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Numerical and algebraic methods for computer-aided design of linear and piece-wise linear systems 1967
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Title | Numerical and algebraic methods for computer-aided design of linear and piece-wise linear systems |
Creator |
Sarkar, Basanta |
Publisher | University of British Columbia |
Date Created | 2011-09-13 |
Date Issued | 2011-09-13 |
Date | 1967 |
Description | A method is presented for linear control system design using functional relations between system parameters and system response. The functional relations are obtained by frequency domain evaluation of an integral performance criterion. The performance criterion is defined as a correlation measure between the responses of a known reference system and the system to be designed. A method is also presented for obtaining algebraic expressions relating the time-domain response of linear and piece-wise linear systems with system parameters. By means of a rational fraction approximation to the exponential e and through use of a known technique for evaluating time-domain convolution integrals, it becomes possible to obtain the time-domain response without the necessity of first having to determine the poles of the system. The time-domain response is obtained as a ratio of polynomials in t with the coefficients as algebraic functions of the system parameters. The extension of the linear design theory to cover non-linear and multivariable systems is given. Several examples are given to illustrate the usefulness of the proposed techniques. |
Subject |
Numerical Calculations Algebra, Linear System Amalysis |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2011-09-13 |
DOI | 10.14288/1.0104713 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/37265 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0104713/source |
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The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of BASANTA.SARKAR.. B.Tech., Indian I n s t i t u t e of Technology, 1958 M.E., Indian I n s t i t u t e of Science, 1959 M.Sc., U n i v e r s i t y of New Brunswick, 1963 IN ROOM 402, MACLEOD BUILDING WEDNESDAY, MAY 10,. AT 10:30 A.M.- COMMITTEE IN CHARGE Chairman: I. McT. Cowan E. V. Bohn R. W. Donaldson A. D. Moore Y. N. Yu A. C. Soudack G. F. Roach External Examiner: R. J. Kavanagh Uni v e r s i t y of New Brunswick Fredericton, N'.B. Research Supervisor: E. V. Bohn NUMERICAL AND ALGEBRAIC METHODS FOR COMPUTER-AIDED DESIGN OF LINEAR AND PIECE-WISE LINEAR SYSTEMS ABSTRACT A method i s presented for l i n e a r control system design using functional r e l a t i o n s between system param- eters and system response. The fu n c t i o n a l r e l a t i o n s are obtained by frequency domain evaluation of an inte - g r a l performance c r i t e r i o n . The performance c r i t e r i o n i s defined as a c o r r e l a t i o n measure between the response of a known reference system and the system to be designed. A method i s also presented for obtaining algebraic expressions r e l a t i n g the time-domain response of l i n e a r and piecewise l i n e a r systems with system parameters. By means of a r a t i o n a l f r a c t i o n approximation to the expo- nent i a l e s t and through use of a known technique for evaluating time-domain convolution i n t e g r a l s , it -becomes possible to obtain the time-domain response without the necessity of f i r s t having to determine.the poles of the system. The time-domain response i s obtained as a ratic of polynomials i n t with. the. c o e f f i c i e n t s as algebraic functions of the system parameters. The extension of the linear.design theory to cover nonlinear and mu l t i v a r i a b l e systems i s given. Several examples are given to i l l u s t r a t e the.usefulness of the proposed technique. GRADUATE STUDIES F i e l d of Study.: E l e c t r i c a l Engineering Nonlinear Systems E l e c t r o n i c Instrumentation Network Theory Servomechanisms D i g i t a l Computers Related Studies: • Numerical Analysis Mechanics and S t a t i s t i c a l Mechanics A. C. Soudack F. K. Bowers A. D. Moore E. V. Bohn E. V. Bohn C. A. Swanson B. L. White NUMERICAL AND ALGEBRAIC METHODS FOR COMPUTER-AIDED DESIGN OF LINEAR AND PIECE-WISE LINEAR SYSTEMS by BASANTA SARKAR B. Tech., Indian Institute of Technology, Kharagpur, 1958 M.E., Indian Institute of Science, Bangalore, 1959 M.Sc, University of New Brunswick, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Members of the Committee ................. Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA MAY, 1967 In presenting this thesis in pa r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia,, I agree that the Library shall make i t freely avail able for reference and study„ I further agree that permission., for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of E l e c t r i c a l Eng ineer ing The University of B r i t i s h Columbia Vancouver 8, Canada Date 1 0 May, 1 9 6 7 ABSTRACT A method i s presented for l i n e a r control system design using functional relations between system parameters and system response. The functional relations are obtained by frequency domain evaluation of an i n t e g r a l performance c r i t e r i o n . The performance c r i t e r i o n i s defined as a correlation measure between the responses of a known reference system and the system to be designed. A method i s also presented for obtaining algebraic expressions r e l a t i n g the time-domain response of l i n e a r and piece-wise l i n e a r systems with system parameters. "By means of s t a r a t i o n a l f r a c t i o n approximation to the exponential e and through use of a known technique for evaluating time-domain convolution i n t e g r a l s , i t becomes possible to obtain the time- domain response without the necessity of f i r s t having to determine the poles of the system. The time-domain response i s obtained as a r a t i o of polynomials i n t with the coefficients as algebraic functions of the system parameters. The extension of the l i n e a r design theory to cover non-linear and multivariable systems i s given. Several examples are given to i l l u s t r a t e the usefulness of the proposed tech- niques . i i TABLE OP CONTENTS Page ABSTRACT i i LIST OP ILLUSTRATIONS v i LIST OP TABLES v i i i ACKNOWLEDGEMENT . i x 1. INTRODUCTION 1 1.1. The Control Problem 1 1.2. Mathematical Models 1 1.3. Trial-and-error versus A n a l y t i c a l Design 1 1.4. Possible Design Methods 5 1.5. Statement of the Problem 5 2. FUNCTIONAL RELATIONS BETWEEN TIME DOMAIN RESPONSE AND SYSTEM PARAMETERS 9 2.1. Outline 9 2.2. Generalized Performance Integral 10 2.3. Generalized Performance Indices 19 2.3.1. Generalized Performance Index Based on System Error 20 2.3-2. Generalized Performance Index Based on the Correlation Between the Response of Two Systems 21 2 . 3 . 3 . Minimization and Maximization Procedure 22 3 . A PERFORMANCE FUNCTION APPROACH TO LINEAR SINGLE VARIABLE SYSTEM DESIGN 2 3 3 . 1 . Outline 2 3 3.2.1. Design of a Third Order System . 2 3 3.2.2. Design of a System With and Without Time Weighting 3 0 i i i Page 3 . 3 . Methods of Obtaining Approximate Values of Design Parameters 3 2 3 . 3 - 1 . Routh Array Approximation 3 3 3 . 3 . 2 . Correlation Function Approximation . 3 6 3 . 4 . I l l u s t r a t i v e Example 4 1 3 . 4 . 1 . Routh Array Approximation 4 2 3 . 4 . 2 . Correlation Function Approximation . 4 3 3 . 4 . 3 . Remarks 4 4 4 . ALGEBRAIC EXPRESSIONS RELATING THE TIME DOMAIN RESPONSE WITH SYSTEM' PARAMETERS 4 6 4 . 1 . Outline 46 4 . 2 . Generalized Time Domain Design Method 4 7 4 . 3 . The Derivation of Algebraic Relations Between System Response and System Parameters 4 9 4 . 3 . 1 . I l l u s t r a t i v e Example 5 4 4 . 4 . Applications to the Time-Domain Analysis of Linear Time-Invariant Systems 5 6 4 . 5 . Method of Residues 60 5 . NONLINEAR SYSTEM DESIGN 6 5 5 . 1 . Outline 6 5 5 . 2 . The Design P r i n c i p l e 6 6 5 . 2 . 1 . Choice of q for the Popov Line and the Range of K 6 8 5 . 3 . Time-Domain Analysis of Piece-wise Linear Systems 7 2 6 . MULTIVARIABLE CONTROL SYSTEM DESIGN 7 7 6 . 1 . Outline 7 7 6 . 2 . Design Method Based on Performance Functionals 7 7 6 . 3 . Time Domain Design Method 80 iv Page 6.4 An I l l u s t r a t i v e Design Example 83 7. CONCLUSIONS .' 91 APPENDIX A 93 APPENDIX B 103 REFERENCES 3_06 v LIST OF ILLUSTRATIONS Figure Page 2.1 Block Diagram of a Feedback Control System ... 9 3.1 Phase-Lead Compensated Position Control Servomeonanism ..t 23 3.2 Unit-Step Responses of the System Shown i n Figure 3.1 and the Reference Second Order System 28 3.3 Second Order Position Control Servomechanism . 30 3.4 Third Order Control System with Tachometer Feedback 33 4.1 A Feedback Control System 47 4.2 The Gains K and as Functions of Time t f f i of the F i r s t Maximum Amplitude g m 61 4.3 The Exact and Approximate Unit-Step Responses of the System Shown i n Figure 4.1 i n the Case of I n s t a b i l i t y 62 4 . 4 The Exact and Approximate Unit-Step Responses of the System Shown i n Figure 4.1 63 4.5 The Exact and Approximate Unit-Step Responses of the System Shown 'in Figure 4.1 64 5.1 Nonlinear Control System 65 5.2 Characteristic of the Nonlinear Element and Its Linear Bounds 65 5.3 Equivalent Linear System for the System of Figure 5.1 66 5.4 Popov Line and Locus of G*(JOJ) 67 5.5 A Piece-Wise Linear Feedback System 72 5.6 The Exact and Approximate Response v(t) of the System Shown i n Figure 5-5 for a Unit Step ,; Input .... 75 6.1 Multivariable Control System 78 6.2 Block Diagram Representation of Eq. (6.1) .... 79 v i Figure Page 6.3 Block Diagram Representation of Eq. (6.2) 80 6.4 Block Diagram of a Multivariable Control System , 84 6.5 The Gains and a s Functions of Time t for the System Shown i n Figure 6.4 88 B . l Overall Flow Diagram for Minimizing or:. Maximizing the Performance Index on the D i g i t a l Computer 105 v i i LIST OP TABLES Table Page 1.1 A Summary of the Performance Index Measures as Functions of Error 6 3.1 Yalues of K, a, T and P„,„„ for Known max values co and ̂ 27 c 3.2 Comparison of Unweighted and Time- Weighted Error C r i t e r i a 31 3.3 Routh Array for the Characteristic Equation of the Given Control System . 34 3.4 Energy Ratios Defined Prom the F i r s t Column of a Routh Array and the Coefficients of the Characteristic Equation 37 3.5 Results Obtained by Maximizing the Correlation Type Performance Index P on an IBM 7040 D i g i t a l Computer 44 1 4 . 1 Pade Approximations of exp(st) for Various Values of u and v 50 4.2 The Exact and Approximate Solutions of the System of Eq. (4.28) 56 5.1 C r i t i c a l Frequency, Popov Line Slope and Range of Optimum Gain i n Terms of Known Comparison System Parameters 71 A . l Values of I i n Terms of the Transform mn Coefficients 98 A.2 Values of J i n Terms of the Transform m A.3 Values of i n Terms of the Transform Coefficients 100 A.4 Values of J 0 i n Terms of the Transform m2 Coefficients 101 v i i i ACKNOWLEDGEMENT The guidance received from Professor E* V. Bohn, under whose supervision t h i s research work was carried out, i s gr a t e f u l l y acknowledged. The f i n a n c i a l assistance received from the National Research Council of Canada and the f a c i l i t i e s made available by the E l e c t r i c a l Engineering Department and the Computing Centre of the University of B r i t i s h Columbia are sincerely appreciated. Thanks are also due to fellow graduate students, p a r t i c u l a r l y Mr. A. G. Longmuir, Mr. H. R. Chinn and Mr. K. L. Suryanarayanan, for the i r assistance during the course of t h i s work. This work was supported by the National Research Council of Canada under a Grant i n Aid of Research Number 67-3134. i x 1. INTRODUCTION 1 1.1 The Control Problem The modern s c i e n t i f i c approach to engineering i n large measure consists of the formulation of problems so that methods of mathematical analysis may be applied. Engineering design of physical systems involving mathematical techniques have been rapidly developed and extended during the past three decades. One of the most important changes has been the broadening of interest from the frequency characteristics to the performance characteristics with the system excited by transient inputs or by actual t y p i c a l inputs described s t a t i s t i c a l l y . 1.2 Mathematical Models The a n a l y t i c a l complexity that would result from a more or less exact description of a control system i s avoided by s i m p l i f i e d descriptions, called mathematical models, for the physical devices making up the system. Feedback control systems are conveniently c l a s s i f i e d i n terms of the mathematical models that are employed as l i n e a r systems and nonlinear systems. I t i s i n the f i e l d of l i n e a r systems that the greatest advances i n design technique have taken place. However, i n spite of the advanced state of l i n e a r system design there appears to be room for further development. 1.3 Trial-and-error Versus Analytical Design More recently, control engineers have been exploring areas of performance analysis and design beyond the t r i a l - a n d - error design of l i n e a r systems. Exploratory work i s being done 2 i n the f i e l d s of a n a l y t i c a l design techniques to supplement trial-and-error methods for the design of l i n e a r systems. Anal y t i c a l design techniques are i n sharp contrast to trial-and-error design methods since they proceed d i r e c t l y from the problem specifications to the design without the need for human i n t u i t i o n . The trial-and-error design procedure provides no c r i t e r i o n for terminating the sequence of t r i a l s when d i f f i - culty i s encountered i n meeting the specifications. There i s no way of knowing i f the performance demanded i n the s p e c i f i c a - tions can be obtained or not. The a b i l i t y to detect inconsis- tent specifications i s a great advantage for the a n a l y t i c a l design method. I f the performance obtained by a n a l y t i c a l procedure i s not satisfactory, the designer i s certain that either the performance specifications must be relaxed or some of the other specifications must be altered. The design of control systems by application of the methods of mathematical analysis to idealized models which represent physical systems employs a more or less elaborate performance index as the basis on which the system performance i s judged. The objective of the performance index i s to encom- pass i n a single number, a measure for the performance of the system. The specifications that form the s t a r t i n g point of the a n a l y t i c a l design procedure, i n addition to the statement of the performance index to be used, must include a statement of the required property or value that the index must have for the system to be considered satisfactory. The a n a l y t i c a l design 3> procedure requires no e x p l i c i t statement concerning the degree of s t a b i l i t y of the over-all control system. A l l solutions include the twin requirements that the over-all system be stable and that i t be physically r e a l i z a b l e . Ever since the s u i t a b i l i t y of functionals i n engineering dynamical investigations was recognized, many authors have pro- posed various functionals as a quality measure of the performance of a control system. The control system error, e ( t ) , defined as the difference between the actual and the desired value of a controlled quantity or defined as the difference between the input signal and the feedback signal of a feedback control system was used to form various functionals of the general form roo F]_ = I f [ e ( t ) ] d t ( 1 . 1 ) 0̂ where f [e(t)J i s a function of e(t) and e(t) i s a function of time, t. The minimization of such an i n t e g r a l c r i t e r i o n was proposed as the basis of a procedure for the optimum design of a control system. Analog methods for the optimization of E q . ( l . l ) have been proposed by Bingulac and Kokotovic 1. The s e n s i t i v i t y coef- f i c i e n t s , that i s , the derivatives of with respect to the system parameters are obtained through the use of a parameter influence analyser. The parameters of the system are obtained by means of a best match with a second order reference model. This thesis presents a method which can be used to obtain the s e n s i t i v i t y coefficients as algebraic functions of the system parameters. The determination of the functional P -2 ~ e 2 ( t ) d t (1.2) 0 for the case where e(t) has a known Laplace transform which i s p a r a t i o of polynomials was made by P h i l l i p s ; he used Parseval' theorem to replace the i n t e g r a l of Eq. (1.2) by a contour i n t e g r a l and gave tables showing the value of functional F 2 i n terms of the transform c o e f f i c i e n t s . A n a l y t i c a l design theory has since been formulated to implement integral-square-error performance index for transient signals and mean-square-error performance index for stochastic signals. Westcott ^ used a simi l a r technique and gave tables showing the value of the functional 0 i n terms of the transform c o e f f i c i e n t s . Talbot 4 gave a method of computing functionals of the forms (1.3) (1.4) 0 and n=0, 1, 2, (1.5) 0 5 where the functions x(t) and y(t) have known r a t i o n a l Laplace transforms, showing how to determine the value of these func- tionals i n terms of the transform c o e f f i c i e n t s . He gave the solutions i n determinant forms. A summary of the history of the performance index measures as functions of error i s given i n Table 1.1. 1.4 Possible Design Methods Instead of establishing some r i g i d c r i t e r i o n of per- formance and applying i t to the evaluation and design of a l l systems, a more f l e x i b l e c r i t e r i o n may be used which can be adjusted to f i t the pa r t i c u l a r application of each system. The system error suitably weighted can be used to obtain such a f l e x i b l e c r i t e r i o n . The weighted error can be defined as some function of the actual system error, the s p e c i f i c form of the functional relationship depending upon the application of the system. A more f l e x i b l e performance c r i t e r i o n can be established by using the correlation function formed by the responses of two systems, the characteristics of one of the systems being known and taken as a reference. Based on the method of computing functionals of the form P^, given, by Eq. (1.5), i t i s possible to make a t r a n s i - t i o n from the frequency domain to the time domain and obtain the time response '©f the system i n terms of system parameters and - time. - 1.5 Statement of the Problem This thesis deals with the development of a n a l y t i c a l 6 Table 1.1 A Summary of the Performance Index Measures as Functions of Error. Performance Year Index Proposed Author ) e(t)dt 1942 Obradovic 5 n 6 1948 Oldenbourg and Sartorius 1949 Mack 7 1950 Stout 8 ) e 2 ( t ) d t 1943 H a l l 9 0 1943 P h i l l i p s 2 1949 Mack 7 1955 Rosenbrock ^ roo ) t 2 e 2 ( t ) d t 1949 Mack 7 0 1952 Fickeisen and Stout 1 1 12 1953 Graham and Lathrop 1957 Crow 1 5 OO P 14 e (t,7)dt 1949 Aigrain and Williams 15 16 Jr\ 15 1956 Spooner and Rideout OO • 00 1957 Schultz and Rideout 17 \ te(t)dt 1951 Nims 0 1952 Fickeisen and Stout 1 1 \ t e 2 ( t ) d t 1952 Fickeisen and Stout 1 1 i_ 12 1953 Graham and Lathrop 1954 Westcott 5 e(t)|dt 1952 Fickeisen and Stout 1 1 ^ 3 Graham a d Lathr p "1"2 1 8 1953 Caldwell and Rideout 7 Table 1.1 Continued Performance Index '0 t |e(t)| dt 0 0 4) t 2 | e ( t ) | d t oo Jo t n e 2 ( t ) d t , n=0,l»2,-.. oo Jo | e(t,r)|dt ' 0 0 J0 'de(t)' -i 2 dt dt Year Proposed 1953 1953 1954 1959 1957 1957 Oo Jo Author Graham and Lathrop Graham and lathrop 12 12 Vestcott Talbot 4 19 Schultz and Rideout 16 Babister 20 [|e(t)|Jrdt, r=l,2,5,.. 1959 F u l l e r 21 relations between system parameters and the time domain system response. Two methods are proposed to determine a n a l y t i c a l relations suitable for design purposes. One method i s based on the use of a correlation function as a generalized performance function. The system parameters are chosen to obtain a maxi- mum correlation between i t s response and the response of a reference system. One of the distinguishing features of this approach compared with other techniques i s that the reference system has a specified configuration but i s otherwise arbitrary, As a consequence, the use of t h i s c r i t e r i o n does not place undesirable constraints on the system pole-zero locations which 8 may be d i f f i c u l t to s a t i s f y . Also proposed i s a generalized time domain design method for l i n e a r and piece-wise l i n e a r control systems which allows an easy and rapid t r a n s i t i o n from the pole-zero locations or frequency domain to the time domain. The mathematical theory has been applied to the design of l i n e a r control systems. The application of the proposed methods to the design of a certain class of nonlinear system and multivariable sys- tems i s given. 2. FUNCTIONAL RELATIONS BETWEEN TIME DOMAIN RESPONSE AND SYSTEM PARAMETERS 2:1 Outline Figure 2.1 shows the block diagram of a feedback con- t r o l system configuration. This i s a rather general block diagram i n the sense that more complex configurations can be manipulated into t h i s form. As far as the control system designer i s concerned, he seldom has a completely free choice for the system. Usually he i s faced with a system that i s p a r t i a l l y specified. Actuating Signal Actual Output Input Compensating Elements Fixed Elements Ideal Feedback Signal "Feedback Elements + Output Error Figure 2.1 Block Diagram of a Feedback Control System. It i s a common practice i n the design of practical- systems to i d e a l i z e i t i n one or more ways, to reduce excessively complicated mathematics, by a simpler model which retains some of the more important features of the o r i g i n a l s p e c i fications. In a n a l y t i c a l design methods using performance indices based on system error, the error i s defined as the difference between the actual output and i d e a l or desired output«- The concept of actual output and ide a l or desired output w i l l be 10 used here to define a generalized performance index for the a n a l y t i c a l design of control systems using transient input signals. As a f i r s t step towards the above objective, a generalized performance i n t e g r a l w i l l be derived. 2.2 Generalized Performance Integral Let u=u(t) and v=v(t) be the actual and desired sys- tem outputs, respectively, of a feedback control system. A functional P can be defined by the i n t e g r a l which i s a measure of the correlation between u and v. By introducing weighting factors F can be modified to a functional I as follows where c n, c-, , c are functions of time. Defining the (2.1) 0 (2.2) weighting functions c 0' c as follows c Q = (qt) =1; c± = - ( q t ) 1 ; c 2 = (qt) 2/2 c = ( - l ) n ( q t ) n / n l where q i s a positive number, Eq. (2.2) becomes 11 or I = S 00 *Oo uvexp(-qt)dt , uvdt 1 - qt + for large values of n (2.3) + (-i)Vt n/n:] 0 The functional I i s the generalized performance in t e g r a l which w i l l be studied and i t may be used to define other functionals. D i f f e r e n t i a t i n g the right hand member of Eq. (2.3) with respect to q the performance i n t e g r a l I-, i s obtained as 0 Di f f e r e n t i a t i n g the right hand member of Eq. (2.3) with respect to q k times the performance i n t e g r a l I, i s obtained: Eq. (2.5) i s s i m i l a r i n form to the functional given by Eq. (1.5) of the previous chapter. It can be seen that Eqs. (2.3) and (2.4) are par t i c u l a r cases of Eq. (2.5) for values of k equal to zero and k equal to 1, respectively. Eqs. (2.3) and (2.5) w i l l be denoted i n the following forms: (2.4) (2.5) 0 r-ca uv exp(-qt)dt = (u,v) = I. (2.6) J mn 0 1 2 and (-l ) k J r V*'t kuv exp(-qt)dt = (u,v) k = 1 ^ (2.7) 0 where the subscript k i n I , denotes the kth derivative of I ̂ mnk mn with respect to q. The meaning of the subscripts m and n w i l l be explained l a t e r i n this chapter. For k'= 0, Eq. (2.7) reduces to J™uv exp(-qt)dt = (U,V) q ; .=. I ^ Q 0 = (u,v) : = 1 ^ (2.6) where, for reasons of convenience, the subscript zero has been omitted. Though Eq. (2.7) appears to be the most general form from which Eq. (2.6) :and other performance integrals having various forms of time weighting can be obtained, Eq. (2.6) w i l l be considered as the equation giving the generalized performance i n t e g r a l I . The reason for doing so w i l l now be considered. Linear system design i s often carried out i n the domain of the complex frequency variable s. The functions considered are then the Laplace transforms'U(s) and V(s), and i n the majority of cases, these are r a t i o n a l functions of s. The int e g r a l forms can be evaluated by using Parseval's theorem to replace the int e g r a l by one taken along the imaginary s-axis, the integrand being a product of transforms. By using Parseval's theorem, Eq. (2.6) can be written as 1 3 roo 0 = 2^j\ u(s)V(-s+q)ds (2.8) Since I , i s the kth derivative of I , mnk mn' W = d k lmn/ d^ = j E 2 i j | U(e)V(-s + q)ds (2.9) Thus i t i s only necessary to evaluate Eq.,(2.8) to be able to express ^•rm̂ . i n a suitable algebraic form i n terms of the coefficients of U(s) and V(s). Hence, I i s chosen as the generalized performance i n t e g r a l instead of I ^ ^ * When the integrand i n Eq. (2.8) i s i n a symmetrical form, the known properties of symmetrical functions make the desired result possible. This may be achieved by properly selecting the contour of integration so that the in t e g r a l taken along the imaginary s-axis i s replaced by one taken along a contour C ̂ » 4. This i s j u s t i f i e d as long as U(s) and V(s) have poles i n the l e f t - h a l f plane only. Thus, I can be written as Znm = g i j T U ( s + P ) V ( - s + P ) d s (2.10) C where p = q/2. It i s now possible to derive standard forms by solving the i n t e - g r a l I i n terms of the coefficients of U(s) and V(s) and the to ^ 14 parameter p for any given order of denominator polynomials of U(s) and V(s) with the obvious r e s t r i c t i o n that the order of the numerators must be one less than that of the denominators. F i n a l l y , performing the required operations on I , • ^ - M N ] £ ^ S obtained. The di f f e r e n t i a t i o n s w i l l now be performed with res- pect to p instead of q. If the Laplace transforms of the response functions U(s) and V(s) are expressed as rat i o s of two polynomials U ( S ) = f f f } (2.11) where m-1 ^ * k A'(s) = > a' ks k=0 m C'(s) = ) c ' k s k and where n-1 • * B'(s) = k=0 n D'(s) = ) (2.12) k=0 V(-s) = (2.13) (2.14) k=0 then U(s+p) and V(-s+p) can be expressed as °<-+p> - * H f $ = ( 2 - 1 5 ) 15 where and where m-1 A.1 (s+p) = ) a^ (s+p)" C(s+p) A(s) k=0 n y, k=0 m-1 r k=0 n c k(s+p)' a ks C ( B ) = c, s k k a, k=0 k+m-1 E 1=0 k+m 1 = 0 Trf \ B' (s+p)' B(s' n-1 B' (s+p) = ) b» (s+p)- k=0 n D'(s+p) = > d k(s+p) k=0 (2.16) (2.17) (2.18) (2.19) (2.20) 16 B(s) = n-1 k=0 n b ks (2.21) D ( B ) = d ks b, = k=0 k+n-1 L i=0 k+n k+i P" (2.22) i=0 It i s shown i n Appendix A that the solution of the inte g r a l I i s given by I = A'/c A (2.23) mn ' m where A i s the determinant of the (m+n)-rowed square matrix M and A' i s the determinant obtained from M on replacing i t s l a s t column by the column F, where M = '0. '1 m 0 "0 ' • . 0 '0 0 (2.24) n 0 0" m n (2.25) 17 and where i f . = > a.b. .. , for 0 = i^m+n-2 i=0 In the int e g r a l I , the subscripts m and n denote the orders of the denominator polynomials U(s) and V(s), respectively Letting v = u i n Eqs. (2.6) and (2.7), yields j u 2 exp(-qt)dt = (u,u) = 1 ^ = J m (2.26) "0 X oo t ku 2exp(-qt)dt = (u,u) k = 1 ^ = J m k (2.27) u Letting u = v i n Eqs. (2.6) and (2.7), yields roo v 2 exp(-qt)dt = (v,v) = I = K (2.28) no r~oo and ( - l ) k j t kv 2exp(-qt)dt = ( v , v ) k = I n n k = K n k (2.29) The subscripts m, n, and k have the same meaning as before and again for convenience the subscript zero i n Eqs. (2.26) and (2.28) has been omitted. Using Parseval's theorem and properly choosing the contour of integration, <J"m can be represented i n the following symmetrical form rV+i00 J m = 2^\ U(s+p)U(-s+p)ds (2.30) P-J°° where p = q/2 and Eqs. (2.11), (2.12), and (2.15) to (2.18) hold for U(s+p); and the solution of the in t e g r a l J f f l i s obtained as (see Appendix J = ( - l ) m + 1 A'/c A m ' m 18 (2.31) where A i s the determinant of the (mxm) square matrix M1 and A' i s the determinant obtained from M' on replacing i t s l a s t column by the column L, where and M1 '0 '0 '2m-2 (L Q. L 2f » L2m-2^ 0 '1 '0 'm-1 where Since J 2L. = 3 (-1) 3 1a.a. .. , for 0^ i ̂ m-1 i=0 m-1 (-1) i=j-m+l a. a . • , for m = j =2m-2 . mk i s the kth derivative of J m with respect to q, replacing q by p and performing the required operations on J , i s obtained. The solution of Z has the same form as J where a n m and c are replaced by b and d, respectively. Thus, there are three performance integrals, a gener- alized performance in t e g r a l I , and two derived from 1 ^ , J m and K . Starting from these performance integrals other per- formance integrals 1^^' m̂k' a n d *Sik c a n ^ e Grained • These performance integrals when evaluated appear as algebraic func- tions of system parameters. 19 2.3" Generalized Performance Indices In designing a feedback control system for a s p e c i f i c application, the designer usually has a d e f i n i t e goal i n mind. The control system i s to perform some given operation subject to physical constraints on i t s response. The designer i s , therefore, faced with the problem of translating t h i s e s s e n t i a l l y physical information into a mathematical d e f i n i t i o n of the desired performance which then becomes a c r i t e r i o n for synthe- s i z i n g the system. There sometimes arises the problem of designing the best possible control system of a given order of complexity to meet a given requirement. However, i t i s not easy to give a precise c r i t e r i o n of best performance. I f i t i s desired to design a best ( i n some sense) possible control sys- tem, i t i s necessary to define mathematically a c r i t e r i o n of performance. One commonly used c r i t e r i o n i s the minimum i n t e - g r a l of error-squared c r i t e r i o n f or transient inputs. While mathematically convenient, t h i s c r i t e r i o n has the disadvantage of giving too great an emphasis to large momentary errors. However, for comparison of performance between widely d i s s i m i l a r systems, this comprehensive error c r i t e r i o n i s l i k e l y to be much more consistent than any empirical ones. The question then arises of the p o s s i b i l i t y of devising a c r i t e r i o n of a compre- hensive type that w i l l correspond to the accepted empirical c r i t e r i a i n straight forward cases, but w i l l be available for wider use as systems become more complicated and diverse. To be convenient to apply, such a c r i t e r i o n must also be capable of allowing a simple algebraic representation of the error measure d i r e c t l y i n terms of system parameters. This i s possible for the generalized performance functionals chosen since they are a l l capable of algebraic representation i n the desired form. 2.3*1 Generalized Performance Index Based on System Error The performance integrals J and K and J , and K . r to m n mk nk can be used as performance indices when u = u(t) and V = v(t) are replaced by the system error function e = e ( t ) . Then Eqs. (2.26) and (2.28) give the performance indices as quadratic measure of error and Eqs. (2.27) and (2.29) give the perfor- mance indices as time-weighted measure of error. Any desired performance index based on error measure can be obtained by proper choice of q and k. For the sp e c i f i c case of q=0, k=0 Eqs. (2.27) and (2.29) reduce to the fa m i l i a r form of the quadratic measure of error, the integral-squared-error (ISE). The time-weighted measures of error, the integral-time multiplied- squared error (ITSE) and the integral-squared time multiplied- squared error (ISTSE) are obtained for the cases q=0, k=l and. q=0, k=2, respectively. Since J f f i and K have si m i l a r forms, i t i s s u f f i c i e n t to consider only one of them. The following form of J i s considered here as the generalized performance m o x . index based on system error. J = e 2 ( t ) exp(-qt)dt (2.32) 0 Minimization of J f f i or i t s derivatives with respect to q can be used as the c r i t e r i o n for optimum or best design. 21 2 . 3 . 2 Generalized Performance Index Based on the Correlation Between the Response of Two Systems I t has been shown that when u and v are replaced by the system error, e, 1^, J f f l and a l l reduce to the form given by Eq. ( 2 . 3 2 ) and can be used as a performance index, the basis on which the system performance can be judged. It w i l l now be shown that I , J , and K , given by Eqs. ( 2 . 6 ) , ( 2 . 2 6 ) and mn' m' n' to J ^ ' ( 2 . 2 8 ) , respectively, can be used to define a generalized per- formance index based on the correlation between the response of two systems. Regarding Eqs. ( 2 . 6 ) , ( 2 . 2 6 ) and ( 2 . 2 8 ) as the cor- r e l a t i o n equations between (u,v), (u,u) and (v,v), respectively, and using Schwarz's inequality yields [(u,v)] £ (u,u)(v,v) or (u,v)| £ J (u,u)(v,v) . ( 2 . 3 3 ) Rearranging Eq. ( 2 . 3 3 ) , and defining P = |(u,v)| / J(u,u)(v,v) £ 1 ( 2 . 3 4 ) a performance index r e l a t i n g I , J , and K i s obtained. The * to mn m n performance index P given by Eq. ( 2 . 3 4 ) can be regarded as a normalized measure of the correlation between the two responses u and v. P can be regarded as a performance index i n the best match sense when maximization of P i s considered as the design objective. Maximization of P, therefore, i s a meaningful ap- proach and can be used as a basis for the optimum or best design of a control system. When evaluated, P appears as an algebraic function of the system parameters. By using a suitable maximi- zation procedure, values for the unknown parameters which maxi- mize P can be found. Using time-weighted forms (u,v) k, (u,u) k 22 and ( v , v ) v , P takes the new form P. given by P k - (u,v) k|/ J(u,u) k(v,v) k i 1 m (2.35) 2.3.3 Minimization and Maximization Procedure The performance index, chosen on the basis of either the error measure or the best match measure, when evaluated, w i l l be an algebraic function of the system parameters. In minimizing or maximizing the performance index the usual procedure i s to di f f e r e n t i a t e the performance index with respect to each of the k design variables, equating each derivative to zero. It i s evident from the form of Eq. (2.23) that the derivatives, which are the s e n s i t i v i t y c o e f f i c i e n t s , are expressed as algebraic functions of the system parameters. That i s , e x p l i c i t relations are obtained for the s e n s i t i v i t y coefficients which are determined by analog means by the method of Bingulac and Kokotovic However, the solution of the k simultaneous nonlinear equations for the parameter presents some computational d i f f i c u l t i e s . The procedure selected here avoids the d i f f e r e n t i a t i o n problem and i t s associated d i f f i c u l t i e s and has the advantage of being an automatic method for dealing with the problem of minimization or maximization of the performance index with the aid of a d i g i t a l 22 computer 3. A PERFORMANCE FUNCTION APPROACH TO IINEAR SINGLE VARIABLE SYSTEM DESIGN 23 3.1 Outline The following examples demonstrate the use of the per- formance indices P and J^. i n the design of l i n e a r single input - single output control systems. In the f i r s t example, a l i n e a r t h i r d order system i s designed on the basis of a response cor- r e l a t i o n ,with the response of a known second order system. In the second example, a comparison between unweighted and time- weighted error c r i t e r i a i s given for the case of a simple second order system. 3.2.1 Design of a Third Order System A position control servomeonanism having two time con- stants and an integration and compensated by a phase-lead net- work,;1 as shown i n Figure 3 . 1 » i s designed on the basis of maxi- mizing the correlation of i t s response with that of a reference second order system for unit impulse input to both systems. R(s) ?! a(l+Ts) • K C ( B ) _ V 1+aTs s(l+T 1s)(l+ ,l' 2s) Figure 3.1 Phase-Lead Compensated Position Control Servomechanism. The closed-loop transfer function of the reference 2 4 second order system i s a)2 H (s) = -p 2 p ( 3 . 1 ) s + 2?co s + co„ c c where w , the natural frequency of o s c i l l a t i o n of the system i n c rad./sec., and the damping r a t i o of the system, are known. The closed-loop transfer function of the system shown i n Figure 3 . 1 i s KaTs + Ka H(s) = [aTT 1T 2s 4+ {aT(T 1+T 2)+T 1T 2}s 3 + (aT+T-j+T^s' (KaT+l)s + Ka] ( 3 - 2 ) where T^=4 sees, and T 2=l sec. and K, a, and T are regarded as the design variables with the constraint on a that a ^ 0 . 1 . The values of K, a, and T are chosen to obtain the maximum correlation between the two system responses for a unit impulse input to both systems. Denoting the output response of the reference second order system as V(s) and that of the unknown system as U(s), for a unit impulse input, Eqs. ( 3 . 1 ) and ( 3 . 2 ) y i e l d V(s) = - 5 2 p ( 3 . 3 ) s + 2?w s + GO c c , T T / \ KaTs + Ka and U(s) = aTT xT 2s 4 +faT(T 1+T 2)+T 1T 2}s 5 + (aT+T^T^s 2 (KaT+l)s + Ka] ( 3 - 4 ) The maximum correlation between the two system responses i s obtained by maximizing the performance index P such that P - , i ^ v ) l - ̂ i ( 3 . 5 ) m a x j(u,u)(v,v) where 25 "00 (u,v) = uvdt -oo (u,u) = u dt ^0 and (v,v) = v 2dt ' 0 Using Eqs. (2.24) and (2.32) and regarding q = 0, that i s p = 0, (u,v), (u,u) and (v,v) can be written i n the following determi- nant forms. (u,v) = 7- c o 0 w 2 c 0 0 2 a 0 W c c l c0 c co 2 c 0 2 a l w c c2 c l 1 -2<?w c 0 °3 c2 0 1 -2§<o c 0 C4 C3 0 0 1 0 0 c4 0 0 0 0 c0 0 ^c 0 0 0 c l c o -2?o)c co 2 c 0 0 c2 c l 1 -2?o>0 < 0 C3 c2 0 1 -2Sa> c co2 c c4 °3 0 0 1 -2fu c 0 C4 0 0 0 1 26 (u,u) (v,v) = where c0 0 0 2 a o C2 c l c o C4 °3 C2 0 (-1) 0 0 c4 0 2 c4 "co 0 0 0 C2 c l c o 0 c4 c3 C2 c l 0 0 C4 °3 u>2 c ( - D 1 0 2 2 CO c 1 0 2% = Ka C l = (KaT+l) • 9 '0 &q =-. Ka ; = KaT L l " r x 2 ' aTT xT 2 Results obtained by substituting (u,v), (u,u), and (v,v) i n Eq. (3.5) and maximizing P for various values of co are shown 2 2 i n Table 3.1. A b r i e f summary of the method used for maximizing P i s given i n Appendix B. This method avoids d i f - f e r e n t i a t i n g P with respect to each of the design variables, equating each derivative to zero, and solving a set of non- l i n e a r algebraic equations. The method works d i r e c t l y with the expression for P and searches for the maximum automatically on 23 a d i g i t a l computer once the search i s i n i t i a t e d . In terms of the notations explained i n Appendix B values of a = 3-0, S = 0.5 and e = 0.1 were used, along with s t a r t i n g values of 27 K _= 3.0, a = 0.2, and T = 3 . 0 , to initiate..the search for maximizing P. Given i n Section 3 . 3 are methods of obtaining approximate values of design variables to i n i t i a t e a search on the computer. The method i s fast and r e l i a b l e especially when a large scale d i g i t a l computer i s used. The meaning of the number of t r i a l s and the number of stages given i n Table 3 . 1 i s explained i n Appendix B. Table 3 - 1 Values of K, a, T and P for Known Values of to ' max c and f . CO c rad/sec P max K a T No. of Tr i a l s No. of Stages Appr.ox. Time of Solution 5.0 0.5 0.070 2.917 0.100 0.235 1972 60 200 sec. 4.0 0.5 0.092 11.936 0.100 1.860 1990 60 200 sec.. 3.0 0.5 0.239 52.827 0.100 1.763 1738 60 175 sec. 2.0 0.5 0.505 24.609 0.100 2.517 2231 60 235 sec. 1.0 0.5 1.000 7.115 0.156 2.825 2 2 3 7 .25 sec. The step response of the system i l l u s t r a t e d i n Figure 3.1 for values of K, a, and T corresponding to P__v =1.0, as given i n Table 3.1, i s shown i n Figure 3.2. The step response of the reference second order system for coc = 1.0 and ? = 0.5 i s also shown i n Figure 3.2. From the results given i n Table 3.1 i t can be inferred that the performance index P could be used to provide a f l e x i b l e c r i t e r i o n capable of being adjusted to suit system performance. A performance index based on system error when minimized becomes insensitive to parameter v a r i a t i o n but the time domain response of the system, such as the overshoot, could s t i l l be sensitive to parameter variations. The minimization of such a fixed 28 (a) Compensated Third Order System Response (b) Reference Second Order System Response Figure 3.2 Unit-Step Responses of the System Shown i n Figure 3.1 and the Reference Second Order System. 29 performance index, therefore, cannot always ensure meaningful results and can lead to system designs that are unstable or 23 physically unrealizable . Furthermore, i t i s usually more - important to minimize the time domain response s e n s i t i v i t y to parameter changes than i t i s to optimize the performance index. To overcome these disadvantages, i t i s desirable to allow a degree of f l e x i b i l i t y i n the choice of a performance function. The performance function P used here, since i t i s based on the response correlation with a class of reference systems, gives this f l e x i b i l i t y . I f a = ( a ^ , a ) represents the parameters of the system to be designed and (3 =(P-j_» » P m) represents the parameters of a reference system having a known response, then i t i s possible to determine the parameters a and |3 by the operation Min S Max P(a,p) W (3.6) where S represents some suitably chosen time-domain s e n s i t i v i t y function for the system and where, i n general, the parameters P k have specified upper and lower bounds. For example, S could be the s e n s i t i v i t y of the maximum overload with respect to variations i n K, Further improvement could be made i f the class could, for example, be the class of systems having a delayed second order response for step inputs. The reference system variables then become the time delay, ̂ , the damping ratio,: and ooc, the natural frequency of o s c i l l a t i o n of the system. The use of this.idea w i l l be considered i n the next Chapter. By using a class of reference systems, i t becomes 3 0 possible to investigate the parameter s e n s i t i v i t y of the time domain response, for example the overshoot to a unit step input. Thus, instead of choosing the system given by Eq. (3.6), i t may be more meaningful to choose the system which has least s e n s i t i v i t y to variations i n i t s time domain response. 3 - 2 . 2 Design of a System With and Without Time Weighting To compare the merits of different performance indices a second order position control servomechanism w i l l now be designed on the basis of an unweighted and two time weighted error c r i t e r i a . The system i s shown i n Figure 3 . 3 . The closed-loop transfer function of the system i s given by - S ^ f - T — 1 (3-7) R(s) s + as + 1 R(s) Figure 3.3 Second Order Position Control Servomechanism, where a i s regarded as the design variable. I t may be noted that i f the natural frequency of o s c i l l a t i o n of the system i s considered as unity, a equals twice the value of the damping r a t i o , •? , of the system. Consider that the system i s subjected to a unit step input and that the value of a i s to be found so as to minimize 31 the i n t e g r a l of error squared, Z^, the f i r s t time moment of error squared, a n <^ ^ e second time moment of error squared, J 2 2> i n that order. The error transform of the given system for a unit- step input i s given by E(s) = 2 s + a (3.8) s + as + 1 Using the standard form from Table A.2, Appendix A, for p = 0, J 2 = (1 + a 2)/2a J"2 i s a minimum for a = 1, when J 2 = 1. Using the standard form from Table A.3, Appendix A, for p = 0, J 2 1= (2 + a 4)/2a 2 J 2 ^ i s a minimum for a = 1.19, when Z^ = 0.718. Using the standard form from Table A.4, Appendix A, for p = 0, J"22= (eft - a 4 + a 2 + 4)/a^ J" 2 2 i s a minimum for a = 1.334, when J 2 2 = 1.737. The results are summarized i n Table 3»2 Table 3.2 Comparison of Unweighted and Time-weighted Error C r i t e r i a , m̂k Minimum value of J , mk a J2 1.000 1.000 0.500 J21 0.718 1.190 0.595 J22 1.737 1.334 0.667 32 This example shows how the i n t e g r a l performance c r i t e r i a can be used to determine unknown system parameters. In general, however, one weakness of these c r i t e r i a i s that there i s no direct relationship between the performance integrals and the time-domain response. To overcome this weakness a method for determining algebraic relationships between system parameters and the time-domain response w i l l be developed i n Chapter 4- 3.3 Methods of Obtaining Approximate Values of Design Parameters The search procedure selected for minimizing or maximizing a chosen performance index on a d i g i t a l computer requires a s t a r t i n g value to i n i t i a t e the search. While i t i s possible to arrive at a reasonable i n i t i a l guess for simple systems, the problem becomes increasingly d i f f i c u l t as more complex systems are encountered. The following sections deal with two methods of a r r i v i n g at reasonable s t a r t i n g values for the design variables to i n i t i a t e a d i g i t a l computer solution. The methods to be discussed use a comparison system and give relationships between the design variables and known parameters of the system to be designed and the comparison system. The comparison system i s assumed to be a second order system. However, this i s not a r e s t r i c t i o n imposed on the method; other comparison systems could be used. The theoretical development w i l l be i l l u s t r a t e d by an example i n Section 3.4- 24 33 3.3.1 Routh Array Approximation Consider the system of Figure 3.4 where K and are regarded as the design variables. The method to be'discussed i s based on choosing values of K and to give an optimum correlation with respect to the response of the comparison system H c(s) = - CO (3.9) s~ + 2§co s + co R(s) Figure 5.4 Third Order Control System With Tachometer Feedback. The closed-loop transfer function of the system shown i n Figure 3.4 i s given by H(s) = 1 (3.10) r i r 2 3 V r 2 2 l+K, T K s + 1 The Routh array for the characteristic equation of t h i s system i s shown i n Table 3.3. The .characteristic equation i s obtained by equating the denominator of Eq. (3.10) to zero. If an approximating transfer function, called an associated function, of second order i s constructed by using the l a s t 34 Table 3-3 Routh Array for the Characteristic Equation of the Given Control System. f i r 2 K r±+r2 K 1+KT K r 1 + r 2 i+z a ~K~" 1 .J three elements of the f i r s t column of the Routh array, then this transfer function w i l l have the same in t e g r a l squared impulse response as the system transfer function,Eq. (3.10). The i n t e g r a l squared impulse response i s computed from the l a s t two elements i n the Routh array. The associated function i s given by A(s) = (3.11) r1+r2 2 — r ~ 3 + 1+K G "TT~ r±z2 S + 1 The natural frequency of o s c i l l a t i o n of the system given by Eq. (3.1l) i s the same as that of the system given by Eq. (3.10). As a f i r s t approximation i n designing the unknown system, i t s natural frequency of o s c i l l a t i o n i s equated to 3 5 that of the comparison system. E q s . ( 3 . 9 ) and ( 3 . 1 1 ) y i e l d Z _ As a f i r s t approximation, therefore, z = ( r x + r 2 ) w 2 > ( 3 . 1 2 ) The energy r a t i o of the impulse responses of the two systems H(s) and H (s) w i l l now be considered. This r a t i o c i s computed from the f i r s t order coefficients of H (s) and C' A(s) which y i e l d the energy r a t i o E l = 1 + Z T Z r i r 2 This r a t i o w i l l be unity for a correlation match i n the id e a l sense. Therefore, as a second approximation, the following equation i s obtained. 1 + Kn K 2? ^_r 2 C 1 2 ( 3 . 1 3 ) Eqs. ( 3 . 1 2 ) and ( 3 . 1 3 ) are the required equations from which Z and are obtained i n terms of the known parameters X j , and co . c Additional energy r a t i o s can be defined between the 3 6 elements of the f i r s t column of the Routh array and the coefficients of the characteristic equation to. obtain further relationships between the system parameters. These energy ratios are l i s t e d i n Table 3.4. The energy r a t i o method i s based on the concept of obtaining a simple response approximation for a system by comparing i t with a known comparison system. The approximation i s r e s t r i c t e d to systems described by a lumped-constant l i n e a r d i f f e r e n t i a l equation whose transform i s of the following form. H(s) = — °- a s + + a-, s + a A n 1 0 The response approximation i s performed by placing a constraint on the r a t i o between the integrated square of the system impulse response and the corresponding i n t e g r a l of the comparison system. Since the response of a second order system i s easy to v i s u a l i z e i t i s convenient to choose a second order comparison system. However, by means of a Pade approximation, a delayed second-order response could also be used. Many systems of high order have responses which can be accurately approximated by such a delayed response. 3 . 3 . 2 Correlation Function Approximation For the two system transfer functions H (s) and H(s), c given by Eqs. (3.9) and (3.10), the following three relationships are obtained using Parseval's theorem. 37 Table 3.4'Energy Ratios Defined Prom the F i r s t Column Elements of a Routh Array and the Coefficients of the Characteristic Equation. Energy Ratio Routh Array E „ z= a o/R-7 -I n-2 n-2' 3,1 • a o e o « e • © o E3 = a 3 / R n - 2 , l E 2 = a 2 / R n _ 1 A Ê = an/R n 1 1' n , l i a n a 'n-2 .. a 2 a Q j n-1 S.1 a n-3 S,2 Rn-2,1 Rn-2,2 / / R T T R ., - / n-1,1 n-1,2 / R h , l R n+1,1 l i = H c(s)H c(-s) ds -2oo (3.14) 1 2«;j H ( S ) H ( - B ) ds (3.15) and hh = 1 2*3\ H ( B ) H C ( - S ) D S (3.16) Each of the above three equations can be solved from a set of 3 8 l i n e a r equations. Corresponding to Eq. ( 3 . 1 4 ) the set of li n e a r equations i s 2 ' 4 , N <Vl = " V 2 z 1 + 2?co cz 2 = 0 ( 3 . 1 7 ) where z = h 2 2 c Corresponding to Eq. ( 3 . 1 5 ) the set of l i n e a r equations i s ( r x + z 2 ) j i + ( 1 + K r ) y 2 + ' K^3 = 0 T1Z2*Z + (Tl+r2^3 = 0 (3.18) 2 where y ? = h Corresponding to Eq. ( 3 .16) the set of l i n e a r equations i s w 2x x + Kx, • = Kto2 2 c ~ 2 ? w c X l + w c x 2 + ^ 1 + K t ) x4 + Kx 5 = 0 x l " 2 § ( 0 c x 2 + w c x 3 + ( ? i + 2 2 ) x 4 + ( 1 + K T ) x 5 = 0 (3.19) x2 " 2 ? a j c x 3 + r i ^ 2 x 4 +( r 1+ 22 ) X 5 = 0 x 3 + \r27i5 = 0 where x = hh~ 5 c In the above equations the variables z-^,Z2,y-^,y2> etc., represent 39 the coefficients of powers of s occurring i n Eq. (A.9) i n Appendix A. • „ . . — —2 Assuming that h = h , that is,. y 3 . z 2 = (0 o/4S (3.20) Eqs. (3.18) and ( 3 . 2 0 ) y i e l d 1 + K T = K l u2 + 2 t -2 c Solving for Xj-, from the set of Eq. (3 .19), (3.21) AK + BK(1+KT) 5 ~ ^2 K + EK + PK(1+ZT) + G(1+KT) + H(l+Z T)^ + J (3-22) where A B E = 2 o ) 2 ( r 1 + r 2 ) ( 2 ^ 2 - i ) + 2 ? o > ^ r 1 r 2 ( 4 ? - 3 ) F = 2?w .G. H 2 ^ r 1 r 2 ( 2 ^ - i ) + 2 ^ ( r 1 + ^ ) 2 and J = + ^ ( r ^ z , ) * + 2 ? ^ r 1 r 2 ( r 1 + ^ ) To maximize the correlation function given by Eq. (3 .16), Eq. (3.22) i s differentiated with respect to K. Equating the 40 derivative to zero yields A+B(l+KT) -K 2+G(l+K T)+H(l+K T) 2 + J = 0 Therefore, either A + B(l+K T) = 0 which i s a t r i v i a l solution, or K 2 = G(l+K T) + H(l+K T) 2+ J (3.23) Substituting Eq. ( 3 . 2 3 ) . i n Eq. (3.22), yields A + B(1+KT) '5 ~ 2K + E + E(1+KT) (3.24) Assuming that hh = h , c c ' Letting x 5 = (o0/« a = to. (3.25) (3.26) and substituting Eqs. (3.21), (3.25) and (3.26) i n Eq. (3.24) yields K = a^a + a^ (3.27) where a. (^,+^) 3 3 S o (5?-12?:') L l c 2 4 1 a. (2" 1 +r 2) 1 6 2 (6S 2-2) and a 4 = i + e f r ^ I f coc of the comparison system i s regarded as an unknown quantity, instead of assigning an arbitrary fixed value to i t , Eq. (3.27) can be used to obtain an approximate s t a r t i n g value. Prom Eq. (3.27) i t i s seen that for K to be positive, a-̂oc - a 2 > 0 or a>a 2/a- L that i s , tt> 6 < g " 2 (3.28) Eq. (3.28) provides the relationship between the known parame- ters of the system to be designed and ^, the known parameter, and co , the unknown parameter, of the comparison system. It c w i l l now be shown how the results obtained i n this and the previous section can be used to obtain approximations to system parameters which can then be used as i n i t i a l estimates for the d i g i t a l computer solution. 3.4 I l l u s t r a t i v e Example Consider the system of Figure 3»4 for which the system transfer function i s given by Eq. (3.1 0 ) . l e t the comparison 42 system be a second order system and i t s transfer function have the form given by Eq. (3.9). I t i s assumed that the time constants V and ^ i n Q̂.. (3.10) and t i n Eq. (3.9) are known and have the following values. T = 1.174 sec. 2̂ = 0.426 sec. = 0.6 Based on these choices and considering K and as unknowns, values of K and w i l l be calculated using the approximation methods described before. 3.4.1 Routh Array Approximation To obtain a suitable value of co consider that the c r i s e time of the unknown system i s the same as that of the comparison system and from physical considerations l e t t h i s , for example, be 2.02 seconds. Therefore, for the comparison system, the maximum value of the impulse response i s h m a x = l/(Rise Time) = 0.495 In terms of ^ and co , h i s given by C ULclJC CO c hmax = T T ^ e x p I ? VJ^y 1 - 7j--§ tan 1( J l-"§7<?) 32 ^ 180 (3.29) Substitution of the value of h into Eq. (3.29) yields max co = 0.786 rad./sec. c 43 For this value of co , Eqs. (3-12) and (3.13) y i e l d K = 0.99 1 + Krp = 1.82 Results obtained using these values of K and K̂ , as i n i t i a l estimates for the design variables and maximizing the correlation type performance index P are shown i n Table 3«5. The results correspond to the comparison system parameters t = 0.6 co = 0.786 rad./sec. c 3.4.2 Correlation Function Approximation For t = 0.6, Eq. (3-28) yields a > 0.392 From Eq. (3-27) i t i s seen that for a = 0.392, K = 0 and for small values of a, K i s also small. For example, i f a = 0.4 K = 0.0034, and Eq. (3.26) gives co = 1.28 rad./sec. Choosing a larger value of a, for example cc = 1, Eqs. (3.26) and (3.27) y i e l d co = 3„ 2 rad/sec. K = 1.051 Substitution of these values of co and K i n Eq. (3.21) yields c 1 + K T = 0.723 The negative value of K^, as an i n i t i a l estimate, i s the result of the choice of and 2^ a n d ^ . The value of coc also depends on the choice of these parameters. 44 Maximizing the performance index P for the above i n i t i a l estimates of the unknown parameters K, and u>Q i t i s found that the value of P increases as w i s reduced from the c estimated value of 3.2 rad./sec. For comparison with the Routh array approximation r e s u l t s , obtained i n Section 3-4.1, values of K and K^, corresponding to wc = 0.786 rad./sec, obtained by th i s method are given i n Table 3.5. Table 3-5 Results Obtained by Maximizing the Correlation Type Performance Index P on an IBM 7040 D i g i t a l Computer. Routh Array Approximation Correlation Function Approximation Approximate Values K 0 . 9 9 1.051 K T 0.82 -0.277 0) c 0.786 3.2 Maximum.Correlation Values K 1.00074 1.00083 K T 0.91208 0.91198 0) c 0.786 0.786 p max 0.98733 0.98733 3.4.3 Remarks Results given i n Table 3.5 indicate that^irrespe.ctive of the i n i t i a l value used to obtain a maximum correlation between the unknown system and the comparison system the 45 d i g i t a l computer program yielded almost i d e n t i c a l r e s u l t s . I t can, however, be seen that the Routh array approximation gave i n i t i a l estimated values closer to the maximum correlation values of the design variables. Furthermore, the correlation function approximation method becomes d i f f i c u l t to handle as the complexity of the unknown system increases. From t h i s point of view the Routh array approximation has an advantage over the correlation function approximation. 46 4. ALGEBRAIC EXPRESSIONS RELATING THE TIME-DOMAIN RESPONSE WITH SYSTEM PARAMETERS 4.1 Outline. A continuing problem i n systems design i s to determine the relationships between time-domain response characteristics and system parameters. The root-locus and the parameter-plane methods are graphical means of establishing numerical r e l a t i o n - ships between the characteristic roots of li n e a r time-invariant 25 systems and system parameters. Computer solutions of system d i f f e r e n t i a l equations, whether obtained by analog or d i g i t a l means, are ess e n t i a l l y numerical i n nature and can only be used to determine empirical relationships, between system parameters and system response by curve f i t t i n g techniques. I t has been suggested that the i n i t i a l value theorem and a Taylor series 2 6 could be used to obtain a n a l y t i c a l relationships. However, l i t t l e use has been made of t h i s suggestion because of the poor convergence of the Taylor series. Better methods for the eval- 27 uation of system response are the state-space approach and 28 the use of moments and special sets of polynomials. However, these methods are again numerical i n nature. The following sections deal with a technique which determines algebraic r e l - ations between system parameters and the time-domain system response which i s superior to the Taylor series approach both i n the accuracy achieved with a given number of terms as well as i n i t s computational convenience. Application of this technique to the analysis of piece-wise l i n e a r systems w i l l be 47 discussed i n Chapter 5. 4.2 Generalized Time Domain Design Method Consider the feedback control system of Figure 4.1 with the closed-loop transfer function : having the general form n-1 a_ -, s +. + a-, s + a„ H ( S ) = JOZL 1 0 c s n + + c ns + c A n 1 0 K Ts Figure 4.1 A Feedback Control System. Taking the inverse Laplace transform of H(s) yields h(t) = J_ C H(s)exp(st)ds • (4.1) Eq. (4.1) i s inconvenient i n control system design because of the fact that i t does not y i e l d any direct r e l a t i o n - ship between system parameters and system response. One such relationship i s obtained by r e l a t i n g a Taylor series of H i n powers of l / s i n the frequency domain to a Taylor series of h ,,1 , v, 0 48 26 i n powers of t i n the time domain . The expansion of h i n the Maclaurin series form yields h(t) = h(0) + th'(O) + t 2h"(0)/2: + + t k h ( k ) ( 0 ) / k i where h(0) = a-̂ h'(0) = a 2 - 0 ^ ( 0 ) h ( k ) ( 0 ) = a k + 1 - c ^ ^ ^ ^ O ) - - c kh(0). The usefulness of t h i s approach of time domain design i s , however, l o s t because of the necessity of employing a large number of terms even for simple systems. A new method for obtaining direct algebraic relations between system response and system parameters through use of Eq. (4-1) w i l l be presented i n Section 4.3. I t appears from Eq. (4.1) that i f a system with a transfer function H (s) = exp(-st) could be used, i t would be possible to obtain h(t) i n terms of the system parameters and t following the procedure for evaluating I i n Eq. (2.23). This, however, requires H (s) to be a r a t i o of polynomials i n s, and that the denominator c of H (s) be a Hurwitz polynomial. The required representation c for exp(-st) can be obtained by means of a r a t i o n a l f r a c t i o n a l approximation. A Taylor series approximation to exp(-st) i s one such p o s s i b i l i t y . However, i t i s commonly accepted that Pade approximations are superior to a Taylor series approxi- mation. Pade approximations for exp(st) are given by 49 P w = exp(st) = lim F (st) uv (u+v)-oo Q. ( G T ) 2 where P / , \ -, u(st) u ( u - l ) ( s t ) ' UVvSt j — -L + + u+v 2J(u+v)(u+v-l) u + u(u-l) (2)(l)(st) ui (u+v) (v+1) and G u v ( s t ) = 1 - v ( s t ) + v ( v - l ) ( s t ) 2 + v+u 21(v+u)(v+u-1) + (-D T v ( v ~ 1 ) . . ( 2 ) ( l ) ( s t ) V v! (v+u). (u+1) Pade approximations of exp(st) for various values of u and v are shown i n Table 4.1. 4.3 The Derivation of Algebraic Relations Between System Response and System Parameters The technique to be discussed can be applied to time- invariant l i n e a r and piece-wise l i n e a r systems and i s based on Parseval's ide n t i t y u(t)v(t)dt = _1_ j ' V(s)U(-s)ds (4.2) and on a method given by Talbot 4 for evaluating Eq. (4.2). Talbot's method requires that u(t) and v(t) be the output response of stable time-invariant l i n e a r systems and i s a generalization of the well known technique for evaluating mean-square integrals. 50 Table 4*1 Pade Approximations of exp(st) for Various Values of u and v . u V 1 1 1 2 2 2 2 3 3 3 3 4 4 4 P u v = exp(st) 2 + st 2 - st 6 + 2st 6 - 4st + s 2 t 2 12 + 6st + s 2 t 2 12 - 6st + s 2 t 2 60 + 24st + 3 s 2 t 2 60 - 36st + 9 s 2 t 2 - s 3 t 3 120 + 60st + 1 2 s 2 t 2 + s ^ t 3 120 - 60st + 1 2 s 2 t 2 - s 5 t 3 840 + 360st + 6 0 s 2 t 2 + 4 s ^ t 3 840 - 480st + 120s 2t 2 - l 6 s 3 t 3 + s 4 t 4 1680 + 840st + 180s 2t 2 + 20s^t 3 + s 4 t 4 1680 - 840st + 180s 2t 2 - 2 0 s 5 t 5 + s 4 t 4 Let where V(s) = A(s cTs U(-s) = Bis DTs A m-1 v :. n-1 , A(s) ~ JZZ a v s k ; B(s) 4 £Z b k s k 0(s) k I k=0 m k=0 n k=0 c ks D(s) i X Z ^ k=0 (4.3) 5 1 and where C(s) and D(-s) are Hurwitz polynomials. The evaluation of E q . ( 4 . 2 ) could be performed by a p a r t i a l f r a c t i o n expansion of the form A(s)B(s) = Q(s) + R(s) ( 4 . 4 ) C(s)D(s} D(F) cTs) where Usl = I Z A ( s k ) B ( s k } . 1 ( 4 . 5 ) C W k = 0 D(s k)C'(s k) s-s k and by completing the path of integration along an i n f i n i t e l y large semicircle i n the l e f t - h a l f s-plane. Thus, m i = r^ A ( s k ) B ( s k ) ( 4 > 6 ) k = 0 D(s k)C'(s k) = lim sR(s) ( 4 . 7 ) s—•oo C(s) However, the evaluation of I by E q . ( 4 . 6 ) requires the numerical determination of the characteristic roots, which can be avoided i f suitable use i s made of E q . ( 4 . 7 ) . l e t ,m-I, . n - l t k R(s) = } _ r , s K ; Q(s) = )_2 % s ( 4 . 8 ) k = 0 K k = 0 * n+m-2 , P(s) £ A(s)B(s) = I _ f v s k ( 4 . 9 ) k = 0 K I t follows from E q . ( 4 . 7 ) that I = rm-l ( 4 . 1 0 ) c m and from E q . ( 4 . 4 ) that E(s) = Q(s)C(s) + R(s)D(s) ( 4 . 1 1 ) 52 Substituting Eqs. (4-3) and (4 .8 ) into Eq . ( 4 . 1 l ) and comparing l i k e coefficients of s yields o o + dOrO •o c i * o + c o q i + + d ^ + d 0 r x = f 1 c q~ + m̂ -0 + d r„ + n 0 = f m (4.12) c q -, + m^n-1 d r -i n m-1 = 0 where f k = 0 i f k>n+m-2. The system of Eq . (4 .12) can be solved for rm_-|_ and substituted into Eq . ( 4 . 1 0 ) . Thus where I = 1_ c m A' A (4.13: '0 L0 • 0 0 A = m '0 n l 0 (4.14) 0 0 m n and where A' i s obtained from A by replacing the l a s t column by 53 the right hand column of Eq. (4.12). The result given by Eq.(4.13) expresses I i n terms of system parameters. Consider now the p o s s i b i l i t y of expressing I i n terms of a time-domain response. I f = £(Z"-t), where S(t) i s the unit impulse, then U(-s) = exp(st) ; (t>0) (4.15) and Eq.(4.2) reduces to the conventional inverse Laplace transform. However, Talbot's method does not apply for Eq.(4.15). On the otherhand, i t i s known that Eq.(4.15) can be approximated by means of r a t i o n a l f r a c t i o n s , f or example, _ 1 29 the Fade approximation P 2 3 ( s t ) = 60 - 24st + 5 ( s t ) 2 (4.16) 60 + 36st + 9 ( s t ) 2 + ( s t ) 5 may be used to approximate the ideal delay exp(-st). Let umn('Z') be the impulse response of a system whose transfer function i s P ( s t ) . Por the all-pass case where m = n, mn r u n n ( r ) = ("D^^) + S n n ( r - t ) } ' ( t > 0 ) ( 4 ' 1 7 ) and for the low-pass case where m = n-1 = S m ( r - t ) , ( t > o ) (4.18) I t i s a consequence of the Pade approximation that lim S (r) =S(T) (4.19) n—•oo Por the all-pass case, l e t B(s) and D(s) be poly- nomials i n s which have no common di v i s o r and which are defined by P (-st). = ( - l ) n + B(s) (4.20) ^ D(sT Substituting Eq.(4.17) and Eq.(4.20) into Eq.(4.2) yields 54 where /-do v n n ( t ) a V ( T ) S n n ( r - t ) d r , (t>0) (4.22) For the low-pass case, l e t B(s) and D(s) he polynomials i n s which have no common di v i s o r and which are defined by P f-st) = B(s) (4.23) D(s Substituting Eqs.(4.18) and (4.23) into Eq.(4.2) yields v n n ( t ) = ^ r 3 " a. (4. 24) where vmn ( t ) = j v ( r ) £ m n ( r - t ) d r ' ( t > 0 ) ( 4 * 2 5 ) The integrals i n Eqs.(4.21) and (4.24) can be expres- sed i n the form of Eq.(4.13), consequently v ft) = 1,A' . (t>0) (4.26) c A m I t follows from Eqs.(4.19), (4.22) and (4.25) that lim v (t)•= v(t) , (t>0) (4.27) n—*co Thus, Eq.(4.26) gives the desired algebraic r e l a t i o n between the time-domain response of a system and i t s parameters. 4.3.1. I l l u s t r a t i v e Example To i l l u s t r a t e the proposed method consider V(s) = 2s 2 + 5.5s + 1.75 = A(.s) (4.28) s 5 + 3s 2 + 2.75s + 0.75 55 and choose Eq.(4.l6) so that Eq.(4.23) i s used. The choice of Eq.(4.28) i s made so that a comparison can be made with the state-space method proposed by Liou which i s claimed to be superior to c l a s s i c a l methods. Eq.(4.14) yields A = 0.75 0 0 60 0 0 2.75 0.75 0 -36t 60 0 3 2.75 0. 75 9 t 2 -36t 60 1 3 2. 75 - t 5 9 t 2 -36t 0 1 3 0 - t 5 9 t 2 0 0 1 0 0 - t 3 and Eq.(4.9) yields f Q = 105 , f-L = 210 + 42t, f 2 = 120 + 8.4t + 5.25t 2 f = 48t + 10.5t 2 , f^ = 6 t 2 , f 5 = 0 (4.30) The response Vg-^t), given by Eq. (4.26), i s expressed as the r a t i o of a f i f t h order polynomial i n t and a s i x t h order polynomial i n t . Table 4.2 compares the results given by Liou with Eq.(4.26). A direct comparison i s not possible. However, Liou's method requires the computation of ninth-order matrix products and i s e s s e n t i a l l y based on a Taylor expansion which includes terms up to the ninth order. For the i n i t i a l portion of the response Eq.(4.26) i s not only a simpler representation, but has the further advantage that system parameters enter i n a simple way. This i s readily seen by replacing the numerical entries i n Eq.(4.29) by parameters. The response given by Eq.(4.26) then consists of a r a t i o of polynomials i n t with the 56 Table 4.2 The Exact and Approximate Solutions of the System of Eq.(4.28). t = nT v(t) v 2 5 ( t ) Exact Solution 0 2.00000 2.000000 2.00000 0.1 1.76781 1.767809 1.76781 0.2 1.56775 1.567742 1.56774 0.5 1.39515 1.395146 1.39515 0.4 1.24604 1.246038 1.24604 0.5 1.11701 1.117022 1.11700 0.6 1.00515 1.005196 1.00515 0.7 0.907982 0.908084 0.907979 0.8 0.823383 0.823582 0.823379 0.9 0.749542 0.749889 0.749538 1.0 0.684914 0.685474 0.684912 system parameters entering i n a simple algebraic manner. I t i s evident from t h i s representation that time-domain response s e n s i t i v i t y to parameter variations can be readily evaluated. This as well as other p o s s i b i l i t i e s w i l l now be discussed. 4,4 Applications to the Time-Domain Analysis of Linear Time- Invariant Systems Consider a feedback system whose closed-loop transfer function i s given by (Figure 4.1) H(s) = K (4.31) ^ r 2 s 5 + ( r i + ^ ) s 2 + (l+KrjJs + K I t i s of interest to determine how the gain K and the tachometer 57 feedback parameter Z^ affect the maxim"um overshoot for a unit step input. Let h(t) and g(t) be the unit-impulse and uni t - step response, respectively. The maximum overshoot occurs at the f i r s t zero of h ( t ) . This can be found from Eq.(4.26) by choosing V(s) and, for example, U(s) = P 2 ^ ( s t ) : ^25^ t ) = - i C1L2 Z 0 0 60 0 60Z z 0 -36t 60 24tK 1+KT z 9 t 2 -36t 3 t 2Z 1+1^ - t 5 9 t 2 0 0 r 1 + r 2 0 - t 3 0 0 0 0 0 0 z 0 0 60 0 0 K 0 -36t 60 0 r 1 + r 2 z 9 t 2 -36t 60 r i r 2 - t 3 ' 9 t 2 -36t 0 r x r 2 0 - t 3 9 t 2 0 0 * i Z 2 0 0 ' - t 3 (4.32) Equating the numerator determinant of Eq.(4.32) to zero yields t 6 Z 2 - 8t 5Z(l+K T) - 1 2 8 t 4 K ( r i + r 2 ) - 7 4 4 t 3 K r i r 2 + 2 0 t 4 ( l + K T ) 2 - 840t 2 ( l+K T ) r i e ' 2 + 12 0 t 3 ( l + Z T ) ( 2 ' 1 + r 2 ) + 1200t 2(?' 1+2 2) 2 2 2 + 96oot(^n + r „ ) r 1 r 0 + 3600ZT J- d ± <L l 2 = 0 ( 4 . 3 3 ) I f , and t = t are specified, Eq.(4.33) i s a quadratic m form i n Z and 1+Z^ and shows how these parameters must be r e l a - 58 ted i f the maximum overshoot i s to occur at the instant t m The maximum output amplitude = g(t ) can also be expressed i n the form of Eq.(4.26). To obtain the desired result l e t A(s) and C(s) be polynomials i n s which have no common factor and which are defined by G(s) = 1 + H(s) - 1 = 1 + A(s) (4.54) s s s C(s) If V(s) = A(s)/C(s), the inverse Laplace transform of Eq. (4.34) yields g(t) = 1 + v(t) , (t >0) (4.35) and from the previous discussion i t follows that v(t) can be approximated by Eq.(4.26). Thus «mn ( t ) = 1 + V ( t ) 5 ( t > 0 ) ( 4 ' 3 6 ) i s an approximation to the unit step response where v (t) = 1 mn — z^2 K 0 0 60 0 f 0 1+Krp K 0 -56t 60 f l 1+^ K 9t 2 -36t f 2 1+Krp - t 5 9t 2 f 3 0 T1T2 r1+z2 0 - t 5 f 4 0 0 r i r 2 0 0 f5 K 0 0 60 0 0 I + K T K 0 -36t 60 0 r 1 + r 2 K 9t 2 -36t 60 r x r 2 r i + r 2 - t 5 9t 2 -36t 0 7r t 1-2 V*2 0 - t 5 9t 2 i 0 0 0 0 - t 5 . ( 4 . 3 7 ; 59 and where f Q = -60(1+^) ; f = -60(^+2^) - 2 4 ( l + % ) t ; f 2 = -60 ̂ - 24(^L+^)t - 3 ( l + K T ) t 2 ; f 3 = - 2 4 r 1 r 2 t - 3 ( r 1 + r 2 ) t 2 ; ±A = - 3 r ± ? 2 t 2 ; 0. Figure 4.2 i l l u s t r a t e s the type of data that can be obtained from Eqs. (4 .53) and (4 .56) where the choice ?^ = 1.174, ^ = O.46 has been made. By choosing 1+K^ and t m , Eq . (4 .33) can be solved"for K and Eq . (4 .36) can be solved for the maximum output amplitude g m < The time-domain s e n s i t i v i t y of g m and t m to v a r i a t i o n i n K and K̂ , can be determined from Figure 4 . 2 . The s t a b i l i t y boundary (SB), defined by the values of K and which result i n an unstable system, i s also shown i n Figure 4 . 2 . I t i s interesting to note that with suitable r e s t r i c t i o n s , the proposed method determines the i n i t i a l response of unstable systems as shown i n Figure 4 . 3 . To discuss the method for an unstable system, direct use must be made of the inverse Laplace where the l i n e c+jco i s chosen so that a l l poles of H(s) are to the l e f t . Provided that the poles of P m n(~ s"t) a r e ^° ̂ e right of this l i n e , the exponential function i n Eq . (4 .38 ) can be approximated by P (-st) and the int e g r a l transform g(t) (t >0) 60 ^ n ( t ) = . j M S i a i P m n ( - s t ) d s ; (t>0) 2JCJ I s mn c-joo (4.39) evaluated by the method of residues, that i s , Eqs. (4.6), (4.7) and consequently Eq.(4 .13) then remain v a l i d and can be used as an alternative method for evaluating Eq.(4.39)« Figure 4.4 i l l u s t r a t e s a plot of Eq.(4.36) for the case K = 1 , = 1.7, compared with the exact response. The closed-loop transfer function for this•case i s H(s) = 2 (4.40) s 5 + 3.2s 2 + 3.4s + 2 The accuracy can be improved by choosing a larger value of n. However, even for the choice m = 2, n = 3, i t i s seen that reasonable accuracy i s maintained up to the f i r s t over- shoot . 4.5 Method of Residues I f most of the system parameters are specified numeri- c a l l y eq.(4.26) can be readily evaluated by a d i g i t a l computer, even for systems of high order. However, i f most of the system parameters are i n i t i a l l y unspecified, the algebraic forms obtained from Eq,(4.26) could become unwieldy. An alternative approach, based on the method of residues applied to a form such as Eq.(4«39)» could then be considered. The conventional method for evaluating Eq.(4.38) i s to complete the path of integration i n the l e f t - h a l f s-plane and requires that the poles of H(s) be determined. However, i f Eq.(4.39) i s used, the path of 61 62 2.5 ^ Figure 4.5 The Exact and Approximate Unit-Step Responses of the System Shown i n Figure 4.1 i n the Case of I n s t a b i l i t y . 63 integration can be completed i n the right-half s-plane and (4.41) t Figure 4.4 The Exact and Approximate Unit-Step Responses of the System Shown i n Figure 4.1. where the form, of Eq.(4.23) has been used and where are- the poles of ^ m n ( - s " t ) ' f o r example, n = 3 i s chosen, Eq.(4.4l) contains only three terms irrespective of the order of the system. Figure 4.5 i l l u s t r a t e s the response obtained from Eq.(4.4l) where H(s) i s given by Eq.(4.40). Figure 4.5 The Exact and Approximate Unit-Step Responses of the System Shown i n Figure 4 .1. 65 5. NONLINEAR SYSTEM DESIGN 5.1 Outline ... Any system with any number of loops and l i n e a r elements can be reduced to an equivalent system having the block diagram representation shown i n Figure 5.1 provided that the system contains only one nonlinear element. The characteristic of the nonlinear element i s taken to have the form shown i n Figure 5.2 so that i t l i e s i n the R(s) -^Xh Figure 5.1 Nonlinear Control System. N y C(s) Figure 5.2 Characteristic of the Nonlinear Element and i t s Linear Bounds. 66 sector formed by the x-axis and the l i n e y = kx where k > 0. The l i n e a r part, given by G-(s), i n Figure 5.1 can be designed on the basis of an optimum output correlation of the closed loop system with respect to the output of a closed loop comparison system where G i s replaced by G . The parameters of the comparison system are assumed to be known except for the optimum gain K. K can, however, be expressed as a function of the slope k and the parameters of the comparison system. For example, for a second order system K could be expressed as K = K(k, ?c,wc) where f i s the damping r a t i o and coc i s the natural frequency of o s c i l l a t i o n of the comparison system. 5.2 The Design P r i n c i p l e Replacing the nonlinearity i n Figure 5.1 by k and the l i n e a r part by the comparison system G (s) the following equivalent system of Figure 5.3 i s obtained. C(s) Figure 5.5 Equivalent Linear System for the System of Figure 5.1. 67 In order to define the optimum gain i n terms of •30 k and the known parameters of the comparison system, Popov's c r i t e r i o n for absolute s t a b i l i t y w i l l be used. Popov's c r i t e r i o n for absolute s t a b i l i t y for the system of Figure 5,. 3 requires that Re [(l+jwq) G + l / k > 0 ( 5 . 1 ) Im[G*(jw)] Figure 5 . 4 Popov Line and Locus of G (jw). where q, an arbit r a r y r e a l parameter, determines the slope of the Popov l i n e shown i n Figure 5 . 4 . The system i s stable pro- vided that the locus of G*(jw) l i e s to the right of the c Popov l i n e passing through the point ( - l / k , 0 ) . In Figure 5 . 4 the dotted l i n e represents the tangent to the locus of G*(jco) at the c r i t i c a l frequency.WQ and Re [G*(ju)J = RefG c(ju)] and , Im'••'••[ G* ( j to)] = wlm [GQ( ju)] If the arb i t r a r y variable q i s chosen so that the 68 Popov l i n e passing through the point (-l/k,0) i s p a r a l l e l to the tangent to the locus of G*(jco) at the c r i t i c a l frequency, then the s t a b i l i t y of the system i n the Popov sense i s ensured, The above value of q when substituted i n Eq. (5.1) w i l l then define a range of the optimum gain Z of the comparison system. 5.2.1 Choice of q for the Popov Line and the Range of K Consider the following comparison system given by G c ( s ) = - £ (5.2) (s+b)(s 2+2f co s+co2) c c • c Substituting s = jco and separating the r e a l and imaginary parts yields r n KpbW2," w2(k+2S>„)] Re[GJ(jco)] = Re[Gc(jco)] = — L - 5 ^ and ( W 2+b 2)[(co 2-a) 2) 2 +4^a) 2co 2 J (5.3) r- • r Kco2 Rco2-co2)-2^co bl Im[G*(ju)] = w l m ^ C ^ ) ] L ° Q- e- J" (co 2+b 2)[( W 2-co 2) 2+4^^co 2 J (5.4) The locus of G*(jco), shown i n Figure 5.4, cuts the r e a l axis i n the l e f t half plane at co = W Q , the c r i t i c a l frequency. Equating Eq. (5.4) to zero yields CO = COQ = 1 co2 + 2 ? co b c c c The positive value of WQ i s taken since the locus of G*(jto) i s plotted for positive values of to only. The slope of the tangent to the locus of G*(jto) at (OQ i s given by the expression to2 + 2 ? u b -2 £ - S - ( 5 . 5 ) 2?w + b c c Substituting the value for G*(jto) into the inequality (5.1) yields K[bco2 - to2((b+2?ca)c) - q(to 2 +2^to cb) + qco2j] _ ̂ (to2+b2) [ ( ( / - t o 2 ) 2 + 4<?2 to2(o2 ] ( 5 . 6 ) Choosing the slope of the Popov l i n e equal to the slope of the tangent to the locus of G-*(jto) at tô and using Eq. ( 5 . 5 ) results i n the following expression for q: 2 f u + b • ' c c q = -p to + 2 ? to b c c c Substituting t h i s value for q into Eq. ( 5 . 6 ) and evaluating i t at the c r i t i c a l frequency yields + i7 > o 2 ? u> (co2 + 21? to b + b 2) k c c c C C ' Thus the range of K i s given by 0 <kK <2 ^ cw c(to 2+ 2? cto cb + b 2) 70 where k, § , co and b are known quantities. The l i n e a r part G(s) can now be designed on the basis of an optimum correlation with respect to the comparison system G (s) by maximizing the performance index P, c given by Eq. (2.35), where U(s) = kG(s) 1 + kG(s) and kG (s) V(s) = Q 1 + kG c(s) The c r i t i c a l frequency U)Q, the slope of the Popov l i n e ^, and the range of the optimum gain K for several comparison systems are given i n Table 5.1. The s t a b i l i t y of a system designed by the above technique cannot be unconditionally guaranteed. However, since the responses of the l i n e a r i z e d systems are s i m i l a r which i s a consequence of maximizing P, i t follows that G(jco) must approximate G (jco) over a range i n values of co. Thus i f G (jco) c c does not cross the Popov l i n e i t can be anticipated that G(jco) w i l l not cross the Popov l i n e . This approach' cannot, therefore, guarantee a suitable response i n the time-domain. It does, however, result i n a comparatively simple way-for choosing system parameters. This method could be used to determine suitable i n i t i a l estimates for the time-domain approach discussed i n the following section. Table 5,1 C r i t i c a l Frequency, Popov l i n e Slope and Range of Optimum Gain i n Terms of Known Comparison System Parameters. Comparison System G c ( s ) C r i t i c a l Frequency to, '0 Popov Line Slope 1/q Range of Optimum Gain K K 2 2 s + 2 ?OKS + io c c c 2 c c 0 < kK < oo K 2 2 s(s + 2 ? co s + co ) c c c to. c c 0<i kK< 2 <?co -c c K (s+b) ( s 2 + 2? a s+w-) c c o co + 2 ? co b c c c < + 2 % " c b 2 ? c w c + b 0 < kK< 2 <? co (co2+2 ? co b+b2) c c c c c K s(s+b) (s2+2 f co s+co2) c c c' bco b(oc(2fccoc+b) 2 f cA(co 2 + 2^co cb-fb^) (2%% + b ) 2 . 2 t ^ c + b 2§ (co2+2 ?co b+b2)-bco c c c c ' c 0< kK< K(s+a) (s+b) (s 2+2f co s+co2) c c c (a-b)o£+2-Sc ocb a - (b+2?ccoc) {(a-b)(o2+2Sccocbj .{a-(b+2?ccoc| (a-b)co +2 ^cocb (a(b+2?ccoc); .{a-(b+2fa ) c)} 0<kK<- 2 ?co c c L 2 - i 1 Q _-b[a-(b+2 l^coc)}_ a - b - 2f cco c 72 5.3 Time-Domain Analysis of Piece-wise Linear Systems Consider the piece-wise l i n e a r feedback system i l l u s t r a t e d i n Figure 5.5 where the nonlinear element has a saturation type of c h a r a c t e r i s t i c . This type of non-linearity approximates the saturation characteristics of many energy conversion devices at high signal l e v e l s . The saturation l e v e l i n t h i s example i s taken as v Q = 1.4. The method to be discussed i s , however, also applicable to the case of amplifier saturation where the nonlinear element precedes the l i n e a r element. Figure 5.5 A Piece-Wise Linear Feedback System. Let v(0) = 0 = v(0) be the i n i t i a l conditions and l e t the input be a unit step. Let v.̂ = v, v 2 = v. Before saturation occurs, the state equations are v 1 - v 2 = 0 (5.7) v 2 + v 2 + lOv-^ = 10 Solving Eq. (5.7) by Laplace transform methods yields 10 V u ' — sU+sJ o * v1 ~V4 73 V t ) \ \\^{0)) + r h i " h 2 i (5.8) v 2 ( t ) y V - i o h 1 h J \ v 2 ( o ) y \ lOb^ where i e x P ( s t ) ds- h - — 1 " 2*d J . s 2 + s + 1 0 d S ' h2 " 2«3 3 s exp(st) h ^ a _ i _ x c ^ S P u , ; 0 = 7 T r r \ " 2 ^ ^ •- ds „ , „ , _ s +s+10 (5.9) Por the given i n i t i a l conditions Eq. (5.8) reduces to v 1 ( t ) \ / l - h ^ h g v 2 ( t ) / V 101^ , 0 i t § t x (5.10) At the instant t ^ of saturation, the state equations change to v-̂ - v 2 - 0 (5.11) v 0 + v Q = y where y = 10(l-v o) and where v i s the saturation l e v e l of the nonlinear element. s Solving Eq. (5„ll) by Laplace transform methods yields y(t-l+h 3) N 1 • + 1 = _ „ 2 + ; t t .y(i-n.3) (5.12) 7 4 where J- joo At t = t ^ , the state equations are again given by Eq„ (5.7) and Eq. (5.8) can be used provided that v-^(O) and V2(0) are replaced by v ^ ( t 2 ) and v 2 ( t 2 ) , respectively. The functions h^, h 2, h^ can be evaluated to any degree of accuracy i n the form of Eq. ( 4 . 2 6 ) by means of a r a t i o n a l f r a c t i o n approximation to -s t e . The choice of a r a t i o n a l f r a c t i o n approximation i s based on a compromise between the desired accuracy over a given period of time and computational s i m p l i c i t y . I f , for example, P 2^(st) i s used and i f v 0 = 1.4, Eq. (4.26) yields s h ^ 500t 5-240t 4-5780t 3+560t 2+5600t 1 1000t6+900t5+4500t4+2640t5+2700t2+2l60t+3600 h ^ 5100t4+480t3-15660t2-1440t+5600 2 1000t6+900t5+4500t4+2640t3+2700t2+2l60t+3600 h g ?t 2 -24t+60 ( 5 > 1 4 ) 5 t5+9t2+36t+60 Eigure 5.6 shows the response v(t) obtained from Eqs. (5.8), (5.10), (5.11) and (5.14) compared with the exact response. Since the matrices i n Eqs. (5.8) and (5.12) are state-'transition matrices, i t i s seen that the elements of the stat e - t r a n s i t i o n matrices can be represented i n the form of Eq. (4.26), that i s , the elements can be expressed as the r a t i o of polynomials i n t with coefficients which are algebraic 75 76 functions of the system parameters. By means of these algebraic forms, the parameters of a piece-wise l i n e a r control system can be d i r e c t l y related to i t s time-domain response. The application of these forms to system design and to the determination of response s e n s i t i v i t y to parameter variations i s s i m i l a r to that given i n Section 4.4 and w i l l not be discussed further. 77 6. MULTIVARIATE CONTROL SYSTEM DESIGN 6.1 Outline The design of multivariable control systems u t i l i z i n g 31 32 33 matrix formulation has been considered by many authors. ' ' While some of them have been concerned with the question of physical r e a l i z a b i l i t y , the problem of r e l a t i n g system parameters to time-domain response and interaction i n the time-domain are not considered by these authors. Interaction within a multi- variable control system may, i n some applications, be desirable, the interaction being controlled rather than removed. Recognizing that the physical construction of a completely noninteracting control system i s impossible, a root- 34 locus design method^ applying the techniques of single-variable system design has been suggested. However, the p r a c t i c a l advantage of the root-locus method applied to multivariable control system design could be realized only i f a rapid t r a n s i t i o n from the pole-zero locations to the time-domain characteristics could be made. Two methods of designing multivariable control systems, based on the methods of l i n e a r single-variable control system design discussed i n Chapters 2 and 4, are given i n the following sections. 6.2 Design Method Based on Performance Functionals Consider the system of Figure 6.1 which represents an interacting plant with f a c i l i t y for compensation to be inserted i n as G-̂ and G-^0 Assume that E±1 = l / ( s + l ) , H 1 2 = H 2 1 = l/(s+20), H 2 2 = l/(s+2), G l l = K i ( s + a i ) / ( s + n ] _ a i ) > nj< 1 G22 = K 2 ^ S + a 2 ^ ^ S + n 2 a 2 ^ ' n 2 < ; 1 * The group of design, variables K^, K 2, o^, a 2, and n 2 are positive r e a l numbers. The equations describing the block diagram of the 78 system are n G 1 1 H 1 1 ( 1 + G 2 2 H 2 2 ) " G l l G 2 2 H 1 2 H 2 1 r A G22 H12 r °1 = A V A R2 G l l f 2 1 p G 2 2 H 2 2 ( 1 + G 1 1 H 1 1 ) " G11 G22 H12 H21„ ' 2 ~ A 1 A n2 where A = (l+G^H^) (l+G 2 2H 2 2) - G 1 1 G 2 2 H 1 2 H 2 1 R l - ^ Figure 6.1 Multivariable Control System. Consider the case when R2= 0, C 1 11 R l " 1 + A l l (6.1) where A-^ = ^]_]_[^]_]_- ^12^21^-G22'//^+G22'^22^ 79 Eq. (6.1) can be represented by the block diagram shown i n Figure 6.2. Si m i l a r l y , i f R^ = 0, '22 R, 1 + A, (6.2) where A - G-A22 ~ 22 2 ~22 H22 " H 1 2 H 2 l { G l l / ( l + G l l H l l ) B then the block diagram representing Eq. (6.2) i s shown i n Figure 6.3. Considering Eqs. (6.1) and (6.2) as the transfer functions of single variable systems i t becomes possible to determine two sets of parameters of the compensating networks ^11 a n d ^22 maximizing the correlations of C^ i n Figure 6.2 H l l Figure 6„2 Block Diagram Representation of Eq. (6.1) and C 2 i n Figure 6.3 with the output responses of two known reference systems. The design technique has been described i n Section 2.3=2 of Chapter 2. The choice of which set of parameters to be used may be made on the basis of s a t i s f y i n g 80 Figure 6.3 Block Diagram Representation of Eq. (6.2) 34 a given interaction constraint-^ such as 2i R. 2 R.= 0 l w= 0 R.= 0 J w= 0 (6.3) or on the basis of minimizing a suitable time domain s e n s i t i v i t y function for the system sim i l a r to that given by Eq. (3-6) i n Chapter 3° 6.3 Time Domain Design Method If the design specifications are given i n terms of the transient response of the multivariable system then the design technique described i n Section 6.2 cannot "be employed. A method w i l l now be described which employs time domain specifications for the design of a multivariable control system. Consider the following design specifications for the system shown i n Figure 6.1. Here u(t) represents the unit 81 step and a l l i n i t i a l conditions are zero, c 1 ( t ) reaches a maximum r x ( t ) = u(t) r 2 ( t ) = 0 overshoot of x-̂ % i n t ^ seconds, c 2 ( t ) reaches a maximum r x ( t ) r 2 ( t ) 0 u(t) overshoot of x 2$ i n t 2 seconds. 2l 3 o p i •1*2 ! R. R 1 = 0 io= 0 1 R2= 0 co= 0 4. c 1 ( t 1 ) r 1(t)=0 r 2(t)=u(t) c 2 ( t 2 ) r 1(t)=u(t) r 2(t)=0 where t-^ and t 2 are the times defined i n ( l ) and (2) above. The l a s t design s p e c i f i c a t i o n defines an additional interaction constraint. In the general form i t i s defined as 34 c. 1 =- e! ., for 0 £ i r.=f(t) 1 J J r.=0 x at some time, for example at the time for which the response c^ i s a maximum. 82 In terms of the above d e f i n i t i o n the l a s t design s p e c i f i c a t i o n yields e j ^ = £ 2 i * For the set of design specifications outlined above the required design equations -will now be obtained. Recognizing that the time of maximum overshoot for a step input corresponds to the time when the impulse response equals zero, the design specifications ( l ) and (2) y i e l d h 1 ( t 1 ) = 2 ^ J \ V s ) exp(st 1)ds = 0 h 2 ( t 2 ) = 2^3" \ H 2(s) exp(st 2)ds = 0 ( 6 . 4 ) where H^(s) and H 2(s) are given by Eqs. (6.1) and ( 6 . 2 ) . For the system under consideration H^(s) and H 2(s) have s i x t h order denominator polynomials and f i f t h order numerator polyno- mials i n s. Using Fade approximation of the desired order for the exponential functions i n Eq. ( 6 . 4 ) and evaluating Eq. ( 6 . 4 ) two design equations, i n terms of the known and unknown system parameters and r e a l time, are obtained. The interaction constraint defined by the design s p e c i f i c a t i o n ( 4 ) yields -, ( d c 1(t- L) = 2rij \ [ H i ( s ) / s exp(st 1)ds = e-[2 (6.5) i j C [H 2(s)/s]exp(st 2)ds k £' C 2 ( t 2 } = 2k, , . , , 2 1 where £^2 = e^. The design s p e c i f i c a t i o n (3) yields the 83 (6 .6) interaction constraint relations given by Eq. (6 .6 ) = £ 2 - ^ 2 ( 2 0 ^ + 1 9 . 9 ^ ) 2 K 2n 1 = e 12K 1(40n 2+19.9Z 2) where £21 = e ] _ 2 ' The set of relations given by Eqs. ( 6 . 4 ) , (6 .5 ) and (6 .6) constitute the required design equations for the evaluation of the s i x design variables K^, K^f oĉ , oc^, n^, and n 2. The above method i s also applicable to the design of noninteracting multivariable control system by making e. . = 0 and e! . = 0. An i l l u s t r a t i v e example i s given i n the following .section. 6.4 An I l l u s t r a t i v e Design Example Consider the system shown i n Figure 6.4 where H l l = - 2 / ( s + 1 ) » H i 2 = V ( s + 1 ) , H 2 1 = 4/(s+l), H 2 2 = (8s+2)/(s+l) The equations of the system are ° i = ( V ° i ) G i i H i i + ( V c 2 ) G 2 2 H i 2 C 2 = (R 1+C 1)G 1 1H 2 1 + (R 2-C 2)G- 2 2H 2 2 In matrix form the above equations can be written as (6 .7 ) where A l l = [ G11 H11 ( 1 + G22 H22 ) ~ G l l G 2 2 H 1 2 H 2 l ] / A A 1 2 = (G 2 2H 1 2)/A A21 = ( G 1 i H 2 1 ) / A A22 = [ & 2 2 H 2 2 ( 1 - G 1 1 H 1 1 ) + G l l G 2 2 H 1 2 H 2 l ] / A and where A = (l-G^H-^H 1+& 2 2H 2 2) + &]_]_&22H12H21° Taking r ^ ( t ) , r 2 ( t ) as impulses at time t=0 of areas and r 2 respectively, the inverse Laplace transform of Eq. (6.7) yields ^ ( t A / a i ; L ( t ) \ a 2 ^ ( t ) a 2 2 ( t ) Figure 6<,4 Block Diagram of a Multivariable Control System. 85 where a,, . a.r 3° , ail Hllt^ G22 H2 2)- Sli a22 Hl 2 H21 e ] c p ( s t ) a s ^G H h\ - £ 2- J i 2-exp(st)ds (6.8) a12 - 2jtj ) A A 2 1 = w\ ^ i e x p ( s t ) d s a ? ? = A C J " 6 2 2 H 2 2 ( 1 - 6 1 1 H I 1 ^ g ; i S 2 2 H 1 2 H 2 1 ( s t ) f l s 22 2 3 t J ) . Let G^ = and G^2 = K^' where and are the design variables. Substituting the values for G^ and G 2 2 into Eq. (6.7) yields 2 2 A = (a+bs+cs )/(l+s) A x l = -22^(1+8)(1+8K2)/(a+bs+cs2) A 1 2 = 3K 2(l+s)/(a+bs+cs 2) 2 (6.9) A 2 1 = 4^(1+3)/(a+bs+cs ) A 2 2 = 2K 2 [(1+8^) +8(5+8^)]/(a+bs+cs 2) where a = 1+2K 1 +2K 2+16X^2 b = 2(1+^+5X2+8^2) c = 1+8K2 86 Substituting Eq. (6.9) into Eq. (6.8) and using the Pade approximation for exp(st) ,a-^, a-^2, a 2 i » a n ( ^ a 2 2 c a n b e expressed as ratios of polynomials i n t with coefficients as algebraic functions of the system parameters. Using the f i f t h order Pade approximation ,st _ 60 + 24st + 5 s 2 t 2 60 - 36st + 9 s 2 t 2 - s 3 t 3 solving for a-^ a n ^ a 2 1 from Eq, (6.8) and equating the results to zero, the following relations are obtained: 7 2 OK. 960Kn A+BK1+CK2+DK^+EK2+PK3+GK1K2+HK1K2 L +JK 1K^+1K 2K 2+MK 2K 2+M^K 3 "A+BK1+CK2 +DK2+EK2 +PK^+GK1K2 +HK1K2 L +JK1K|+LK2K2+MK2K2+NK2K5 = 0 (6.10) = 0 where A = B .= c = D = E = E = Gf = H = - 0 . 0 1 2 5 t 5 - 0 . 1 1 2 5 t 4 + 0 . 4 t 5 + 0 . 6 t 2 - 3 t - 1 5 - 0 . 0 5 t 5 - 0 . 5 5 t 4 + 0 . 6 t 3 + 4 o 2 t 2 + 1 2 t - 0 . 1 5 t 5 - l . 0 5 t 4 + 7 . 8 t 5 + l . 8 t 2 - 1 0 8 t - 3 6 0 - 0 . 0 5 t 5 - 0 . 6 5 t 4 - 1 . 4 t 5 - 3 t 2 - 0 o 4 5 t 5 - l ' . 0 5 t 4 ' + 3 5 . 4 t 5 - 1 1 3 . 4 t 2 - 1 1 5 2 t - 2 8 8 0 i - O . 4 t 5 + l o . 2 t 4 + 2 9 6 t 3 + 5 5 2 t 2 - 3 8 4 0 t - 7 6 8 0 - 0 . 9 t 5 - 9 . 3 t 4 + 2 2 . 8 t 3 + 1 1 8 . 8 t 2 + 2 8 8 t - 4 . 8 t 5 - 4 3 . 2 t 4 + 1 4 7 . 2 t 3 + 6 7 2 t 2 + 2 3 0 4 t 87 J = -6 ,4 t 5-32t 4 +844o8t 5+3302.4t 2+6l44t L = -1.2t5-15.6t4-33«6t3-72t2 M = -9.6t 5-125.4t 4-268,8t 3-576t 2 N = -25.6t5-332.8t4-7l6„8t5-1536t2 Figure 6.5 i l l u s t r a t e s the type of data that can be obtained from Eq. (6.10). By choosing IC, and t , Eq. (6.10) can be solved for K^ and Eq. (6.7) can be solved for the maximum output amplitude for specified inputs. The time-domain s e n s i t i v i t y of the maximum output amplitude to va r i a t i o n of Z^ and Z 2 can be determined from Figure 6.5. This information i s s i m i l a r to that obtained from Figure 4.2 of Chapter 4 . A l l a n d A22 "*"n 9) c a n ^ e expressed as follows: -2Kn(1+8Z9) A - A A l l - 3Z 2 A12 2Z,(l+8Z n) 8Z„s A n^ = d r7jr A. • 22 _ 3K 0 "12 -i 2 J ° a + bs + cs When A^2= 0, A-^ = 0, and A, 8Z 2s 22 ~ 2 a + bs + cs Solving for a 2 2 , using the f i f t h order Pade approximation for exp(st), and equating to zero yields 2400K2 A' +B 1K 1 +C ' Z2+D' K2+E »Z2+F' z|+G' Z ^ +H»KXZ2+J' ZjK^+L' Z 2Z 2 +M' Z2Z2+N' Z 2Z 3 (6.11) =0 88 Figure 6.5 2h© Gains and 'Kg as Functions of Time t , for the System Shown i n Figure 6 .4. 89 where A' = 0.17t 4+0.l6t 3-2.88t 2-9.6t+12 B' = 0.68t 4+0.48t 5-5.76t 2-9.6t C» 2.04t4+2.4t3-51.84t2-201.6t+288 D' 0.68t 4+0.32t 3+2.4t 2 E' - 6. l'2t4+10.2613-254.88t2-1382. 4t+2304 p. = 5.44t 4+12.83t 4+12.83t 5-1209.6t 2-•3072t+6.144 Gr' = 12.24t4+9.6t3-138„24t2-230.4t H' 65o28t4+6l.44t3-998„4t2-1843.2t J' = 87 .04t4+122.88t3-3870.72t2-4915- 2t L 1 = 16.32t 4+7.68t 5+57.6t 2 M' = 130 . 56t 4+6l.44t 5+460.8t 2 If' = 348.16t 4+l63.84t 3+1228.8t 2 The relationship between K^, K 2 and the instant of time when a 2 2= 0, given by Eq. (6.1l) for t=5, i s also shown i n Figure 6.5» The point of intersection of the two curves gives the values of K^ and K 2 for which the impulse responses ei^^ $ ^12 * ̂ 21 "̂22 reach zero value at t=5. In other.words, the amplitude of the outputs for unit step inputs reach th e i r respective maximum values at that instant of time. This could be considered a desirable effect i n some applications. The example discussed above only i l l u s t r a t e s the pr i n c i p l e of the proposed design technique and the p o s s i b i l i t i e s of getting useful information from the algebraic r e l a t i o n s . In general, the output and input signals of a multi- variable control system can be related by the following matrix representation. where the c^(t) are the outputs and the r ^ are the areas of impulse inputs or the amplitudes of step inputs and where each &j| c(t) i s a r a t i o of polynomials i n t with coefficients which are algebraic functions of system parameters. It i s interesting to note that the interaction 34 constraints e and e'., , discussed i n Sections 6.2 and 6.3 can e a s i l y be expressed as r a t i o of polynomials i n t with coefficients which are algebraic functions of system parameters. I t , therefore, becomes possible to investigate the interaction effects of a multivariable control system with parameter variations i n the time-domain along with a s e n s i t i v i t y investigation. 7. CONCLUSIONS 91 A method has been presented for obtaining algebraic relations between system parameters and system response based on the frequency domain evaluation of an i n t e g r a l performance c r i t e r i o n . The performance c r i t e r i o n P, defined-as a correlation measure between the responses of a known reference system and the system to be designed, provides a f l e x i b l e c r i t e r i o n . Unlike the minimization of performance c r i t e r i a based on error measures, th i s method allows a choice of different values of P to be made. Within the class of systems defined by the maxi- mization technique, the par t i c u l a r system with the smallest para- meter s e n s i t i v i t y can then be chosen. This i s often more important than minimizing or maximizing a fixed performance function. A method has also been presented for obtaining alge- braic relations between the parameters of l i n e a r and piece-wise l i n e a r systems and t h e i r time-domain response characteristics. Since the method i s based on the solution of systems of l i n e a r equations, the computations required are e a s i l y performed, and the d i f f i c u l t problem of r e l a t i n g characteristic roots to several system parameters i s avoided. The algebraic relations obtained or the systems of l i n e a r equations used are well suited for time-domain s e n s i t i v i t y calculations by d i g i t a l computer means. 35 As i s done i n the s e n s i t i v i t y analysis of networks ^ , the unspecified parameters can be tagged and derivatives with respect to these parameters obtained by simply deleting the parameters i n the systems of l i n e a r equations. This i s 92 possible since the parameters enter the equations i n a l i n e a r manner. A graphical display of the algebraic relations allows one to see the effect of several parameters on the time-domain response. The method augments very e f f e c t i v e l y other parameter plane methods since i t avoids dealing d i r e c t l y with the characteristic roots which i s an essential feature of these other methods. APPENDIX A 93 Evaluation of Performance Integrals I and J ° mn m A.l Outline and The evaluation of performance integrals Xmn = 2 ^ f U(s+p)V(-s +p)ds (A.l) C Jm = 2 ^ 1 ^ U(s+p)U(-s+p)ds (A.2) C can be reduced to the solution of a system of l i n e a r algebraic equations. The following derivations are based on a proof given by Talbot 4. . A.2 Evaluation of Performance Integral I mn When U(s+p) and V(s+p) are r a t i o n a l functions of s vanishing at i n f i n i t y , with the poles of U(s+p) a l l to the l e f t of the poles of V(-s+p), the contour C may be completed by an i n f i n i t e semicircle on either side of C. Taking i t to the l e f t and expressing U(s+p) and V(-s+p) i n the form given by equations (2.15) and (2.19), respectively, equation (A„3) i s obtained from Cauchy's residue theorem. V = 0 F ( s ) / C ' ( s ) D ( s ) ] s = s . ( A ' 3 ) X 1 where P(s) denotes A(s)B(s) and where s^ are the zeros of C(s). Equations (2.16) to (2.18) and (2.20) to (2.22) give the numera- tor and denominator polynomials of U(s+p) and V(-s+p), respec- t i v e l y . Since a l l zeros of C(s) are to the l e f t of the zeros of 94 D(s), C(s) and D(s) have no common factors. In order to evaluate I i n equation (A.3), the mn u following i d e n t i t y i s considered: Since B(s) and D(s) have no common factor and the degree of F(s), or A(s)B(s), i s less than the degree of C(s)D(s), R(s)/C(s) i s the sum of those p a r t i a l fraction- terms of F(s)/C(s)D(s). . which belong to C(s); and s i m i l a r l y for Q(s)/D(s). Thus, R(s) _ [F(s)/0'(s)D(s)3: CTs) ~ ^ S - 3±- (A.5) s=s i I t follows that I i s the coe f f i c i e n t of l / s i n mn ' R(s)/C(s) i f th i s i s expanded i n descending powers of s, that i s , ! = l i m sR(s) mn s—-oo C(s) Thus, i f m 1 R(s) = rm_2_s +........+ T Q (A. 6) and Q(s) = ^ . i S 1 1 - " 1 " + ........ + q_Q (A.7) then I = ̂ 2=1 ( A o 8) mn c m Equation (A.4) i s equivalent to the polynomial equation F(s) = R(s)D(s) + Q(s)C(s) . - (A.9) By equating terms containing the same powers of s i n equation (A.9), a set of simultaneous equations are obtained for the 95 coefficients i n R(s) and Q(s) which may be written as M w = F (A.10) where, w i s the (m+n)-rowed column (q~, . 10 ; F i s the (m+n)-rowed column (fg, q n - l ' r0' f k , 0, ... 0) r -.) m-1 where J , f. = ̂ > a.b. . , for 0 ̂ j ̂ m+n-2 i=0 and M i s the (m+n)-rowed square matrix given by equation (A.11) M = '0 1̂ 0 L0 . 0 d l *• m. 0'- c n m 0 n (A . l l ) The solution of I i s given by mn C = A — mn c A m (A.12) where A i s the determinant of M and A i s the determinant of M on replacing i t s l a s t column by F. Eqs. (A.8) and (A.12) remain v a l i d even i f the zeros of C(s) are not a l l simple. The procedure above holds for an arbit r a r y numerator F(s) and arbitrary denominator fac- tors C(s) and D(s) having no common factor, provided the degree of F(s) i s less than that of C(s)D(s). The roles of U and V may always be interchanged. A does not vanish since the poly- nomials C(s) and D(s) have no common factor. C(s) and D(s), however, must be Hurwitz polynomials. 96 A. 3 Evaluation of Performance Integral J m To evaluate the in t e g r a l J m i t i s noted that equation (A.3) becomes J = m A(s)A(-s) C (s)C(-s) (A.13) J s=s. and i n place of equation (A.9) equation (A.14) i s obtained. R(s)C(-s) + R(-s)C(s) = A(s)A(-s) = 2L(s) (A.14) J m i s the coe f f i c i e n t of l / s i n R(s)/C(s), and i f Ms) = l2m-2 + J - m o o o e + L 0 (A.15) (A.16) In terms of determinants the f i n a l solution i s ,m+l A J = (-1) m v • ' c A m (A.17) where A i s the determinant of the (mxm) square matrix M = '0 0 °1 0 c3 c2 c l C0 (A.18) '2m-2 ° 0 ' V l A i s the determinant of M on replacing i t s l a s t column by J0' ~2' °° ° ° J'2m-2̂ ,' the column L = (L,~, I>0, o ° ° . l 0 r y i 0) and where 2 i = YZ ( - D ] " V J i=0 J 1 m-1 for 0 = j = m-1 = I (-1)*3 xa.a. . , for m =• j 4 2m-2 . (A.19) i=j-m+l 97 A,4 Table of Integrals Solution of the integrals I for values of m from 1 D mn to 2 and n from 1 to 3 are given i n Table A . l . Solution of the integrals J and t h e i r derived form J , are given i n Tables to m mk to A.2 to A.4 for values of m from 1 to 4 and k from 1 to 2. The int e g r a l forms of I and J are fa mn m I = d s UsMsl (A. 2 0) mn 2%j J C(s)D(,s) C J m = ^ r J (A.2i) c where . / \ m-1 m-1 0 n—1 B(s) = t>n_^s + ........ + bQ and D(s) = d^s31 + ............ + dQ (A.22) 98 Table A . l Values of I i n Terms of the Transform Coefficients, mn 12. '22 a0 bQ c0 11 c ^ d Q - C Q ^ ) a 0 ( c l b 0 " C 0 b l ) c0 d2 " c l c O d l + c l d 0 a l b l ( c O d l " C l d 0 } + a 0 b 0 ( c l d 2 " c 2 d l } ' + (a^Q + a 0 b 1 ) ( c 2 d 0 - c Qd 2) ( c 0 d 2 - c 2 d 0 ) + ( c 1 d 2 - c 2 d 1 ) ( c 1 d 0 - c 0 d 1 ) a l b 2 °0 ( c0 d2- c2 d0 ) " c l ( c 0 d l - C l d 0 ) 2 + ( a Q b 2 + a 1 b 1 ) ( c 2 ( c 0 d 1 - c 1 d 0 ) - GQ&^J + (a 0 b 1 + a 1 b 0 ) { c 2 d 0 - c 0 ( c 2 d 2 - c 1 d 3 ) } + Q - Q ^ Q I 0 ! ( c 2 d 2 - c 1 d 3 ) + c 2(c Qd 3-c 2d- L)} "23 c 0 { c 2 c 0 ( d 2 - d 3 d 1 ) - c 2 c 1 ( d 2 d 1 - d 3 d 0 ) + c 2(d 1-d 2d Q) 2 + G Q ^ G Q ^ - C ^ ) + d 3 d 1 ( c 1 - c 2 c 0 ) } + d Q { ( c 1 d 3 - c 2 d 2 ) ( c 2 c 0 - c 1 ) + c 2 c 1 ( c 0 d 3 - c 2 d 1 ) + O2 4o} 99 Table A.2 Values of J i n Terms of the Transform Coefficients m a J, = 0 2 C 1 C 0 a l c 0 + aQ c2 2 c 2 c l C 0 2 n ( 2 Q x 2 ^2^10 + i a ~ £-&2&Q) Q-^GQ aQC^Cg 2C ^ C Q ( — C ^ C Q + CgC^) J, = " 2 2 2 a^(—C ^ C Q + ^2*^1^0^ ( a2 2a^a-j_)^^^2.^0 2 2 2 + (a-ĵ - 2 a 2 a Q ) c 4 c ^ c Q + a Q ( - c 4 c 1 + c^c^Cg) 2 c 4 c 0 ( - c | c 0 - c 4 c 2 + c 5 c 2 C l ) 1 0 0 Table A. 3 Values of i n Terms of the Transform Coefficients J 1 1 a 0 2 c 0 J 2 1 1 0 . 2 c ' 2 2 2 a l a 0 . a l C 0 + a 0 C 2 C 0 0 C l ° 0 2 2 C 1 C 0 J 3 1 J 4 1 2 a 0 a 2 a l + a l a 0 C 2 / / c 0 2 c 0 a 0 C 2 C 1 c 3 ° 0 • 2 2 2 2 3 ' a 2(c 1+c 2c 0) + a 0(c 5c 0+C2)/c 0 2 2 + ( a 1 - 2 a 2 a Q ) ( C ^ H ^ ) 2 c 0 (CgC-j^ - C ^ C p , ) 3 0 ' 2 2 2a^Cg + a^a2C-^ + (a2-2a^a.^) C2 + ( a 2 a - ^ - 3 a ^ a Q ) c^ + 2 (a-^-^agaQ) c^ 2 2 + a 1 a Q ( c 5 c 2 ~ c 4 c i ^ / c o + a o ( c 4 c 2 + c 5 ^ c O . 2 _ 2 C 3 C 2 C 1 C 4 C 1 C 3 ° 0 + ( C ^ C - ^ C ^ C Q + C ^ ) f a 3 c i + ( a 2 - 2 a ^ a 1 ) c 3 c 1 2 2 + ( a 1 - 2 a 2 a 0 ) c 5 2 + a 0 ( c 3 c 2 - c 4 c 1 ) c 3 / c Q } _ ( - 2 _ n2 \ 2 \O^C2C-^ ^ 4 ^ 1 3 ^ 0 101 Table A.4 Values of J 0 i n Terms o f the Transform Coefficients J. 12 2 a0°l '0 J 22 a o c i 4 2 2 C1 C0 2 2 2 4 a 1 a 0 c 2 - 4 a 0 c 2 / c 0 4a-Lc2 — 2 — + 3 C1 C0 °1 J 32 a o c i '0 r 2 2 Q 2 2 2' a2 C0 + ^ a2 a0 C2 C0 + al C2°0 + a0 C2 2a-^aQC2c^ 4~ 2a^aQC3CQ C0^°2 C1 ~ C3 C0^ 2 3 2 2 2 4a 2a 1c 0(c 2+c^c 1) + 4 a 1 a Q (C ^ + C ^ C Q ) + 4a Qc, 5c ? C Q ( C 2 C ^ "" C ^ C Q ) 2 3 2 2 3" 4 a 2 ^ C 2 c 0 + C 3 C 2 c l c 0 + C 3 C 0 + C 3 C l ^ - 8a 2a 0(c 2+2c|c 2c 0+c 2c 2) + 4a 1(c 2+2c 2c 2c 0+c 2.c 2) ^ C2 C1 ~ °3C0^ Table A.4 (Continued) 1 0 2 42 2 V i + 2 2 2 2 2 Sa^c^CQ + 4a^a2C2CQ + (5a2-6a^a^) CJI°Q 2 2 + 8 ( a 2 a 1 - 3 a ^ a 0 ) c 4 c 0 + (a-L+2a2a0) ( c . ^ - c ^ . ^ ) c 2 2 + a 1a 0(4c 4c 2c 0+6c 3c 0-2c 3C2C 1+2c 4c- L) . + a o ^ c 4 C 3 C Q + 8 c 4 c 2 c Q - c 4 c 2 c 1 - c 2 c 2 ) 0 2/ 2 2 x c0 3°2C1 C 4 C 1 ~ °3 0 2 ("2 ( c 2 + c 5 c 1 - 4 c 4 c 0 ) { a : 5 c 0 ( 2 c 5 c 0 + 4 c 2 C 1 ) + 3 a 5 a 2 C 5 c 1 c 0 + (a 2-2a 5a 1) U c ^ - ^ c ^ ) c Q + 3 (a2a-^—3a^aQ) C ^ C Q 2 + 10(a- L-2a 2aQ) c 4 c ^ c 0 + 3a-^aQC^ (Crzl<^2~ ̂ 4 ^ 1 ̂ + a 2 ( 6 c 4 c 3 C 2 C 0 - 4 c 2 c l C o + 3 c ^ c 0 3 2 C^C2C3_"'"C4C3C2_","C4G3C2 2 2 + 8c 5c 2{ 2 2 2 + (a 1-2a 2a Q)c3C Q + a g C ^ c ^ - c ^ ) } / 2 2 N2 C0^ C3 C2 C1 " °4C1 °3 C0 ; 2 2 2 2 2 '4c 5(c2+c^c 1-4c 4c 0) f a 5 c 1 + ( a ^ a ^ ) c^c-^' + (8L-^~2.SL23-Q) .̂ + ag(c 3C2-c 4c 1)c 3/c Q} _ r _ 2 _ ^2 N3 ^ c 3 c 2 c I c 4 c i c3 ° o ^ APPENDIX B 103 MINIMIZATION AND MAXIMIZATION PROCEDURE B.l Outline The search procedure used i n th i s thesis leading to the minimum or maximum value of the performance index, was 22 f i r s t suggested by Rosenbrock and works with n orthogonal directions i n which the search progresses at each stage. A stage i s defined as the set of t r i a l s made with one set of directions and the subsequent change of these directions. Each attempt to find a new value of the performance index i s called e. I f the step i s successful, e i s multiplied by a, where a > l . I f the step i s unsuccessful, e i s multiplied by -|3, where 0<|3<1. Success i s defined to mean that the new value of the performance index i s less than or equal to the old value when a minimum i s sought or i s greater than or equal to the old value when a maximum i s sought. steps are taken the value of e i s altered u n t i l at least one t r i a l i s successful and one t r i a l i s unsuccessful i n each of the n directions. Suppose that D̂ i s the algebraic sum of a l l the successful steps z-. , i n the dire c t i o n V, , etc., and i f Each stage i s started with a step of arbitrary length To change the dire c t i o n of a vector V i n which the A l = D 1 V 1 + D2 V2 + A 2 = D2y0 + (B.l) * 0 D V U n n 104 then A-̂ i s the vector joining the i n i t i a l and f i n a l points ob- tained by use of orthogonal unit vectors V^, V®> •«•»• > V^, Ag i s the sum of a l l the advances made i n directions other than the f i r s t , etc. The orthogonal unit vectors V^, V^, V^, are then obtained as follows: = B 1 / | E J Bg = A 2 - A g - V ^ (B.2) - B2/|B2| n-1^ B = A - ) \ -vW n n / , n j j V 1 = B /|B | n n' 1 n' The above algorithm ensures that V_̂ l i e s along the dir e c t i o n of fastest advance, V 2 along the best di r e c t i o n which can be found normal to V^, and so on. An obvious advantage of this method i s that no p a r t i a l derivatives of the performance index with respect to the design parameters need be calculated. B.2 IBM 7040 D i g i t a l Computer Program The computer program incorporating the above mentioned ideas and written i n the FORTRAN IV language for the IBM 7040 d i g i t a l computer includes the evaluation of the performance index from the determinant form. The overall flow diagram i s shown i n Figure B . l . 105 Input - output stage VECTOR Performance index minimization or maximization stage FUNXON Orthonormal d i r e c t i o n finding stage Determinant — evaluation stage by Gauss' method DETUV GAUSSB Performance index evaluation stage Overall Flow Diagram for Minimizing or Maximizing the Performance Index on the D i g i t a l Computer, REFERENCES 106 1. 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