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Steady-state ocsillations and stability of on-off feedback systems Mohammed, Auyuab 1965

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The U n i v e r s i t y o f B r i t i s h C o l u m b i a FACULTY OF GRADUATE STUDIES B . Se . , M . S c , PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of AUYUAB MOHAMMED The U n i v e r s i t y of M a n i t o b a , 1954 The U n i v e r s i t y of M a n i t o b a , 1956 .Y, MAY 21, 1965, a t 9t00 A . M . ROOM 410, MacLEOD BUILDING COMMITTEE IN CHARGE C h a i r m a n : I . M c T . Cowan E . V . Bohn E . L e i m a n i s S. W. N a s h F . Noakes L . Young Y . N . Yu E x t e r n a l E x a m i n e r : R. A . J o h n s o n U n i v e r s i t y o f M a n i t o b a W i n n i p e g , M a n i t o b a STEADY-STATE OSCILLATIONS AND STABILITY OF ON-OFF FEEDBACK SYSTEMS ABSTRACT Methods for studying the behaviour of on-off feed-back systems, with the emphasis on steady-state periodic phenomena, are presented i n th i s t h e s i s . The two main problems analyzed are (1) the determination of the periods of s e l f and forced o s c i l l a t i o n s i n si n g l e - , double-, and multiloop systems containing an a r b i t r a r y number of on-off elements; and (2) the i n v e s t i g a t i o n of the asymptotic s t a b i l i t y - i n the small of single-loop systems containing one on-off element which may or may not have a l i n e a r region of operation. To study the periodic phenomena i n on-off systems, methods of determining the steady-state response of a single on-off element are f i r s t described. Concepts per-tain i n g to the steady-state behaviour are then introduced: i n t h i s respect i t has been found that generalizations of the concepts of the Hamel and Tsypkin loci, and also of the phase c h a r a c t e r i s t i c of Neimark are useful i.n the study of s e l f and forced o s c i l l a t i o n s , Both the Tsypkin loci, and the phase c h a r a c t e r i s t i c concepts are used to determine the possible periods of s e l f and forced o s c i l l a t i o n s i n single- and double-loop systems containing an a r b i t r a r y number of on-off elements; these concepts are-also applied to multiloop systems. On-off elements containing a l i n e a r region of opera-t i o n c a l l e d a proportional band, are then describedj both the transient and periodic responses are presented. An approximate method for determining the periodic response i s given. The concept of the Tsypkin l o c i i s used to determine the possible periods of s e l f and forced o s c i l l a t i o n s in. a single-loop system containing one on-off element with a proportional band. The asymptotic s t a b i l i t y ' i n the small, or l o c a l s t a b i l i t y , of the periodic states of single-loop systems containing one ide a l on-off element has been considered by Tsypkin. In t h i s t h e s i s , Tsypkin 1s r e s u l t s have been generalized to include the cases on on-off elements containing a proportional band. The s t a b i l i t y of such systems i s determined by the s t a b i l i t y of equivalent sampled-data systems with samplers having f i n i t e pulse widths. F i n a l l y , t h i s s t a b i l i t y problem i s solved by a d i r e c t approach, one that makes use of the physical d e f i -n i t i o n of l o c a l s t a b i l i t y ; the r e s u l t s obtained by t h i s method agree with..those derived by the sampled-data approach. GRADUATE STUDIES; F i e l d of Study: E l e c t r i c a l Engineering. Servomechanisms Ele c t r o n i c Instrumentation . Network theory ' E. V. Bohn F. K. Bowers A. D, Moore Related Studies: P r o b a b i l i t y and S t a t i s t i c s Numerical Analysis ' D i f f e r e n t i a l Equations Modern Algebra Real Variable Noise i n Physical Systems Advanced El e c t r o n i c s F l u i d Mechanics R. R. S. W.. Nash C„ Froese A. Swanson B, Chang D. Derry E a Burgess E„ Burgess R. W. Stewart PUBLICATIONS 1. "An Investigation of the Performance of Barium. Titanate Sandwich Transducer Elements excited by High Power", Naval. Research Establishment Technical Memorandum. No„5, 1959, ( T i t l e only u n c l a s s i f i e d ) . Also presented at the USN Underwater Acoustics Symposium, 1958„ 2. "Comments on 'The Dependence of D i r e c t i v i t y Patterns on the Distance from the-Emitter' by J„ Pachner", Jour. Acoust. Society of America, 35, 1963, pp, 1666-67. 3. "On the Determination of F a r - f i e l d D i r e c t i v i t y Patterns from N e a r - f i e l d Measurements", Naval Research E s t a b l i s h -ment Technical Report No. 1, 1964. STEADY-STATE OSCILLATIONS AND STABILITY OP ON-OFF FEEDBACK SYSTEMS by AUYUAB MOHAMMED B . S c , The Univers i ty of Manitoba, 1954 M . S c , The Univers i ty of Manitoba, 1956 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 Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t freely-a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that, copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of ^ ^ c t U . c ^ J i 7 ^ t j j ^ v J t - ^ The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date tPAoJUL, 2-1 } l^ C-b ABSTRACT Methods for studying the behaviour of on-off feedback systems, with the emphasis on steady-state per iodic phenomena, are presented i n t h i s t h e s i s . The two main problems analyzed are ( l ) the determination of the periods of s e l f and forced o s c i l l a t i o n s i n s i n g l e - , double- , and multi loop systems containing an arbi t ra ry number of on-off elements; and (2) the i n v e s t i g a t i o n of the asymp-t o t i c s t a b i l i t y i n the small of s ingle - loop systems containing one on-off element which may or may not have a l i n e a r region of operation. To study the per iodic phenomena i n on-off systems, methods of determining the steady-state response of a single on-r-off element are f i r s t descr ibed. Concepts per ta ining to the steady—state behaviour are then introduced: i n th is respect i t has been found that general izat ions of the concepts of the Hamel and Tsypkin l o c i and also of the phase c h a r a c t e r i s t i c of Neimark are useful i n the study of s e l f and forced o s c i l l a t i o n s . Both the Tsypkin l o c i and the phase c h a r a c t e r i s t i c concepts are used to determine the possible periods of s e l f and forced o s c i l l a t i o n s i n single— and double-loop systems containing an a r b i t r a r y number of on—off elements; these concepts are also applied to multi loop systems. On-off elements containing a l i n e a r region of operation, c a l l e d a proport ional band, are then described: both the t ransient and per iodic response are presented. An approximate method for determining the per iodic response i s g i v e n . The concept of the Tsypkin l o c i i s used to determine the possible i i periods of s e l f and forced o s c i l l a t i o n s i n a s ingle - loop system containing one on—off element with a proportional band. The asymptotic s t a b i l i t y i n the small , or l o c a l s t a b i l i t y , of the per iodic states of s ingle - loop systems containing one i d e a l on-off element has been considered by Tsypkin. In t h i s t h e s i s , Tsypkin's resu l t s have been generalized to include the cases of on-off elements containing a proport ional band. The s t a b i l i t y of such systems i s determined by the s t a b i l i t y of equivalent sampled-data systems with samplers having f i n i t e pulse widths. F i n a l l y , t h i s s t a b i l i t y problem i s solved by a d i rec t approach, one that makes use of the physica l d e f i n i t i o n of l o c a l s t a b i l i t y ; the r e -sul ts obtained by t h i s method agree with those derived by the sampled-data approach,, i i i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS • > * * * « # » * 0 e « * » * s « * * * » * * » « * « » * « iX-9-L l ST OF TABLES ©oa*»*»»««»»**»oooee«©o«ooo«e©»»#««e XI ACKNOWLEDGrEMENTS ©.**»a**»»»©«>»«©o*»©c>©©;oee*©*©*»©»» x i i 1 * INTRODUCTION ©•••«««••« 9« « o * « « e » e«««*9« a o e « « * « « - I PART I : FUNDAMENTAL CONCEPTS OF ON-OFF ELEMENTS 2• ON—OFF ELEMENTS • ©•••-•••*£>*©«>©oe©©«>©©o©o©o©©©©© 4 3. RESPONSE OF ON-OFF E L E M E N T S . . . . . . . . . . . . . . . . . . . . 10 3.1 The Response for an A r b i t r a r y Input 10 3.2 The Steady-State Response 13 4. CONCEPTS PERTAINING TO THE STEADY-STATE RESPONSE OF ON-OFF ELEMENTS . . . . . . . . . . . . . . . . . . . . 32 4.1 Generalized Concepts of the Hamel and Tsypkin Loc i 35 4.2 Concept of the Phase Charac ter is t i c 37 4.3 Conditions for the Existence of Periodic O s c i l l a t i o n s i n Single and Mult i loop Systems 49 PART II ; ON SELF AND FORCED OSCILLATIONS IN ON-OFF FEEDBACK CONTROL SYSTEMS 5. SINGLE-LOOP SYSTEM CONTAINING AN ARBITRARY NUMBER OF ON-OFF ELEMENTS 53 6. DOUBLE-LOOP SYSTEM CONTAINING AN ARBITRARY NUMBER OF ON-OFF ELEMENTS 63 6.1 A p p l i c a t i o n of Tsypkin 1 s Method to a Double-loop System -with Two On-off Elements 63 6.2 A p p l i c a t i o n of the Phase Charac ter i s t i c Method to a Double-loop System containing an A r b i t r a r y Number of On-off Elements . . . . 68 T m MULTILOOP SYSTEMS • *»»o«»a»o©e©o«©*a*« ©©*•«•«•© 80 iv Page PART III s ON-OFF ELEMENTS WITH A PROPORTIONAL BAND 8. ON-OPP ELEMENTS WITH A PROPORTIONAL BAND . . . . . . 89 8.1 Transient Response of a Single- loop System containing One On-off Element with a Proport ional Band 89 8.2 Periodic O s c i l l a t i o n s i n a Single- loop System containing One On-off Element with a Proport ional Band 97 PART IV : THE STABILITY PROBLEM 9. STABILITY OF PERIODIC STATES IN ON-OFF SYSTEMS WITH OR WITHOUT A PROPORTIONAL BAND 110 9.1 The Concept of S t a b i l i t y of Periodic States 110 9.2 V a r i a t i o n a l Equation for a Single- loop System containing an Element with a Saturation Charac ter i s t i c . . . . . . . . . . . . . . . . 113 9.3 An Approximate Solut ion to the Asymptotic S t a b i l i t y of Per iodic States 126 9.4 A Direct Approach to the S t a b i l i t y Problem 131 X O • CONCLUSIONS « o © o « » * © * © » * « i * > o e o © » > © « . ? » > e 4 > » © » * > o * . * e « © 143 RSF£iH£]NC£j S • © • © © © © • • • • • • • e > © t > « © © © © o o © © © © © © © » « * > » » » » o © X 4 5 V LIST OF ILLUSTRATIONS Figure Page 2.1 Conventions and notations for the relay S y S "b Gill o o o o e « « » » « 0 0 * > o » o o o « o « o o o e o e 6 o o o o o o o o o 4 2.2 I n i t i a l conditions i n the l i n e a r part referred to the output 9 3.1 (a) On-off c h a r a c t e r i s t i c with dead zone and h y s t e r e s i s ; (b) Control s ignal x ( t ) ; (c) Correct ion s ignal y( t ) 11 3.2 (a) General form of control s ignal x(t) (b) General form of correct ion s ignal y ( t ) , i n the case of complicated o s c i l l a t i o n s . . . . 14 3.3 Form of y ( t ) f o r n = 2, with p1 and 0"2 absent . • . . . » o . « » « « o • • • • » « • • . . . • • . . . . O . . . . . . 23 4.1 (a) Block diagram of unit system (b) Charac ter i s t i c of on-off element 33 4.2 (a) Input to l i n e a r part of Figure 4 .1(a) , (b) Output of on-off element of Figure 4.1(a) 33 4.3 Sketches of general form of the Hamel and Tsypkin l o c i • » • • • » • • » . . . . 36 4.4 (a) Block diagram of System Is x(t) = v( t ) (b) Charac ter i s t i c of N i n Figure 4.4(a) . . . 41 4.5 Phase Charac ter i s t i c for H(s) = l / s 41 4.6 Phase Charac ter i s t i c for H(s) = l / s 42 4.7 Phase Charac ter i s t i c for H(s) = l / (fs+l) . . 42 4.8 Phase Charac ter i s t i c for H(s) = l / ( f s - l ) . . 43 4.9 Phase Charac ter i s t i c for H(s) = s/ (s+<x) , (a) a>o, (b) a<o • 44 4.10 Phase c h a r a c t e r i s t i c for s/ (s-Kx)(s+(3) where oc,|3 are r e a l s , a^P , a>|3>o 45 4.11 Phase Charac ter i s t i c for s/ (s+oc)(s+p) where a and B are complex conjugates . . . . . . . 45 4.12 Phase Charac ter i s t i c for H(s) = e - s 7 " 46 v i Figure Page 4.13 (a) Block diagram of System II (b) Charac ter i s t i c of N i n Figure 4.13(a) . . 46 4.14 (a) Block Diagram of System III (b) Charac ter i s t i c of N . . . . . . . . . . . . . . . . . . . . 47 4.15 Phase Charac ter i s t i c f o r H(s) = l / s . . . . . . . . 48 4.16 Phase C h a r a c t e r i s t i c for H(s) = l / s 48 4.17 Phase Charac ter i s t i c for H(s) = l / (Ts+l ) . . . 49 4.18 (a) S ingle- loop system containing n on-off elements (b) C h a r a c t e r i s t i c of Kh 49 4.19 Decomposition of system i n Figure 4.18 into n sub-systems 0 . » « . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1 Graphical procedure for determining possible h a l f - p e r i o d s of s e l f o s c i l l a t i o n s . . . . . . . . . . 53 5.2 On the determination of possible values of T that permit the occurrence of forced o s c i l l a t i o n s . . . » . » o o o . 56 5.3 On the determination of possible values of T that permit forced o s c i l l a t i o n s . . . . . . . . . . . . 58 5.4 Influence of A upon the number of values of T that may permit forced o s c i l l a t i o n s . . . . . . . . 60 6.1 (a) Double-loop system containing two on-off elements (b) Charac ter is t i cs of and ^ 63 6.2 (a) and (b) Outputs of and N 2 64 6.3 The Tsypkin l o c i ^(ocyT), 72(a,T) 66 6.4 Curves of a=f 1 (T) and oc=f2(T) 66 6.5 On the determination of values of T that permit forced o s c i l l a t i o n s 68 6.6 (a) Double-loop system containing an a r b i t r a r y number of on-off elements; (b) Charac ter i s t i c of i t h on-off element . . . 69 6.7 Open-loop system as a composition of uni t systems . . . . . . . . . . o ^ * ^ « . » » « . . » . . « » » . » * . » • • • • « 6.9, v i i Figure Page 6.8 Sketches of possible plots of © 1 j © 5 ^ , © 3 , © * 3 71 6.9 Relationships i n the n.^1 th sub-system . . . . 72 6.10 CTn + 1 -plane . . . . . . . . . . . . . . . . . . . . . 75 6.11 A double-loop system containing two N . elements . . . . » . • • • • • • . 76 6.12 Open-loop system of Figure 6.11 showing uni t systems »«•• . o o . . . . . . . . . . . . . . . . . . . . . . . . 76 6.13 Phase c h a r a c t e r i s t i c of the system shown i n Figure 6.12 « • • * . . . . . . . • 79 7.1 Basic uni t systems under consideration . . . . . 80 7.2 Phase c h a r a c t e r i s t i c notations and conventions for the type III uni t system . . . 81 7.3 Curves of Y=f 1 (T) and Y=f 2(T) 83 7.4 Four-loop system containing an a r b i t r a r y number of on—off elements 85 7.5 Curves of *=f.(T) f o r i = 1,2,3 shoving range of poss I b l e half—periods of o s c i l l a t i o n s i n loops 1, 2, and 3 87 8.1 C h a r a c t e r i s t i c s of some on-pff elements with proport ional band (a) Without hysteresis and dead zone (b) -With hysteresis and without dead zone (c) Without hysteresis and with dead zone (d) With hysteresis and with dead zone . . . . . 89 8.2 Block diagram of s ingle - loop system containing one on-»off element with proport ional band 90 8.3 System equivalent to that of Figure 8.2 . . . . 91 8.4 Equivalent system for the i n t e r v a l o<t<h^ . . 92 8.5 Equivalent system f o r the i n t e r v a l T^<-t<T^+h2 94 8.6 (a) Exact output of N i n Figure 8*2 i n the case of simple symmetric o s c i l l a t i o n s , (b) Corresponding approximation when H(s) has a f i l t e r i n g ac t ion • 99 1 v i i i Figure Page 8.7 Exact and approximate outputs of N for a sinusoidal input . . . * * » • * •«•«o...... •• » . o • 100 8.8 Construction f o r the determination of the possible half—periods of s e l f o s c i l l a t i o n s . 104 8.9 Construction for the determination of the possible half—periods of s e l f o s c i l l a t i o n i n the case of saturat ion with hysteresis . . 105 8.10 Construction to determine values of h and -r that may give r i s e to forced o s c i l l a t i o n . . . 107 9.1 A s ingle - loop system containing one on-off element . . . . . . . . . .••<>.. . . . . . . . 114 9.2 (a) Saturation c h a r a c t e r i s t i c , (b) Its der ivat ive . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.3 Transfer diagram f o r the graphic determination of $>'LxCt)j when x(t) i s a simple symmetric per iodic o s c i l l a t i o n of h a l f - p e r i o d T 116 9.4 Linear system equivalent to Equation (9.9) or (9.10) . . . . . . 117 9.5 Form of der ivat ives (J>'(x) for various types of saturat ion c h a r a c t e r i s t i c s . . . . . . . . . . . . . . 120 9.6 Transfer diagram f o r the graphic determination of (£>pLx(t)J when 5c(t) i s a simple symmetric per iodic o s c i l l a t i o n of h a l f - p e r i o d T * • • • • » . • • • • • . . . • • . . . 121 9.7 Transfer diagram f o r the determination of cjb' I]Sc(t)j when x(t ) i s a simple symmetric per iodic o s c i l l a t i o n of h a l f - p e r i o d T . . . . . . 123 9.8 Linear system equivalent to Equation (9.17) or (9.18) 9.9 Transfer diagram f o r the determination of ^[^("Oj where cj)(x) i s the simple saturation c h a r a c t e r i s t i c , and. Sc(t) i s a complicated per iodic waveform of period 2T . . . . . . . . . . . . . 125 9.10 Linear system determining the s t a b i l i t y of a complicated per iodic state x(t) for the saturation c h a r a c t e r i s t i c <$>(x) . . . . . . . . . . . . . 126 ix Figure Page 9.11 A s i n g l e - l o o p system containing one on-off element . . . • • • » • « o • • • » » . . . . . 131 9.12 Per iodic and modified outputs of N . . . . . . . . . 132 9.13 Deviat ion i n the output of N . . . . . . . . . . . . . . . 132 9.14 Equivalent sampled—data system for the s t a b i l i t y problem 136 x LIST OF TABLES Table Page I . C l a s s i f i c a t i o n of on—off elements . . . . . . . . . . 6 I I . Charac ter i s t i cs and Equations of on-off elements 8 x i ACKNOWLEDGEMENTS I wish to acknowledge my indebtedness to D r . E. V» Bohn for suggesting the topic of t h i s t h e s i s , and for his invaluable advice and c r i t i c i s m s throughout i t s preparation, I also wish to thank the Defence Research Board of Canada, and i n p a r t i c u l a r the Naval Research Establishment, Dartmouth, Nova S c o t i a , for their encouragement and substantial assistance, without which the comple-t i o n of t h i s thesis would have been d i f f i c u l t . x i i 1. INTRODUCTION The study of on-off feedback control systems having a single loop with one on-off element has been developed by many authors during the l a s t three decades. Many of the techniques for inves t iga t ing the steady-state behaviour of such systems resort to approximate methods, of which the best known is that of 1 2 3 the descr ibing f u n c t i o n . ' ' On the other hand, the best known 4 5 6 exact methods are those of D . A . Kahn, B. Hamel, J . Z . Tsypkin, 7 and E . V , Bohn. Concerning the determination of the periods of se l f o s c i l l a t i o n s i n a s ingle - loop feedback control system containing g two symmetric r e l a y s , Tu Syui-Tan and Tei-Lui—Vy gave both an exact s o l u t i o n , using the method of the Tsypkin L o c i , and an approximate s o l u t i o n , using the method based on harmonic balance. A l s o , Y u . I . Neimark and L . P . Shilnikov studied the symmetric per iodic motions of a multistage relay system by means of Neimark*s concept of the phase c h a r a c t e r i s t i c . Nevertheless, to the knowledge of the author, no study of multi loop automatic control systems containing an a r b i t r a r y number of on—off elements has been attempted* The main purpose of the f i r s t two parts of th is thesis i s to investigate the complicated forms of o s c i l l a t i o n i n a s ingle - loop system con-t a i n i n g a single on-off element and the simple symmetric modes of s e l f and forced o s c i l l a t i o n s i n s i n g l e - , and double-loop control systems having an a r b i t r a r y number, of on-off elements. Part I of t h i s thesis gives the fundamental concepts and formulae required i n the study of the various systems con-sidered i n Part I I . The working p r i n c i p l e , c l a s s i f i c a t i o n , and 2 equations of on-off elements are reviewed i n Chapter 2. The response of these elements to an a r b i t r a r y input and to the general per iodic input , and the methods of c a l c u l a t i n g the response are given i n Chapter 3. Next, i n Chapter 4, the con-cepts per ta ining to the per iodic response of on—off elements, namely, the concepts of the Hamel and Tsypkin l o c i (or hodograph), are reformulated so as not only to make evident the r e l a t i o n s h i p s ex is t ing among these concepts, but also to f a c i l i t a t e the study of s e l f and forced o s c i l l a t i o n s i n the multi loop systems considered i n Part I I . The conditions for the existence of s e l f and f o r c e d o s c i l l a t i o n s for the various multiloop systems are then determined with the help of these concepts. Methods of solving for the simple symmetric modes of o s c i l l a t i o n i n s i n g l e - , and double-loop systems are given i n Chapters 5, 6, and 7. Feedback control systems with proport ional bands are con-sidered i n Part I I I . The problem of determining the per iodic states of feedback control systems having a single nonlinear element with a r b i t r a r y piecewise l i n e a r c h a r a c t e r i s t i c has received rigorous at tent ion i n the l a s t few years . M.A. Aizerman and P.R. G a n t m a k h e r " ' ' ^ ' s t u d i e d the piecewise l i n e a r c h a r a c t e r i s t i c consis t ing of segments p a r a l l e l to two 12 given s t ra ight l i n e s , whereas L . A . Gusev dealt with an a r b i t r a r y piecewise l i n e a r c h a r a c t e r i s t i c * Their methods of solving the problem d i f f e r , but i n both cases the solutions take into account a l l the harmonics. Part III deals with an exact method for the determination of the transient state i n a 3 system containing one nonlinear element having the saturation c h a r a c t e r i s t i c with h y s t e r e s i s . A simple method of solving the simple symmetric o s c i l l a t i o n s i n such a system i s presented. The method i s approximate, but s u f f i c i e n t l y accurate for systems possessing l i n e a r parts with a f i l t e r i n g a c t i o n . An exact s o l u t i o n i s then formulated i n the form of a set of l i n e a r V o l t e r r a in tegra l equations of the second k i n d . F i n a l l y , Part IV of the thesis deals with the s t a b i l i t y of the per iodic states i n control systems having one on-off element with or without a proport ional band. An exact so lut ion shows that the "asymptotic s t a b i l i t y i n the small " of such systems reduces to a consideration of the s t a b i l i t y of f i n i t e pulse width sampling systems with feedback. The resul ts obtained are a general iza t ion of those of T s y p k i n . ^ An approximate method applicable to systems with nonlinear elements having c h a r a c t e r i s t i c s other than the on—off type, with or without a proport ional band, i s also presented. In contrast to the sampled-data approach, a d i r e c t method of i n v e s t i g a t i n g the s t a b i l i t y of s e l f and forced o s c i l l a t i o n s i n s i n g l e - l o o p systems having one on-off element i s presented. This method i s d i r e c t l y re la ted to the physical d e f i n i t i o n of s t a b i l i t y : a disturbance i s appl ied , and the ensuing devia t ion from the state of equi l ibr ium i s s tudied . P A F U N D A M E N T O N - O F F R T I A L C O N C E P T S 0 F E L E M E N T S 4 2. ON-OFF ELEMENTS According to t h e i r working p r i n c i p l e , on—off control systems are e s s e n t i a l l y nonlinear . Therefore i t i s evidently impossible to analyze t h e i r behaviour by the well-known l i n e a r methods of the theory of feedback control systems. Neverthe-l e s s , the s p e c i f i c p e c u l i a r i t y of on-off systems, namely that they are piecewise l i n e a r , permits t h e i r i n v e s t i g a t i o n by comparatively simple mathematical methods. In general , the on-off or re lay element may be regarded as consis t ing of the on-off component followed by a l i n e a r par t , which i s composed of the actual l i n e a r part of the relay plus the l i n e a r part fol lowing the r e l a y . Figure 2 i l gives the convention and notations for the re lay element. The symbol N represents the on-off (nonlinear) component, whereas On-off Linear Element Part x(t) N yet) H(s) vOt) Figure 2*1. Conventions and notations for the relay element. H(s) denotes the t ransfer funct ion of the l i n e a r par t , where s i s the complex frequency v a r i a b l e . The quanti t ies x ( t ) , y ( t ) , and v ( t ) are respect ively the input to the on-off element, the input to the l i n e a r part , and the output of the l i n e a r par t , and are a l l functions of the time var iable t . 5 In the f i e l d of automatic control x(t) i s referred to as the control s i g n a l , and y( t ) as the correct ion s i g n a l . In on-off control systems the correct ion signal y ( t ) changes by jumps at every instant when the control s ignal x(t) passes through cer ta in f i x e d values known as the threshold values* Hence the l i n e a r part of the system H(s) i s sub-jected to r e c t i l i n e a r pulses of f i x e d height , the s i g n , duration and r e l a t i v e d i s t r i b u t i o n of which depend both upon the external e x c i t a t i o n and upon the i n i t i a l conditions e x i s t i n g i n the l i n e a r part of the system. In general , on-off elements may be c l a s s i f i e d as symmetric or asymmetric with respect to the o r i g i n of the coordinate axes x and y , where x = x(t) i s the control s i g n a l , and y = y( t ) i s the correct ion s i g n a l . Furthermore, i n each of these two classes a dead zone may or may not be present. In addi t ion these elements may or may not possess hys teres is , that i s , y ( t ) may be a single or multivalued funct ion of x ( t ) . Table I gives t h i s c l a s s i f i c a t i o n of on-off elements. Equations and c h a r a c t e r i s t i c s of on-off elements The output y( t ) of the on-off symmetric component N i s a funct ion both of x(t) and i ( t ) , where i ( t ) = . Con-sequently, the equation of the on-off symmetric component can be wri t ten i n the form y( t ) = <£> (x( t ) , x(t ) ) where <$) (x ( t ) , x ( t ) ) i s a nonlinear f u n c t i o n . For s i m p l i c i t y we w i l l use the notation y = <$(x) (2.1) TABLE I . CLASSIFICATION OF ON-OFF ELEMENTS ON-OFF ELEMENTS S y m m e t r i c A s y m m e t r i c Without Dead Zone With Dead. Zone wl Without With Without With Hysteresis Hysteresis Hysteresis Hysteresis o, * 3 JJ n: Without DectcL Zone, With Dead Zone Without Wrt hi Without With l Hysteresis Hysteresis Hysteresis Hysteresis r 1 7 o 'x ON 7 The plot of y vs , x i s c a l l e d the c h a r a c t e r i s t i c of the on-off component N. In the case of asymmetric on-off elements the character-i s t i c can be expressed i n the form y = y Q + <£>(x - x ), (2.2) that i.sf cj)(x - x ) i s symmetric with respect to the point ( x a , y a ) » The c h a r a c t e r i s t i c s and corresponding equations for asymmetric on-off components are given i n Table I I . If the elements are symmetric we merely put x = y = 0, Prom Table II we observe that the f i r s t three c h a r a c t e r i s t i c s can be regarded as special cases of the four th . In f a c t , = 0 3 ( x - x a ) , 3>4 U " x a ) J X=l 3>4 ( x - X a ) X=-l and f i n a l l y <P4U - x a ) = ^ l ( x " X a } J x = 0 o The l i n e a r part of the system can best be analyzed by means of the Laplace transform. In the case of zero i n i t i a l condi t ions , the output of the l i n e a r part i s determined by V(s) = H(s) I (s) (2.3) where V(s) = d ! [v( t ) ] and T(s) = ^ ( y ( t ) } 8 TABLE I I . CHARACTERISTICS AND EQUATIONS OP ON-OFF COMPONENT N Charac ter i s t i c Equation M y - y a = c ^ i ( x - x a ) = M s i g n ( x - x a ) ->-x M M y - y a = 0 2 ( x - x a ) " M sign(x-x -x ) , for x>o £L O ->-X ^ M sign(x-x +x ) , for x<o 8/ O y T M JL. ~? -*-X y - y a =cJ^(x-x o) a | [ s i g n ( x - x a - x o ) + sign(x-x a +x Q ) ] M Jt— — T _ Jr. fM | ^sign(x-x a -x o )+sign(x-x a +Xx o)J for x>o sign(x-x +x )+sign(x-x ->x ) L a o a o-J f o r x < o Remarks* 1 N 1 , for x > a, 2» sign (x-a) = <{ o , for x = a, -1 , for x < a. i 3s In the case of a symmetric c h a r a c t e r i s t i c put x a = y a = o. Equation ( 2 » 3 ) may be rewrit ten as V(s) = H(s) &(y& + Cj)(x - x a ) ] . Now suppose that non-zero i n i t i a l conditions exist withi the l i n e a r part H(s) . By means of the Laplace transform, the output V(s) can always be expressed as V(s) = H(s)T(s) + V Q ( s ) , where V o ( s ) i s the output r e s u l t i n g from the i n i t i a l con-d i t i o n s w i t h i n H(s ) . Consequently, the e f f e c t of the i n i t i a l conditions may conveniently be referred to the output of the l i n e a r part i n the manner shown i n Figure 2*2, S i m i l a r l y , any external influence f ( t ) applied to the system may be referred Y(s) H(s) H(s)Y(s)^ VCs) = HCs)Y(s)-r-V CS) ^_ o >.-r V 0 (s) Figure 2*2. I n i t i a l conditions i n the l i n e a r part referred to the output. to the output of the l i n e a r p a r t . 10 3. RESPONSE OF ON-OFF ELEMENTS In on-off elements the correct ion signal y ( t ) changes by jumps at every instant when the control s ignal x( t ) passes through the threshold values with x ( t ) > o i n cer ta in cases and x ( t ) < o i n others . Consequently, the i n v e s t i g a t i o n of the response of on-off control systems i s reduced to the i n v e s t i -gation of the behaviour of the l i n e a r parts of the system to a sequence of r e c t i l i n e a r pulses , the parameters of which depend upon the form of the control signal and upon the threshold values of the on-off elements. Hence, the basic method of determining the response of the system i s through the a p p l i c a t i o n of the superposition p r i n c i p l e to the l i n e a r p a r t s . For any one on-off element, the response i s determined by the equation V(s) = H(s) ^ y a + c £ ( x - x a ) } + V q ( S ) . 3.1 THE RESPONSE FOR AN ARBITRARY INPUT The most general on-off c h a r a c t e r i s t i c , that i s , the case of the asymmetric on-off element with hysteresis and dead zone i s represented by the equation: y - y a = ^ 4 U " x a } * Without loss of g e n e r a l i t y , and for d e f i n i t e n e s s , we w i l l assume that the control s ignal x(t) passing through the f i r s t threshold value at the instant i s decreasing, that i s X (T ^ ) < O . The general forms of the control and correc t ion s i g n a l s , together with the on-off c h a r a c t e r i s t i c are shown i n Figure 3.1. 11 3 =X+4>0s-X.) Figure 3*1.(a) On-off characteristic with dead zone and hysteresis; (b) Control signal x ( t ) ; (c) Correction control sign signal y ( t j . The switching conditions x(t, ) = x + (-1) x , k' a ' o' x ( t k ) ( - l ) K > o .(k = 1,2,...) (3.1) correspond to the switching instants t ^ , t2»«»«. along the / \k threshold values x + (-1) x ; whereas the switching conditions £L O x(T t) = x + ( - l ) k + 1 \ x k a / o x ( T k ) ( - D K > o > (k = 1,2,...) (3.2) 12 correspond to the switching instants T -y »7~2» • • • along the threshold values x + (-l)^l+"'"Xx . It may happen that the £t 0 switching instant t i s absent, i n which case the switching instant T w i l l also be absent. m+1 The input to the l i n e a r part i s given by n y ( t ) = y a u ( t ) + M ^ ( - l ) 1 1 - 1 [ u C t - t ^ ) - u(t - T k ) ] , ( T n < t < t n ) k=l (3.3) = Right-hand side of (3.3) + M ( - l ) n u ( t - t ), ( t n< t<'^ 1 + 1) (3.4) where t = o, and u(t-a) i s the uni t step funct ion i n i t i a t e d at the time t = a . Let g(t - a) be the response of the l i n e a r part to the uni t step u(t - a) , that i s 2 ( g ( t - a ) ) = H i s l e - s a with the understanding that g(t - a) = o for t < a . Then the expression for the response of the on—off element to an a r b i t r a r y input with switching instants T-^jt^y 7-2*^2'"* * s n r v Q ( t ) + y a g ( t ) + M XI(-l)1""1 [s(*-^-l)~Z{t-\)] ' v( t ) =< k=l (T < t <t ) (3.5) n — n .Right-hand side of (3.5) + M ( - l ) n g ( t - t ), n <T N + 1) (3.6) where V Q ( t ) represents the response due to the i n i t i a l con-13 d i t i o n s ; that i s v ( t ) = < ' v Q ( t ) + y a g ( t ) v 0 ( t ) + y a g ( t ) v Q ( t ) + y a g ( t ) v Q ( t ) + y a g ( t ) ( 6 < t < T 1 ) ( T 1 < t < t 1 ) ( t 1 < t < T 2 ) ( T 2 < t < t 2 ) v Q ( t ) + y a g ( t ) + g(t + M[ g ( t ) + g(t g t t - r ^ - g U - t ^ T 2 ) + g ( t - t 2 ) J , ( t 2 < t < T 3 ) In general , the response may be constructed g r a p h i c a l l y by means of the superposit ion p r i n c i p l e . the general case of an on-off c h a r a c t e r i s t i c represented by In the case of complicated forms of o s c i l l a t i o n s , s e l f or forced , the input to the l i n e a r part of the system y( t ) repeats i t s e l f , i n general , a f ter 2n commutations, where n i s an even i n t e g e r . In the absence of a dead zone there are, i n general , n commutations, where n i s even. The general forms of the per iodic control s ignal x( t ) and of the per iodic correc t ion s ignal y ( t ) , . c o r r e s p o n d i n g to the on-off charac-t e r i s t i c under considerat ion, are shown i n Figures 3*2(a) and 2(b), r e s p e c t i v e l y . 3.2 THE STEADY STATE RESPONSE Various methods of evaluating the steady-state output response of the l i n e a r part of the system are now presented for 14 1 I I I I I I I J I L J L III I 1 I i I I L j C<rn. 2-OT « ^ r i ) T | (<rn-i)T (b) T p+ Figure 3*2.(a) General Form of Control Signal x(t) (b) General Form of Correct ion Signal y ( t ) , i n the case of complicated o s c i l l a t i o n s . I t may happen that p^T i s absent. In such a case i t follows from the c h a r a c t e r i s t i c of the on-off element that C T + 1 T i s also absent. The correc t ion signal y ( t ) can be expressed as the sum of a f i x e d component y , and a sequence of r e c t i l i n e a r pulses r e l a t i v e to y and denoted by y, ( t ) ; that i s €1 -1-y(t ) = y a + y x ( t ) , (3.7) where, l e t t i n g A u k , i = [ u ^ - v k + p ^ T ] - u [ t - ( k + q + 1 ) T ] J , (3.8) 1-1 - o o n-1 y i ( t ) =M[(-1) U [t-(m+ft)T] + X A u m , i + Zi 2 A u k , i ] 1=0 k=m-l i=o (3.9a) (m+ /^)T<t<(m+q + 1)T, m=o, + 1 » + 2, ... , £ =o , 1, ... , n-1 ; l-l oo n-1 = M [ E A V i + E Z!Auk,il (3-9b) i=o k=m-l i=o (m+Gf)T^t < ( m + p i ) T , m=o y + 1» + 2• ooo , •E —I • 2 « O O A « n» A l t e r n a t i v e l y , expressions (3.9a) and (3.9b) can be written as ^ -<*> £-1 n-1 Y l ( t ) =M[(-1) u [ t - ( m + p £ ) T ] + J ] ( X | A u k > i + 2 A u k - l , i > ] k=m i=o i=l (3.10a) (m+fy)T.<t <(m+CJ + 1)T; - o o H-l n-1 y,(t) = M A u k , i + S A u k - l , i > • ( 3' 1 0 b> k=m i=o i=£ (m+<^)T <t <(m+ f^T, respectively. In the case of dead-zone only, the above expressions retain the same form, except that the CT's change values, whereas i n the absence of a dead zone we have A = -1 and X01> o» s o "that we simply replace <j\ by p^ for a l l i . The output v(t) of the linear part of the system i s 16 determined as f o l l o w s . Let g(t) be the response to a unit step input i n i t i a t e d at time t = o i g(t) = < 0 , t < o (3.11) Then the response of the l i n e a r part to the input y-^(t) i s given by -oo £-1 -jtt) = M [ ( - l f g [ t - ( m + a ) T ] + E < E A ^ , i + E ^ - l . i ' J k=ra i=o i=d n-1 (3.12a) (m +p^)T <t <(m+CT;+|T, m=o, + 1, + 2 , . . . I - 0, 1 , . . . , n-1 ; - o o £-1 n-1 T i < * > = M Z < E A e k , i + E A « k - i , i > • k=m i=o i=l (m +OpT <t <(m+^)T, m=o, + 1, + 2 , . . , i = i » 2»• • •» n > (3.12b) where A g k , i = [g[t-(k+ft)T] - g[t-(k-K^ + 1)T]] (3.13) Since £ (g(t-T)) = H i s l e - s f s then * K , i ) = ^ <e"SplT - . " " ^ V * 1 , so that l - l ^ ( t ) ) = Me" s m T S k i sT n-1 1=0 1 - e sT (3.14a) 17 (m +C£)T <t<(m+ p£)T, £-1 s T n-1 ( - D e ' L + 1 - e s T (3.14b) (m +/C|)T <t <(m+OJ + 1)T , where |. = ( - l ) 1 (e r i - e 1 + 1 ) (3.14c) The response of the lin e a r part to a fixed component y a i n the steady state i s v a = y a g ( o o ) = y a H ( o ) » 0.15) which i s f i n i t e i f the linear part of the system i s stable. Consequently, the t o t a l output of the linear part of the system can be expressed as v(t) = v a + v 1 ( t ) T T / \ , M 0) H(s) x / \ —smT st , = y H(o) + j-_T y I (s)e e ds , a ^ J C, or C„ s 1 1 2 (3.16) (m+0£)T<t <(m+p^)T, where i-1 ™ n-1 V S ' = — : > < 3 - " > 1 - e where C-^  i s a path enclosing only the poles of H(s)/s, where i s a path enclosing only the poles of I^( s ) , and where the contour integrals along C^ and C^ are taken i n the mathematically positive and negative sense respectively; whereas 18 iliai x ( s) e-s™V t d s s 2 C l 0 r C2 (3.18) where (m+fy)T <t <(m+C^ + 1)T , (3.19) In general , v^(t) i s asymmetric, and v 1 (t+T) = v 1 ( t ) . (3.20) I f , however, the condit ion v 1 ( t + § ) = - v ^ t ) i s s a t i s f i e d , then the funct ion v^(t) i s said to be symmetric. This necessar i ly means that = odd integer , Pa Pn 2 +k 1 2 ' f * = | + P k » (k = 1» 2 , . . . , |) = 7 + C T t , . (k=l, 2 , . . . , § ) (3.21) Thus, i f we are considering the response v^(t) for mT <t < we get <(m+ ^ )T , then, subst i tut ing conditions ( 3 » 2 l ) into (3.17), 19 £ - 1 n-1 £ - 1 T 2 i=o i=£ i=o i = £ I-i (s )— m — _m 1 1 - e s i 1 - e S i _ i=o i=t = 1 + e s T / 2 (3.22) Consequently, i n the case of symmetric but complicated forms of o s c i l l a t i o n s , the response of the l i n e a r part of the system i s given by v( t ) = y a H(o) + j S i f i l I ^ s j e - ^ 1 e s t d s (3.23a) C l o r C2 (m+Cf)T<t <(m+^)T, m~o f + 1, . « . j £ = 1 j 2 , * . « , 2~ v ( t ) = y a „ ( 0 ) + i | » M [ ( - D V ' A ^ . ) ] . - » V * d . (3.23b) C l o r C2 (m+^)T <t < ( m + C £ + 1 ) T , m=o, + l , . . . ; £ . = o , 1 , . . . , ^ ~1 where I-^(s) i s now given by (3.22). Methods of Calcula t ing the Periodic Output Waveform So f a r we have set up very general expressions for the per iodic output v( t ) of the l i n e a r part of the system. Let us now turn our a t tent ion to the various methods of c a l c u l a t i n g the shape of the per iodic s ta te . We w i l l c l a s s i f y these methods as f o l l o w s : 20 1. The g-Method, which uses the unit step response g(t) of the l i n e a r part of the system; 2. The C^—Method: We derived an integra l representation of v^(t) i n the form V l ( t ) = 2^J ^ I ( s ) e S + d s » ( 3 - 2 4 ) C l where i s a contour enclosing only the poles of H ( s ) / s . By the residue theorem, of the theory of functions of a complex v a r i a b l e , v , ( t ) = M X + Residues of ^ i s i l ( s ) e s t l — a x TJ / \ s Poles of 31*1 (3.25) Thus, t h i s method uses the transfer f u n c t i o n , H(s) , of the l i n e a r part of the system. 3. The Method: An alternate in tegra l representation of V-^(t) was found to be T i ( t ) = 2^3- $ ^ I ( s ) e S t d s » ( 3 - 2 6 > where i s a contour enclosing only the poles of l ( s ) . Thus, by the residue theorem, v, (t) = -M 2 Residues of l ( s ) e s t 1 Poles of I(s) s (3.27) Since the poles of l ( s ) a l l l i e along the imaginary axis of the complex s—plane, we are e s s e n t i a l l y using H ( j « ) , the s o - c a l l e d frequency response of the l i n e a r part of the system, i n the evaluation of v ^ ( t ) . For th is purpose we w i l l f i n d i t more 21 convenient to rewrite H(jft)) as H ( j » ) - HQ(fl>) e j©(») where H 0( f l >) = JH(j<o) , and 0(<o) = arg H(j(o) • The g—Method of Determining the Periodic Output Waveform R e c a l l i n g that v (t) = y g(oo) we f i n d the t o t a l output v ( t ) , i n terms of g ( t ) , to be ••(t) =y ag(«>) + ( .3.12a), (m+|^)T <t <(m+CJ + 1)T, I = o, 1 , . . . , n-1 ; (3.28) v ( t ) = yag(oo) + ( ,3.12b), (m+or)T <t <(m+p£)T> £ = 1, 2 , . . . , n (3.29) ( m = o , ' + l , + 2 , . . . ) . Hence the construction of the per iodic state reduces to the superposit ion of the responses of the l i n e a r part of the system to pulses of height ( - l ) 1 M and of duration (d". - O. -, ) , i = l , . . . , n , plus the steady component y g(oo). This method i s convenient i f Ag, . — o as k—>-©o , that i s , i n those cases where the l i n e a r part of the system i s s t a b l e . The C^—Method of Determining the Periodic Output Waveform Let us suppose that the transfer funct ion H(s) i s a f r a c t i o n a l r a t i o n a l f u n c t i o n , i . e . 22 and that the degree of the numerator does not exceed that of the denominator. Furthermore, l e t us assume that H(s) has poles at S q = o of m u l t i p l i c i t y T Q - 1, s^ ^ o of m u l t i p l i c i t y r^,, (U= 1, 2 , . . . , p ) The sum of the m u l t i p l i c i t i e s of the poles i s equal to the degree of the denominator of H(s) , i . e . r - l + r . +r^. + . . . + r = N, say . o 1 2 P Let us put d ^ ^ " (r, , - n-Dl Q(s)s ^ (3.30) - 1 s=s. V •* • t y " ds R e c a l l i n g that V a ^ = y a g ( ° ° ) = y a H ^ ° ^ ' and using Eqs . (3.30) and (3.25), we get the t o t a l output of the l i n e a r part of the system i n the form P ^ L 1 CL, d^l(s,.)e ^ ^ a G o o + M E E ^ — I s V ^ a x v s ^ , e (3.31) 1/ =0 (A=0 Ve now evaluate specia l cases of ( 3 . 3 l ) . Suppose that H(s) has only simple poles* a l l d i f f e r e n t from zero . Then r Q = r± = . . . = r N = 1, p=N, u.=o , so that ( 3 « 3 l ) becomes N t v<*> = y a c o o + M Z W V 6 ^ (3-32) i^=0 23 where C = , and C oo Q(o) ' Therefore, i n the case where y( t ) i s asymmetric, we have from (3.16), (3.17) and (3.32) N s,,t v ( t ) = y C + M y n C , 1 , ( 3 . , ) , (3.33) v ' J a oo Z i i/o 1 v V ( G £ T < t < f y T ; |=1, 2 , . . . , n ) , and from (3.18), (3.19) and (3.32) N v( t ) = (y a + M ( - l ) ) C o o + M]TV0 I 2 ( 8 ) e s^t 1^=1 (3.34) In the simplest case where n = 2, and p ^ and CT^ are absent, I . e . the input has the shape shown i n Figure 3 » 3 , we obtains ft™ q T ftT.T Figure 3 » 3 . Form of y ( t ) for n = 2, with p i and 0 " 2 absent. 24 N s ^ ( l - q ) T s t v(t ) = ( y a + M ) C o o + M 2 % 0 1 " 6 S „ T 6 " ( 3 ' 3 5 > l>0 ays V=l 1 - e " (o<t<CT 1 T), N - s ^ T g t V ^ = ^ a C 0 0 + M ^ C ^ 1 - 6 s T e " (3.36) V=l 1 - e " (CT 1 T<t<T) . In the other simple case where dead zone is absent and n = 2 we have o-x =p l f cr2 =p 2 = 1, so that equation (3.34) reduces to v( t ) = ( y a + M ) C o o + 2M 2 <Vo " ° s?yT e V=l 1 - e U (3.37) (o <t<p xT) , N " V / ° L T s t v ( t ) = ( y a - M ) C o o + 2 M 2 < V o 1 " 6 s,yT e " <3-38> 1^ =1 1 - e ^ (P 2 T < t < T ) . Let us now consider the complicated forms of symmetric o s c i l l a t i o n s , the general formulas of which are given by (3.23a) and (3.23b). Special cases of these f o l l o w . Case Is H(s) has simple poles a l l d i s t i n c t from zero, so that r Q = r± = . . . = r N , p = N, |i = o . In th is case we get 25 N v ( t ) = y C + M y 1 C J a oo Z i i t-1 s,.-x 2 i=o i=l  T SI/ 2 1 + E ( 3 . 3 9 ) e (07T<t< p T ; t =1, 2 , § ) , v - nt and v( t ) = M(-l) C + M V V ( - l ) e ^ ™ e ^ + Right-hand side of Eqn. (3.39) (3.40) (p^T <t<CJ + 1 T; t =o, 1 , . . . , | - 1) . In the simplest case when ^ = 1 ( r e c a l l that ^ must be an odd number for symmetric o s c i l l a t i o n s ) , equations (3.39) and (3.40) reduce to = ^ o o + M X] %o 1 " 6 1 e ^ <3"41> ^=1 ! + e S ^2 (CT 1T < t<iT = p x T ) , and N s ^ ( i -QJ)T g t v(t ) « ( y a + M ) C o o + M ^ f • " ^ 1 + e V 2 (3.42) (o <t <CT1T), r e s p e c t i v e l y . Case 2; H(s) has one pole equal to zero, and the other N-1 poles are simple, i . e . 26 Then r = 2, = r 2 = . . . = r ^ = 1, so that from E q . (3.3l) ve obtain (t) - ( y a + K o ) ) C 0 0 + C 0 1 st" N-1 V Js=o 1^ =1 (3.43) Computing E q . (3.43) i n the case of (3.23a) and (3.23b), i . e . for complicated but symmetric o s c i l l a t i o n s , ve obtain l-l — -1 1=0 e-i N-1 M E ^=1 Vo .=1 T n _ 1 2 W - e 8 ^ 2 2 y ; ^.(sj,) i=o i ^ T 1 + e * V 2 (3.44) ( 0 - T < t < Q T ; £ =1, 2 , . . . , |) , N-1 v ( t ) = M [(-1) C o Q + C o l ( - l ) ( t - f t T ) + 2 C ^ ( - l ) e "*.y J (3.45) ^=1 + Right-hand side of Eq . (3.44) ( P £ T<t<CT + 1 T; I =o, 1 , . . . , vhere C = oo ds s=o (3.46) Furthermore* i f ^ = 1, that i s ve have simple symmetric 27 o s c i l l a t i o n s , Equations,,, (3.44) and (3.45) reduce to N-1 " V ° i T v(t) = y .cA n +cwl ^ cr, I - e a oo 1 o l 2 w 1 ' " Z_J "i/o o T ^=1 1 + e ^ 2 e ^ (3.47) (CT 1 T<t<|) , and v( t ) = ( y a + M ) C o o + C o l M ( t - |oi) t M ^ t j ^=1 N-1 , , - l / i - O i T ) 1+e  Vo _ T V1 1 + e 2 ( o ^ t ^ T ) . (3.48) Case 3; H(s) has two poles equal to zero, whereas the other N - 2 poles are simple, i . e . Then r = 3, r, = r_ = . 1" x 2 r N - 2 ~ 1 * Equation (3 ,3l) then becomes x l i l = + i(o))c + c n ^ f l a M M ' oo ol ds st' s=o 4 . J2l d 2 l ( s ) e s t 9T 9 2 * ds^ N-2 + 2 <Vo I (V> s^t -J s=o i/=l (3.49) The computation of (3.49) i n the case of (3.23a) and (3,23b), i . e . for complicated but symmetric o s c i l l a t i o n s , y i e l d s £ - 1 T r v £ _ i 2 1 i=o i = £ 28 2. _ i t-1 2 + ¥ ( - i ) i ( a " i + i - / 3 i ) - z i ^ ^ t o i+i-Pi)] i=o i=Jt - -1 t - 1 2 1 + ( f ) 2 [ E (-D> i - q + l)(2ft+ 2 C T i + 1 + l ) i=o i = £ - -1 i^=l i=o i=i ( O i T < t < a T ; ^ = 1, 2 , . . . , § ) , ( 3 e 5 ( ) ) whereas ^=1 + Bight-hand side of Eq . (3.50) (3.5l) where oo _ 1 d f _ f F ( s ) -I _ d_ [P is ) I s=o s=o (3.52) C .= ^ l r , and n In the case of simple symmetric o s c i l l a t i o n s 9 i « e « n T TJT = 1, p n T = p^T = 2 > equations (3.50) and (3.5l) reduce to 2 29 y j t l M y Coo + C o l 1^ 1 + -f l^ i t 2 * - ¥2°i+ 1}] and N -2 + Z c 1 - e Po s, ,J_ ^=1 1 + e ^2 (3.53) M (C^T < t < § ) ^ + ^Coo* c o i ( t - § ° i > + - ^ P - t T q - t f f c r . d - z c r , ) ] N -2 + s w ( i - C T J T , S P 1 + e ^ 1 SU% C,._ — r e Z =^l 1 + e T~ (3.54) (o<t<O^T) . Cases 1, 2 and 3 dealt with above are the ones usual ly encountered i n p r a c t i c e . Other cases may be s i m i l a r l y evaluated by an a p p l i c a t i o n of equation ( 3 . 3 l ) . The C^—Method (or Frequency Response Method) of Determining the Periodic Output Waveform Here we apply formula (3.27) to equations ( 3 » 1 4 a ) and (3,14b). The poles of I-^(s) and ^ ( s ) , given by equations (3.17) and ( 3 « 1 9 ) , are the same, and occur at j k » , (k = 0, + 1, + 2, a = Zjji) Consequently, v x ( t ) = -M y ~ j + 0 0 k=-«» H(,jko) 3^ ~~7ji i-1 n-1 1=X 1=0 -T e sT st 30 Nov sT n-1 1 = 0 - T e sT n-1 1 = 0 ) Let us put n-1 ikn Z _ l v jk27tpi -jk2ixc^ + 1 jk  i=o and substitute - e ) = ^ H(jtt) = H q(W) e j © ( » ) (3.55) (3.56) where Then H ( « ) = H(jtt) , and 0(co) = arg H(jtt) . v*> = S ^^(ktt) j [kttt-<^+ ©(ktt)] k=-oo which can be rewri t ten as r x ( t ) = | C o H o (o ) + 2 | C k | H o ( k t t ) c o s [ k t t t - « ^ + ©(kt t ) ] k=l (3.57) If v^(t) has the addi t ional property of symmetry, then from equation (3*22) n T 2 s 77 -1 T S S 5 , = [ ! ] ? , - • • 5 i=e i = 0 i=* 1 = 0 so that the poles at 31 are e l iminated . Hence, i n the case of symmetric o s c i l l a t i o n s v^(t) becomes 0 0 / V j f t ) = 2 | C k| E 0 ^ ^ cosjlaot - < £ k + 0 ( k » ) ] (3.58) k=l where >^ j means the summation with respect to odd numbers only . Also C .^ i s now given by S -1 ) 1 = 0 Equation (3.58) may be conveniently rewrit ten as r i ( t ) = 2 l ° 2 k ~ l | H o ( ( 2 k - l ) w ) c o s[(2k - l)i . t - ^ 2 M 0 ( ( 2 k - l ) « ) ] (3.59) k=l + 32 4 . CONCEPTS PERTAINING TO THE STEADY-STATE RESPONSE OF ON-OFF ELEMENTS Before proceeding to the study of s e l f and forced o s c i l l a -t ions i n on—off feedback control systems, we w i l l f i r s t introduce concepts per ta ining to the steady-state response of such systems. In t h i s respect , the Hamel and Tsypkin l o c i (or hodograph, or c h a r a c t e r i s t i c ) ^'^haye been formulated to f a c i l i t a t e the s o l u -t ions of per iodic o s c i l l a t i o n s i n s ingle - loop systems containing q one on—off element. Furthermore, Neimark used the concept of the phase c h a r a c t e r i s t i c to determine the simple symmetric s e l f -o s c i l l a t i o n s i n a s ingle - loop system containing an a r b i t r a r y number of on—off elements, but no mention was made as to how i t may be adapted to the problem of forced o s c i l l a t i o n s . In t h i s chapter we redefine the above—mentioned concepts i n order ( i ) to include the effec ts of i n i t i a l conditions and of external i n f l u e n c e s , ( i i ) to show the re la t ionships ex is t ing among these concepts, but moreso ( i i i ) to extend t h e i r sphere of a p p l i c a t i o n to the so lut ion of the possible per iodic motions i n m u l t i -loop control systems, containing an a r b i t r a r y number of on-off elements. For t h i s purpose i t w i l l be convenient to regard any given system as a composition of simple uni t systems, or sub-systems, shown i n Figure 41(a) the charac ter i s t i cs of which can be r e a d i l y ascer tained. Let us assume that the c h a r a c t e r i s t i c of the on-off element i n Figure 4,1 (a) i s symmetric with hysteresis and dead zone, as depicted i n Figure 4.1(b), The i n i t i a l conditions are referred to the output of the l i n e a r part and are designated by v ( t ) , 33 Linear HCs) v(t) fCt) On-off -r xCt) Part N — Element v0Ct) - — M (a) (b) Figure 4*1(a) Block diagram of uni t system (b) Charac ter i s t i c of on-.off element whereas f ( t ) accounts for any external a c t i o n . Let the input to the l i n e a r part of the system be a steady per iodic waveform of symmetric rectangular pulses as shown i n Figure 4 .2(a) .Then the output v ( t ) of the l i n e a r part w i l l also be a per iodic waveform with the same p e r i o d i c i t y as the input y t ( t ) . - 2 T J -T 0 — M i 2.T 3T ' (a) -> ocT [< T v H ~T >- L (b) Figure 4 » 2 ( a ) Input to l i n e a r part of F i g . 4*1 (ft), (b) Output of on-off element of F i g . 4.1 (a) . 34 In fac t s ( l-p x)T f_M_ j H M 1 + e ^ e s t d g j ( 0 < t < p i T ) (t) = • 2itj r _ P 1 + e " C l o r C2 ^ 2TXO r- J n 1 + e ^ C l o r C2 (4.1) where i s a contour enclosing only the poles of H ( s ) / s , where s T C 2 i s a contour enclosing only the poles of l / ( l+e ), and where the contour integra ls along and are taken i n a mathematically p o s i t i v e and negative sense r e s p e c t i v e l y . Now the input x(t) to the on-off element i s given by x(t) = f ( t ) + v ( t ) + v Q ( t ) (4.2) In the case of simple symmetric per iodic responses, that i s y(t+T) = ~*y(t), the only switching conditions are x [(cc+k)T] = ( - l ) k x Q = £ x [ (a+p+k)TJ (4.3) x [ ( a + k ) T ] ( ^ l ) k > o > x [ ( a+p+ k ) T ] ( - l ) k (4.4) (k=o, + 1 , + 2 , •••) where a i s taken as>0 and 0<p<l. Consequently $ the output of the on-off element i s also per iodic with half period T ; i t has a pulse duration pT which i s i n general d i f f e r e n t from the pulse duration p^T of the input y ^ ( t ) ; and i t i s s h i f t e d to the r i g h t by an amount ocT. The condit ion expressed by Eq.(4 .3) i s referred to as the condit ion for the proper switching i n s t a n t s , 35 whereas that given by Eq»(4.4) i s the condition for the proper d i r e c t i o n of switching. If a dead zone i s absent then we put A — — 1 » 0 = 1 so that the switching conditions reduce simply to x [(a + k)T] = (-l) kx * [ ( a + k)T] (-l) k>o J (4.6) (4.5) > (k = O j -1, -2,...) D k > o Furthermore, i f hysteresis i s absent then X q i s set equal to zero, 4.1 GENERALIZED CONCEPTS OP THE HAMEL AND TSYPKIN LOCI From the above we note that the quantities x(ctT) and x(ocT), together with x [(a +/3)TJ and x [ ( a + JO)TJ i n the presence of a dead zone, completely charaterize the parameters Y = <a, the frequency of the-periodic response* p the r e l a t i v e pulse duration, and a the s h i f t to the right r e l a t i v e to y^(^) of the output of the unit system. Hence we are led to the following concepts of a " c h a r a c t e r i s t i c " of a unit system of the type shown i n Figure 4*1s 1. Generalized Hamel Loci. The generalized Hamel Loci are defined by 3-1 (a,<a) = x(aj) + j x(aj) (4.7a) and CH,(a,p,W) = x [(a +p)J] + j x [(a + p ) j ] (4.7b) 2. Generalized Tsypkin L o c i . The generalized Tsypkin Loci are defined by 3(a,«) = + 3 *(«J) (4.8a) and J («,p..) - J i [ (« + p)jf ] + 3* [ (a • p)j] where J-t(oc,p,tt) and J (oc,p,«) . are required i n addi t ion to J-t(oc,«) and £T(oc,G>) i n the case of a dead zone. It i s i n t e r e s t i n g to note that for a given <a as a var ies from 0 to 1 , the quantity Im J ( a , « ) or ReJ{(oc,fl>) determines the per iodic waveform x ( t ) , since t i n x(t) takes on a l l values between 0 and T; s i m i l a r l y , the quantity Re J(oc,a>) weighted by the factor l/a or ImJ-((cx,a>) determines the der ivat ive x ( t ) . The Hamel and Tsypkin l o c i are convenient graphical representations of the input s ignal conditions at the switching i n s t a n t s . They are therefore useful i n the study of per iodic phenomena i n on—off systems. Sketches of the general form of the Hamel and Tsypkin l o c i are shown i n F i g . 4*3. Rett CH(.rf.,cO)-plane Im J J<C«=<-iP)U))-plane ^ ( o c ^ u ) ) - plane Figure 4 .3 . Sketches of general form of the Hamel and Tsypkin L 37 Quite obviously, the Hamel and Tsypkin l o c i are equivalent except that Hamel's x i s replaced by i n the case of Tsypkin and that the coordinates are interchanged, Hamel*s c h a r a c t e r i s t i c i s advantageous from the point of view that ( i ) i t uses the phase-plane var iables x and x which describe the system's behaviour, and ( i i ) a der ivat ive control introduced into the system i s very e a s i l y s tudied . On the other hand, the Tsypkin representation i s generally very close to the t ransfer locus H (j<») i n the high frequency region. 4.2 CONCEPT OF THE PHASE CHARACTERISTIC In the preceding sect ion we observed that the output has the same general features as the input y ^ ( t ) . In f a c t , i t has the same p e r i o d i c i t y , but i t i s s h i f t e d to the r ight by an amount ocT as shown i n Figure 4*2* The curve ocT vs T w i l l be referred to as the phase c h a r a c t e r i s t i c of the uni t system. To emphasize the fac t that ocT i s a funct ion of T* we w i l l denote i t by O(T) . C l e a r l y the instant ©(T) of switching from -M to +M that i s c losest to the instant t = 0 i s a non-negative root of the equation x(t) = X Q (4.9) Obviously, the phase c h a r a c t e r i s t i c represents the information concerning the switching instants given by the i n t e r s e c t i o n of the Hamel l o c i with the s traight l i n e X q , o r , a l ternat ively# by the i n t e r s e c t i o n of the Tsypkin l o c i with the s traight l i n e 38 The Hamel and Tsypkin l o c i are very convenient concepts i n the study of the s ingle - loop system containing one on—off element^ but are very cumbersome i n the case of s ingle or multiloop systems with more than one on—off element* It w i l l be seen l a t e r that the phase c h a r a c t e r i s t i c i s better suited for determining the periodic modes of o s c i l l a t i o n s i n multi loop systems containing an a r b i t r a r y number of on—off elements* The i n v e s t i g a t i o n i s considerably s i m p l i f i e d i n those cases where an analyt ic expression for the phase c h a r a c t e r i s t i c i s a v a i l a b l e . In the case of on-off elements with dead zone i t i s neces-sary to know pT, the duration of the output pulse corresponding to a f i x e d input pulse duration £^T» Consequently, i n such cases the concept of the pulse duration c h a r a c t e r i s t i c * which i s a curve of pT vs T with p^ as the parameter* has to be introduced. We now proceed to the computation of the phase character-i s t i c ©CT) f o r a few simple systems, i n which a dead zone i s absent* We f i r s t l i s t formulas f o r v ( t ) * the output of the l i n e a r part of the system? for commonly encountered specia l cases of H (s)s Case Is H(s) has simple poles , a l l d i s t i n c t from zero* Then T ( t ) = 2 M x [ - f + £ « V « , 2 - = ? ] < 4 ' 1 0> where C__ = Trfy , and C, V=l 1+e " ( o < t < T ) * ( s „ ) V oo Q(o) * *Vb Case 2.8 H(s) has one pole at the o r i g i n , and the remaining N—1 poles are simple, that i s , 39 Then N-1 Sjyt _ T ( t ) = M [ C O O + C o l ( t - f ) + 2 - ^ . J (4.11) (o <t <T) where G oo ds P 'P(s) 1 p P (o) , n  Q^TtyJ > C o l = Q^ToT > a n d ° W = Q M s )s • s=o Case 3 8 H(s) has a second order pole at the o r i g i n , and the remaining N — 2 poles are simple, i . e . H(s) =£14 = ^ 1 - . s Q 2 ( s ) Then v ( t ) = M ^ o Q + C o l ( t - § ) + C o 2 t ( t - , T ) $ N-2 s^t (4.12) ZA=1 l+e ^ ( e < t < T ) where n 1 • r p ( s ) 1 p d rp(s) i n p(o) c o o = .2 "Tr [o^TtTJ ' c o i = dT [ ^ T s T J > Ge2 = ^ T o T * s=o s=o 40 and Case 4s H(s) has a second order pole at s^(^ o) j and the remaining N — 2 poles are simple and d i s t i n c t from zero, i « e . H( 8 ) = m = — — 2 ( s ) ( S - S l ) 2 Q 3 ( s ) Then r s i T s i t t<*> - 4 ° o . + ( c i o + °ii* - c n 2 H ] X -l+e l+e N-1 s^t + ^ 1 f/O S j . W=2 l+e ( o < t < T ) where s=sx oo ~ QTOT * U l o ~ ds [s Qjll) ] ' u l l ~ F~QjR7J ' P(s..) <Vo = i - # ( ^ y We now turn our at tent ion to the computation of the phase c h a r a c t e r i s t i c 0(T) for a few systems* System Is x (t) = -fv(t)t hysteresis and dead zone absent in_JI This system i s shown i n Figures 4<*-4 (a) & (b) . 41 (ft-1) H(s) S 0 ^ N —>— (a) M o - M (b) Figure 4.4 (a) Block diagram of System Isx(t) = v ( t ) (b) Charac ter i s t i c of N i n F i g » 4 « 4 ( a ) # Let us consider the fol lowing representations fpr H(s )« (1) H(s) = - s We use E q . ( 4 . 1 l ) . Here Vtf)\ = 1, so that — >L_ y^\.s; C = o, C _ = 1, C = o ( a l l u) oo o l ' l/o Hence x( t ) = M^( t - | ) . Sett ing x(t) = X q = o we get the phase c h a r a c t e r i s t i c 0(T) = ? Figure 4* 5« Phase character-i s t i c f o r H(s) = l / s . (2) H(s) = l / s ; We use Eq» (4.12). The only non-zero c o e f f i c i e n t i s C ^ which i s equal to 1. 42 M, Hence x(t ) = ( t -T) t , ( o < t < T ) Thus ©(T) = T Pigure 4»^6. Phase character-i s t i c f o r H(s) = l / s 2 (3) H(s) == l / ( X s + l ) : Ve use E q . (4.10)* Here C s x = - l / T . Therefore ( o < t < T ) Set t ing x(t ) = o we get © ( T ) = f £ n 2 ? f 7 f 1+e 7 oo 1* C 10 = -1, Pigure 4*7* Phase character-i s t i c f o r H(s) = l / (Ts+l ) (4) H(s) = l / ( T s - l ) s Referring to case (3) above we simply replace T by — f and H(s) by -H(s) to get t / T -1+e (o <t <T) 43 Therefore 9CT) O(T) = Tin 1+e T/T 0 ( T ) = f £ n - L ± ^ / ^ - S l o p e =1 Pigure 4*8, Phase character-i s t i c for H(s) = l / ( T s - l ) (5) H(s) = l/Cs(s+a)!]: Ve use E q . ( 4 . 1 l ) . Here _ c _ I 2 ' o l ~ a a Therefore C o o - " ~   C  = a ' C l o = \ * s l - -a . x ( t ) = M £ [ - ^ + l ( t - i ) + 1 2 e -a t I a 2 l + e - a T J ( o < t < T ) No analy t i c expression can be found for © ( T ) . But given a, we can solve for 0(T) g r a p h i c a l l y or numerically, (6) H(s) = (s+a)(s+ p ) ( i ) Suppose a ^ 0 , a ^ o, |3 ^ o. Using E q . (4.10) we get x ( t ) - Mt [k + A ( i fr^r ~ JT ( o < t < T ) . ( i i ) Suppose a = p ^ o. Then, using E q . (4.13), we get 44 (t) = 2 + ( - % t , T e - a T a a 1 + e - a T (o<t<T) 2e -at l+e -aT No analytic expression i s available for 0(T) for this case. But, given a and p, we can solve for 0(T) either graphically or numerically. (7) H(s) = ( s + * ) { a +Qy. ( i ) Suppose a p ^ o. Then, using Eq. (4.13) we get Hence x(t) = M,(t - Te -aT, r) 2e -at J v " , -aT y - -aT ** l + e 1-4- e ( o < t < T ) . ©(T) = Te -aT l + e T -aT l + e aT Figure 4.9. Phase characteristic for H(s) (s + a ) 2 ' a and p pure reals 5 Q< = (3 =£0. ( i i ) Suppose a ^ p, a ^ o , p ^ o , & a and p pure r e a l s . Then, using Eq. (4.10), we obtain 2M. r e - p t e - a t -j ( t ) = <* - > L i T F ? 1 " 7 7 7 ^ J ( o < t < I ) . 45 Putting x(t) = o we obtain 1 - 1 + e ~ P T OCT) ©(T) = tn a - 3 i K l + e which may be rewritten as -aT ' 0(T) = tanh -1 sinh—^-T x e ° L_±_1 T 2 + cosh S-f-^T ] oc-(S l+-e - a r This phase c h a r a c t e r i s t i c i s plotted i n Pig. 4.10 for the case a > P > o . Figure 4.10. Phase character-i s t i c for H ( s ) = (s + a ) U + B where a, p are reals,-a ^ p, a >B> o>. Ciii) On the other hand,, i f a and P are complex then they are complex conjugates* Let a = a+jb, then P = a-jb, and a - g a + 8 In t h i s case we get sinbT e m 9(1) = I t a n - i [ - a i a W L - 1 . b U a T+cosbTJ F i g . 4.11 shows a sketch of t h i s phase c h a r a c t e r i s t i c . o 9 (T ) = X t a n - | r ^ n b T 1 b L e~ T_ hcosloTJ 2.TT Figure 4.11. Phase character-i s t i c for H(s) = ( s + a ) ( s + p) where a and P are complex conjugates. 46 (8) H(s) = e - ^ Obviously x(t) = y^ (t -7") Hence the phase characteristic i s given by O(T) = T — [ £ ] 2T where {[^rjj denotes the integral part of T/ ( 2T). ' i Figure 4*12* Phase Character-i s t i c f or H(s) = e • System I I : x(t) = - v ( t ) ; hysteresis and dead zone absent i n N, This system i s shown i n Figure 4.13. <flrO HCs) v f t U . x c t ) H g ) — > N ->-M O *x (b) Figure 4-13. (a) Block diagram of system I I . (b) Characteristic of N i n F i g . 4.13(a). Let O j ( T ) be the phase char a c t e r i s t i c of system I . Let © J J ( T ) be the phase char a c t e r i s t i c of system I I , corresponding to system I , i . e . same H(s) and same N but with the change x(t) = - f ( t ) . Then* i n terms of the phase char a c t e r i s t i c of system I , O J ( T ) , the phase ch a r a c t e r i s t i c of system I I i s given by G N ( T ) = ©j(T) + T - ( [ 1 2 T + T ] 2T , (4.14) 47 where |£ Jj denotes the integral part of i t s argument. As i l l u s t r a t i o n s consider the following cases: (1) H(s) = l / s : Ve obtained Oj(T) = T/2. Therefore, by Eq, (4.14) t O j j d ) = 3T/2 (2) H(s) = l / s : In t h i a case ©j(T) = T f so that Oj j l T ) = 2T - [||] 2T = o. System I I I : x(t) = +v(t); N has hysteresis, but no dead-zone, This system i s shown i n Figure 4.14. Cp £ - 0 H(s) ^g) •* (a) N y(t) M A X 0 O (b) Figure 4 .14. (a) Block diagram of System I I I ; (b) characteristic of N. For t h i s p a r t i c u l a r system, the phase characteristic i s found as the least positive root of the equation v(t) = x Q. Ve now compute 0(T) for the cases of H(s) considered i n connec-ti o n with system I. 48 (1) H(s) = ± Putting v( t ) = M t ( t - § ) = x ( we get 0(T) T 2 + M £ where i t i s understood that Figure 4 » 1 5 * Phase character-i s t i c for H(s) = l / s . (2) H(s) = \ s_ In t h i s Instance we have M x(t) = f- t (t - T) = -x Q Provided that x < x ( t ) o max i . e . *0<\T commutations w i l l occur. The phase c h a r a c t e r i s t i c i s given by ©(T) = r 0 8x 3^ - [T 2 - - j f ] Figure 4.-16, Phase characteris-t i c for H(s) = l / s 2 . 49 (3) H(s) Ts + 1 Here r " t / T i x(t) = M< L1 - f+r^J ( o < t < T ) . Provided that T x < x(t) = M ntanhTn= o max £ 2T commutations w i l l occur . . The phase c h a r a c t e r i s t i c i s given by 2 0(T) = Tin (1 + . - * / » - ) (1 v a l i d for T > 2 T t a n h -1 __o M„ ©CT) !6W=rJ n ( l + e T / T ) 0 - ^ ) zTtanK" 1 -** M 4 Pigure , 4*17«»Phase character-i s t i c for H(s) = l / ( T s + 1) . 4.3 CONDITIONS FOR THE EXISTENCE OF PERIODIC OSCILLATIONS IN SINGLE AND MULTILOOP SYSTEMS Let us f i r s t examine a single—loop system containing an a r b i t r a r y number of n on-off elements. The system under considera-t i o n i s shown i n Figure 4.18. ——*~o< N, 3 , N 2 A (a) Figure 4.18. (a) Single loop system containing n on-off elements; (b) c h a r a c t e r i s t i c s of N . . H„<s) I - * o i C O L For the purpose of inves t iga t ing the possible periods of o s c i l l a t i o n s , s e l f or f o r c e d , we decompose the above system into n sub—systems or uni t systems as shown i n Figure 4.19. The 50 HL(s) x L + i „ ' V . ( i = i , a , ...5YI-I) Hn(s) Vn Tt -f N, if i —> e.cr) Figure 4 .19. Decomposition of system i n F i g . 4.18 into .n sub-systems. phase c h a r a c t e r i s t i c associated with the system containing the on-off element N. i s denoted by © . ( T ) . I J I Let ' ©*(T) 4 ; | O.(T) - [ i = 1 2 T 1 ] 2T (4.15) n The quanti t ies £ 0 . (T) and ©*(T) w i l l be refer red to as the t o t a l i= l 1 phase c h a r a c t e r i s t i c and the reduced phase c h a r a c t e r i s t i c respec-t i v e l y of the open-loop system (opened at any connection between and E L ( s ) ) . C l e a r l y , the closed-loop system w i l l exhibi t simple symmetric o s c i l l a t i o n s with h a l f - p e r i o d T i f the reduced phase c h a r a c t e r i s t i c i s equal to zero, that i s , ©*(T) = o, (4.16) and i f [ © . ( T ) + kT] = (-1)1 0 1 ( i = 1, . . . , n ; (4.17) *1 [°i< T ) + k T ] ( ~ 1 ) k >° J k = o, - 1 , . . . ) are the only switching conditions s a t i s f i e d i n the separate subsystems. Equations (4.16) and (4.17) are the conditions 51 required for the existence of per iodic o s c i l l a t i o n s i n a s i n g l e -loop system containing n on-off elements. In the simplest case where n = 1, i . e . the s ingle - loop system contains only one on-off element, the conditions for the existence of per iodic o s c i l l a t i o n s reduce simply to the f a m i l i a r expressions *-(kT) = ( - D k x , I I o l X l ( k T ) ( - l ) k > 0 (k = o, - 1 , . . . ) (4.18) In the more general case of multi loop systems the require conditions fo l low n a t u r a l l y from the above. Suppose that the system under consideration has S. loops, where the mth(m = 1, 2 , . . . loop contains an a r b i t r a r y number n of on-off elements. Some or m a l l of these loops may have elements i n common. Furthermore, assume that a l l the on-off elements are without dead zone. Let x.. m be the input to the i t h nonlinear element ( i = 1, 2, nm) I ,  i n the mth loop (m = 1, 2, . . . , £ ) . Ve consider each loop i n t u r n . Let ©*(T) be the reduced phase c h a r a c t e r i s t i c of the mth m open loop . Then the multi loop system w i l l exhibi t simple symmetric o s c i l l a t i o n s with h a l f - p e r i o d T i f the reduced phase c h a r a c t e r i s t i c s of a l l the loops are simultaneously zero, that i s G*(T) = o , (m = 1, 2, I ) (4.19) and i f the proper switching instants and switching di rec t ions are also sa t i s f ied? 52 x< ™ m ( T ) + k T l = ( " l ) k x . - ; m l ( i = 1, 2, n ; itm |_ i ,  J oi ,m m ra = 1, 2, £ s i$m L l,jn J K = o, — if •*•) (4.20) where ©. (T) i s the phase c h a r a c t e r i s t i c associated with the a. >m subsystem containing the i t h on-off element i n the mth loop, and x . i s r e l a t e d to the hysteresis width of t h i s on-off element, oi ,m Another way of s ta t ing the conditions expressed by Eqs. (4.19) and (4.20) i s that the existence conditions expressed by Eqs. (4.16) and (4.17) must hold simultaneously for each loop of the multi loop system. P A R T I I O N S E L F A N D F O R C E D O S C I L L A T I O N S I N O N - O F F F E E D B A C K C O N T R O L S Y S T E M S 53 5. SINGLE-LOOP SYSTEM CONTAINING AN ARBITRARY NUMBER OP ON-OFF ELEMENTS Let us f i r s t consider the system shown i n Figure 4.18, that i s a single loop system containing n on-off elements without dead zone, and investigate the possible half-periods of self and forced o s c i l l a t i o n s . S e l f — o s c i l l a t i o n s A simple graphical procedure for ascertaining the possible half-periods of s e l f o s c i l l a t i o n i s as follows* ( i ) the phase characteristics0.(T) VS T of the individual sub-systems ( i = 1, 2, n) are f i r s t evaluated; ( i i ) n the t o t a l phase c h a r a c t e r i s t i c , £ vs T, i s then i=l 1 plotted; ( i i i ) f i n a l l y , we apply the condition (4.16) that the reduced phase char a c t e r i s t i c must equal zero; thus, the values of T at which the straight l i n e s 0 = 2kT, (k = o, 1, 2, ...) intersect the t o t a l phase char a c t e r i s t i c curve give the possible half-periods of s e l f o s c i l l a t i o n . The construction i s shown i n Figure 5.1. .Sft(T)J Figure 5.1. Graphical procedure for determining possible half—periods of s e l f o s c i l l a t i o n s . T^, T^, T^, •*« represent the possible half-periods of s e l f o s c i l l a t i o n . 54 Forced o s c i l l a t i o n s Let us assume that the input f ( t ) to the system, shown i n Figure 4.18, i s simple symmetric with h a l f - p e r i o d equal to T q , i . e . f ( t ) = - f ( t + T Q ) . R e s t r i c t i n g ourselves to the consideration of simple symmetric o s c i l l a t i o n s , and excluding the case of sub-harmonics, the system var iables y i ( i = 1> •••• n) , v n w i l l eventually a l l be per iodic with h a l f - p e r i o d T Q . Consequently, the phase c h a r a c t e r i s t i c s of the i n d i v i d u a l u n i t systems e i + l ^ o 5 * ( i = l f 2> n _ l ) which are real non-negative q u a n t i t i e s , are known (or can be calculated by the methods presented e a r l i e r ) . The only var iable at our disposal i s 0 ^ ( T Q ) which i s a funct ion both of the " a m p l i -tude" of f ( t ) and of the "phase s h i f t " T of f ( t ) r e l a t i v e to v n ( t ) . Let us write f ( t ) = Af ( t - T ) o A = max J f ( t ) j max I f (t - T) I = 1 o ' , o < T < 2 T Q (5.1). Thus, given A and f Q ( t ) , the sought—for quantity i s the value (or values) of T that w i l l permit forced o s c i l l a t i o n s to occur i n the 55 system. The procedure for determining the values of T that permit forced o s c i l l a t i o n s to occur i s as follows? n ( i ) The t o t a l phase c h a r a c t e r i s t i c £ 0 . ( T ) between points i=2 1 A and B ( in Figure 4.18a) i s computed. ( i i ) The reduced phase c h a r a c t e r i s t i c between A and B, namely n IT .§_©, (T ) 1 Q * <To> = 2 A ( T o ) " L 1 2T ° J 2 T o ( 5 ' 2 ) i=2 o i s evaluated. For forced o s c i l l a t i o n s to occur, the reduced phase c h a r a c t e r i s t i c of the entire loop must equal zero . Let us define the complementary phase charac ter i s t i c of ©*(T ), with respect to 2T , as 2T - ©*(T ) , for ©*(T ) >o O 0 0 . o , for © * ( T Q ) = o Then forced o s c i l l a t i o n s may occur i f the phase character-i s t i c of the f i r s t sub-system (between B and A ) i s equal to the complementary phase c h a r a c t e r i s t i c ©* between A and B % that i s , V T o > = 9 c ( T o>-( i i i ) The phase c h a r a c t e r i s t i c © q ^ ^ ) i s a funct ion of T and w i l l be denoted by ©, (T T") : i t i s determined as the smallest JL 0 y non—negative root of the equation X l ( t , r ) ] T = T = A o f H - r ) . - v n ( t ) ] T = T = x o l o o (5.3) 56 (iv) The values of T s a t i s f y i n g © x ( T o T) = © * ( T o ) give r i se to forced o s c i l l a t i o n s , provided that the only switching conditions are x, ["©*•(T ) + kT I = ( - l ) k x , 1 L c o o J o l |~©*(T ) + kT 1 ( - l ) k > o 1 L c o oJ ' (k — o, -I,•••) (5.4) and ( i = 2» 3 , . . . , n ; x. TO. (T ) + kT 1 = ( - l ) k x . ' i L i o oJ ' o i I * i [ W + k T o l ( ~ 1 ) k > ° ' k =• ° » * ! » • • • ) (5.5) and these can be v e r i f i e d from plots of x^(t) and x^(t) as functions of t . The construction corresponding to steps ( i i i ) and ( iv) above i s shown i n Figure 5.2. e,CT0) = e,(T0JT) »-T Figure 5 » 2 . On the determination of possible values of T that permit the occurrence of forced o s c i l l a t i o n s o Another method for determining the values of T that may permit forced o s c i l l a t i o n s u t i l i z e s the Tsypkin approach i n the l a t t e r part of the procedure. The steps i n the procedure are as follows? 57 ( i ) As above, the reduced phase c h a r a c t e r i s t i c © * ( T ) between o points A and B ( in Figure 4.18a) i s f i r s t computed, and then the complementary phase c h a r a c t e r i s t i c © * ( T Q ) i s found, ( i i ) For forced o s c i l l a t i o n s to occur at a p a r t i c u l a r value of T , two conditions must be s a t i s f i e d : f i r s t , X l ( t ) ] t = ©*(T ) = A f o ( t " T ) - V n ( t ) ] t = ©*(T ) C O C O (5.6) = x o l for the proper switching i n s t a n t s ; then x l ( t ) ] t = ©*(T ) > 0 c o' for the proper switching d i r e c t i o n s . The Tsypkin plane J = i k + jx can be used to represent these two conditions graphica l ly i n the f o l l o w i n g manner. ( i i i ) The contributions - v n [ © £ ( T Q ) ] and - v n [ © j ( T Q ) ] to x± and x, r e s p e c t i v e l y , are f i r s t p lot ted on the 3—plane; these are denoted as coordinates (a ,b) , as shown i n Figure 5.3 . (iv) The remaining contributions A f Q [G*(T Q) ~ T ~ \ a n d A f Q ^ © * ( T Q ) — TJ to x^ and x^, . r espec t ive ly , are added to those of part ( i i i ) . These contr ibut ions , however, are functions of T and therefore , as T var ies between o and 2 T q , they give r i s e to a curve ^ E^ C^ O^ 'T] ' c a ^ ^ e d the hodograph of f [©J(T0)]» about the point (a ,b ) , where 3i[e*(To),T] = f (t _T) + ( t _ T )] _ C O (5.7) 58 (v) To s a t i s f y the condit ion of the proper switching i n s t a n t , the hodograph <3? must in tersec t the s t ra ight l i n e j x ^ * A l s o , to obtain the proper switching di rec t ions [ j ^ (^0 il^0» the points of i n t e r s e c t i o n must l i e i n the r i g h t - h a l f CT—plane. Furthermore! the values of T at these points of i n t e r s e c t i o n ( of 3* with jx Q ^) w i l l allow forced o s c i l l a -t ions to occur, provided that there are no a d d i t i o n a l commutations i n the i n t e r v a l ©*(T ) < t < © * ( T ) + T . c o c o o Im Z J - pi a Hociograpn o"f using O'as origin Figure 5 . 3 . On the determination of possible values of T that permit forced o s c i l l a t i o n s . It i s obvious from E q . (5*7) that the non-negative r e a l quantity A , c a l l e d the "amplitude" of f ( t ) , i s a s ca le - fac tor f o r the hodograph of 3f|©*(T O), T J g that i s , the r e l a t i v e shape of t h i s hodograph remains the same for various values of A ? and an increase or decrease i n the value of A merely magnifies or contracts the curve of <ffr[©*( T q ) ,TJ about 0 T as o r i g i n . Hence the value of A, i n general , determines the number of values of T at which forced o s c i l l a t i o n s may occur. The e f f e c t of varying A i s i l l u s t r a t e d i n Figures 5.4 (a) to ( f ) . In Figure 5.4 (a) the value of A i s too small to allow 59 forced o s c i l l a t i o n s with h a l f - p e r i o d equal to T q . In th is case sub-harmonic o s c i l l a t i o n s are p o s s i b l e . As A i s increased to the c r i t i c a l value A - ^ c r the l i n e J X q ^ becomes tangent to the hodograph of <3« ( T Q ) fTJ i n the r i g h t - h a l f J - p l a n e . A further increase i n A brings us to Pigure 5.4 (c) for which forced o s c i l l a t i o n s may occur at T= (for the hodograph as drawn). For very large values of A forced o s c i l l a t i o n s w i l l be possible at the one value of T, namely T = i n Figure 5.4 (d) . In Figures 5.4 (e) and ( f ) , 0* l i e s i n the l e f t - h a l f J - p l a n e . At A = A 2cr the hodograph of ^ [ ^ ( ^ Q ) 7"J passes through the i n t e r s e c t i o n of the j Im2T—axis and jx .. whereas a further increase i n A may allow o l , forced o s c i l l a t i o n s at the one value as shown i n Figure 5.4 (f) . For A = A ^ c r j w e have from Figure 5.4 (b)s Im ^ [ 0 ; ( T o ) , T o l ] b - x ol By using E q . (5.7) the above equal i ty can be wri t ten as l c r = | fo [ eJ<V - T o l ] | S i m i l a r l y j from Figure 5.4 (e) we have M ^ V ' To2.1 I = V a 2 + ( b - x n 1 ) 2 (5.8) o l ' and by using E q . (5.7) we obtain 2cr . a 2 + (b - x o l ) 2 -2- f (O*(T ) - T J j + If (O*(T ) - r J J _ •% o c o o2 - J L o c o o2 J 1/2 (5.9) 60 with A<A Icr (a) jlm J Re J Cb) w i t h A=Aj c r ,1m J .0' Re? vyitb A>A l c r CO ilm J "* I^ C^elC-r^ rl ' with A»A lcr Cd) $01 I m J To2 with A-A 2cr i I m j f V7* + / 1 d 1 Retf w'.+h A>A2cr Remarks? 0* = (a,b) : hodographs ^£(©*(T ) j-T) drawn about 0' c o as o r i g i n . Figure 5.4. Influence of A upon the number of values of T t h a t may permit forced o s c i l l a t i o n s . 61 Obviously, for A > A l c r or * h e d e s i r e < * v a l u e s o f T can be determined from the equal i ty b - x I f o [ W -T]l= A "' < 5 - 1 0 ) or, by making use of Equation (5.8) and (5*9), t h i s equal i ty becomes I f o [eJ<To> _ T] I = " A ^ I f o IW ~Tol] I «'•») f o r A > A 1 C J . and P 2 -N1/2 • [ ' . ^ V - To2>] " l l - ] (5-l2) A 2cr for A >A_ 2cr , r e s p e c t i v e l y . In the specia l case where f Q ( t ) = s i n « t , we have f (t - T ) = s i n « ( t - T ) , £ f Q ( t - T ) = cos » ( t - T ) , so that the hodograph of (TQ) » r ] i s given by $ [ ° S ( T o ) < T ] = A [ C ° S <°o [ G J ( T o ) " T ] + j S i n "o [ 0 ? ( T o } - T ] ] jto r© * ( T ) - r l = A e 0 L c 0 J (5.13) 62 where «>O = T T / T q . Hence the hodograph of ^ | © £ ( T O ) rj i s a c i r c l e of radius equal to A. By making use of E q . (5.13), e q u a l i t i e s (5.11) and (5.12) become s i n <o fo*(T ) - r l I = -7 0 L c 0 J I A cr s i n « o [ e * ( T o ) - f o l ] | (5.14) for A >A, , l c r ' and s i n « |~0*(T ) - r l I = iSZIZ 0 l_ c o J I A (5.15) for A >A 2cr, r e s p e c t i v e l y . 63 6. DOUBLE-LOOP SYSTEM CONTAINING AN ARBITRARY NUMBER OP ON-OPF ELEMENTS Ve mentioned e a r l i e r that i n more complex systems the a p p l i c a t i o n of the Tsypkin method to the determination of the possible periods of simple symmetric o s c i l l a t i o n s becomes very cumbersome. In th is chapter we f i r s t show that the Tsypkin approach can be used i n the study of the double-loop system i n which each loop contains one on-off element. This p a r t i c u l a r case points out the d i f f i c u l t i e s that would be encountered i n any contemplated extension of the Tsypkin method to the study of systems with three or more on-off elements. We then indicate how the possible periods of simple symmetric o s c i l l a t i o n s i n a double—loop system, containing an arbi t ra ry number of on-off elements, may be determined by the phase c h a r a c t e r i s t i c method. 6.1 APPLICATION OP TSYPKIN'S METHOD TO A DOUBLE-LOOP SYSTEM WITH TWO ON-OFF ELEMENTS •Ht)-t-x x, The system under consideration i s shown i n Figure 6 . 1 . 2i N , / — ^2. H?fe) v^Ct) H 3 « -*oL \ X . 'x, OL * Cb) Ca) Figure 6.1 (a) Double-loop system containing two on-off elements, (b) Charac ter i s t i cs of N-^  and N,>. In the case of simple symmetric o s c i l l a t i o n s , the outputs of N^ and N 2 are, i n general , as shown i n Figure 6 . 2 . In f a c t , the expressions for y-^(t) and y 2 ( t ) are 64 -2T - T M , O - M , E T Ca) ocT cpH-iyr - M a . Cb) Eigure 6*2. (a) and (b) Outputs of and Ng. y x ( t ) = 2MX I ( - l ) k u(t - k T ) , for - «* < t <(n+l )T k=-eo y 9 ( t ) = 2M, f ( - l ) k u f t - (a + k ) T l , for - o o <t<(n+l+a)T, * ^ k=-oo L J where i t i s assumed that a > o and o < a < 2. From the resul ts of Chapter 3 the response of the l i n e a r part H^(s) i s 2M 1 C l 0 r C 2 1 $ H l ( s ) ( - l ) n e - s n T . s t sT 1 + e ( n T < t < ( n + 1)T, n = o, - 1 , - 2 , , . . ) e s x d s + K± (6.1) where i s a constant re la ted to the i n i t i a l condi t ions . S i m i l a r l y , the outputs of the other l i n e a r parts are given by T i ( t ) 1^2 § M i l ( - D e - 5 ' " " ' 1 e s t d s 2id „ J „ s , . sT e as + ^ C l o r C 2 1 + e K i (6.2) for (a + n ) T < t < ( a + n + l ) T , i = 2, 3, 4; n = o , - 1 , - 2 , . . . , where 65 L 2 ( s ) = H 2 (s ) L 3 ( s ) = H 2 ( s ) H 3 ( s ) L 4 ( s ) = H 2 ( s ) H 4 ( s ) , and K are constants related to the i n i t i a l condi t ions . 1 The conditions for symmetric o s c i l l a t i o n s of the above type are and x l = x o l ' x i ( ° ) > 0 x 1 ( t ) > - X q 1 for o < t < T x 2 (aT) = X q 2 , i 2 ( a T ) > o 1 x 2 ( t ) > - x Q 2 for aT <t <(a + l ) T Self O s c i l l a t i o n s (6.3) (6.4) Following Tsypkin's method, we introduce the Tsypkin l o c i J ^ o c . T ) = | X l ( o ) + j x1(o) J 2 ( a , T ) = | x 2 (aT) + j x 2 (aT) (6.5) Using a as the parameter (o<oc<2) and T as the v a r i a b l e , we construct these l o c i as shown i n Figure 6.3. The straight l i n e s J'X q ^ and J X q 2 are next inserted on the J ^ - and ^ " " P l & n e s » respec-t i v e l y . The points a^, b^ , c ^ , . . . of i n t e r s e c t i o n of the •3^(oc,T) l o c i with the s t raight l i n e j x ^ i n the f i r s t quadrant of the 3^— plane correspond to pairs of values (oc,T) that s a t i s f y the conditions x^(o) = X Q ^ , X^(O)>O: s i m i l a r l y , the points 66 i^i Coc.T)-plane X ^ O C ^ T ) - plane Figure 6.3. The Tsypkin l o c i ^ ( o c , T ) , J 2 ( a , T ) . a 2 , b 2 , c 2 , , . . i n the f i r s t quadrant of the J 2 - p l a n e correspond to pairs of values (oc,T) that s a t i s f y the conditions x 2(aT) = X q 2 , x 2(aT) > o * We now plot these points of i n t e r s e c t i o n as curves of a = f-^T) corresponding to the points a^* b ^ , c ^ , . . . of the plane* and a = f~(T) corresponding to the points 1 2. | a 2 , b 2 > c 2 * . « » of the O^-plane. Any pai r of values (<x,T) at the i n t e r s e c t i o n of the curves f-^T) and f 2(T)> such as (a*, T*) shown i n Figure 6.4, may give r i s e to s e l f o s c i l l a t i o n s . C C Figure 6.4. Curves of a = f (T) and a = f « ( T ) . 67 Forced o s c i l l a t i o n s . Let the input to the system, f ( t ) , be per iodic with half -period equal to T Q = The conditions for the existence of forced o s c i l l a t i o n s are again expressed by equations (6.3) and (6.4) with T set equal to T q , but now, instead of x x ( t ) = - v 4 ( t ) , we have X ]_(t) = f ( t ) - v 4 ( t ) . A l s o , instead of a and T the sought-for quanti t ies are a and <f> where <p i s the phase s h i f t of f ( t ) r e l a t i v e to some a r b i t r a r y reference phase ^>q. For convenience, we write or f (tt t) = A f (tt t - d>) o 0 0 0 ~ f ( t ) = A Q f o ( t - T) where T = <^>/<o , A = max o f ( t ) | and max f (t) = 1. From the curve of a = f ^(T!) w e l 0 0 8 - ^ 6 "the value a = a Q at which T i s equal to T q . ¥e next inser t the point 0 ' = - ^2. v ( 0) _ v ( 0 ) 1 TC 4 ' 4 a = a , o ' T = T o on the «J^~plane . With the point 0 ' as o r i g i n , we construct the hodograph of as T v a r i e s from 0 to 2 T q i n c l u s i v e l y , as shown i n Figure 6.5. 68 Hodograph o"f CT) w i t h O' Rs J, ~-plane The value(s) of T corresponding to the i n t e r s e c t i o n of the 3^(T) l o c i with the s traight l i n e jx Q ^ and l y i n g i n the f i r s t quadrant of the CT^-plane, together with the value of a determined above, o ' are the sought-for value (s) of (oc,T) which may allow the occurrence of forced o s c i l l a t i o n s . Figure 6,5. On the determina-t i o n of the values of T that permit forced o s c i l l a t i o n s . The conditions x, (t) >-x for o <t <T and x (t)">-x 0 for a T < t <(a + 1 )T must be v e r i f i e d . o o o • -o In p r i n c i p l e , the Tsypkin approach can be applied to the study of the periods of o s c i l l a t i o n s i n a double-loop system containing an a r b i t r a r y number of on—off elements. But the extension to cover the cases of more than two on—off elements i s d e f i n i t e l y awkward. Such complicated cases are best solved by the method of the phase c h a r a c t e r i s t i c . 6.2 APPLICATION OP THE PHASE CHARACTERISTIC METHOD TO A DOUBLE-LOOP SYSTEM CONTAINING AN ARBITRARY NUMBER OP ON-OFF ELEMENTS Consider the double-loop system containing an a r b i t r a r y number of on—off elements as shown i n Figure 6.6 (a) . Assume •fct) *> V4 * 2 . * n , n.+i A --Xoi t * >-X . Cb) 'n<ti N, V H. Ca) H Figure 6.6 (a) Double-loop system containing an a r b i t r a r y number of on-off elements; (b) Character is t ic of i t h on-off element. Unit No. I 1 H, ryH i _ . Input Un i t No.in.-t-2. 'n,-H N . J r U n i t No. r » 2 + ) l _ i T j Uni t " No. T n 3 + i H, L Unit No .n, Unrr i N O . Ylj+I j r u n 3 1 ^"s l V 3 ^ — ' A f" U n i t No. 1 l _ V I N, U n i t No.a 1 1 i N 2 1 1 I -1 I U n i t 1 N o . m Figure 6.7. Open-loop system as a composition of uni t systems. 70 that the c h a r a c t e r i s t i c of the i t h on-off element has the form shown i n Figure 6.6 (b), i . e . with or without hysteresis so that x . > o « 01 — S e l f — o s c i l l a t i o n s . In order to determine the possible periods of s e l f o s c i l l a t i o n , we open the system i n Figure 6.6 (a) at the point 0. The r e s u l t i n g open-loop system can be regarded as a composi-t i o n of uni t systems as shown i n Figure 6.7. The i t h uni t (or sub—system) consists of the i t h on-off element and the l i n e a r system or systems immediately preceding i t . Let 0^(T) be the phase c h a r a c t e r i s t i c of the i t h sub-system. The functions ©^(T) for i = 1, 2, n^ except for i = n^ + 1 (we are assuming a t o t a l of n^ on-off elements i n our system) are a l l known, or can be calculated by the methods indicated i n Chapter 4, We now evaluate the t o t a l phase c h a r a c t e r i s t i c s , © ^ a n d , between the points 0 and A and between 0 and B, r e s p e c t i v e l y , i n Figures 6.6 and 6.7: 3 n^ + 2 n-^  + 3 n 2 + 1 3 ® i = ° T 1 , o + © . ,+ . . .+ ©„ + ©„ , , + . . . + © „ } (6.6) 1 n^ + .2 n^ + 5 xv^ n-j + 1 n^ + 0.. +...+ Q 1 n. where, for s i m p l i c i t y , we have wri t ten 0^ for © ^ ( T ) . Next we determine the reduced phase c h a r a c t e r i s t i c s 71 © * 3 = © 3 " fl^] 2T ©*! = ©! " I ^ J 2 T (6.7) Sketches of possible plots of © , © * ^ > ©3* ©*3 a s functions of T are given i n Figure 6.8. Observe that O<(H^< 2T ( i = 1 ,3) . Figure 6.8. Sketches of possible plots of ®!» © V © 3 > ©* 3-Consider now the n^ + 1 th sub-system. Figure 6.9 i l l u s t r a t e s the general forms of the inputs and output of th is uni t system. Since the functions (H)*^ and are known, we can therefore compute ©*, 2M. (t) = 1 H n . ( s ) ( - l ) V < k + T 1 ) 1 B t ~ ^ - 7p e ds 2 ^ C 1 o r C 2 S 1 + e s T (6.8) ( i = 1,3| © J + k T < t < © * + (k + 1)T; k = o, - 1 , . . . ) . 72 Consequently, \ + l ( t ) = v l ( t ) " v 3 ( t ) Hn3 h -M. H "rt,+i —>-Figure 6.9. Relationships i n the n^ + 1 th sub—system. can be determined for any time i n t e r v a l . In p a r t i c u l a r , we can determine the time @ * = © * ( T ) , o ^ © * < 2 T , at which the output y . >r (t) of th is uni t system f i r s t jumps from -M . , to + ^ i n f a c t , @ * i s the least p o s i t i v e root of the equation x , , (t) = x . The quantity © * = © * (T) n-^  T x o 9 u^ *r J-i s the reduced phase c h a r a c t e r i s t i c of the entire open—loop system i n Figure 6.7. Hence the values of T f o r which © * ( T ) = o are the possible h a l f - p e r i o d s of s e l f o s c i l l a t i o n of the closed-loop system. The method described above automatically guarantees that the condi t ion expressed by Eqs. (4.19) and ( 4 » 2 0 ) are s a t i s f i e d ? that i s , that the reduced phase c h a r a c t e r i s t i c of each loop i s zero . 73 Forced o s c i l l a t i o n s The procedure for the determination of the conditions that permit forced o s c i l l a t i o n s i s as f o l l o w s : Let T = T t / « > o be the h a l f - p e r i o d of forced o s c i l l a t i o n . The t o t a l phase c h a r a c t e r i s t i c between 0 and B ( in Figure 6.7), minus the contr ibut ion due to Unit No- 1, i s denoted by © 2 ( T ) and the corresponding reduced phase c h a r a c t e r i s t i c by © * ( T ) so that ® 2 ( T ) = © X ( T ) - 0 X (T) and ©$(T) = ©,(T) - IL—|T—J 2 T (6.9) The reduced phase c h a r a c t e r i s t i c between 0 and A i s denoted by © * ( T ) . For per iodic phenomena of h a l f - p e r i o d T q > the quanti t ies © ^ ^ o ^ a n ( * © f ^ 0 ^ a r e f i x e d non-negative numbers less that 2T » o Forced o s c i l l a t i o n s of h a l f - p e r i o d T q may occur i f the N element switches over at time t = o and i f the slope of the input to t h i s element i s p o s i t i v e at t h i s i n s t a n t : that i s , x , , (o) = x , i and x , ( o ) > o n + 1 o,n^ + 1 n^ + 1 Because of the reduced phase s h i f t of © * ( T q ) between 0 and A , the input to H at point A i s s h i f t e d to the r ight by n 3 © * ( T q ) r e l a t i v e to the input at point 0. Referr ing to Unit ) 74 No. 1 to which the f o r c i n g funct ion f ( t ) i s a p p l i e d , we l e t f ( t ) = A Q f ( t - T) where T i s the phase s h i f t of f ( t ) r e l a t i v e to the input to H n . 4 The phase c h a r a c t e r i s t i c of Unit No. 1 i s a funct ion of T and i s denoted by 0,(T T ) • Therefore the phase c h a r a c t e r i s t i c l o» between 0 and B i s also a funct ion of T • i t i s determined by <S>l ( T o, T ) = © * ( T O ) + 0 L ( T 0 T ) Thus, r e l a t i v e to the input at 0 , the input to H n i s s h i f t e d to i s the r i g h t by © (T T ) . Consequently, the output of N , x n x + l ( t ) = " V 3 ( t " ® 3 ) + v l ( t ~ ® 1 ) where v (t) and v , ( t ) are the outputs of H and H when 3 ^3 there i s no phase s h i f t of the waveforms between 0 and A and between 0 and B r e s p e c t i v e l y . The conditions that x -. (o) and x , (o) s a t i s f y can be represented on the Tsypkin J q + ^—plane T J n± + 1 ~ % X n 1 + 1 + + 1 F i r s t the contr ibut ion due to -v^ i s p l o t t e d : i t i s the point °' = - \r V - ® 3 ( T o ) : ) " d v 3 ( - ® 3 ( T o n which i s independent of T . Next the contr ibut ion due to -v^ i s 75 added to the point O 1 ; th is contr ibut ion depends on Tand there-fore y i e l d s the curve T ^ i ( T ) V - ® * ( T 0 , r ) ) + J v i ( - ® J ( T O T ) ) with the point 0* as i t s o r i g i n , a s T v a r i e s from 0 to 2T Q. Figure 6.10 shows the "3 , -p lane and the two contributions n^ T J. to k , , and x . , . n^ + 1 n^ + 1 <3*(T) loci with O'as origin n,-H v . —-Figure 6.10. J , , - p l a n e . n^ + 1 The values of T l y i n g i n the f i r s t quadrant of the J " n + -j^-plane and corresponding to the points of i n t e r s e c t i o n of the l o c i of <^(T) with the s t ra ight l i n e J X Q N + ^ § determine the conditions that are necessary for forced o s c i l l a t i o n s . I l l u s t r a t i v e Example of the A p p l i c a t i o n of the Phase Character- i s t i c Concept to the Determination of the Periods of Self  O s c i l l a t i o n s i n a Double-Loop System A double-loop system containing two N elements i s shown i n Figure 6.11. The method ( of solving for the possible half—periods of s e l f o s c i l l a t i o n ) given i n sect ion 6.2 i s used: that i s , the phase c h a r a c t e r i s t i c of the system i s evaluated 76 and the points of i n t e r s e c t i o n with the s t raight l i n e s 0 = 2kT(k o, 1, 2 , . . . ) give the possible ha l f -per iods of s e l f o s c i l l a t i o n s . N. —4 k KCS) N. -H — k, -i S v 3 HaC6) +1 - 1 x Hte) 3 k 3 s Figure 6*11* A double-loop system containing two N elements. As indica ted i n Figure 6.6 and 6.7, the double-loop system i s opened at the point X, and the system i s redrawn as shown i n Figure 6.12* The open-loop system consists of two unit systems, one (unit n o . l ) of the type shown i n Figure 4.13 and Unit No.2 X Unit N o . l r H2Cs) HACs) kj, k 4 X r- • "1 rUs) rL(s) H,(s) +1 1 1 K, — - 1 1 1 J 1_ V. +1 -1 Figure 6*12* Open-loop system of Figure 6.11 showing unit systems. the other (unit no.2) of the type shown i n Figure 6.9. From E q . (4.11), the output of ^ ( s ) H^(s) i s given by 77 V4 (t) = k 2 k 4 |^ -T 2 + (t - § ) + 2T 2 e _ T y ^ ] , o < t < T (6.10) 1+e Let the smallest non—negative value of t for which v^(t) = o be denoted by t * Therefore the phase c h a r a c t e r i s t i c 0 , (T) of o 1 uni t no. 1 i s ' t , i f v . (t ) <o o 4 o ^ ( T ) = J (6.11) . t Q + T , i f v 4 ( t Q ) > o From E q . (4.10) and Figure 6.9, the output v^(t) of H^(s) i s determined by _ ( t + T - t o ) / T v ^ t ) = ± k x | " l -— zfjr ' ] ' ° < t < t 0 (6.12a) 1+e 1 where the plus sign before k^ i s used when v 4 ( t Q ) > o , and the minus sign when V . ( t ) <o: and 4 o v x ( t ) = H & i ^ 1 - ^ _ ° y T j , t Q < t <T (6.12b) 1+e 1 where the minus sign before k ^ i s used when v 4 ( t Q ) > o, and the plus sign when v 4 ( t Q ) < o . The output v,j(t) of H ^ s J l L ^ s ) i s determined by E q . (4.10): - t / T 2 V 3 ( t ) = k 2 k 3 |^ 1 - 2e_T/r j , o < t < T (6.13) 1+e 2 The t i m e © * ( T ) , o < ® * ( T ) <2T, at which the output of uni t no.2 f i r s t jumps from —1 to +1 and at which v^(t) - v 3 ( t ) = o, i s the phase c h a r a c t e r i s t i c of the open-loop system. The values of T 78 for w h i c h © * ( T ) = o are the possible h a l f - p e r i o d s of s e l f o s c i l l a t i o n of the closed-loop system. For reasons of s i m p l i c i t y , the parameters k^, k 2 k 3* a n d 71 are kept f i x e d * the values used are Three d i f f e r e n t values of 7^, 7^ = 0*125, 0.25, and 0»50, are used to i l l u s t r a t e the e f f e c t of the parameter 7^ on the phase c h a r a c t e r i s t i c © * ( T ) of the system. Figure 6.13 shows the p l o t of © * ( T ) vs T f o r the above-mentioned values of k^ , 2^^ 3* 7*2' and also shows the e f fec t of varying T-^. The possible h a l f -periods of s e l f o s c i l l a t i o n are = 1 = 1 T = 0.725 for T-L = 0.125, T = 0.925 for Tn = 0.25, and T = 1.025 for 7-, = 0.50. shown i n Figure 6.12. 80 7. MULTILOOP SYSTEMS In the preceding chapter we presented a method using the phase characteristic concept for the determination of the possible periods of symmetric; o s c i l l a t i o n s i n a double—loop system containing an arbitrary number of on-off elements. This method may also be applied to any multiloop system, containing any number of on-off elements, and i n which a l l the loops can be opened simultaneously by opening the system at one point. If there exists no one point which can open a l l the loops usimultaneously, then an e n t i r e l y new method of attack must be developed. This chapter w i l l be devoted to systems composed of the three types of unit systems shown i n Figure 7 . 1 » Methods of finding the phase characteristic of the basic units designated HCs) N T y p e I -»j HCs) |-^g)^[Nn[->-Type II ^ T y p e i l l Figure 7*1. Basic unit systems under consideration. type I and type II are indicated i n Chapter 4. The manner of describing the phase characteristic patterns of the type III basic unit w i l l now be discussed. If a l l the on—off elements are without a dead zone, then the general forms of the inputs to and output of the type III 81 unit system are as shown i n Figure 7.2 . , Let YT be the phase lag of y.j("k) v i t h respect to y ^ ( t ) . C l e a r l y , as Y varies between the l i m i t s o < Y < 2 we generate the possible s i tuat ions that - T M Mi ST* . % Hi YT =77 N, M, +T> V 7 % 4 f Figure 7.2. Phase c h a r a c t e r i s t i c notations and conventions for the type III uni t system. w i l l occur i n the presence of simple per iodic phenomena with h a l f - p e r i o d T . Let 0^(T ,Y) be the phase c h a r a c t e r i s t i c of the output y^ of r e l a t i v e to the input y^ to, H . ; s i m i l a r l y , G^(T,Y) w i l l denote the phase c h a r a c t e r i s t i c of y^ r e l a t i v e to y . . For any f i x e d value of Y i n o < Y < 2 we can determine J ©?;(T,Y)« Since the phase r e l a t i o n s h i p between y . and y . i s K 1 J given, t h i s means that e£(T ,Y) i s known once 0^(T,Y) has been determined. In f a c t , ' e j ( T , Y ) o j ( T , Y ) = < . e k (T ,Y) - Y t , for ©£(T ,Y) >YT - YT + 2T, for © £ ( T , Y ) < Y T (7.1) 82 Consequentlyj by allowing Y to take on f i x e d values i n the i n t e r v a l o ^ K < 2 we can determine the phase c h a r a c t e r i s t i c s for both © ^ ( T , Y ) and ©jjj.(T,Y) with Y as the parameter. For def ini teness we w i l l use the notation 6^(T,Y"^) to represent the phase c h a r a c t e r i s t i c of y, r e l a t i v e to y. when y . lags y. by YT. K 1 J 1 Having examined the phase re la t ionships i n the type III uni t system, we can now determine the possible periods of s e l f o s c i l l a t i o n for the double-loop system i n Figure 6.6 (a) by the fol lowing new approach. For s e l f o s c i l l a t i o n s of h a l f - p e r i o d T to occur, the reduced phase c h a r a c t e r i s t i c of each loop must be zero s i m u l -taneously. The new approach uses the information concerning the reduced phase c h a r a c t e r i s t i c s of a l l the loops . The system i n Figure 6.6 (a) consists of basic units of type I and one basic uni t of type I I I . (The various basic units are shown i n Figure 6.1.) The phase c h a r a c t e r i s t i c s of the i n d i v i d u a l u n i t s , namely ©^(T) for a l l units except i = n^ + 1 and n 1 n, n » n G / ... i (T ,Y ) and © ^ . , (T,Y ) for values of V * x 3 1 3 i n the range o < Y < 2, are determined by the methods presented i n Chapter 4. With both the inner and outer loops (of Figure 5.6 (a)) open at A and B, the t o t a l phase c h a r a c t e r i s t i c of the inner loop i s 83 n~ 1 3 • • .« (T) i ^ + 2 and that of the outer loop i s (7.2) © 2 ( T , Y ) = n 2 n 4 n l ' °n' + i (« .Vj ' + I Z M 1 ' + g , ° i ( T ) + g e i< T > i ^ + 2 i=n3+l where o < Y < 2TQ . The corresponding reduced phase charac ter i s t i cs are then evaluated: n-©, (T , V)-n © * ( T , Y ) = © i ( T f t ) - { - ^ J 2T, ( i = 1, 2) (7.3) The values of Y and T at which the reduced phase character-i s t i c s ©*(T ,Y) = O ; are now plot ted on a Y-T plane as curves of i ' Y= f i ( T ) , ( i = 1, 2) , as shown i n Figure 7.3. The reduced phase c h a r a c t e r i s t i c s of the two loops are simultaneously zero f o r values of T at the i n t e r s e c t i o n of the f -^T) and f 2 ( T ) curves. These values of T are possible h a l f - p e r i o d s of s e l f •> o s c i l l a t i o n f o r the closed-loop system. yC—Y--F((T) — y / / 1 / / 1 / 1 >— Figure 7.3, Curves of Y= f ^ T ) and Y= f 2 ( T ) 84 Possible periods of self o s c i l l a t i o n i n a more complex system As the multiloop system increases i n complexity, so does the procedure f o r the determination of the possible periods of o s c i l l a t i o n s . Nevertheless, a solution i s possible i n every case provided that we are w i l l i n g to carry out the necessarily increased labor. For i l l u s t r a t i v e purposes we consider the four-loop system as shown i n Figure 7»4, The steps i n the determination of the sought-for values of T are as follows: ( i ) Ve f i r s t decompose the system into unit systems of the types I, II and I I I , ( i i ) The phase characteristics of these unit systems are then evaluated. Let these be denoted by ^ ( T ) for i = 1, 2,..., n g but i ^ nj+1, n 3+l, n 5+l , n, n, n_ n, n_ n 0 n K n 0 e „ ' + l < T ' \ 3 > ' •n'+l<*'\j>' •nj+l<*'\*>' <>n|+l<*.*n*> • 5 7 5 7 Instead of a single value of Y (the quantity Y i s the r e l a t i v e phase s h i f t between the two inputs to a type III unit system), as i n the case of the system of Figure 6.6 (a) with one type III unit system, we now have three values of Y because there are three type III unit sys-tems, Ve therefore proceed thus: ( i i i ) Ve open loops 1, 2, and 3 at A, B, and C, as shown i n Figure 7,4. The t o t a l phase characteristics © n ( T , Y ) , © 2 ( T , Y ^ N , 11-^. H, X , * n . H, Loop 4 Loop I "3 IN -xoi X^  Characteristic, erf i+h on-off element ( i s i , 2 , . . . , n g ; -n, < n 2 < • • • < n 8 ) N, H, N, H, Loop 2. N. V 1 H_. — -X N . H. 5L l - O O p 3 ^ H. e 7.4. Pour-loop system containing an a r b i t r a r y number of on-off elements. 86 and ©j(T,V) of the open loops 1, 2, and 3, respectively, are determined, with the input phase s h i f t variable Y (b <Y< 2) as a parameter i n each case* n n n3 1 3 i =n-^  +2 n . ©_(T ,Y) = ©* 5 +,(T , Y * 2 ) + * * i ( T ) ^ n 3+i n 5 i = n 3 + 2 x n„ © O ( T , Y ) = QI\AT,)L4) + I 7 © (T) n 5 + i n ? i = n 2 l 5 (7.4) (iv) Prom these we obtain the reduced phase characteris-t i c f o r loops 1, 2, and 3* i r© (T , Y ) - n ©*(T ,Y) = © ^ (T,Y) - [ 1 2 T J 2T , ( i = 1, 2, 3) (7.5) If we now open loop 4 at D and close loops 1, 2, and 3, then the values of T corresponding to the zeros of ©*(T,Y)» ( i = 1» 2, 3), may permit periodic o s c i l l a t i o n s to occur i n loops 1, 2, and 3 simultaneously. The problem remaining i s to determine from these values of T those that w i l l allow periodic o s c i l l a t i o n s to occur simultaneously i n a l l loops when loop 4 i s closed. We solve this problem as follows: The pairs of values (Y,T) corresponding to the zeros of the reduced phase characteristics ©*(T ,Y) of loops 1, 2, and 3 are plotted as curves of Y= f ^ ( T ) , ( i = 1, 2, 3), as shown i n Figure 7.5. 87 2. •• An i n t e r v a l of f over which Y= t'^(T) exists simultaneously for i = 1, 2, 3» Figure 7 . 5 » Curves of Y = f ^ T ) for i = 1, 2, 3 showing range of possible h a l f - p e r i o d s of o s c i l l a t i o n s i n loops 1, 2, and 3. Ve consider only those i n t e r v a l s of T ( in Figure 7 » 5 ) for which a l l f^(T) ex is t simultaneously; th is means that on any v e r t i c a l l i n e through the Y-T p l o t , there exis ts a t r i p l e t of Y that determine a value of T such that o s c i l l a t i o n s are possible i n loops 1, 2, and 3. However, i f at a p a r t i c u l a r value T , the quanti t ies Y, = f (T ), Y 0 = f~(T ) exist but Y.j = f ^ ( T Q ) does not, then o s c i l l a t i o n s of half—period T Q are possible i n loops 1 and 2 but not i n loop 3. ) Sequences of values of T, say , l ^ , . . . , T m , c o v e r i n g the i n t e r v a l s of T i n which f^(T) exis t simultaneously for i = 1, 2, 3 are se lec ted . At each value of T . ( i = m) we n. n. n. r e a d o f f the c o r r e s p o n d i n g t r i p l e t Y " * " , Y 2 , a n d Y ^ from Figure 7«5» From the s e t o f phase c h a r a c t e r i s t i c ' s o b t a i n e d 88 i n step ( i i ) we f i n d the values of the phase charac ter i s t i cs n n, n_ n_ of three type III unitss namely ©„ , , ( T , V ), ©„ M , ( T , Y ) r, 1 3 3 5 and © ^ , ( T , Y ) for the above T. and t r i p l e t s of Y . n^ + 1 ' n ^ ' I ^ ( v i i ) Ve now open loop 4 at D and form the t o t a l phase character-i s t i c of t h i s loop for the above !L\ and t r i p l e t s of Y t ®4<Ti> + , V l " ' . ^ * ° » 5 « ( , » ^ ) n 2 =4 "6 n 8 + E V T i ' + z ; W + E V V + E W k=n1+2 k=n3+2 k=n^+2 k=n^+2 (7.6) At t h i s stage we know that o s c i l l a t i o n s of h a l f -period T (where T belongs to the above-chosen i n t e r v a l s ) are possible i n loops 1, 2, and 3. Prom among these values of T, we f i n d those that w i l l make the reduced phase c h a r a c t e r i s t i c © ^ ( T ) of loop 4 equal to zero; s e l f o s c i l l a -t ions may occur at such values of T for which © £ ( T ) = o, when loop 4 i s c losed. Forced o s c i l l a t i o n s The possible periods of forced o s c i l l a t i o n s are determined i n p r e c i s e l y the same manner as the e a r l i e r indicated methods. More complicated systems may be studied by the above-mentioned method or s l i g h t modifications of i t . P A R T I I I O N - O F F E L E M E N T S W I T H P R O -P O R T I O N A L B A N D 8* ON-OFF ELEMENTS VITH PROPORTIONAL BAND In Parts I and II we considered ideal on-off elements. Let us now turn our at tention to on-off elements with a propor-t i o n a l band. Examples of some of the c h a r a c t e r i s t i c s of such elements are shown i n Figure 8.1 . (c) (d) Figure 8.1* Charac ter i s t i cs of some on-off elements with proport ional band. (a) Vithout hysteresis and dead zone. (b) V i t h hysteresis and without dead zone. (c) Vithout hysteresis and with dead zone. (d) V i t h hysteresis and with dead zone. 8.1 TRANSIENT RESPONSE OF A SINGLE-LOOP SISTEM CONTAINING ONE ON-OFF ELEMENT VITH PROPORTIONAL BAND The system under consideration i s shown i n Figure 8.2. 90 -PC*) xCt) _N_ 1ST —;> 0K+) S l o p e s A Figure 8*2* Block diagram of s ingle - loop system contain-ing one on-off element with proport ional band. Suppose that the error s ignal x(t) remains i n the l i n e a r regions for a l l times t i n the i n t e r v a l s T < t < T + h ,,,(11 = 0,1, 2,...) n— — n n + 1 where, without loss of g e n e r a l i t y , we take T q = o, and stays i n the saturat ion regions for the remaining i n t e r v a l s \ + h n + x < t < T n + 1 , (n = o, 1, 2,...) Let the transform of the i n i t i a l conditions referred to the out-put of the l i n e a r part H(s) be denoted by V Q ( s ) . Then, an e q u i -valent system, shown i n Figure 8.3, consists of a number of samplers operating i n p a r a l l e l ; the number of samplers depends on the number of times the error s ignal passes through the l i n e a r region of N. The samplers that correspond to operation i n the l i n e a r regions have inputs denoted by X n ( s ) , where X n ( s ) = X(s) for n = o, 1, 2, . . . , ; t h e sampler with input X n ( s ) i s closed during the i n t e r v a l T < t < T + h , and open otherwise. The quanti t ies X (s) are n — — n n+r ^ np v ' the p-transforms of ^n(s)*"'"? The sampler with input —M is closed during the saturat ion i n t e r v a l s and open otherwise; 91 A n p(s) i s the p—transform of the output of t h i s sampler. l T " 2 Figure 8 .3 . System equivalent to that of Figure 8.2. Let us now evaluate the response of the above system for the d i f f e r e n t time i n t e r v a l s (T , T + h _,_ , ) and (T + h , n n n + 1 n n + 1' T n i^» ( n = °> 1» 2 , . . . ) . Figure 8.4 gives the equivalent system for the time i n t e r v a l o < t < h ^ . (Note that T q = o.) The input X q ( S ) to the sampler i s given by X Q ( s ) = F Q (s ) - C o (s) that i s , = F q ( S ) - X o ( s )AH(s ) ; F D ( s ) X o ( s ) = 1 + AH(s) ' o < t < h r 92 V0C9> Fes) t F0&t X„Cs) Xp/O V 0 CS) AH(s) VCs) — v -Figure 8 . 4 , Equivalent system for the i n t e r v a l o < t < h , , Now at t = h^ the sampler i s opened and the input to H(s) i s equal to zero for t > h ^ , i . e . we can define ( x Q ( t ) , for o < t < h 1 * o p ( t > = < k o , for t > h 1 Therefore op (t) = x Q ( t ) [u(t) - u(t - 1^)] , for t > o . Using the complex convolution i n t e g r a l we get the Laplace t rans-form of x ( t ) j Jr X (s) = ^ \ X (V) op 2-rc.T *J o - ( s -v ) h x i r j s - y dV where C i s a contour enclosing a l l the poles of X Q ( v ) or j- - ( s - v ) ^ - . |_1 - e J / ( s - V ) i n a mathematically p o s i t i v e or negative sense r e s p e c t i v e l y . Using the p-transform notation of the theory 13 of sampled-data systems, namely 93 T +h -(s-v)T -(s-V>) (T +h , ) n n+1 ,_ _ , A l p x / n n n+1 , . , e - e dy PT ^ [E(s)] = IE(J)  n C (8.1) we get the transform of the component of the output from the f i r s t pulses C o ( s ) = X o p ( s ) A H ( s ) =AH(s) P J [1 ' ^ j , ) ] , t > o . Consequently, the output of the system i s V(s) = V Q ( s ) + AH(s) P* [ 1 /1H(!)] ' ' o r O S t ^ y For the duration h ^ < t < T , we have the addi t ional component , „ / \ - s h , - s T , B (s) = ±M S i s i ( e 1 - e 1 ) , for t > h , o s ' 1 due to the saturat ion e f f e c t . Hence the t o t a l output of the system i s h, r F (s) - i 4. r w A - s h , - s T , V(s) = V o ( s ) + AH(s) Pj [ , + ° A H ( s ) ] - M ^T1 ( e " e h 1 < t < T 1 (8.2) or , i n shorter notat ion, V ( s ) = V q ( S ) + C Q ( s ) + B q ( S ) , h 1 < t < T 1 = D ( s ) s a y . o Since F ( s ) , H(s) are known and V q ( S ) i s known or can be determined, the output v( t ) may be evaluated from the inverse of V(s) for the i n t e r v a l i n quest ion. 94 Let us now consider the output for the duration T ^ < t < T ^ + h^* The equivalent system for this period i s shown i n Figure 8.5. I>0Cs)= V0Cs)+C.cs)+Vs) AHCs) C.Cs) Figure 8 .5 . Equivalent system for the i n t e r v a l T^< t<T^+h2< Since the sampler i n Figure 8.5 i s open during o < t <T^, the input f^(t ) = *C ^F^(s)J to th is sampler has no e f f e c t on the output component c^(t) for o < t < T ^ . We can therefore replace f^( t ) by a new funct ion f ^ ( t ) : , for o < t < T l f L f ^ t ) , , for t > T 1 , f n ( t ) = i which may also be wri t ten as f x l ( t ) = f 1 ( t ) u(t - T±). In terms of the p-notat ion , the Laplace transform of f ^ ( t ) i s P l l ( s ) = ^ ^ ( 8 ) ] = ? T I [ F ( s ) - D 0 ( S ) ] Consequently, the error for t h i s durat ion, namely X ; L ( t ) = f x ( t ) - C ]_(t) 95 may likewise be replaced by x n ( t ) = f n ( t ) - c 1 ( t ) , which states that the ef fec t ive error may be regarded as zero for the equivalent system during the i n t e r v a l o < t < T ^ » In order to calculate X^^(t) conveniently, we l e t t^ represent a new time axis such that t x = t - T 1 # Therefore f 11 (t) = f n ( t x + T x ) , c x ( t ) = c 1(t ]_ + T x ) X l l ( t ) = x n ( t 1 + T x ) , x l p ( t ) = x l p ( t 1 + T x ) . The int roduct ion of the new time axis t^ renders the s i t u a t i o n i d e n t i c a l to that of the equivalent system for the i n t e r v a l o < t<h^; that i s , the input f ± ^ ( t ) i s sampled for the period o<t^< and i s fed to a system with zero i n i t i a l condi t ions . Consequently, X ( c l ( t ) ) . ^ . ^ [ j ^ & ^ i . ] . By making use of the r e l a t i o n s h i p 3 ! ( g ( t ) ) = X{g(\ + T x ) ) , — s T that i s , G(s) = e 1 ^ [ g ( t 1 ) ] , and replacing F ] _ 1 ( s ) by P° T [ P ( S ) - V q ( S ) - C Q ( s ) - B q ( S ) ] = 1 OO i -P T i [ P ( s ) - D o ( s ) ] 96 we f i n a l l y get the Laplace transform of the component c^(t) of the output to be sT 1 \ [ * < • > - 3 > 0 < - > ] i + AH(s) - s T , h0 C.(s) = e 1 AH(s) P* The t o t a l output transform for the i n t e r v a l T ^ < t < T ^ + h^ i s V(s) = V (s) + C (s) + B (s) + CL(s) . O O O l For the duration T^ + h 2 < t < T 2 we have the a d d i t i o n a l component + r -s(T,+h ) - s T -j , B 1 ( s ) = ±M Le 1 2 - e 2 J , t ^ + h , , due to the saturat ion e f f e c t . Thus the t o t a l output transform i s V(s) = V o ( s ) + C Q (s) + B Q (s) + C ^ s ) + B 1 ( a ) , for T x + h 2 < t < T 2 . The genera l iza t ion to the t o t a l output transform for any time i n t e r v a l i s now obvious. In f a c t , n n-1 2 Ms) + £ B t ( s ) , f o r T n < t < T T i + h V(s) = V o ( s ) +i k=l n k=l n n+1 (8.3) 2 C k (s ) + B k ( s ) , for T n + h n + 1 < t < T n + 1 Lk=l where V o ( s ) represents the i n i t i a l conditions referred to the out-put of the o r i g i n a l system under considerat ion, where the component C^(s) i s given by C k (s ) = e -sT, AH(s) x P Ts+l sT, P T [ F ( s ) - V o ( s ) - C o ( s ) - B o ( s ) - . . . - C k - 1 ( s ) - B k_ 1(s3 1 + AH(s) (8.4) 97 for t > T k t and where the saturat ion component B^(s) i s given by for t > T k ( 8 i 5 ) Analogous equations can be developed for the t ransient r e -sponse i n the case where the nonlinear element incorporates a dead-zone. We have demonstrated above how the superposit ion p r i n c i p l e (as applied to the l i n e a r part of the system) and some properties of the p-transform can be used to evaluate the exact response of the system under consideration by means of a step-by-step a n a l y s i s . 8.2 PERIODIC OSCILLATIONS IN A SINGLE-LOOP SYSTEM CONTAINING ONE ON-OFF ELEMENT WITH PROPORTIONAL BAND The determination of the per iodic states i n automatic control systems having a single nonlinear element with piecewise l i n e a r c h a r a c t e r i s t i c has already received wide at tent ion i n the l i t e r a t u r e . 10 11 M. A . Aizerman and F» R» Gantmakher r determined the per iodic states i n nonlinear s ingle - loop systems with a piecewise l i n e a r c h a r a c t e r i s t i c consis t ing of segments p a r a l l e l to two given s t raight l i n e s * In making use of t h i s method i t i s necessary to integrate the equations of a l l the l i n e a r systems* into which the system under consideration can be decomposed. 98 The per iodic solutions are then constructed with the help of these i n t e g r a l s , 12 L . A , Gusev also dealt with the determination of the per iodic states of a broader class of s ingle - loop nonlinear control systems, namely, those with nonlinear elements having an a r b i t r a r y piecewise l i n e a r c h a r a c t e r i s t i c . His method does not require the i n t e g r a t i o n of the respective l i n e a r equations into which the system may be decomposed. The per iodic solutions are determined i n the form of a complete Fourier series without neglecting harmonics. The problem here i s reduced to solving a set of simultaneous transcendental equations that determine the behaviour i n each segment of the c h a r a c t e r i s t i c . In t h i s sect ion we w i l l r e s t r i c t our a t tent ion to a c o n s i -derat ion of simple symmetric o s c i l l a t i o n s i n the system as shown i n Figure 8.2 . We w i l l present two new methods of solving the per iodic states i n such systems: 1. an approximate method which i s v a l i d for the s u f f i c i e n t l y large class of systems i n which there i s some f i l t e r i n g ac t ion by the l i n e a r part of the system. I t has the advantages of being just as simple as but much more accurate than the descr ibing funct ion method i n the majority of cases of p r a c t i c a l i n t e r e s t . 2. the second method i s through the s o l u t i o n of l i n e a r V o l t e r r a i n t e g r a l equations. Reasonably accurate solutions may be found by the method of successive approximations• 99 1. The "Trapezoidal" Approximation* Assume that the system i n Pigure 8.2 has attained a simple symmetric steady state such that x(t) i s i n the lin e a r regions of the saturation characteristic (with or without hysteresis) for durations of length hT as shown i n Pigure 8.6 (a). Figure 8.6. (a) Exact output of N i n Figure 8.2 i n the case of simple symmetric o s c i l l a t i o n s $ (b) Corresponding approximation when H(s) has a f i l t e r i n g action. If the f i l t e r i n g action of the linear part of the system H(s) i s good and the system input f ( t ) has a predominant fundamental com-ponent, then we can replace the portions of the waveform y(t) i n the intervals n T ^ t < n T + h(n = o, —1, ...••) by straight l i n e segments as shown i n Figure 8*6 (b). 100 The precision of this approximation can be best judged by comparing i t with that made by the describing function method,' For this purpose, assume that the input to the nonlinear element i s sinusoidal* Then the t y p i c a l output y(t) i s a clipped sinusoid as shown i n Figure 8,7, where i t i s assumed that M<1, The exact output of N i s Input to N i ~" = sin cot M \-T 0 Approximate Output erf N 2J... . / \ /1 1 V / ' i i \T / •• -2T hT Exact Outpu'tof N. V Figure 8,7* Exact and approximate outputs of N f o r a sinusoidal input, f s i n wt, for (n - h) T < t ^ (n + h) T y = < (n = o, -1, -2,»»*) v ( - l ) 1 ^ = (sin u h ) ( - l ) n , for (n + h ) T < t < ( n + 1 - h) T and i t s Fourier series expansion i s , \ ~ n«t L n-1 + n+1 J i n odd 2 T sln(n-l)-n:h , sin(n+l)-rch 1 s i n nfl) / Q ,\ y = nZ, L n-1 + n+1 J n ( 8 - 6 ) n=l The approximate output, using straight l i n e segments^ i s described by 101 ' ( - l ) n sin_2ph ( t _ n T ) 9 f o r ( n _ h ) T < t < ( n + h ) T (n = o, i l , . , , ) . ( - l ) 1 ^ = ( - l ) n s i n Tth , for (n+h) T < t < ( n + l - h ) T and i t s Fourier ser ies expansion i s y a p 4 s i n un r c 2 h S n=l n odd s i n nTch s i n nttt n n (8.7) The f i r s t few terms of the expansions (8.6) and (8.7) for various values of h are y = 0.944 s i n » t + 0.046 s i n 3ttt - 0.028 s i n 5 « t ^ y = 0.916 s i n ttt + 0.000 sir* 3 « t - 0.036 s i n 5 » t +*••, 6b p y = 0.817 s i n cot + 0.106 s i n 3 » t - 0.021 s i n 5 » t +••• y = 0.814 s i n fi>t + 0.091 s i n 3 « t - 0.032 s i n 5 » t +**• y = 0.475 s i n « t + 0.128 s i n 3a>t + 0.047 s i n 5«ot + . . . h = 8 ap 0.475 s i n a t + 0.128 s i n 3fi>t + 0.046 s i n 5 « t +..• (8.8) The descr ibing funct ion method ignores a l l the harmonics and considers only the fundamental component. The trapezoidal approx-imation, however* takes a l l the harmonics into considerat ion . An inspect ion of Equations (8.8) indicates that the l a t t e r approximation i s superior to that of the descr ibing funct ion method for inputs c l ipped to about eighty-seven percent of t h e i r amplitudes* 102 Let us now analyse the per iodic states of the system for the shape of the per iodic output and the possible periods of o s c i l l a t i o n * Consider y ( t ) as shown i n Figure 8*6 (b) . approx Let Then y o ( t ) l )M [u(t) - u(t - hT)] y x ( t ) = M [u(t - hT) - u(t - T ) ] f I y o ( t + n T ) ( - l ) n + 2 y ( t + n T ) ( - l ) n n=o 0 n=l 1 for + <o <t <hT * approx = < 2 ( - l ) n T y Q ( t + nT) + y } ( t + n T ) ] for + hT<t<T n=o Now !(>„<*>) - I„(.) = T ^2 [4 - (2 H- shT)( l • .-•>*)] and M8.9) X ( y i ( t ) ) = I x ( s ) = f(e so that the output v ( t ) i s given by -shT -sT\ — e ) , i £ Y (s)-I,(s)e s T , v( t ) = 9 H(s) i-^Tf e s t ds , f o r o <t <hT 7 1 3 C l o r C 2 l + e sT , r T (s)+T 1(s) , W ^ J „ H ( s ) ° * ds , H8.10) C l 0 r C 2 for hT <t <T l + e where C ^ encloses only the poles of H ( s ) ^T q (S) - Y^(s )e S ' * ' J or 103 H(s) [[^ 0( s) + ^ i ^ s ^ J * a n ^ ^2 e n c - L o s e s o n l y the poles of s T l / ( l + e ) . The contour in tegra ls along C^ and are evaluated i n a mathematically p o s i t i v e and negative sense r e s p e c t i v e l y . This w i l l be implied f o r a l l contour integra ls occurring i n this chapter. Since ^ Q (s) and Y-^(s) are known (Eq. (8.9) ), and H(s) i s given, the per iodic output i s determined by (8.10)* Consider the c h a r a c t e r i s t i c s i n Figures 8,1 (a) and 8.1 (b) . The conditions for the existence of per iodic o s c i l l a -t ions are, under the assumption that t = 0 as shown i n Figure 8.6 (a) , x [(n + h ) T J = ( - l ) n x c = x [ ( n + 1)T] (8.11a) x [(n + h)T] ( - l ) n > o > x [(n + 1 )T ] ( - l ) n (8.11b) (n = o , — l , —2,...) i n the case of saturat ion without hysteresis or dead zone, and are x [(n + h)T] = ( - 1 ) % , x [(n + 1 )T ] = ( - l ) n ( - X l ) (8.12a) x Rn + h ) T l ( - l ) n > o > x [(n + 1 )T ] ( - l ) n L J L J (8.12b) (n - o, - 1 , —2,...) in the case of saturat ion with hysteresis and without dead zone. In order to determine the possible h a l f - p e r i o d s of o s c i l l a t i o n , we introduce the concept of the Tsypkin l o c i . These are defined by J(T) = J x(T) + jx(T) 1 % I (8.13) a n d J(M)= I x(hT) + jx(hT) J Since x(t) i s determined by x(t ) = f ( t ) - v ( t ) , as shown i n Figure 8.2, and v ( t ) i s a funct ion of h and T, as given by Eqs. (8.9) and (8.10), i t follows that the Tsypkin l o c i J (T) and J(hT) are each functions of h and T. 104 Two Tsypkin l o c i are required because the system i n Pigure 8.2 has two switching instants wi thin the half—period T . The imaginary parts of the Tsypkin l o c i determine the switching i n s t a n t s , and the r e a l parts determine the switching d i r e c t i o n s . The proper switching instants occur at the intersec t ions of the Tsypkin l o c i with the l i n e ; jx ( in the case of saturat ion c without hysteresis and dead zone); a l s o , from Pigure 6*6 (a), the proper switching d i r e c t i o n s must be i n the left—half plane f o r the J ( T ) l o c i f and the r i g h t - h a l f plane for the CT(hT) l o c i . Self o s c i l l a t i o n s i n the case of the saturat ion c h a r a c t e r i s t i c . The Tsypkin l o c i are p lo t ted with the help of Eqs . (8.10). possible h a l f - p e r i o d s of s e l f o s c i l l a t i o n . 105 Using h as the parameter and T as the variable. The straight lines jx are next inserted on the J(hT) and J(T) planes* c The values of h and T corresponding to the points of intersection of these l o c i with jx are then plotted on the h-T plane. The c construction i s shown i n Pigure 8 . 8 . Any pair of values, such as (h , , T ,) and (h T „), occurring at the intersection of o l ' o l o2 7 o2 1 & the resulting curves i n the h-T-plane may give r i s e to se l f o s c i l l a t i o n s . Self o s c i l l a t i o n s i n the case of saturation with hysteresis. The construction i n this case proceeds i n precisely the same way as the above, except that instead of the straight lines j x Q we introduce the straight lines - j x ^ and j x 2 on the J(T) and 3(hT) planes respectively, as shown i n Figure 8 . 9 . Figure 8 .9. Construction for the determination of the possible half—periods of se l f o s c i l l a t i o n i n the case of saturation with hysteresis. 106 Forced o s c i l l a t i o n s i n case of saturation* In the case of s e l f o s c i l l a t i o n s x(t) = -v ( t ) and the unknown quanti t ies are h and T. But i n the case of forced o s c i l l a t i o n s x(t) = f ( t ) - v ( t ) , T q the h a l f - p e r i o d of o s c i l l a -t i o n i s known, and the sought-for quanti t ies are now h and the phase s h i f t T of f ( t ) r e l a t i v e to v ( t ) . As i n E q . ( 5 « l ) , we l e t f ( t ) = A o f Q ( t - f ) where A = max o f ( t ) , and max f ( t ) = 1. The procedure for determining h and T i s as f o l l o w s . As mentioned e a r l i e r , the imaginary parts of the Tsypkin l o c i deter-mine the switching instants of x(t) and the rea l parts the switching di rec t ions A ( t ) » We now have two contributions to x(t) and x(t)> because x(t) consists of two par ts , -v ( t ) and f ( t ) , where v( t ) i s determined by E q . (8.10). The h parameter, o < h < l , i s v a r i e d by choosing a sequence of values , o<h^<h2 . . . < h n = 1. The contr ibut ion of —v(t) to x(t) for a f i x e d h a l f - p e r i o d T q and for h = h^ appears as the points T 0 T . = - - ° - v(T ) - Jv(T ) i n the . J (T) -plane, 1,1 % o o and the points T ° h T , i = ~ "T ^ r ( h T o ) ~ 3 ' v ( h T 0 ) i n t h e J (hT) -plane , f o r i = 1, 2 , . . . , n . Using the points 0^ , ^ and 0^ ,^ ^ as o r i g i n s , we next add the contr ibut ion due to f ( t ) = A f (t - f ) : these v ' o o ' ' contr ibut ions , denoted by 107 appear as closed curves, as T v a r i e s over the range o < T < 2 T Q » The pairs of values (h,7") at the inters e c t i o n of the 3*— curves with the straight l i n e s jx , (such pairs must be i n the l e f t - h a l f J(T)-plane and i n the right-half J(hT)-plane to s a t i s f y the proper switching instants and switching directions) may give rise to forced o s c i l l a t i o n s * The (h,T) values are plotted i n the h-Tplane, as shown i n Pigure 8,10, to give two curves corres-ponding to each of the $-planes. The points of intersection of the h-T curves y i e l d pairs of values (h,T) for which forced o s c i l l a t i o n s may occur. Pigure 8*10* Construction to determine values of h and T that may give r i s e to forced o s c i l l a t i o n . 108 We observe that we may get more than one h— T curve from each J—plane, depending upon the complexity of f ( t ) . An analogous procedure can be used to determine pairs of values (h, 7") that may give r i s e to forced o s c i l l a t i o n s i n the case of the saturation characteristic with hysteresis. 2. The Integral Equation Approach. Referring to the exact output y(t) of the nonlinear element, l e t y 2 ( t ) = A x(t) [u(t) - u(t - hT)] Then the Laplace transform of the output of the linear part of the system, V(s)j has, by an argument analogous to that used i n deriving Equation (8*10), the form V(s) = 1 fp 2 (s) - Y ^ s ) e s M M J H(s) , for o<t<hT L 1 + e rl (s) + T (s) I M ± _ H(s) , for hT<t<T . L 1 + e I (8.14) where and Let T 2 ( s ) = A ^ [ x ( t ) [u(t) - u(t - h T ) ] ) v / s M / -shT - S T N T 1 ( s ) = - ( e - e ). 2 * j C l o r C 2 1 + e s T 6 * where i s a contour which encloses only the poles of H(s)Y^(s) s T and C 0 encloses only the poles of l / ( l + e ). This expression 109 for v^(t) can be evaluated by the methods described i n Chapter 3. Furthermorej l e t «<*> = ^ J | e s t ds C 1orC 2 l + e s x where C^ encloses only the poles of H(s) and encloses only sT the poles of l / ( l + e )• Recall that v(t) = f ( t ) - v ( t ) . By using the real convolution i n t e g r a l , and the expressions f o r t i f ( t ) , v(t) and ^ ( s ) above, the inverse Laplace transform of Eq. (8.14) y i e l d s * upon rearrangement of i t s terms, t x(t) = f ( t ) + v^t+T) - A J * x(T) [u(t) - u(t-hT) J«(t-T)dT (8.15) o for o <t <hT, t x(t) = f ( t ) - v x ( t ) - A ^ x(T) [u(t) - u(t-hT) ]«(t~T)dT for hT<t<T. These equations are l i n e a r Volterra integral equations of the / \ 14 second kind with x(t) as the only unknown. Such equations are readily solved by Picard's process of successive approxima-tions. P r a c t i c a l solutions of such equations may be found 15 by means of a repetitive d i f f e r e n t i a l analyzer. P A R T I V T H E S T A B I L I T Y P R O B L E M 110 9. STABILITY OF PERIODIC STATES IN ON-OFF SYSTEMS WITH OR WITHOUT A PROPORTIONAL BAND The investigation of the possible periods of the periodic states, including both self and forced o s c i l l a t i o n s , was considered i n the preceding chapters. Now the question of the s t a b i l i t y of these periodic states acquires considerable importance. Only when stable can these periodic states be observed i n systems ph y s i c a l l y . Before investigating the s t a b i l i t y problem, l e t us f i r s t review the concept of s t a b i l i t y that w i l l be used* 9.1 THE CONCEPT OF STABILITY OF PERIODIC STATES In t h i s study we w i l l consider the concept of s t a b i l i t y i n the sense of Lyapunov,^^ and i n particular asymptotic s t a b i -l i t y i n the small, or> as i t i s sometimes c a l l e d , l o c a l s t a b i l i t y . Let x(t) define a periodic state, the s t a b i l i t y of which is to be investigated. According to A. M. Lyapunov, we determine the s t a b i l i t y of the periodic state by studying the behaviour of the neighbouring non—periodic states. The non—periodic states close to the periodic one are excited by the introduction of a s u f f i c i e n t l y small disturbance; such a non-periodic state may be represented by x(t) = Sc(t) + $ ( t ) , (9.1) where £(t) i s the deviation from the periodic state* D e f i n i t i o n 1. If the deviation 2j(t), after the removal of the s u f f i c i e n t l y small disturbance, approaches zero asymptotically as time increases, that i s lim ^ ( t ) = o, (9.2) t~>-oo I l l then the periodic state investigated i s said to be asymptotically stable i n the small or i n the sense of Lyapunov* This means that as time increases a l l s u f f i -c i e n t l y close non-periodic states approach the periodic state asymptotically. I f * however, under the above-mentioned conditions |^(t) increases i n d e f i n i t e l y as time becomes i n d e f i n i t e l y large, then the periodic state under consideration i s said to be unstable* D e f i n i t i o n 2* In this case we consider any non—periodic state; a l l states other than the periodic state investigated are referred to as non-periodic states. The quantity £(t) i s now the deviation (from the periodic state) caused by any disturbance, regardless of s i z e . If |^(t)j approaches zero as time increases, no matter what the disturbance may be, then the periodic state investigated i s said to be asymptotically stable i n the large or globally stable. In this thesis we w i l l be concerned with only the problem of asymptotic s t a b i l i t y i n the small. For s i m p l i c i t y , whenever we speak of s t a b i l i t y i n the remainder of this chapter we sha l l always mean asymptotic s t a b i l i t y i n the small. To investigate the asymptotic s t a b i l i t y i n the small of the on-off systems considered, we w i l l use one of the c l a s s i c a l methods of Lyapunov* In this method we form the equation of motion with respect to the deviation £(t) by replacing, i n the general equations governing the behaviour of the system, the periodic solution x*(t) by x(t) = x(t) + £(t) and rejecting i n these 112 equations a l l terms containing powers of £(t) exceeding the f i r s t * Consequently, a li n e a r equation i n £(t) i s obtained; this equa-t i o n i s referred to as the equation of the f i r s t approximation or the v a r i a t i o n a l equation. Moreover, i n the case under consideration this equation has periodic c o e f f i c i e n t s * According to a theorem of A. M. Lyapunov, i f the solution ^ ( t ) of the v a r i a t i o n a l equation approaches zero as time approaches i n f i n i t y * then the periodic state investigated i s asymptotically stable, regardless of the nonlinear terms neglected i n the i n i t i a l equation. In the case of an unbounded increase o f |$(t)| the periodic state i s said to be unstable. It may happen that the solution £(t) of the v a r i a t i o n a l equation neither approaches zero nor approaches i n f i n i t y i n absolute value as time increases i n d e f i n i t e l y , but merely remains bounded i n absolute value. In such cases i t i s impossible, i n general, to ascertain the s t a b i l i t y or i n s t a b i l i t y of the system by means of the v a r i a t i o n a l equation. But i n the IT 18 systems under consideration, a theorem of I. G. Malkin * shows that i n thi s c r i t i c a l case the v a r i a t i o n a l equation s t i l l gives an answer to the s t a b i l i t y problem. Lyapunov 1s method applies to equations containing con-tinuous nonlinear and linear functions. On—off systems, however, are usually described i n terms of discontinuous functions. Hence, a rigorous investigation i n such cases requires that a l l arguments be conducted with continuous functions which approximate the discontinuous functions with any degree of accuracy* and uses the l i m i t i n g process to obtain the behaviour of the system described by discontinuous functions. 113 Without claiming mathematical r i g o r , we w i l l use a method which makes use of the uni t step and del ta functions for the systems under considerat ion . This method, besides leading to the very same r e s u l t s as the rigorous but cumbersome approach, possesses the advantage that , from the physical point of view, i t i s very graphic , 9.2 VARIATIONAL EQUATION FOR SINGLE-LOOP SYSTEM CONTAINING AN ELEMENT WITH A SATURATION CHARACTERISTIC For the purpose of i n v e s t i g a t i n g the s t a b i l i t y of a given per iodic state i n a single—loop system containing an on-off element with a proport ional band, l e t us f i r s t form the, v a r i a t i o n a l equation. Without loss of g e n e r a l i t y , we assume that the nonlinear c h a r a c t e r i s t i c (y = ( £ ( x ) ) i s an odd f u n c t i o n . Let us suppose that Sc(t) = f ( t ) - v ( t ) (9.3) corresponds to the per iodic state of frequency a .. The quantity x( t )* d e f i n i n g the per iodic control s ignal to the nonlinear element, s a t i s f i e s the equation j ( x ( t ) ] = l ( f ( t ) ) - H ( s ) X ( ^ ( x ( t ) ) ) (9.4) Suppose that somewhere i n the system at time t = o, there ar ises a s u f f i c i e n t l y small disturbance (for example, a change i n i n i t i a l condit ions , or the a p p l i c a t i o n of some external a c t i o n ) , which breaks the per iodic state x(t ) and excites the neighbouring non-periodic state x(t ) = x(t) + £ ( t ) . The small 114 •fc-to On-erf Element L Lnear P a r t N Sfct) HCs} v(t) W l ith "Proportional Band Figure 9.1. A s ingle - loop system containing one on-off element. disturbance can be t ransferred to the input of the system, where i t w i l l be designated by f ^ ( t ) . Equation (9.4) now becomes £ ( x ( t ) +$(t)) =X[f ( t ) + f d ( t ) ) - H ( s)X ( ^ [ x ( t ) + $(t}]) . (9.5) The difference between Equations (9.5) and (9.4) gives the equation for the devia t ion $(t) from the per iodic states I($<*0' = l [ f d ( t ) ] - H(s )X(cJ ) [x(t ) + $<t)] - <£(*(t))). This equation i s nonlinear i n X^§ ( ^ ) ) • Assume that £ ( t ) i s s u f f i c i e n t l y small ; then 4>[s?<t) +?(t)] - <p(5E(tj> * ^K(t) + ||t|] -WW) 5 ( t ) = [x(t)] £ ( t ) + higher order terms, where cj>'. denotes the der ivat ive with respect to i t s argument. Disregarding terms i n £ ( t ) of degree higher than the f i r s t , we obtain the v a r i a t i o n a l equation for the system under considerat ion: l [ j ( t ) ) = l ( f d ( t ) ) -H(s)2(<|>' [x(t)J | ( t ) ) (9 .6) 115 This equation i s l i n e a r i n £ ( t ) and has periodic c o e f f i c i e n t s by v i r t u e of the presence of <J>' j"x(t)J . As indicated e a r l i e r , the behaviour of the solut ion of t h i s equation determines the asymptotic s t a b i l i t y of the per iodic state 5c ( t ) . In the general case of an a r b i t r a r y 4>(x) the i n v e s t i g a t i o n of the exact solutions of t h i s v a r i a t i o n a l equation meets with insurmountable d i f f i c u l t i e s . By v i r t u e of the s p e c i f i c charac-t e r i s t i c s cj)(x) under considerat ion, i t i s possible to carry out the i n v e s t i g a t i o n of the s t a b i l i t y of the per iodic states by comparatively simple and well-known methods. Let us f i r s t consider the case where cj)(x) i s the saturation c h a r a c t e r i s t i c , as shown i n Figure 9.2 (a) . The derivat ive of t h i s c h a r a c t e r i s t i c i s c£>'(x) = A [u(x+xc) - u(x-x c )] (9.7) so that <J? [x*(t)] = A [ u ( x + x c ) - u ( x - x c ) ] where x = x(t) i s a per iodic solut ion of frequency t O Q . >-x Figure 9.2. (a) Saturation c h a r a c t e r i s t i c , (b) Its d e r i v a t i v e . The expression for (J)' £x(t)J i s e a s i l y and g r a p h i c a l l y determined by means of the transfer diagram with the help of (jy ^x j as shown i n Figure 9.3. Furthermore, l e t us assume that x( t ) i s a simple symmetric p e r i o d i c state of hal f "^period T» With 116 no loss i n generali ty* we can choose the time axis t such that x(o) = -x and K* (o )>o . Let x(t) be equal to x at t = h < T . c c Then [x(t)] = A 5 [ u (t " k T ) " At - kT - h) ] (9.8) where u(t) i s the unit step f u n c t i o n . A<£>'Cx) lO 'CxCt)] O h T T+h 2-T 2T+h 3T 3T+h Figure 9.3. Transfer diagram for the graphic determina-t i o n of c £ [ £ ( t ) ] when x(t) i s a simple symmetric per iodic o s c i l l a t i o n of h a l f - p e r i o d T . Consequently, the v a r i a t i o n a l equation for the system under consideration becomes X = l ( * d ( * > ) - AH(s) T(5 ( t ) ^ [u(t -kT)-u(t -kT-h)]) (9.9) k=o 117 Using the notation X($(t)) = H(s) , X [ f d ( t ) ] = P d ( s ) , X($(t)]T [u(t -kT) - u ( t -kT-h) ] ) = P h, T[H(s)] , k=o where the symbol ^ «p[ J represents the p-transform notation used by Parmanfarma and J u r y , ^ * Eq.(9 .9) takes the form H(s) = F d ( s ) - P h ) T [H(s)] AH(s) . (9.10) ¥e now make the observation that equation (9.9) or (9.10) corresponds to the l i n e a r feedback f i n i t e pulse width sampling system, as shown i n Figure 9.4, i n which £ ( t ) i s sampled p e r i o d i c a l l y with period T for f i n i t e durations of length h and then fed to the l i n e a r t ransfer funct ion AH(s) . Hence the asymptotic s t a b i l i t y of the periodic state x(t) can be deduced ^ XT kT AH(s) Figure 9.4. Linear system equivalent to Equation (9.9) or (9.10). from an i n v e s t i g a t i o n of the s t a b i l i t y of the equivalent f i n i t e pulse width sampled—data system depicted i n Figure 9.4. The s t a b i l i t y of the l a t t e r system i s well-known, and an excellent 19 discuss ion of th is topic can be found i n Farmanfarma and i n Jury . 118 The above solut ion of the (asymptotic) s t a b i l i t y problem i s a genera l iza t ion of that given by Tsypkin. It i s of in teres t to consider the l i m i t i n g cases of the above system: 1. h = T, In t h i s case u (t) = £ Tu(t-kT) - u( t -kT-h) 1 p k=o L becomes the uni t step funct ion u ( t ) . This means that opera-t i o n i s confined to the l i n e a r port ion of the c h a r a c t e r i s t i c , and the problem i s reduced to a consideration of the s t a b i l i t y of a simple l i n e a r feedback system. 2. h = o and u^(t) has any f i n i t e amplitude. In t h i s case ^£|\ip(t)J = o, so that the sampler output i s zero , and the system remains at r e s t . This case would be possible i f x(t) were a square wave of amplitude >x c with h a l f - p e r i o d T. 3. h = o but u (t) becomes c* T ( t ) , a sequence of unit impulses. Under these condit ions , x = o, and the nonlinear ' c ' c h a r a c t e r i s t i c <J)(x) becomes the ideal on-off element without a proport ional band. This i s the case considered by Tsypkin. We now obtain & [x(t)]= 2M£ [ £ ( ! ) ] . Since u [x-(tj] = | ( - l ) k 5(t - k T ) , k=o and u [ £ ( t ) J =S [ x ( t ) ] 5f(t), 119 i t follows that the de l ta funct ion of a per iodic argument can be expressed as JK.—O where kT (k = o, 1 , . . . ) are the roots of the equation x(t) = o, assuming, of course, that x(o) = o. Because of the p e r i o d i c i t y of x(t) we have it—0 = |*(T) | ^ T ( t ) . Consequently, E q . (9.10) reduces to H ( s ) = F d ( s ) - r f f f j - f H * ( s ) (9.11) where 5»(s) = i ( $ ( t ) S T ( t ) ] . Hence, the problem of the asymptotic s t a b i l i t y i n the case of the simple on-off c h a r a c t e r i s t i c i s reduced to a considerat ion of a simple l i n e a r feedback sampled—data system corresponding to the system i n Figure 9.4, but i n which A i s now replaced by l/|x*(t)|. 4. h i s small compared to the time constants of the system. This s i t u a t i o n ar ises i f T > > h , i . e . the magnitude and p e r i o d i c i t y of x(t) are such that , e f f e c t i v e l y , the nonlinear c h a r a c t e r i s t i c possesses an exceedingly narrow pro-por t ional band. The output of the n o n l i n e a r i t y , due to the input over t h i s durat ion, can be approximated by replacing the f i n i t e pulses by impulses of equivalent area . Let us 120 remark that i f H(s) has a discontinuous impulse response the modified ^.—transform, and not the ^- t ransform, may be used to give a true approximation of the component of the response for the time duration nT + h < t < ( n + l ) T a r i s i n g from the input component 5(t) [u ( t - nT) - u(t - nT - h)J ; whereas i f H(s) has a continuous impulse response, we may use e i ther the ^—transform or the modified ^—transform for t h i s purpose* But the true approximation of the response during the i n t e r v a l n T < t < n T + h cannot be estimated. On the other hand this e f fec t w i l l be n e g l i g i b l e when h i s s u f f i c i e n t l y small and H(s) has a continuous impulse response, The exact behaviour, however, can be evaluated by means of p-transform methods. So f a r we have considered only the case of the saturat ion c h a r a c t e r i s t i c shown i n Figure 9.2 (a) . Let us now consider the (asymptotic) s t a b i l i t y problem for various types of saturation c h a r a c t e r i s t i c s * The other types of c h a r a c t e r i s t i c s considered and t h e i r der ivat ives are shown i n Figure 9.5. Slope=A ^rX - X , O X , X .NI o / h ^ P f . X, x 2 •NI A<£> 3 <X) A r - i A x - X^ -X , O X, X a -*-x -x z-x i A K-4 V -H A K-X, x 2 Slope = A X a - A x Figure 9*5* Form of der ivat ives c£>' (x) for various types of saturat ion c h a r a c t e r i s t i c s . 121 Case of(J>2(x) For the saturation c h a r a c t e r i s t i c with h y s t e r e s i s , i l l u s t r a t e d i n Figure 9.5 (a), we have 3>»(x) = A [\i(x - x )^ - u(x - x )^ J , for i >o A £u(x + x 2 ) - u(x + x )^ J , for i < o (9.12) The t ransfer diagram for the determination of c£>2 £ x ( t ) J Figure 9.6. Transfer diagram for the graphic determination of (£> 2 [ x ( t ) J when x(t) i s a simple symmetric periodic o s c i l l a t i o n of h a l f - p e r i o d T . gives (p'2 [ x ( t ) ] = AJ? [ u ( t - t Q - k T ) - u ( t - t Q - k T - h ) ] (9.13) 122 Since the choice of the i n i t i a l time instant i s a r b i t r a r y , then the displacement t does not influence the form of the v a r i a -t i o n a l equation, which i s thus given by Equations (9.10) and (9.14) are the same, except that the values of h are, i n general , d i f f e r e n t . Hence, the s t a b i l i t y of, the system containing a c h a r a c t e r i s t i c with saturat ion and hysteresis can again be deduced from the behaviour of the simple feedback sampled-data system with f i n i t e pulse width. Cases ofc£5i(x) and<$>.(x) without hysteresis w i l l y i e l d v a r i a t i o n a l equations of the same form - just as the cases of c h a r a c t e r i s t i c s without dead zone and with or without h y s t e r e s i s . Consequently, i t i s s u f f i c i e n t (9.14) The cases of c h a r a c t e r i s t i c s with dead zone and with or for x >o (9.15) u(x - x + A ) for x<o By subst i tu t ing A = o i n E q . (9.15) we getcj)^(x) In t h i s case (9.16) 123 i . e . ( J ) ^ £"5c(t)] corresponds to the sum of two sequences of pulse f u n c t i o n s . The p e r i o d i c i t y of each sequence i s the same and i s equal to T the h a l f - p e r i o d of the per iodic state x(t) (we are assuming simple symmetric o s c i l l a t i o n s for x( t)X The second i s displaced r e l a t i v e to the f i r s t by a f i x e d time i n t e r v a l YT. The geometric transformation into the indicated sequences of pulse functions i s shown i n Pigure 9 . 7 with the help of the der ivat ive of the c h a r a c t e r i s t i c <J>^ (x) . By an appropriate choice of the i n i t i a l time instant (set t = o) , -*-x - T "5 ""I o L Vl x X x * xT x "x x Sift) -*-YT T Figure 9 . 7 . Transfer diagram for for the determination of Cj54 [x(t)j when £ ( t ) i s a simple symmetric per iodic o s c i l l a t i o n of h a l f -period T . the v a r i a t i o n a l equation for t h i s p a r t i c u l a r c h a r a c t e r i s t i c Cj>4(x) has the form 124 ~ (s ) = p ( s) - AH(s) X($(t) I Tu ( t -kT) - u(t*-kT-h ) ^ k=o (9.17) + u(t-kT-<fT) - u ( t - k T - Y T - h 2 ) ] ^ Using the p-notat ion P h , T [ H ( s ) ] = X ( 5 ( t ) 1 [ u ( t - k T ) - u ( t - k T - h . ) ] ) , x k—o E q . (9.17) can be rewri t ten as H ( s ) = F d ( s ) - AH(s) [H ( s ) ] + ^\2y: [ H ( s ) e s Y T J (9.18) Equation (9.17) or (9.18) corresponds to the l i n e a r f e e d -back f i n i t e pulse width sampled-data system i n Figure 9.8. It consists of two samplers i n p a r a l l e l and a feedback l i n k con-t a i n i n g a l i n e a r t ransfer funct ion AH(s) . The samplers close synchronously and t h e i r outputs have uniform pulse widths h^ and h 2 » However, the second sampler operates with a delay YT with respect to the f i r s t . Even though t h i s system contains an kT-t-h, ( k + W AHfe) (k-rX)T+-k Figure 9*8* Linear system equivalent to Equation (9.17) or (9.18). a d d i t i o n a l sampler* as compared to that for the case without dead zone 9 the analysis of the behaviour of the former i s no 125 more d i f f i c u l t than that of the l a t t e r , because of the f a c t that the samplers operate synchronously. The Case of More Complicated Forms of Per iodic O s c i l l a t i o n s , The method described above can be extended e a s i l y to the study of the s t a b i l i t y of any given complicated form of per iodic o s c i l l a t i o n . As a example, l e t us consider the case of the simple saturat ion c h a r a c t e r i s t i c , Without deducing the v a r i a t i o n a l equa-t i o n i n £ ( t ) , we make use of the t ransfer diagram shown i n Figure 9*9. The der ivat ive of the per iodic funotion S(t ) of period 2T now consists of n sequences of p u l s e s . The duration of the pulses i n the successive sequences, i n i t i a t e d at times o, Y 1 2T, Y 2 2 T , , * * , * n - i 2 T w i t h respect to the f i r s t , are i n general d i f f e r e n t , and are denoted by h Q , h ^ , h ^ , , , , * n n . i ]_ r e s p e c t i v e l y . carton L L <k 2 T --t Figure 9.9. Transfer diagram for the determination of c£)T £ x ( t ) J where ($)(x) i s the simple saturat ion c h a r a c t e r i s t i c , and x(t) i s a complicated per iodic waveform of period 2T. 126 C l e a r l y , the l i n e a r system corresponding to the v a r i a t i o n a l equation i n t h i s case w i l l consist of n samplers i n p a r a l l e l of uniform pulse widths h Q , h ^ , h b-n_^ and a feedback l i n k containing the l i n e a r t ransfer funct ion AH(s) . The samplers close synchronously with p e r i o d i c i t y 2T, but are not i n phase. This system i s shown i n Figure 9.10* 2.kT-+-h, (zk-rY,)T-th, AHCs) Figure 9.10* Linear system determining the s t a b i l i t y of a complicated per iodic state 3c(t) f o r the sa turat ion c h a r a c t e r i s t i c cj)(x). 9.3 AN APPROXIMATE SOLUTION TO THE ASYMPTOTIC STABILITY OF PERIODIC SOLUTIONS In the preceding sect ion we formulated an exact method, which reduces to well—known solved problems i n sampled—data systems, for the determination of the asymptotic s t a b i l i t y of per iodic states* We now present an approximate s o l u t i o n to the above problem but without resor t ing to the sampled—data approach. Let us assume that the l i n e a r t ransfer funct ion H(s) i s a f r a c t i o n a l r a t i o n a l f u n c t i o n , which may be wri t ten as 127 and that the degree of P(s) i s less than that of Q(s ) . Then the v a r i a t i o n a l equation (9.6) can be expressed i n d i f f e r e n t i a l equation form thus: Q(p) |(t) +P (p) d>» |"x(t)l £ ( t ) = Q(p) f (t) . a (9.19) where p = ^ » and P(p) and Q(p) are d i f f e r e n t i a l operators. Since the der ivat ive of the c h a r a c t e r i s t i c Cj5* |>(t)] i s per iodic with period T, we can write i t as an exponential Fourier series thus : & [ x ( t ) J = fc ^ , I =-oo where °i = T [*(*)] e _ ; j l f l , t d t (c = constant) •(9.20) and (6 = 2n/T We now seek a general s o l u t i o n of the homogeneous equation Q(p) 5(t) + P(p)0 ' [ x ( t ) ] § ( t ) = o (9.21) of the form £ ( t ) = 5 B e ( a + J * " * * , (9.22) k=-oo where the B*s aire the complex amplitudes and a i s the so—called c h a r a c t e r i s t i c exponent which i s to be determined. C l e a r l y , i f the r e a l parts of the values of a are found to be negative, then the system i s asymptotically s t a b l e . Subst i tut ing (9.20) and (9.22) into (9.21), we obtain [Q(P) + C O P ( P ) ] l\-ia + ^ H + P ( P ) I f k B ^ e ^ ^ ^ ^ t C B J o c + j ( k - 0 « ) J t". = o # 128 This l a s t equation can be rewrit ten as [U<P> +C oP(p)] 1 B k . < « + J h " > t -+ ? ( p ) l i J L M-*°-t B^ e < a + ^ H = ° ( 9 - 2 3 ) By using the r e l a t i o n P(p) e 5 t = e 5 t P(S) , and equating the c o e f f i c i e n t s of l i k e frequency components, we obtain A + c o p ( s k > ] + | , + c - e B k + J p ( V = 0 (9.24) (k = o, - 1 , - 2 , . . . ) where 5fc = a + jktt Equation (9.24) i s an i n f i n i t e system of equations, each of which contains an i n f i n i t e number of terms i n B .^ (k = O J *1, - 2 , . . . ) . The c h a r a c t e r i s t i c equation of the system i s obtained by equating the determinant of E q . (9.24) to zero . As i t stands, t h i s c h a r a c t e r i s t i c equation i s of I n f i n i t e degree i n a . Let the roots of the c h a r a c t e r i s t i c equation be oc^(i = 1, 2 , . . . ) . Then a necessary and s u f f i c i e n t condit ion that the system be stable i s that the rea l parts of ou l i e i n the l e f t -hal f s -plane . A P r a c t i c a l Approximation. In prac t ice* the l i n e a r parts of the systems considered are such that the frequency components l y i n g outside c e r t a i n f i n i t e bandwidths can be regarded as n e g l i g i b l e . This can 129 always be achieved by choosing the pertinent bandwidths s u f f i c i e n t l y l a r g e . Let us assume that a l l frequency components larger than (A are n e g l i g i b l e . Then a l l complex amplitudes for which c + « c < I m S± < - « c (9.25) may be neglected. Unfortunately, the values of are unknown. However, by choosing s u f f i c i e n t l y large values of k i n S = a + jkw, say |k| > M , condit ion (9.25) can usual ly be f u l f i l l e d . Thus a l l complex amplitudes for |k|>M may be neglected. Consequently, i n place of the i n f i n i t e system of equations (9.24), each containing an i n f i n i t e number of terms, we now r e s t r i c t our a t tent ion to the fol lowing f i n i t e system of equations, each containing a f i n i t e number of terms: J a i k B k = 0 ( i = o, - 1 , . . . , -M) where (9.26) < Q(S.) + CQP(5 ) , for i a i k = < C k - i P ( 5 i } , for i ^ k The c h a r a c t e r i s t i c equation i s now given by the determinant of the system (9.26), i . e . a_.,_| = o , I k | which i s polynomial of degree 2M + 1 i n a . If a l l the roots ( i = o, - 1 , . . * * ^M) of t h i s polynomial l i e i n the l e f t - h a l f s-plane, i . e . they a l l have negative r e a l parts, then the periodic state under consideration i s stable. 130 In the case of the saturat ion c h a r a c t e r i s t i c , with or without hysteresis and without dead zone, <Jp' £ x ( t ) j has the form Cj>1 [ x ( t ) J = A 2 [ u ( t - t -kT) - u ( t - t - k T - h ) ] k=o 0 0 when x(t ) i s a simple symmetric per iodic o s c i l l a t i o n of half period T . The Fourier series for t h i s sequence of rectangular pulses i s where tt = 2TI/T* By choosing t Q - ^ = o, the exponential form for t h i s series i s & [x(t)] = n Z j ^ ^ ^ Similar expressions for the saturat ion c h a r a c t e r i s t i c with dead zone can be found* When the c h a r a c t e r i s t i c of the nonlinear element <J>(x) ceases to be of the on-off or saturat ion type, the question of the s t a b i l i t y of the per iodic states cannot, i n general , be reduced to a consideration of the s t a b i l i t y of sampled-data systems. Under these conditions the present approximate method can s t i l l y i e l d an answer to the s t a b i l i t y problem i n most cases of p r a c t i c a l i n t e r e s t . 131 9.4 A DIRECT APPROACH TO THE STABILITY PROBLEM The method to be presented below w i l l be c a l l e d the d i r e c t approach, i n contrast to the sampled-data approach, because i t i s d i r e c t l y related to the physical d e f i n i t i o n of s t a b i l i t y ; that i s , a disturbance i s a p p l i e d , and the deviat ion from the state of equil ibr ium i s s tudied. If the deviat ion dies out the system i s said to be s table ; otherwise, i t i s unstable. This approach w i l l be applied both to forced and s e l f o s c i l l a t i o n s i n the system shown i n Pigure 9.11. N Hcs) Figure 9.11. A s ingle - loop system containing one on-off element. Let f ( t , T ) be the per iodic input with h a l f - p e r i o d equal to T, i n the case of forced o s c i l l a t i o n s . Let y ( t , T ) and v ( t , T ) be the corresponding outputs o f J and H(s) , r e s p e c t i v e l y . The input to N i s denoted by x ( t ) . S t a b i l i t y of Forced O s c i l l a t i o n s The system i n Figure 9.11 i s assumed to be i n a state of forced o s c i l l a t i o n s with half—period equal to T» Let a random disturbance AT"Q occur i n the zero—crossover at t = o as shown i n Figure 9.12, so that the response v ( t , T ) for t > o i s modified to v ( t ) . We take AT <<T, and neglect higher order terms m I o I i n A7T. Let y (t) be the modified output of N, and l e t i t s m devia t ion from y(t*T) be denoted by y^("t): that i s y d ( t ) = y r a ( t ) - y ( t , T ) . +1 ytt) /-3fCtjT) m T 2 . T +4 K 3 T Figure 9.12. Per iodic and modified outputs of N. 2 T t 2 A T , - 2 T I T , AT 0 ->1 K-3 T t 'a Figure 9.13. Deviation i n the output of N. The quantity y^(t) consists of a series of impulses as indicated i n Figure 9.13. The devia t ion i n the system response, v , ( t ) = v (t) - v ( t , T ) f i s the response of H(s) to y ^ ( t ) . Let y m ( t ) = 0 for t = t Q , t^f 1*2* *"•*"' and o < t < 00 AT = t - n T , n = o , 1, 2, . . . n n 133 The quanti t ies AT^(n = 1, 2, •••) are now determined i n terms of A T . o The change i n the f i r s t crossover past the o r i g i n , AT^, can be found by solving f ( t 1 , T ) - v m ( t 1 ) = o (9.27) where v m ( t ) = v ( t , T ) - 2h(t) A T (9.28) and h(t) i s the u n i t impulse response corresponding to the t ransfer funct ion H ( s ) . Subst i tu t ion of (9.28) into (9.27) gives f ( t .T) - v ( t , T ) = -2h(t) A T (9.29) A Taylor series expansion of (9.29) about t = T y i e l d s f (T,T) - v ( T , T ) + [f (T,T) - v ( T , T ) ] A ^ = -2h(T)AT Q , where f ( T , T ) 4 ^ 1 and v(T ,T) k 5 t Jt=T d t Jt=T But f ( T , T ) - v ( T , T ) = o, so that A T = 7 ] h(T) A T Q (9.30) where TJ = 2 ( - f ( T , T ) + v (T ,T) ) _ 1 (9.31) The change i n the next crossover A T ^ i s determined by f ( t 2 , T ) - v f f l ( t 2 ) = o (9.32) where v m ( t ) i s now given by v m ( t ) = v ( t , T ) - 2h(t) A T q + 2h(t-T) A ^ (9.33) 134 Substitution of (9»>33) into (9.32), and expansion about t= 2T y i e l d f(2T,T) - v(2T*T) + [f(2T,T) - v(2T,T)] A T 2 = -2h(2T) A T q + 2h(T) LT± (9.34) Since f(2T,T) - v(2T,T) = o and f ( t , T ) - v(t,T) = - f(t-T,T) + v(t-T,T) , equation (9.34) y i e l d s A T 2 = Y| [-h(2T) A T + h(T) A T J (9.35) This equation for A T - may be written i n terms of A T using (9.30) o but t h i s i s not necessary as w i l l be shown later. In general, the expressions for A T N are given by A T 1 = 1 [h<T> A T o ] A T 2 =y) [-h(2T) A T q + h(T) A T ^ ] A T 3 = 1) |h(3T) A T - h(2T) A ^ + h(T) A T 2 ] (9.36) A T 4 = 1 [~ h( 4 T) A T 0 + h(3T) A T - h(2T) A T 2 + h(T) A T ^ J etc . The deviation i n the response i s L ( t ) = -2h(t) A T q + 2h(t-T) A T ^ - 2h(t-2T) A T , + . . . V or V d ( s ) _ . -Ts ^ \ ^ -2Ts A T 2 -3Ts A T 3 , -2H(s) A T - 1 ~ E A T + 6 A T " A T ~ + * * * o o o o (9.37) 135 S u b s t i t u t i o n of (9*36) into (9.37) y i e l d s zsJSr; - i - T^'[h(.)] + *1 e " 2 T s [-h(2T) + h(T) ^ ] o - TJ e " 3 T s [h(3T) - h(2T) ^ + h ( T ) - | ^ ] o o + ... - 1 - r , [ f h(nT) ] ( ! - . - * • £ £ • e " 2 1 ^ - . . . ) n=l o o (9.38) From (9.37) and (9*38) there resul t s -2H(s) AT V d ( s ) = - 2 (9.39) 1 +nf h ( n T ) e - n T s *n=l where Y| i s given by E q . ( 9 . 3 l ) . S t a b i l i t y requires that a l l the poles of (9*39) l i e i n the l e f t - h a l f s-plane or that a l l the zeros of 1 +Tj ^ h(nT)d n=l l i e i n the left—half s—plane. Equivalently- . i f we substi tute Ts 0 0 / \ —n z = e , s t a b i l i t y requires that a l l the roots of 1 +TI £ h(nT)z = o ' n=l are inside the u n i t c i r c l e , with centre at the o r i g i n , i n the z—plane• Comparison with the sampled-data approach: As mentioned by Tsypkin, the study of the above s t a b i l i t y problem i s equivalent to the study of the s t a b i l i t y of the l i n e a r sampled—data feedback system shown i n Figure 9.14, 136 tTT T] H(s)=G(s) Figure 9 » 1 4 * Equivalent sampled-data system f o r the s t a b i l i t y problem. The ^-transform of G(s) = T J H ( S ) i s G(z) =-T|H(z) = if] X h(nT) z ~ n , (z = e T s ) n=o The sampled-data feedback system i s stable provided that a l l the roots of - n 1 + G(z) = 1 + nr) V h(nT) z = o n=o (9.40) l i e inside the u n i t c i r c l e i n the z -plane . The r e s u l t s of the d i r e c t and sampled-data approaches d i f f e r : the term Y)h(o) i n (9.40) i s absent i n (9.39). The sampled-data r e s u l t i n (9.40) was derived on the assumptions that ( l ) x^(t) has small average amplitude as compared to x ( t , T ) and (2) the time der ivat ive of X^(t) does not take too large values . These assumptions imply that h(t) must not be discontinuous at t = -o, or , equivalent ly , that h(o+) = o. Consequently, the r e s u l t derived by the sampled-data approach should be used only i n cases where h(o+) = o; but i t does not say what should be used when h(o+) ^ o» The r e s u l t derived by the d i r e c t approach, E q . (9.39), i s v a l i d both for h(o+) ^ o and h(o+) = o. 137 S t a b i l i t y of Self O s c i l l a t i o n s A s l i g h t modification of the previous arguments w i l l give the desired r e s u l t for the s t a b i l i t y of self o s c i l l a t i o n s . Let the half-period of s e l f o s c i l l a t i o n be T . Let the system o i n Pigure 9.11 be undergoing forced o s c i l l a t i o n s of half—period T, T = T q , up to t = oy after which the input f ( t , T ) i s removed and the ensuing o s c i l l a t i o n periods are compared to T Q » The modified response i s v m ( t ) = v(t,T) - 2h(t) A7\ ( o < t < t 1 ) (9.41) Since v (t,) = v (T + AT. ) = o and v(T ,T ) = o, mv 1 m o 1 oy o ' a Taylor series expansion of (9.41) about ( T O , T q ) y i e l d s where and AT, = - aAT + ?ih(T ) AT (9.42) 1 > o' o A T = T - T Q , YJ = 2 | > ( T Q , T o ) J _ 1 (9.43) A V W a — v(T ,T ) o 1 o and where V„(-T *T ) = ^ v j ^ T ^ ] t = T and T = T T o ' o dT -1 o o For the next i n t e r v a l t ^ < t < t 2 > v (t) = v(t,T) - 2h(t) AT + 2h(t - T ) AT. . m o o 1 Since v ( t j = o = v (2T + AT_) and v(2T ,T ) = o, m 2 m o 2 oT o ' 138 then A T 2 = -.aAT + f) [-h(2T Q ) A T q + h(T Q ) A T ^ ] . In general . AT = -aAT + r\ £ h(mT ) AT ( - l ) m + 1 (9.44) n 1 , - 0 n—m m=l n = l , 2, 3, . . . The devia t ion i n respbnse i s v d ( t ) = v m ( t ) - v ( t , T d ) = v ( t , T ) - v ( t , T Q ) - 2h(t) A T Q + 2h(t-T Q ) A T - 2h(t-2T ) A T + . . . 0 £ ^ v _ ( t , T ) AT - 2h(t) A T + 2h(t-T ) A T T o o o 1 - 2h(t-2T Q ) A T 2 + . . . (9.45) The f i r s t term on the right-hand side of (9.45) i s per iodic with an i n f i n i t e s t i m a l amplitude and therefore can be neglected. Subst i tut ion of (9.44) into the Laplace transform of (9.45) y i e l d s —T s —2T s —3T s *-2H(s) f AT +aAT(e 0 - e " ° +e 0 - ...)1 V , ( s ) = - =L ( 9 . 4 6 ) oo -nT s 1 + *1 X h(nT ) e 0 n=l ° Consequently, the condit ion f o r s t a b i l i t y i s the same as that found i n the case of forced o s c i l l a t i o n s except that Tj i s given by (9.43). oo - T s The zeros of 1 + Tj £ h(nT Q )u , u = e 0 , w i l l be 139 discussed further. Let so P(u) 4 (1 + YJ h(mTo) um)/T| (9.47) Now i n the case where H(s) has n simple poles a l l d i s t i n c t from zero n P ( B . ) Q S k T o (-l) m + 1h(mT ) (9.48) m=l so that (9.47) can be written as P(u) -J^hirnT) [ u m + ( - l ) m + 1 ] m=l A zero of P(u) i s at u = -1, so that F(u) = (1 + u) G (u) . (9.49) The form of G(u) i s derived as follows: ^ -mT s v L i p ( s 1 ) mT (s,-s) ^ h ( m T o ) e o. =X! S - Q T X T T " ° K m=l 0 m=l k=l u v t V e n T , / \ -T (s-s, ) _ p ( y e 0 k " t~i Q' (s. ) -T (s-s. ) k=l * k , o k 1-e Now -, -mT s • (1 + u) G(u) - P(u) = ± +2_J h ( m T 0 ) e ° ' m=l _ P ( s k ) . r e ° k , u e 0 k 1 = fcl L , , T 0 s k T o s k J l+e " 1-u e so that ^ P ( s J V k . G(u) = Z J Qtftj £ - f - s 7 W (9.50) k=l * k . o k , ok k l+e " " 1-u e 140 Since 1+e s k t and since the ^-transfprm of e i s given by s, t- T s, T s, o k i o k z-e 1-u e i t follows that 8 ( u ) . j ( | v ( t , I o ) ) (9.51) o The following p a r t i a l f r a c t i o n expansion i s v a l i d * m=l In the f i r s t term on the right hand side of (9.52), u = —1 corresponds to periodic o s c i l l a t i o n s . Hence, the s t a b i l i t y depends on the zeros of G(u) = G(z~^), and these zeros should be within the unit c i r c l e i n the z-plane. The s t a b i l i t y question may therefore be answered by a Nyquist p l o t . A necessary condition i s that G(-l)> o. Additional notes on the function G(u) are as follows: n v T s, • ^ - i P(s ) o k , ThusT]G(o) = 1, which i s the value for s — > - o o . From rn ^ P ( s J p T o S k 141 1 , - e T ° S k and , ^ » P ( s J V k | v ( T ,T ) = 2j QTTf-T T s ^ 0 0 k=l U V S V , A o k 1 + e i t f o l l o w s t h a t G ( - l ) =H VT ( Tof To } + * < V T o } ] ( 9 ' 5 3 ) Thus f | G ( - l ) = 1 + a where a i s g i v e n by (9.43), Now ^ - 1 P(s, ) T o s k r T s, 2T s, _ 0(+D = E QTrH 6 T . [ l + 2e 0k + 2e 0 k + ...] k = 1 2 U k ^ (1 + e 0 k ) 2 Therefore t} G(+l) = 1 - a + b (9.54) where 0 0 b = - 2 2 V ^ o ' V 7 ^ ( T o ' T o ) m=2 I f b i s s m a l l , then Eq. (9.54) i n d i c a t e s t h a t the *y]G(u) — p l o t does not enclose the o r i g i n f o r l a I < 1. This c o n d i t i o n i s much stronger than the previous one where G ( - l ) > o . I l l u s t r a t i v e Example Consider the simple case where H(s) = l / s . In t h i s case, h ( t ) = 1, t > o + , and h(o+) = 1. 142 The sampled—data equation (9.40) should not be used i n this case because i t i s not v a l i d when h(o+) ^  o. The use of (9.39), however, yie l d s 1 + T) [H(Z) - h(o+)] = 1 + = 0 (9.55) Thus z = 1 - y\ and s t a b i l i t y requires that o <Tf|< 2 (9.56) In the case of forced o s c i l l a t i o n s , y) = 2 [- f(T»T) + v(T,T)] - 1 Since v(T,T) = 1, the condition for the s t a b i l i t y of forced o s c i l l a t i o n s y i e l d s o <-f(T,T) < «> (9.57) For this example, the quantities appearing i n Figure 9«11 have the following description! y(t,T) i s a square wave as shown i n Figure 9.12; v(t»T) i s the integral of the square wave y(t,T) and i s therefore sawtooth i n shape; the waveform f( t , T ) i s such that x(o) = x(T) = o x(o)>o, x(T) <o o <-f (T,T) < « , and, provided that there are no more switchovers i n the interval o < t < T / the shape of f(t , T ) i s otherwise arb i t r a r y . 143 CONCLUSIONS Techniques and concepts for studying periodic phenomena i n on-off feedback systems have been developed. Three methods f o r evaluating the periodic response of the lin e a r part of the on-off element have been presented* the f i r s t method uses the impulse response of the l i n e a r part of the system; the second method i s i n terms of the residues at the poles of H(s)/s» where H(s) i s the transfer function of the l i n e a r part; the t h i r d method i s i n terms of H(j«). the frequency response of the l i n e a r part. Concepts pertaining to the steady-state response of on-off elements are then examined* generalizations of the concepts of the Hamel and Tsypkin l o c i and of the phase character-i s t i c of Neimark have been introduced. These concepts have been found to be useful i n the study of s e l f and forced o s c i l l a t i o n s i n on-off feedback systems* they have been used to determine the possible periods of s e l f and forced o s c i l l a t i o n s i n single-, double-, and multiloop systems containing, i n general* an a r b i -trary number of on-off elements. The behaviour of on-off elements possessing a proportional band has been considered* The response of a single-loop system containing one such element has been determined by means of equivalent sampled-data systems, i n which the samplers have f i n i t e pulse widths* However, i n the study of the periodic o s c i l l a t i o n s i n such a system, an approximate method* ca l l e d the trapezoidal approximation, has been used; i n general} th i s approximation i s more accurate than that of the describing 144 f u n c t i o n , and i s V a l i d when there i s s u f f i c i e n t f i l t e r i n g act ion by the l i n e a r p a r t . The concept of the generalized Tsypkin l o c i has also been found useful i n the determination of the possible periods of s e l f and forced o s c i l l a t i o n s of such systems. The resul ts found by Tsypkin on the asymptotic s t a b i l i t y i n the small of single~*loop systems having one on-off element w i t h -out a proport ional band have been generalized to include the case where the on-off elemen't contains a proport ional band. The inves t igat ions of the s t a b i l i t y of .these systems have been reduced to a consideration of the s t a b i l i t y of equivalent sampled-data systems i n which the samplers have f i n i t e pulse width: mult iple samplers i n p a r a l l e l that close synchronously, but not i n phase, have been found to enter i n the case of h y s t e r e s i s , dead zone and complicated forms of per iodic o s c i l l a t i o n s . F i n a l l y * a d i r e c t approach to the s t a b i l i t y problem has been presented: the d i r e c t use cf the physica l d e f i n i t i o n of asymptotic s t a b i l i t y i n the small has given resul ts that agree with those obtained by the sampled-data approach. 145 REFERENCES 1, G i l le,< J . C«, P e l e g r i n , M. J . , Decauline, P.., Feedback Control Systems r McGraw-Hill Book Company, I n c . , New York, 1959, 2, Kochenburger, R« , "A frequency method for analyzing and synthesizing contactor servomechanisms", Trans* AIEE, V o l . 69, Part I, 1950, pp. 270-284* 3, West, J . C , A n a l y t i c a l techniques for nonlinear control systems, The E n g l i s h U n i v e r s i t i e s Press L t d . , London, 1960. 4» Kahn, D. A o , "An analysis of relay servomechanisms", Trans-. AIEE . V o l . 68* Part I I , pp. 1079-1088. 5. Hamel, B . , "Etude mathematique des systemes a plus ieurs degres de l i b e r t e d^cr i t s par des equations l i n e a i r e s avec un terme de commande d i s c o n t i n u " , Proc . Journees  d T Etudes des V i b r a t i o n s , AERA., P a r i s , 1950* 6. Tsypkin, J . Z . , Theory of r e l a y type automatic control systems« Gostekhizdat, Moscow, 1955 3 (Russian). 7. Bohn, E . V . , " S t a b i l i t y Margins and Steady-State O s c i l l a t i o n s of ON-OFF Feedback Systems", Trans. IRE, PGCT - 8, No. 2,1961, pp. 127-130. 8. Tu Syui -Yan* , Tei L u i - V y , "Sel f o s c i l l a t i o n s i n a s i n g l e -loop automatic control system containing two symmetric r e l a y s " , Avtomatika i Telemekhanika. V o l . 20, No. 1,. 1959, pp. 90-94, (Russian). 9. Neimark, Yu. I . , S h i l n i k o v , L . P. "On the symmetric per iodic motions of multi-cascade re lay systems", Avtomatika i Telemekhanika, V o l . 20, No. 11, 1959, pp. 1459—14669 (Russian). 10. Aizerman, M« A . , and Gantmakher, F . R . , "On the determination of the per iodic states i n nonlinear dynamic systems with piecewise l i n e a r c h a r a c t e r i s t i c " , P r i k l . Mat. Meh. , V o l . 20, 1956* pp. 639-654, (Russian). 11* Aizerman, M. A . and Gantmakher, F . R . , "Determination of the per iodic states i n systems with piecewise l i n e a r c h a r a c t e r i s t i c , consis t ing of l i n k s p a r a l l e l to two given l i n e s " , Avtomatika i Telemekhanikao V o l . 20, Nos. 2 and 3, 1957, (Russian)* 12. Gusev, L . A . ? " D e t e r m i n a t i o n of per iodic behaviour of automatic control systems having nonlinear part with a r b i t r a r y piecewise l i n e a r c h a r a c t e r i s t i c " , Avtomatika  1 Telemekhanika, V o l . 19, No... 10, 1958, pp. 931-944, (Russian)• ~ i 146 13. Jury, E . I . , Sampled-data control systems, John Wiley & Sons, I n c . , New York, 1958, 14. R i e s z , P . B . Sz-Nagy, Functional A n a l y s i s , Frederick Ungar Publ ishing C o , , New York, 1955. 15. Tomovic, R « , Parezanovic, N . , "Solving i n t e g r a l equations on a r e p e t i t i v e d i f f e r e n t i a l analyzer% Trans. IRE, E C - 9 , No. 4, I960* pp. 503-506. 16. Lyapunov, A« M«, Probleme g^ne'ral de l a s t a b i l i t e du mouve-ment, Princeton Univers i ty Press , Pr ince ton . 1947. 17. M a l k i n , I . G . , " 0 n the s t a b i l i t y of the per iodic motions of dynamic systems", P r i k l . Mat. Meh. , V o l . 8, No. 4, 1944, pp. 327-331, (Russian). 18. M a l k i n , I . G.*$ Theory of s t a b i l i t y of motion* Gostekhizdat, Moscow,' 1952$ (Russian). 19. Farmanfarma, G . , "General analysis and s t a b i l i t y study of f i n i t e pulsed feedback systems", Trans. AIEE, V o l . 77, Part I I j 1958, pp. 148-162. 

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