^, but greatly affect Iij^his example, ^ = 30 or -150°. There exists no pole p. such that = 30 or -150°, but pole p. ^ o J Pi *^1 does have S = 56 . One then can expect p.-^ to have more effect on ^ than i 1 other singularities do, and the closer S approaches 30 (l.Ee., when p^ ^1 moves to the right), then more effect p^ will have on C * This is found to be true when the system is simulated on the analog computer, the results of which are presented in the next section. II - 11; Analog computer simulation . The compensated systems as resulting from example 2 and 3 are simulated on the analog computer as shown on figure 13, and step responses for various values of gain and parameters are shown on figure 14 through 15. The following is the equation for analog computer set-up: Let V(s) be the output from the compensator and Y(s) the system output. Then: Y(s) ^ K V(s) (s + p^)(s + pj^)(s + P 2 ) where the nominal values of the poles are = 0, “ 1.43, P 2 “ 3.33. K Y(s) V(s) " _3 s + (Pjj+ Pj^+ p 2 )s + (PqPj^+ Pj^P 2 + P0P1P2 3 2 s Y + (Pq+ Pj^+ P2)s Y + (PqP^, + PiP2+ p2p3)sY + PoPlP2’’^ " = -(p^Pi+ P2 >s^Y -(p^P^+ P^P2 + P2P3^sY - P^PiPz^ + KV s^Y = - \[(Pj+ P^+ P2 >s^Y + (PqPi+ P^P2+ P2P3>sY + P 0 P 1 P 2 Y "KV ] dt Ag s Y + Ag s Y + A^Y - A^ y] dt where A9 ^ Po + Pi + P2 - 33 - *8 • PoPl + n^2 + = V8 *7 “ P 0 P 1 P 2 AjHk = “ 6*6 The above equation leads to plant simulation set-up of figure 13 < Compensatocnls simulated separately, for example 2; C = s + Z s + P where Z = .39 and P = .14 A E s + P Let V = (s + Z) W where W Then V = sW + ZW S = sW + PW or sW = E - PW or -W “ PW ] dt now V = E - PW + ZW V = E + (Z-P) W -V = -[e + (Z - P) w ] The above equations giving W and V lead to set-up of figure 13 for the compensator . For example 3, V = s + P E s + P where P = 0.65 sv = E - PV -V = - \[e - PV ] dt thus one integration with feedback will be needed (see figure 13) < - 34 - ar I • -11 •» wr I • • • «u f T I »>(i 4i I For example 2 14 ®3®3 “ ^ • -7 "3 = ®3 “ ^ ^®4 “ ^ ■7 = Z -P = .25 ^ ^6®6 * ^ (variable) "7 3787 = P^P^P2 = 0 / S 4 = 1 'a^ = .25 85= 1 'a, variable •a7= 0 Vs ' Vl+ PlV P2P3 “ 88 = 10 V9 " P©'^ Pi’’’ P2 ^ J ^9 a^ = .476 89= 10 For example 3: 35 0 #•^1 • I • « I • "'•/ »> «u^‘' « * •< » ^ •• #• • «»«« ^ '4 Pl= .83 '^ 7 = ° '38 = -276 ag = .416 Discussion of results: For example No. 2, step responses are displayed on figure 14 for different values of gain K, varying about the nominal value K = 1.45. It is found that there is essentially no change in damping for values of K between 1.2 and 1.6. Even beyond these values, damping change is rather slow. This agrees with root locus of figure 11. For exampl]^ No. 3, with fluctuations of the pole at origin, step responses for ~.l<'p^^ ,1 are presented on figure 15. Note that the damping is not changed for the above variations in p^, the magnitude of which is not negligible considering the proximity of p^ to the dominant roots. Also note the faster rise time when p increases. This is due o to an increase in co , l.e., increase in bandwidth. n When p moves away from the origin, the system has some steadystate o error. However, the purpose here being the study of S , only changes in p C are of interest. ° - 36 - I For example No. 3 with fluctuations in pole-p^, the step responses for various values of Pj^ presented on figure 16. It is unfortunate that in this example, r ■ ^o is 56'^ (figure 12b), while a sensitivity phase of 30 is needed to make p^ have maximum effect on C (last remark, section II - 10). However, even at 56°, a small variation of .1 in the location of p^ (1.43 to 1.33) changes from 1.15 to 1.20, i.e., C from 0.53 to 0.45, using 2nd order approximation. Although the design was done on the basis of small parame;|^r changes, p- was moved further to the right (toward the position where would o ^1 approach the value 30 ), and as predicted by the tl^ry, the change of damping is more and more violent as the condition = 30*^ is approached. ?1 Therefore, the analog computer study has shown that, for the compensation scheme used, fluctuations of p have very little or no effect on ^ while those of p^ change ^ appr^iably, even for a small vajriation. The results % would be still better if S were equal to -60 and S were + 30 as So Pi computed in section II -10£s. II - 12; Single-stage or multiple-stage compensation ? a) The above designs have been done on the assumption that. oply> one-s|t;age compensators are to be used. The question arises as to whether any im- provments can be obtained by using more than one stage. When the magnitude of 0, phase shift needed to bring Q on the system root locus, is beyond a certain value, then one stage of compensator will not be sufficient. But even when 0 is small enough so that one stage of compensator will do, one still has to ask the same question. Further, in case of multiple stages, stages may be identical or diff- erent. Once the number of stages is decided, if identical stages are used, then the number of degrees of freedom remains the same as before, i.e., only one. But if different stages are used, then additional freedoms are intro- duced and possible improvements may come therefrom. "Improvement" here is used in the sensitivity-design sense, i.e. decrease in sensitivity i.e., increase in the magnitude of the scale vector U. b) It will now be shown that in general, only negligible improvement is introduced by using h identical stages of compensationi (each’ giving a - 37 phase shift of ^ ) instead of one stage (giving 0 ); and most of this negligible improvement is done by taking n = 2. Thus there is no reason \ to take n > 2. Besides, there is a risk of conditional instability in- volved . If non- identical stages are used, conceptually it is possible that for some fortunate choices of compensator stage, an improvement is obtainedo However, this involves cut-and try work, and no rules can be stated nor any definite results predicted. c) A short remark is necessary before the proof of the above statement can be undertaken. It concerns the construction of the (U) circle and the U- locus when compensators have n- (identical stages. Refer to figure 4a. assume that the compensator is double-staged. P is then a double pole. Two sensitivity vectors QM must be drawn, and when added to QI, they give a vector IM twice as long as the one in figure 4a. Thus, radius R of (M) has doubled. On the other hand, the radius r of (U), which was equal to 2R sin 0 for the one-stage compensator, now becomes sin % (the factor ° 2 0 2 comes from the fact that radius R has doubled, the angle ^ is phase shift from each of the two stages.) More generally, for n - identical 2 d becomes n 2d and radius r = sin 0 becomes stages, radius R == n . 0 r = -T sin ^ . n d n d) To prove statement (b), the plant given in previous examples is used. See figure 8 for pole-zero configuration, figure 9a for the U-locus for 1 stage compensation. 0 was measured to be -38^. Now, successively two 0 0 stages giving — each, then three stages giving j each will be used. For each case, the U-locus is drawn, as indicated on figure 17. Values of R and r are given on figure 17 for each case. It can readily be seen that the Increase in the magnitude of U-vector, which, at best, equals the increase in the radius r of the U-locus, is negligible, and thus is not worth the use of additional stages. It has been determined in the preceding paragraph, that for n - identical stages, the radius of (U) is r = ^ sin ^ as compared to r, = -7 sin 0 for non 10 single stage. Elementary trigonometry shows that for small 0, sin 0 = 0 ^ and r, = r. n srn , n = r^^, that is, there is no improvement in increasing the 38 - I «{« * • * fl • M O 0 number of stages. For 0 approaching 90 , n sin — > sin 0 and some im- ^ in. -O provement is possible. For the very best situations where 0 = 90 , 2 sin ^ is 1*^ times sin 0, which gives an increase in r of 40%. However the corresponding percent increase in magnitude of U is less, since U = QI + r, and QI doesn’t change. ^qO The curve of figure 18 shows the values of the ratio sin 90° for n= 1,2,3 .... It shows that most of the improvement is negligible. For 0 > 90°, n sin ^ < sin 0 sn if one stage can do the job. For 0 > 90°, n sin ^ < sin 0 and there is no interest in using many stages SH In conclusion, unless 0 comes out to be very close to 90 - in which case, use of two identical stages may lead to some improvement in sensitivity - it is not worth while to use unidentical stages when one stage can give the necessary phase shift. In addition to cost increase', the introduction of extra roots may be troublesome, while increase in the magnitude of U is insignificant . 39 - ILLUSTRATIONS - 40 - 1 I (a) -B 3 Fig i Construction of the vector diagram Fig .d’b- Vector addition to obtain U 41 mm % y'Htf "1 I i ^ «»if t ^S »| i 42 . Pif ^ • A.n application of the vector diagram. 1 ^ (a) >5Cafy ji ,'A ^J3o|/ fiCa-it |trv \|tctcv dM Qu ~ -<^u. X(^ = Qu'^ J- 2A Fig 4 • Geometric locus of point U a/ Vector diagram and Circle diagram , n . , b/ the U) circle, moved down by QI, gives (M)circle c/IU has constant lengths 2Rsin<t> Fig 5 *: Determination of the limit points of the geometric locus of U. a/ right hand limit; compensator's zero Z is at origin. b/ left hand limit : compensator's pole P is at minus infinity. I % II Fig € ; Determination of P and Z for a given U. a/ obtainO(pando(Hon the circle diagram, b/ obtain location of P and Z on root locus. ^ 5 “ Fl^V: Variations in dominant poles. a,d: stability unchanged, b: stabilizing effect, c: destabilizing effect. Fig S : Design example no 1. (compensation for minimum * sensitivities.) The "uncompensated” vector diagram. MtkPl « • * U *-IAI4> 4lrtDCUi»'’~» I I I ■li jtf I ><s« . 1 " JSli"! «4i« » dl;: « _i •?‘p Q + \ AuJ}’>t/\»'ipl’ (j : C/rrvy tin cJZi^ o^ ■tp- ^v>vxA U ^ • ^4,6 47 .1 giO: Design example no 2. (forS^ sO) . a/Circle diagram. b/Root locus. 48 i \ I I i I CO cO I 45 Fig l( : Design example no. 2: The compensated root locus. so : 1.43 : 1.33 R MARK n CHART" NO. RA 2921 32 Pi : Pi : 1.03 Pi : 0.83 Fig : Step response of compensated system of ex&ra_ pie 3» when plant pole -pi "'^aries. Note the rapid change of damping as approaches 30 ^. 1.23 2921 32 BRUSH INSTRUME .'J, Pig IS : Step response of compensated systeni of example 3» when plant pole _p^ varies a,bout the origin* Note that damping is unchanged. 55 DIVISION OF CLEVITE CORPOR M + :1.2 pt 3 ci 0.45 K : 1.45 (nominal) %t : 1.26 rS ^ 0.40 0.41 • Fig l6 : Step response example 2 when K 1.75 V ^ ‘Si’ 0.37 of compensated system of K varies about its nominal value. 5 "^ U-locus for : Fig \1 : U-locus for 1-stage, 2-stage, and 3-stage compensators. Improvement is negligible. Fig ; Value- of sin90° for different n. This. curve shows that most of the improvement, if any, is given by use of a double stage comp- -ensator. Use of more than 2 stages gives negli- -gible improvement. m ■ .. !l APPENDIX 1 Derivatiomi of an expression for the sensitivity of a multiple-order system root (equation 29) (This derivation is taken from reference 4) Write the expansion of the total differential of L(s) to Include higher- order terms? j ® = [ + 1 ! [ bs ^ ax same bracket ‘‘“j + IIf. j , j = -q. where [1^ ds + || dK ^ (ds)^ + i — (ds)^"^ dK + -* as as -^K a^L i(i-l) a^L ^ +■ nl 4 n y (dK) +• o o o 0+ . as^ ^ aK^^ aK^ Next retain only the lowest order terms for each parameter and not that the first (N-1) derivatives of L with respect to s are zero at s = (equation 27) . Then This suggests the following notation? Comparison of the above 2 equations yields equation (29) of text, which is thus proved. 57 1 APPENDIX II Relationship between S^> the multiple-order system root-case for P, Prove that equations (15) and (16) are still valid for the case of multiple- order root at -q^j From comparison of the last two equation of appendix I, one obtains; i A (-1)^ ^ n; r 5 l 1 “ ^ ” L 9 z , J J s = z . J 9*^1 9 s N But Thus; s' . ^ - L 9z . ” s + J j 4 -1 s= -q. "j ' ‘‘i ■ "i Similar proof shows that that is, equations (15) and (16) are valid for multiple-order root at -q^^ as well as for single-order root. - 58 - REFERENCES 1 - H. W. Bode: "Network analysis and Feedback Amplifier design". Van Nostrand, Princeton 1945. 2 - Hanoch Urs "Root locus properties and Sensitivity relations in control systems". IRE transactions on automatic control. Vol. AC - 5. January 1960. 3 - R. Y. Huang; "The sensitivity of the poles of linear closed-loop systems". Applications and Industry Vol. 77, page 182. 1938. 4 - D. T. McRuer and R. L. Stapleford; "Sensitivity and nodal response for single loop and multi-loop systems". Technical report ASD- TDR-62-812. Air Force Systems Command, Wright Patterson AFB, Ohio, January, 1963. 5 - H. Liu, K. Han, G. Thaler: "An investigation of passive adaptive . systems". Conference paper CP 63-1206. IEEE Pacific annual meeting, Spokane, Washington, August 196 6 - W. F. Horton, R. W. Eisner: "An adaptive technique" Conference paper, C.P. 63-1222 (same as 5). 7- E. R. Ross, T. C. Warren, G. J. Thaler: "Design of servo compensation based on the root-locus approach". Application and Industry, Vol. 79, September 1960. 8 - C. D. Poliak and G. J. Thaler; "s-plane design of compensation for feedback systems". PGAC, Institute of Radio Engineers, vol. A6-6, No. 3, September 1961. ^9 - C. C. Hsu; "a new graphical method for feedback control system com- pensation design". Applications and Industry, July 1962, pp. 160- 165. 10 - I. Horowitz; "Synthesis of feedback systems". Academic press, 1963, New York. - 59 ( I I .^1 iTTITT ^ ^ J AU t'/ 2 Jan* 1 4 U 7 . 68 INTITnLIBR,-VT>Y [Oacse Institute of Tech Cleveland, niiio] ( ^ p tA7 70319 • U6 Rung I no. 41 Feedback control sys- terns: design with re- gard to sensitivity. 10 w AN t7 14 8 7. 2 Jan ’65) INTTPJ.IBILMiY LOAN [Case Institute of Tech, Cleveland, Oliio] TA7 .U6 no. 41 70319 Rung Feedback control sys- tems: design with re- gard to sensitivity.