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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Steady-state ocsillations and stability of on-off feedback systems Mohammed, Auyuab 1965
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Title | Steady-state ocsillations and stability of on-off feedback systems |
Creator |
Mohammed, Auyuab |
Publisher | University of British Columbia |
Date Issued | 1965 |
Description | Methods for studying the behaviour of on-off feedback systems, with the emphasis on steady-state periodic phenomena, are presented in this thesis. The two main problems analyzed are (1) the determination of the periods of self and forced oscillations in single-, double-, and multiloop systems containing an arbitrary number of on-off elements; and (2) the investigation of the asymptotic stability in the small of single-loop systems containing one on-off element which may or may not have a linear region of operation. To study the periodic phenomena in on-off systems, methods of determining the steady-state response of a single on-r-off element are first described. Concepts pertaining to the steady-state behaviour are then introduced: in this respect it has been found that generalizations of the concepts of the Hamel and Tsypkin loci and also of the phase characteristic of Neimark are useful in the study of self and forced oscillations. Both the Tsypkin loci and the phase characteristic concepts are used to determine the possible periods of self and forced oscillations in single- and double-loop systems containing an arbitrary number of on-off elements; these concepts are also applied to multiloop systems. On-off elements containing a linear region of operation, called a proportional band, are then described: both the transient and periodic response are presented. An approximate method for determining the periodic response is given. The concept of the Tsypkin loci is used to determine the possible periods of self and forced oscillations in a single-loop system containing one on-off element with a proportional band. The asymptotic stability in the small, or local stability, of the periodic states of single-loop systems containing one ideal on-off element has been considered by Tsypkin. In this thesis, Tsypkin's results have been generalized to include the cases of on-off elements containing a proportional band. The stability of such systems is determined by the stability of equivalent sampled-data systems with samplers having finite pulse widths. Finally, this stability problem is solved by a direct approach, one that makes use of the physical definition of local stability; the results obtained by this method agree with those derived by the sampled-data approach. |
Subject |
Feedback control systems Automatic control |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-09-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0104850 |
URI | http://hdl.handle.net/2429/37498 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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