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Method of process control with identification Mukerjee, Malay Raj 1965

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A METHOD OP PROCESS CONTROL WITH IDENTIFICATION MALAY RAJ MUKERJEE B.Sc*, University of Allahabad, 1956 D.I.I.Sc., Indian Institute of Science, 1959 M. Tech., Indian Institute of Technology, Kharagpur, I960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE.DEGREE OF MASTER.OF APPLIED SCIENCE' in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Members of the Department o f Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA OCTOBER, 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 o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e 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 , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n , f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s representatives„ I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f , • The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT A general system for controlling an unknown process by adjusting compensation networks is.discussed. The control operation is divided into two parts, that of identification of the unknown process dynamics and ihat of adaption by adjusting the compensation networks. Both operations are carried out by the method of steepest descent, using a suitable error signal as a performance function. A general method of investigation i s considered, which i s applicable to a l l systems of this type. The systems are classified as Type I and Type II; i t i s shown that a Type I system i s stable for a l l input signals and i n i t i a l conditions. The systems which are not of Type I are designated as Type II and i t i s shown that the behaviour of Type II systems can be approximated by that of a Type I system in the neighbour-hood of the optimum point of operation. These results are applied to investigating the behaviour of a number, of controllers and identification systems. TABLE OF CONTENTS Page List of Illustrations .. ....... v 1. INTRODUCTION 1.1 Statement of the Problem 1 1.2 Representation of the Over-all System ..... 3 2. STABILITY AND GENERAL BEHAVIOUR 2.1 The Method of Steepest Descent .. 11 2.2 Mechanization of the Steepest Descent Equations 13 2.3 Stability 18 2.4 Stability of Type II Systems 20 2.5 Tracking Behaviour ., 23 2.6 Effect of Constraints .... 27 2.7 Choice of Performance Criterion. 32 3. IDENTIFICATION SYSTEM 3.1 A General Parameter-Tracking Model 35 3.2 A One-Parameter Tracking System 40 3.3 A Two-Parameter Tracking System 41 3.4 Multiparameter Systems *." 45 4 . DIRECT CONTROLLERS 4.1 Compensation in the Forward Loop • 49 4.2 A One-Parameter System 51 4.3 A One-Parameter' Type II System 55 4.4 n-Parameter Type II Systems .».,..*.....,. 63 4.5 Compensation with Lead-Lag Network 64 4.6 Example of a Two-Parameter, Type II System ..... 69 4.7 . Example of Lead-Lag Network Compensation 70 5. CONTROL WITH IDENTIFICATION. 5.1 Configurations of Indirect Controllers ......... 76 5.2 The Functional Relations f ± ( ft ± , p > 2 » . . ^ n ) = * i 78 5.3 Stability of the General System ............. 81 5.4 A One-Parameter Indirect Controller .».,...».... 85 5.5 Performance of Controller I , 91 5.6 Performance of Controller II 94 6. CONCLUSION 100 i i i Page APPENDIX 1 , . .... 102 APPENDIX 2 106 APPENDIX 3 * .» 108 REFERENCES 110 iv LIST OF ILLUSTRATIONS Page Fig. 1. Block Diagram of the General System 4 Fig, 2. A General Parameter-Tracking System 35 Fig. 3. Computer Circuit Diagram for Two-Parameter Tracking System , , 47 Fig. 4. Response of Two-Parameter Tracking System 48 Fig. 5. System with Compensation in Forward Loop ....... 49 Fig. 6. Different Configurations for a One-Parameter . Controller 52 Fig. 7. Compensation with Lead-Lag Network 65 Fig. 8. Computer Circuit Diagram for Two-Parameter Type II System 72 Fig. 9. Response of Two-Parameter Type II System 73 Fig, 10. Computer Circuit Diagram for Lead-Lag Network Compensation 74 Fig. 11. Response of Lead-Lag Network Compensation ...... 75 Fig. 12. One-Parameter Indirect Controller .............. 87 Fig. 13. Computer Circuit Diagram for Controller I ...... 96 Fig. 14. Response of Controller I ,, 97 Fig. 15. Computer Circuit Diagram for Controller II 98 Fig. 16. Response of Controller II 99 v ACKNOWLEDGEMENT The research work for this thesis was carried out with the financial aid of the Canadian Commonwealth Scholarship Plan and the National Research Council of Canada, whose help i s gratefully acknowledged. The atifthor also wishes to thank Professor E. V. Bonn and the members of the Electrical Engineering Depart-ment of the University of British Columbia, whose many suggestions and discussions are incorporated in the thesis. v i 1. INTRODUCTION 1.1. Statement of the Problem. A major e f f o r t i n t h e f i e l d o f a d a p t i v e system d e s i g n has been concerned w i t h t h e problem o f a u t o m a t i c i d e n t i f i c a t i o n ' *- > 1^* : ) f * i . e . t h e measurement o f the s i g n i f i c a n t parameters i n a c o n t r o l l e d p r o c e s s . The assumption t a c i t l y made i s t h a t changes i n p r o c e s s dynamics can be n u l l i f i e d o r c a n c e l l e d by adjustment o f parameters i n compensation n e t w o r k s . I n o t h e r 4- 6 8 9 cases * f f the e f f o r t has been d i r e c t e d towards r e d u c i n g t h e d i f f e r e n c e between t h e performance o f t h e c o n t r o l l e d system and t h a t o f a model r e g a r d e d as the optimum, by a d j u s t i n g the compensation n e t w o r k s , w i t h o u t o b t a i n i n g any e s t i m a t e o f t h e p r o c e s s dynamics. The t e c h n i q u e s employed i n the two cases a r e e s s e n t i a l l y t h e same; the i d e n t i f i c a t i o n problem i n v o l v e s o b t a i n i n g a model w h i c h approximates t h e dynamics o f t h e c o n t r o l l e d p r o c e s s as opposed t o the c o n t r o l problem o f f o r c i n g t h e system t o approximate t h e dynamics o f the model. The two problems may hence be re g a r d e d as i n v e r s i o n s of each o t h e r . I f a performance c r i t e r i o n P i s s e l e c t e d as t h e measure o f t h e c l o s e n e s s o f t h e d e s i r e d a p p r o x i m a t i o n , i t i s c l e a r t h a t P w i l l be a f u n c t i o n o f the p r o c e s s parameters a^ and the model parameters </^ » A m i n i m i z a t i o n o f P w i t h r e s p e c t t o th e n c o n s t i t u t e s t h e i d e n t i f i c a t i o n p r o c e s s ; w h i l e a m i n i m i z a t i o n w i t h r e s p e c t t o a^ c o n s t i t u t e s the c o n t r o l o r a d a p t i o n p r o c e s s . The g e n e r a l problem o f a d a p t i v e c o n t r o l can be d i v i d e d i n t o t h r e e s e p a r a t e p r o b l e m s . ^ These may be termed 2 a. , I d e n t i f i c a t i o n i . e . obtaining a knowledge of the process to be controlled b. Optimization i . e . constructing an o v e r - a l l system whose response i s optimum according to some pre-specified performance c r i t e r i o n c. Adaption i . e . maintaining the o v e r - a l l system response at the pptimum value i n spite of changes i n the uncontrollable part of the process dynamics. In the ease of control systems using a model as a reference, the model response i s regarded as the optimum value. The control problem hence reduces to the measurement of the unknown process dynamics and the adjustment of suitable compensating networks to reduce the error between the optimum model response and the o v e r - a l l response of the controlled system. It i s the purpose of t h i s thesis to present a method by which these objectives can be achieved f o r controlled processes whose behaviour can be represented by l i n e a r d i f f e r e n t i a l equations. • A scheme i s discussed f o r obtaining an estimate of the coeffifcients: of ther d i f f e r e n t i a l equation; these estimates, together with the values of the parameters of the reference model, are then used to adjust the parameters of compensating networks so that the o v e r - a l l system d i f f e r e n t i a l equation becomes i d e n t i c a l with the d i f f e r e n t i a l equation representing the reference model* 3 1.2. Representation of the Over-all System The general block diagram of the combined identification and adaption system is shown in Pig. 1. The process may be described by the differential equation n m ^ a.pV = ^ b p^x (1.1) i=l D=l where a t 1 This may be written in operator form as y = Hx (1.2) where £ b .p^  £ a ±p It may be noted that the parameters b.., a^ are in general time-varying and hence H is not a transfer function. Under certain restrictions on the behaviour of D j ( t ) a n <^ aj_("t)> i t is possible to define an operator^ £b,(r) pJ H(P,T) = 2 R (1.3) £ a , ( r ) P I In such a case the derivatives "of the operator with respect to b. and a^ may also be defined'. 4 r Compensation Networks Parameters ft Input Vjj Forward Loop Compensa-tion L _ Adaptive Controller I — f f Feedback Loop |^  (pompensa-i o a — Process i ! Parameters a., b. J 1 3 Parameters ^ Identifi-cation Model Identifi-cation Controlled Parameters m. Reference Model Fig. 1. Block Diagram of the General System 5 a a-k i a , ( r ) p i ^ a ( e ) p i ( 1 . 4 ) d H * o b I f a. (t) = a. A + a..,(t) 1 lO i i where a..., b .-, are small and further i f a. and b. are very small i l J l i 3 J i n magnitude, the operator H can be approximated by the following expression n m H = H n + £ a, n 4 | + £ * 4 ? (1.5) where 0 ^ i l d a. ^ j l ^ b 1=1 1 5=1 ^ H 0 = , a i 0 / It should be noticed that i t i s not possible i n general to commute the operators defined above when a^, b-j are time-varying, i . e . , i f H-^ , H 2 are operators, H 1H 2x ± H ^ x The process parameter estimation i s carried out by constructing a model with a d i f f e r e n t i a l equation represented by an operator H» = H'(p,«< i) (1.6) 6 The output of this model is hence given by y' = H'x (1 .7) The error between the model and process response is then y' - y = (H' - H)x - - a.)4^-)x (1.8) If <K ^  is close to a^, the error y' - y may be taken as a measure of the accuracy of process estimation. If an even function of the error is chosen as a performance criterion, the point = a i in parameter space w i l l correspond to a minimum of this criterion, since the error is zero at this point-. The square of the error, is generally chosen as the criterion and the parameters °< ^  adjusted according to the method of steepest descent to enable the system to reach the minimum for this criterion. The equations of the identification controller are d o < i a - 2 , x -rr-± = - A — e i = 1, 2 ..... n (1.9) ax &<< ± where e = y' - y and A is a constant. Substituting for e in (1.9), d* . i• = _ 2Ae ^ H'x i = 1, 2 ...... n (1.10) dt d<x ± The adjustment of the compensation networks is achieved in an analogous manner. The reference model is described by 7 the operator The ov e r - a l l system can also be represented- by the operator % = H 1(p, 3 i,a.,b ) The .adaptive controller operates to make and M i d e n t i c a l . Since the parameters a., b. are unknown, they are replaced by the estimates °< ^  and the operator H^' defined as H 1 » = H ^ C p , (!>±t o < . ) Both H^' and M. are l i n e a r operators. Hence, by putting H 1' =• M a set of relationships may be derived between the parameters P> ^, o< ^  and HK . These may be expressed as fi (-° < 1» * 2' °'' * n ; P 1* 1 2 2''' * ^  m; m i > m 2 ' * ' = 0 i = 1, 2,».....m ( l . l l ) The t o t a l number of such relationships must be equal to the t o t a l number of adjustable parameters . These relationships are, i n general, nonlinear and dependent on the actual Configuration of the compensating networks. However, the. choice of simpler configurations often leads to a set of l i n e a r relationships which may be solved as a set of simultaneous algebraic equations. Depending on the actual method of solution adopted, the following two types of controllers are obtained: 8 Type I Controller. The controller equations are 1 d t + f ± = 0 i = 1, 2,.....m (1.12) The solution f i = 0 i s reached when ^ =0. The stability and convergence properties of this system depend on the equations f^ = 0, i.e. on the conditioning of the problem. Type II Controller. ' The steepest descent method is used, defining a performance criterion m P = £ f. l i=l The controller equations are 2 d fi . ^ p •g-r^ = - A i = 1, 2,. m (1.13) where A is a positive constant. Since P can be considered as a Lyapunov function for the system and since-P is negative definite, the stability of the system does not depend on the form of the equations f^ = 0. The system moves to the minimum of P which is the point at which equations ( l . l l ) are satisfied. The various systems discussed in the literature can be regarded as particular cases of the general system shown in Pig. 1. References (3) and (9) discuss the error in the form —2 1 e 2 7 T J 9 where H = H(s, The integral has no meaning i f the °< ^  are time-varying quantities since the system i s then nonlinear in general. However, the H-operator blocks are comparatively easy to obtain and manipulate. Reference (l) considers a time-domain approach which gives a precise mathematical meaning to the track-ing equations; but the approach leads to very complicated equations and i t becomes d i f f i c u l t to manipulate the operational blocks even for a small number of parameters. The approach used in this thesis is based on an extensive use of differential operators to achieve the best features of both approaches. It may be noticed that (1.8) does not necessarily provide the only error signal that can be used for adjusting; but equation (1.10) is 2 valid whenever an e criterion is being used, whatever the error signal; hence by choosing different signals, i t i s possible to devise a large number of tracking systems, a l l of which can be studied by investigating equation (1.10). In this thesis, an analytical discussion of over-all system performance is considered in order to determine i f the system response converges to the desired response (system sta b i l i t y ) , and, when the system is stable, to determine the time required for the system to achieve the desired response (convergence time). The analysis is complicated by the inter-actions present between the adjustment loops for different parameters and also between the identification and adaption loops. However, i t i s possible to establish stability in the neighbourhood of the optimum point of operation in parameter space. This neighbourhood is of considerable importance, since, 10 in a practical case, the system w i l l "be operating at the optimum point and disturbances w i l l normally be in the form of changes in the process parameters, which w i l l shift the optimum. These changes generally take place very slowly so that, in most cases, the new optimum point and the original operating point w i l l l i e close to each other for a considerable length of time. If the system is stable in this region, the new optimum can then be tracked successfully. In an ideal situation, the changes in a process parameter are cancelled by corresponding changes in the adaptive system before they become significant, so that the over-all system response remains identically equal to the model response. 11 2. STABILITY AND GENERAL BEHAVIOUR 2.1. The Method of Steepest Descent In order to carry out an estimation of the values of process parameters the method adopted is to construct a physical model whose configurational features are the same as that of the process, i t "being t a c i t l y assumed that the latter i s known. In this the problem differs from the "black-box" identification i problem, where no assumption is made regarding the dynamics of the process; in a st r i c t sense, therefore, this should be termed a parameter estimation process rather than an identification process. 1^ The task i s then to control the model in such a manner that i t s transfer characteristics approach those of the process in some predefined sense. This w i l l generally be the case when the values of the model parameters approach the values of the corresponding process parameters. Since the values of the process parameters cannot be measured directly, i t is required to find a function of the error between the process and model parameter values such that i t disappears or is minimized in some sense when the corresponding parameter values are equal. This function may then be termed a performance function for the system and the problem may be regarded as that of minimizing this func-tion. It is clear that this performance function may be written as P = P ( * 2 , . . . o < n ; a 1 , a 2 , . . . a n ) (2.1) where «< ^  are the model parameters and a^ are the process parameters. 12 The minimization, procedure can be carried out by the method of steepest descent or gradient method, which has been 1 7 1 R 1Q discussed extensively in the literature. ' * 3 The instrumenta-tion of this method leads to the adjusting equations do( . \ p d T 1 = " A a? t 1 = 1» 2 » — n (2-2) where A i s a positive constant. It can be shown that i f P is independent of time an absolute minimum of P i s a stable singular point for the above system of equations, and that °< ^  approach this point asymptotically as t -» <*3 . However, in actual systems P i s also a function of time and hence (2.2) must be regarded as a modified form of steepest descent*. Considering an error signal n e-L = ;> (* ± - a i)q i(t) (2.3) i=l and a performance function P = e 2 - 2- n ^ 1 £ c < ± = 2 q . ( t ) | ( -<.-ai)qi(t) b q. + 2(1 ( « < 1 - a 1 ) q 1 ( t ) ) ( £ ( ^ j - a j ) 7 ^ ) (2 . 4 ) i J 1 1 3 Writing 5 i = * i ~ a i i = 1 , 2,«..n ( 2 , 5 ) the steepest descent equations become ^ = - 2 A £ 5 j q j [ q i + g fi ^ — J . ] i = 1, 2„..n j=l -j=l 1 (2.6) 2.2 Mechanization of the Steepest Descent Equations The instrumentation of the equations (2.2) requires op be the signal — or — — . These cannot be obtained directly; but techniques for obtaining an approximation to their values are 2 10 available and may be classified as follows: * I. If the system differential equation i s written as n ± m . £ a < 4 = £ bcL£* ( 2. 7) and the model equation as d°x n . m i = 1 d t 1 . = 1 3 dt J (2.8) and e = y' - y a differential equation may be obtained for J^,e and — ~ -14 ( 2 . 9 ) Writing u = b 6 = -Ail i cx<± 6°t ± _ ^ e _ ^ y' u^ is given by the differential equation n , i , > °< - ^ r - S = - S _ X _ k = 0 , l,....n ( 2 . 1 0 ) ^ 1 d t 1 dt* Similarly, v^ is given by n ^ i £ ^ * ± — - r ^ = ^ - 7 0 , l,....m ( 2 . 1 1 ) i=4 d t d t Equations ( 2 , 1 0 ) and ( 2 . 1 1 ) may be obtained from equation ( 2 . 8 ) by differentiating with respect to °( and @>i respectively and interchanging the order of differentiation. It may be noticed that i f ( 2 . 8 ) i s rewritten as y* = Hx ( 2 . 1 2 ) where H i s an operator as defined in section 1 . 2 , ( 2 . 1 0 ) and ( 2 . 1 1 ) may also be rewritten as 6n u k = "a*~~ x k = ° ' 1 ' n ( 2 . 1 3 ) v^ = ^ x 0 = 0 , l,....m ( 2 . 1 4 ) where -4^—, - r ^ - — are the operators defined in section 1 . 2 . 15 II. An approximation to — , — can also be obtained by expanding P in a Taylor series about the point ( * 0 r * l ••• °< n' 1*1 m^'"^ ' e * ' s * f 0 r * k : ' p ^ o ^ l <*k + S * k } * n ' V *1 = P ^ , ^ ... * k . . . V ^ o ' P l — V t ] + 6 * k » ^ + ° ( ^ k 2 ) If ^ i s small, the higher order terms may be neglected. Then ^ p P ( * + ) - P ( * ) a o c k - <*°<k The f i r s t of the above two methods has been used to b e obtain ^ — for the present investigations. The error signal considered in (2.3) i s not normally available as a signal from the system. The error e = y f - y can, however, be expressed in the same form as e 1 i f 8 ^  i s very small and further i f the changes in S ^  are slow. In that case £^ and higher derivatives of S may be considered negligible quantities. Rewriting (2.8) a S : n , m £ (a + 5 ) i - Z - = £ + 4 ) *2x i=l d t j = l d ^ « J a solution in the form of a series may be obtained for the above equation under the above assumptions. 16 n m y < = H x + 2 S i ^ c + i ^-^-x + 0(8±2) (2.15) i=l 1 3=1 3 The series converges i f f5^| , \S^_ | << 1. Neglecting the second order terms in S , the following expression is obtained for the error. n m e - r - y - ± + Z *3-B-x (2.16) i=i 1 3=1 2 (2.16) is seen to be identical with (2.3) by identifying - T T ^ — x as q.(t) etc. i t may be noticed that q.(t) are not in 0 0 ^ 2 . 1 general independent of 8 ^. If the 8 ^  are varying slowly in comparison with the rate of change of the input signals, i t is possible to consider another approximation to the equations (2.6) by replacing the time-varying quantities q^(t) by their mean values. This makes the analysis of system behaviour considerably simpler and the results obtained from the approximation provide a reasonable estimate of system behaviour even in cases where the assumption of slowly varying S . is not s t r i c t l y valid. Considering the equations (2.6) for the case where 1 = 0, dS n = " 2Aq± 2. &fa i = l , 2,....n (2.17) 3=1 17 These may he rewritten as as at where n n " 2 A 2. 8 . qjg~f - 2A £ 5 ^ ( t ) i = 1, 2,...n j=i 3=1 (2.18) + r i j ( t ) ' r i j = 0 Rewriting (2.18) in matrix form, where g|- = - 2A(B + C(t ) )S 5 = n B = 1*2-^ 2 q2 l*n *2*n q l q n ^ n q-n (2.19) C.(t) = r i : L ( t ) r 1 2 ( ' f c ) ••• r i n ^ r 2 1 ( t ) r 2 2 ( t ) ... ^ ( t ) r n l ^ r n 2 ^ — r n n ( t ) cTtT = o Under certain restrictions on C(t), the behaviour of (2.19) is close to the behaviour of the equation (2.20) 18 In particular, i f C(t) is bounded, (2.19) i s stable provided (2.20) is stable. For the more general error given by (2.15) and (2.16), the matrix equation becomes ||- = - 2A(B + C(t))S + f ( d , t ) (2.21) where f ( S , t ) = 0( S , t ) . The sta b i l i t y and convergence proper-ties of the system may hence be studied by considering the equations (2.19), (2.20), (2.21). 2.3 Stability The expression for the general error may be considered again n e1 = £ (°<±- aj>i(-fc) (2.3'-) i=l The systems for which —r—— = 0 w i l l be referred to as Type I ° 1 systems; a l l other systems w i l l be referred to as Type II systems. In order to examine the sta b i l i t y of Type I systems, a Lyapunov function n V = £ S±2, (2.22) i=l is considered. It can be seen that V ^ 0. The derivative of the function is given by dt' ^~ u i dt i=l 19 dS . Substituting for <^ 1 from equation (2,17) 1=1 j=l n n i=i d«i - 4A e i 2 (2.23) It follows that V Oj hence the system i s stable but not necessarily asymptotically stable. It i s shown in Appendix 2 that V -> 0 as t -> „ For the above system, this implies that e 1-> 0 as t->«*»• from (2,3'), this indicates that S± 0 as t — o c t except i f the equation n 1=1 S ± i ± = 0 has solutions other than S - 0* The functions q^(t) depend on the input signal characteristics and hence the above condition places a limitation on the~kinds -of input signal that can be permitted in order to guarantee asymptotic s t a b i l i t y . For example, for a sinusoidal input and n = 3» p q^ = sin £Ot, q 2 = ^cos <ot, q^ = - ^  sin wt the above equation becomes 6 1 sin wt'+ S^^cos^t - S^.to sin wt = 0 20 p which is satisfied i f <5 n = 0 and S1 = o^.o ; i t follows that no more than two parameters can he identified or controlled i f the input consists of sinusoids only. However, the above condi-tion is not a source of d i f f i c u l t y in controlling actual processes, since the input signals are then V-sually stochastic signals, which generate a set q^ that are linearly independent. An estimate of the convergence time of the system may he obtained from (2.19), which is an exact representation of a Type I system. If the time-varying parameters can be replaced by their mean values T equation (2.20) i s obtained, since C(t) = 0; the solutions of (2.20) can be used, to obtain an approximate idea of the, behaviour of the system. Since the matrix B i s symmetrical, i t s characteristi6 roots are real and negative; hence the convergence i s monotonic., 2.4. Stability of Type II Systems hq When the conditions = 0 are not satisfied^ equations (2.6) have the form a s , * n 5=1 3=1 1 (2.6') The stability of these systems has been discussed for particular types of input signals such as step inputs and sinusoidal 1 3 inputs.- * In order to examine more general inputs, the assumption i s made that h i is very small; in that case, the second term within the parentheses i s small compared to the f i r s t term i.e. 21 • j=i 1 i t being assumed that q^ may be written as q ± = 0(1 + S±) (2.6r) can then be rewritten as d 5 n i = - 2Aq± 2 . «5q + 0 ( 5 ± 2 ) i = ' l , 2 r...n (2.24) d t " ^ i ^ -3^3 It may be expected that the behaviour of (2.24) for small S ^  w i l l be closely related to that of d S n ^r-= = - 2Aq± 5 8 ^ . i = 1, 2,...n (2 .24') 3=1 (2 .24') can be recognized as a Type I system and hence i t s solutions approach 0 as t However, this is not sufficient to guarantee the stability of (2 .24) . The conditions for the convergence of the latter are discussed in Appendix 3; these conditions are that the solutions of (2.24) approach zero as t - » o o provided that the impulse response of the system defined by (2 .24') i s bounded. This cannot be guaranteed i n general and hence the stability of (2.24) cannot be assured for a l l inputs. However, i f 6 i i s again assumed to be varying slowly enough so that the time-varying quantities in (2 .24') can be replaced by their mean values, the impulse response of the system defined by (2 .24') is bounded and the stability of (2.24) i s guaranteed. Hence i t i s possible to construct a stable Type II 22 system by choosing the gain A small enough, so that << 1* Considering the Lyapunov function examined in the previous section n i=l the derivative i s given by di = + 2 < 5 l £ i dt + * ^ i dt i=l n = - AH± S.q.) 2 - 4 A [ 0 ( ^ . 3 ) ] i=l = - AAe±2 - 4 A e 2 ' It may be seen that i f is small, the f i r s t term in the above • expression dominates and hence ? ^ 0 to a f i r s t approximation. In that case, a linear range of operation may be defined where (2.24') is a reasonable approximation to the system equation. The system can become unstable i f V > 0 in this range, which can happen during relatively small time intervals in the neigh-p bourhood of the zeros of q.. and e^, where I eg I > e^.. In these time intervals 6^ i s a second order function of the S * i t can then be argued that the system remains essentially stationary over these intervals. However, because of the presence"of these intervals where V may be positive, unconditional st a b i l i t y cannot be guaranteed for Type II systems as i t can be for Type I 23 systems. In fact, an experimental case has been demonstrated^ where the system is unstable even in the linear range. 2.5. Tracking Behaviour In actual processes, i t is expected that the controlled parameters w i l l normally be operating at their optimum values; these optimum values are changed only because of changes in the parameters of the uncontrollable part of the process. The tracking action is required when the process parameters change; i f these change slowly compared to the convergence time of the; control loops, the control system operates close to the optimum point of operation i.e.. in the linear range as defined above. , It may be added, that, in designing a compensation network to . cancel the unwanted part of the process dynamics, i t i s t a c i t l y assumed that these process dynamics vary sufficiently slowly to permit their representation by operators as defined in section 1 . 2 ; the compensation by exact cancellation of poles and zeros is not valid for systems with rapidly varying parameters. The process parameters may hence be written.as a ±(t) = a i Q + a ± 1 ( t ) i = 1, 2,....n (2.25) and the controlled parameters as * ± ( t ) = a i Q + a ± 1 ( t ) + S.(t) i = 1, 2 , — n ( ^ 6 ) » a.., where I -k— | « 1. In this case, a.n may be regarded as constant l over the interval where 8 . i s sensibly different from zero, and 24 this assumption is used to investigate the transient behaviour of 5 ^ It should be noticed that the stability of the system equations considered in sections 2.3 and 2.4 did not involve this assumption; i t i s useful only in some cases for obtaining an estimate of the transient behaviour, such as the speed of convergence. It is of interest:to consider two special cases cjf a ^ 1 ( t ) . These are a " a-j_]_("k) "-^  0 as t —<*3 . In this case the solutions of the differential equation for the process have the property that y —^ y as- t —v *c , where y i s the solution for a. 1(t) = 0. " " 0 O XX b. |J a ^ ( t ) d t | < . In this case y remains bounded and, i f | a ^ ( t ) | < where £^ are small positive quantities, y can be approximated in series form, as discussed in Appendix 1. The above two types of variations in a^ are the ones commonly encountered in actual processes. The f i r s t one occurs when there is a change in the value, of the parameters from one constant value to another. In that case i t i s not necessary to specify that f a^| << 1 to insure that y -^ v y Q as t -> ©a. An explicit solution may be obtained for the tracking behaviour of a one-parameter, Type I system. The error signal is then e = («<-,_- a 1)q 1(t) 25 Hence the tracking equation i s , from (2.6), dt = = - k(o<i _ a 1 ) q 12 (2.27) A 2 or o < x = a x(t) + (^ 1 0 - a 1 Q)exp(-A J o q 1 dt) (2.28) i f q-^(t) varies rapidly compared to | °<-j_ I t the term q^ in (2.27) can be replaced by i t s mean value q.^  . In that case (2.28) becomes - Aq n 2t o < 1 = & 1 ( t ) + (* 1 0'- a 1 Q ) 4 X (2.29) where ^ - J - Q * a ^o a r e ^ e v a - l u e s of °< ^ , at t = 0 . It can be seen that ©< ^  converges monotonically to a^. An estimate of the convergence time can be obtained from the value of the exponent,;Aq.j^ . It i s convenient to give a more accurate defini-tion of the convergence time; by assuming i t to be the time taken for the error - a]_) to reduce to th of the i n i t i a l Q error, which happens when the exponent i s 3. Hence the convergence time i s given by t„ = "4=^  (2.30) ° Aq The solution for multiparameter Type I systems becomes more complex because of the interaction present between the various parameter loops. In fact, i t is not possible to obtain an explicit solution of the type given by (2.28) for a one parameter system, for the case of a general input. However, i f the assumption i s made that | qjj»|e>< |, the terms involving 26 q^ can be replaced by their mean values, and equation (2.20) obtained as in section 2.2. An explicit solution of (2.20) can be obtained and an estimate of the convergence time found by replacing q.^  in equation (2.30) by the smallest characteristic root of the matrix B defined in section (2.2). If this root i s ^s* (2.31) The problem becomes more complex i f a solution i s attempted for a Type II system. The general equation for this is given by equation (2.6) or (2.21). If the assumption i s made that |q il> > K il , (2.21) becomes ||- = - ABS + f( S ) (2.32) where f( £) = 0( 5 ). In general, an explicit solution cannot be obtained for this equation. However, in the linear range, the second term may be neglected and an approximation to the transient behaviour obtained. Some caution is necessary before proceeding in this manner, since i t can lead to an erroneous conclusion; this arises from the fact that (2.32) is stable in the' linear range, as pointed out in section 2.4, while (2.21) is not necessarily stable. However, i f the system i s known to be stable, the approximation i s found to be reasonably accurate. 27 2.6. Effect of Constraints In a large number of processes, the control adjustment has to be modified by various other considerations; these modifications can generally be expressed as a set of constraints of the type • • • g j ( 5 i r a 2 , . . . 6^t b 1 , S 2,... <5n) > 0 j = 1, 2,,..m, m < n (2.33) The problem then becomes one of minimizing the performance criter-ion P. p = p( s ± f s2,...§n) subject to these constraints. The techniques for solving this problem by steepest descent methods have been discussed in the 17 18 * literature ' for the case when g.. does not depend on The procedure i s to define a new performance function m 2 5=1 P 1 = P + k 2 h , k >> 1 (2.34) where the function h. i s defined by 0 , gj >o 28 The adjustment equations for 5 ^  are then, - _ A .ii at d £ ± 1 j=l 1 (2 .35) Since k >> 1, the f i r s t term in the r.h.s. of the above equation may be neglected as long as the constraints are not satisfiea. It can be shown that the system then proceeas to a point on the curve m 1 - o 2 This follows from the fact that h^ is a positive aefinite function of the 8^ which has a minimum at any point where 2 ^h. =0, ana hence the steepest descent proceaure must approach such a point. Considering any point where h^ ^ 0, d £ h . 2 * dh. J - = 2 1 h dt ^~ j dt 0=1 m ^h, d5 , dh. d £ 0 Jb + 1- • d 3=1 dt dS2 dt *>h. dS a s n dt * d<5. » 3>h2 i=l d=l 29 n m 4k i=l 3h 3=1 as ^2 ( 2 . 3 6 ) The r.h.s. of the above equation i s negative semidefinite; i t can be shown by analogous arguments to those given in section 2 . 3 that §£(£h^ ) H > 0 a s t 4 « ; , which implies that h^ 0 as t ->*c . It i s clear that once the curve £h^ =0 has been reached, the system continues to move along this curve u n t i l the inequality sign in ( 2 . 3 3 ) i s satisfied, i.e. i t continues to "ride the constraint" u n t i l a point i s reached where the S ^ satisfy the inequality sign in ( 2 * 3 3 ) . The procedure may be extended to the case where g. i s a function of 8 ^ also, by defining a set of variables * ± = dt i = 1, 2,...n ( 2 . 3 7 ) In that case the constraints become gj( S 1 . . . S n , €1 . . . 6 n) > 0 j =1, 2 , . . . m The functions h^ may be defined as before and a new performance function found as m P 1 = P + k 2 h^ 2 k >> 1 j=l 30 The equations of the system are then i r 1 - - A c - s « 7 + a ^ ^ - r ^ 1 1 b 1 -a 1 3=1 1 (2.38) i r - = - % <-sSr> + 2 k ^  hd 45^  «- ^ 2 — a 3=1 \ 1 It follows from the previous arguments that the system w i l l again approach the curve ^.k-j =0 and move along i t u n t i l the constraint i s satisfied. As an example, a one parameter system may he considered, subject to the constraint Writing ( f f - ) 2 £ 1 (2.39) a new performance function i s defined as where Pn = P + kh 2, k » 1 (2.41) P = e 2 = [ S q ] 2 - S 2 ^ h = 6 2 - 1, £ 2 > 1 (2.42) =0 , ^ < 1 The adjustment equations are dS , f r - - A t f t (-If) + 2 ^ & = A[|»(-||-) + 4k^( ^ 2 - 1)] € - 2 > l 31 N e g l e c t i n g t h e f i r s t term on t h e r . h . s . s i n c e k > > 1, | f - = - 4Ak e ( 6 2 - i ) 2 1 or 1 - (1 - ) e x p ( - 8 A k t ) ( 2 . 4 4 ) o 2 Hence e 1 a s t A « c , The system hence approaches t h e curve 2 €• =1* Once the c o n s t r a i n t i s s a t i s f i e d , t h e system e q u a t i o n becomes I t i s apparent t h a t , as l o n g as t h e c o n s t r a i n t i s s a t i s f i e d a t the optimum p o i n t o f o p e r a t i o n , t h e a s y m p t o t i c b e h a v i o u r o f t h e system n e a r t h i s p o i n t as t °o i s n o t a f f e c t e d by t h e c o n s t r a i n t . The convergence time,.however, i s a f f e c t e d ; e.g. f o r t h e example c o n s i d e r e d above, t h e convergence time may be e x p r e s s e d as f f = - 2 A * q 2 ( 2 . 4 5 ) t c where time t a k e n t o r e a c h t h e curve £ - = 1 2 time t a k e n a l o n g the curve £ =1 u n t i l t h e c o n s t r a i n t i s s a t i s f i e d 32 This point may he termed a transition point; after i t has been reached, h = 0 and S is found by solving (2.45). S = S t exp[-2Aq2(t - 1 ^ ) ] (2.46) p Along the curve £ =1, S = 8 0 - ( t x - t 0 ) * S Q - t x since t — 0. Hence the time taken to reach S . i s o t t , = & - S , l o t and the subsequent time to reach 0.05 <5 i s , from (2.46) 8. - 0.05 S n  t ^ — S - j - 2 2 2Aq^ Hence t o = t 1 + t 2 = ( l - 2 4 | ) s 0 - d - - ^ ) * t 2Aq^ 2Aq^ which may be compared with t = 0 S without the constraint. 2Aq2 . 2.7 Ohoice of Performance Criterion The discussion in the previous sections has been 2 limited to the choice of P = e as the performance function. However, as indicated in section 2.1, the steepest descent method may be applied to minimize any positive definite function of e. 33 The actual choice i s governed by the configuration being used for control and the ease with'which the required control signals 2 may be generated. The choice P = e leads to the simplest kind of instrumentation; i f a class of polynomial criteria i s consid-ered, i t can be shown that i t also leads to a faster system. Considering a general criterion such as P = e , the convergence time of a one parameter, Type I system may be examined as a function of n. The adjusting equations are If - - * - 2 - . ^ e = qS or H_ = _ 2nA s 2n~l q2n Assuming that q 2 n can be replaced by q 2 n , the solution for an i n i t i a l value 5 i s o S 2 n - 2 t l + 4An(n - l ) ? = t a ^ " 2 ] = S f " 2 As t -><=o , 1 2n-2 [,' 1 ] 4Aq 2 nn(n-l)t , 2ri-2 It follows that the system response becomes slower as n i s increased, the fastest result being obtained for n = 1. 34 The criterion P = I e I has also been considered in the li t e r a t u r e , 1 ^ It can he seen that in the linear range, where S is small, i s of order 8 for P = e 2 and of order 1 for P = I e | , so that the systems produced using the latter can be expected to have a faster response in this range. However, the function i s discontinuous at e = 0 and this results in a tendency for the system to oscillate about this value. This may be corrected by incorporating a small dead zone into the •system at e = 0. :'In theory, this leads to a limitation in the accuracy of system tracking, hut, in practice, this limitation is negligible since the dead zone can be made very small and the tracking model can s t i l l perform effectively. 35 3. IDENTIFICATION SYSTEM 3.1. A General Parameter-Tracking Model The theoretical conclusions arrived at in the previous chapter may now he applied to the study of a particular parameter-tracking system. A general model for this type of system is shown in Fig. 2; the system may he considered as the identifica-tion, block shown in Fig. 1. x(t) Process P = 2 - D_ F i l t e r Gf (s) -<y-e(t) jrCti F i l t e r O f(s) H, Fig. 2. A General Parameter-Tracking System The error signal is written'in operator form as e =H 0G fy - H-^x-and becomes zero when H 0G fy = H-^x (3.1) (3.2) The signals y and x are related by the process dynamics, which may be written in operational form as Dy = Nx (3.3) or 36 H G The operator P is hence approximated by H Q ; here H.^  H 0 are operators containing time-varying co-efficients, and hence not commutable with G^ . Let v^ = Gfx v 2 = Gfy. If the parameters in D and N vary very slowly, G^ Nx = NG^ x and GfD:y = DG^y to a f i r s t order of approximation. Then Dv2 = NV-L (3.4) The equation (3.2) can he written as H 2v 2 = H 1v 1 (3.5) Comparing (3.4) and ( 3 . 5 ) , i t i s seen that the operator I? is H ' estimated as TJ1, when (3.5) i s satisfied, i.e. when e = 0. n 2 Substituting (3.4) into ( 3 . 1 ) , the expression for the error is obtained as e = H 2v 2 - H.^ + ( ^ - Dv2) = (H 2 - D)v 2 - (H x - N ) y i (3.6) In the linear range, when the parameters of H 2 are close to the parameters a^ of D and the parameters £ ^  of are close to the parameters b. of K, the operators (H ? - D) and (H, - N) may be expanded in a Taylor series about the point [a^,a 2.. .an,, b i rb 2...b m] in parameter space. The equation (3.6) then becomes 37 n 0 H~ ™ a>IL i - 3 3 i=i 1 3=1 J Writing q i ="3^ 72 P 3 = - "0771 the linear terms in equation ( 3 . 7 ) can he rewritten as ( 3 . 7 ) n m 3 3' 3 e = $ (°<i-ai)q i+2 ( f W r 1 (5-8) i=l 3=1 The above expression can be identified with the general error equation ( 2 . 3 ) . A special case of interest is when and Hg are realized in the form m H l = ^  <V3l(P) 3=1 n H 2 = <T °<i& 1 2(2) i=l where G ^ s ) , G i 2(s) are specified transfer functions. The process dynamics is then identified in the form of an operator 38 H * ^ V ( P )  H = H^ = n ^ (5.9) i=l If the error signal i s to vanish, i t must be possible to expand the operators D and N in the form n D = 1 a i G i 2 ( P ) i=l m N = § b . ; V ( P ) 3=1 . so that when = a^ and £ . = b^,. =: N, H D. The functions q. and r. may be recognized as q i = G i 2 ( P ) y or Q ± ( B ) = G ± 2 ( S )T ( B ) RJ(S ) - = G (B)X (B) where the capital letters refer to the Laplace, transforms of the functions. Since G-j±(s)» ^±2^B^ a r e l n d e P e n < i e n t of /2>y °<^, dq. d<* ± i - = o 39 Hence the system obtained i s a Type I system, A large number of tracking systems described in the literature may be obtained from Pig. 2 by a suitable choice of 1 2 H^ and H 2. Margolis and Leondes ' have discussed the choice 15 H 0 = 1; Mishkin and Braun describe several systems where G k l(p), & k 2(p) a r e or"thogonal polynomials in p; a simpler choice 7 i s treated by Zitamori and Narendra and McBride, where Gk]_(s), G k 2(s) have single poles or the transfer functions result in orthogonal networks. The choice G^(s) = G j ^ 8 ^ = s ^ results in the representation of H as a ratio of polynomials in p» In this case q i = p l ( J f y r.. = pJ'Gfx Q 2.0 This particular choice has been mentioned by Roberts and Eykhoff, though neither has reported any investigations carried out into i t s behaviour. The approach developed in the thesis and the classification into Type I and Type II systems results in a generalization of the investigations of these authors. The f i l t e r G f is chosen to avoid use of differentiations in the instrumentation necessary to generate the signals q^, r.. If n>m, the denominator of G^(s) must be at least of an order n greater than the order of the numerator. 3 . 2 . A One-Parameter Tracking System 40 A one-parameter tracking system i s obtained when the process is described by the operator p - K P + a o where K i s a constant gain and a Q i s the variable parameter to be identified. Accordingly, the identifying model becomes H 2 = P + *o 0 . = 1 f s + c The system equations are then e = ( < X _ a ) — - — y v o o' p + c^ do< If I a I <^ 1 , the substitution o can be made to yield f f + ce - 5 v = 0 dt d£ ~ = - A 5 [ - 4 —y] 2 dt o p + c ( 3 . 1 0 ) S o = « o - a o ( 3 . 1 1 ) ( 3 . 1 2 ) 41 Putting u = ' 1 "cy» the solution of (3.12) i s obtained as 6 = 5 ( o ) 6 " A ^ u 2 d t o o (3.13) d p If | ^  | i s large compared to | ^  0 |, u may he replaced by i t s mean value i n equation ( 3 . 1 2 ) . In that case SQ = SQ(O) f -Au^t ( % 1 4 ) In order that the above assumption be v a l i d , u, which i s the output of the f i l t e r G„(s) = 1 must be varying rapidly; t h i s i s true i f c i s chosen large enough so that the transients i n the output of the f i l t e r become ne g l i g i b l e before any appreciable change takes place i n the value of 5 Q . 3.3 A Two-Parameter Tracking System Consider a second order process K P = p + a-,p + a^ 1 o where K i s a constant gain. The corresponding i d e n t i f i c a t i o n model i s H 1 = K 2 H 2 = p . + * -LP + * Q G f ( s ) = (a + c j u + c 2 ) The error is then •• given by 42 e = H 2G fy - H-^x p 2 + < * l P + * 0 K (p+c-Jtp-t-Cg/7 " (p+c 1)(p+c 2) x = ( o < l - a l } ( p ^ H p + C g j y + ( < V a o } ( p + c 1 ) ( p + c 2 ) x Writing u i = (p^yfp+cg)^ Uo (p+c 1)(p+c 2) y the expression for the error becomes e = ( c ^ - a . ^ + ( * 0 - a 0 ) u o (3.15) The adjusting equations are a T ~ = - A e u i = ^ [ ( ^ - a ^ 2 •+ ("Va^u^] a^ (5.16) at - = " A£(°Vai>Vo + - ^ o " a o ) u o ] 2 2 Following the procedure aiscussea earlier^ u-^  , u Q and ^2^0 a r e r e P l a c e d hy their mean values. Again, in oraer to do this, the transients in the output of G^(s) must become negligible before any appreciable change occurs in «< , ©< , ; 43 this is true i f 0-^ ,02 ^ a n d m a y ^ e i m l ? r o v e a further by choosing 0.^  = 02 = 0. Then, substituting 5 1 8 8 *1 - a l 5 = <x - a o o o equation (3.16) becomes d a dt &6 dt i = - A [ S l U l 2 .+ S o ^ o ] = - A[ S U U + S u 2 ] L 1 1 o 0 o J (3.17) (3.17) may be written in matrix form as where d£_ dt = - B S 8 = B = A u l u l u o u-,u . 1 o u (3.18) 2 2 ™— 2 Since u.^  u Q > ^ u l u o ^ T s y s ^ e m ^ s stable independent of the gain. The characteristic roots of the matrix B are 44 A U l 2 + U o 2 + A / ( — —2,2 ^ 2' = - A § " 2 V ( u l " uo } + 4 u l u o 2 2 If u Q y u-^  , i t may be seen that the smaller of the two above 2 2 roots l i e s between and U q . The speed of convergence i s consequently less than that of a f i r s t order system, where the "r"~2 root i s U q „ As an example, a sinusoidal input may be considered,, u^ = sin^>t u = k sin(<^t+0), k > %. o o o ' a then u ^ =. ^2/2 u 2 = k 2/2 O 0 1 u 1u Q = k ^ k ^ s i n <*> t sin(wt+0)]av = k 1k o[ ±(cos0(l-cos 2«ot) + i s i n 0 sin 2^ot)]av = "t'k-j:&0 cos0 The characteristic roots are then A 1 , A 2 = -lACk^+k^) + iA ^ ( k l 2 ^ 0 2 ) 2 ~ 4 ^ l \ 2 s i n 2 ^ For 0 = 45°, s i n 2 0 = 4' and The best result i s obtained for 0 = 9 0 ° , when u-,u = 0 ; then 45 A = -=—> A - _ 2 — 1 2 2 2 This is the case for no interaction; for a l l other cases the speed of convergence is reduced. The convergence time may he estimated from the smaller of the two roots* For the system under consideration, u^ = pu Q; hence 0 = 90 and u nu = 0 : there is hence no interaction present in o 1 o * a 2-parameter system* However, for systems with a larger numher of parameters, the interaction i s present and reduces the speed of convergence«, 3 . 4 . Multiparameter Systems Systems with more than two parameters can he investigated hy the procedure outlined in the previous section and an estimate of the convergence time- obtained hy considering the smallest characteristic root of the system matrix obtained by replacing 2 u_^  etc. by their mean values. In actual performance, the estimates prove to be reasonably accurate even when the assump-tio'ri made (i-*e-» | u^|»|<5\jj ) is not satisfied very well. A tracking model for the process P = • 1 0 2 a 0 p + a n p + a^ 2r 1 O was simulated on a Pace 231-R analog computer, using a stochastic input and the following parameter values: a 2 = 1 = c < 2 c = 1 a x = 4 <j_(0) = 0 a Q = 4 °< o (0) = 0 46 The computer circuit diagram i s shown in Fig. 3 and Fig. 4 illustrates the recorded transient response for °< and °< ^ * If ^ ^ 0.6f the response time of a second order system 1 a l to a unit step input i s approximately -r where <^->m = ~ 5 m is the damping co-efficient. It is convenient to estimate the response time of the tracking system in terms of t^m» For the case illustrated, Input K  •+ a, p •+ % K c7 ( P + c ) 2 X X X X (P+cy Fig. 3 o Computer Circuit Diagram for Two Parameter Tracking System 48 Fig. 4. Response of Two-Parameter Tracking System I 49 DIRECT CONTROLLERS 4.1*- Compensation in the Forward Loop The diagram of the over-all system discussed in Chapters 1 and 2 i s shown in Eig, 1. It may he seen that this incorporates adjustable compensation networks both in the forward and feedback loops. A particular modification of this has been widely g Q Q J^Q discussed in the literature ' ' * where the compensation network i s only in the forward loop.- In this case the parameters of the network are used to cancel the unwanted parameters of the process directly, there existing a. direct correspondence between the network and process parameters. One of the advant-ages of the method i s that the identifying process i s made unnecessary; the output of the process i s compared with that of the reference model and the error signal thus obtained i s used to adjust the compensating network without any precise knowledge of the process dynamics being required. The diagram of the modified control system with com-pensation only in the forward loop i s shown in Eig. 5» The expression for the error signal can be written in operator x Compensation Process Ref. Model Fig.- 5. System with Compensation in Forward Loop 50 form as e = "i$p} • ifp} M ( p ) - M ( p ) J x = y - y ' ( 4 a ) N(p) , D(p). K(p), L(p) must be regarded as operators as defined earlier in section 1.2; they are hence non-commutable. let the parameters of N(p), D(p), K(p). l(p) he n^, d^, k^, 1^ respectively. '*, » If n^, d^ vary slowly enough so that n^, d^ are negligible quantities, e may be written as n m e - [ 1 <W |$ . fey- - 1 U r V feW i = i d = i + 0 ( 4 - 2 ) (4.2) where . = 1 . - n. § . = k. - d. i x l (4.2) may be written in the form e = 28±(i± + S s ^ r j + 0 ( s 2 ) (4.3) where _ Ms) I P 1 H = • ! L T P T M ( P ) X ^ = - r ^ ^ - i ^ M ( p ) x (4.4) It may be seen that q^ is independent of L but r. depends on £.; 51 consequently i f the adjustable parameters are only in K(p), £ . = 0 and a Type I system is obtained. This is the case when only unwanted poles of the process are to be cancelled and the reference model is M(p) E. j[jp) * l f there are adjustable parameters in i(p) also, a Type II system results, A general investigation may be performed for these systems, starting with the error expression (4 .3) , along the lines considered in Chapter 2 and similar results obtained for stab i l i t y and convergence. However, a more complete analysis can be performed for systems with one or two parameters only, which gives some insight into the behaviour of systems with a larger number of parameters and also provides confirmation of the general statements made about the behaviour of these systems in Chapter 2 . These are hence considered in the following sections* 4 . 2 . A One-Parameter System A number of one parameter systems are illustrated in Pig. 6(i.)', 6(ii) and 6 ( i i i ) , where the reference model has a transfer function — T = — in each case. The. most important fact s + ni o that has to be considered in connection with these systems is the actual configuration of the process and compensation net-work and the relation of the error signal to the controlled parameter Q> Thus,- superficially, 6(i) and 6(ii) describe the same system except that the positions of the process and compensation network are interchanged; however, i t may be demonstrated that the system of 6(i) i s unconditionally stable 52 1 . —< p + M 1 0 (i) p+m e p + m0 -* X "9 P+<*< I P+ >r\0 ( i i ) P-+ a 0 P + <*c p+nri,. ( i i i ) is P + o<, Fig. 6. Different Configurations for a One-Parameter Controller 53 for a l l inputs while the system of 6 ( i i ) i s not. The actual behaviour of the systems depends on whether the system i s of Type I or not, which, in turn, depends on whether there exists an operator block between the variable c<Q and the point from which the signal e i s derived. It can be seen that for a system to be of Type I e = o< q - a q o^o o o where q Q must be independent of °< q; hence ©< must occur as a gain in the operator block immediately preceding the point from which e i s obtained. If an operator block H(p) exists between the block containing o< and e, the equation for e becomes H(p)e = c< q - a q o^o o o In this case e is no longer a linear function of °( alone and the system becomes of Type II. This can be demonstrated by deriving the error expressions for each of the three systems illustrated; only the system of Fig. 6(i) turns out to be of Type I. System of Fig. 6(i) i . . \ * 1 . 1 1 e = (v+*o) ' pTa^ x " p ^ x If | a Q| « 1, so that a Q is essentially constant over the con-vergence time of °<0> let 54 1_ . l _ V _ P + a o P + mo X " Then ' 2 , ~ * + (a+m)#r + a m u = x - ua ^ 2 o o dt o o o ^-2 + (a +m ) + a m v = x d^.2 v o oy dt o o The term involving a Q can he neglected; hence u = v. The error e = (p+°<o)u - (p+aQ)v = (* 0-a o)v (4 .5) System of Pig. 6(11) e = — ^ ' P + O < 0 ' x - 1 x p+aQ p+mQ p+mQ Let y' = ^ " x ; "t^en (p+aQ)e = (p+°<0)y' - (p+aQ)y* or f| + a Qe = (*0-a0)y (4 .6) Sire t em of Pig. 6 ( i i i ) 1 ' p + a o 1 6 = P+* 0 P+m0X ~ P+m0X or (p+<* )e = (p+a )y' - (p+ <* )y o x o" , •• o o r f t + * 0 e = teo-vy df + a o e = ( ao" * o ) y ' + ( ao- « o ) e ( 4 - 7 ) 55 An examination of equations (4.5), (4.6), (4.7) shows that, since v is independent of o(Q, (4.5) is identical with equation (2,3) i f v is identified with q . Equations (4.6) and derivatives of ot . Thus only the configuration of Fig. 6(i) corresponds to a Type I system. The Type I system has an adjusting equation which may he solved to obtain the transient behaviour and con-vergence time as indicated in Chapter 3. Multiparameter systems of this type may be constructed by placing the compensation network in the same position as in Fig. 6 ( i ) . The conclusions reached in Chapter 3 are valid for a l l these systems and they may be investigated by the same methods. 4.3. One-Parameter Type II Systems A large variety of Type II systems may be obtained by varying the configuration of the process and tandem compensation, as shown above. The configuration of Fig. 6(i i ) may be investi-gated as a typical example. The error i s given by the differential equation (4.6); the parameter °(. i s then adjusted according to the equation o •°- = - k(ot -a )v 2 0 o dt d o t o = - Aeu (4.8) dt where £e 56 The signal u i s obtained as indicated in Fig. 6 ( i i ) . It may be written in terms of operators as 1 1 p + c < o p+X 0 P+aQ p+mQ Substituting y 1 = 1 x, o ^ + ( a + < O f r + a * u = U'+ oC y - uart ^ 2 o o dt o o dt oJ o If | a | 1, the above equation becomes f f + a o u = y > (4 . 9 ) Equation (4.9) can also be obtained from equation (4.6) by differentiating with respect to o< and interchanging the order of differentiation. Writing 2 = oC -a . o o o' the equations of motion for the system are t t + a o e = f | + a o u = y (4.10) d£ °- = - Aeu dt The f i r s t and the third of the above equations can be written in matrix form as f f = B(t)z (4..H) 57 where z = e B(t) = - o Au o The system of Fig. 6 ( i i i ) can he reduced to the same form by rewriting (4.7) as or || + a e = dt o _ S y« - S e o o dt o - y' - e + a u = - y» dt o J - e -d 8 dt~ = - Aeu (4.12) (4.13) (4.14) The equations (4.12) and (4«14) can be rewritten as dz dt = B(t)z + f(z) where z and B(t) are the same as in equation (4.11) and f(z) = (4.15) Equations (4.11) and (4.15) are valid for a large number of Type II systems. As indicated in Appendix 3» the solution of (4.15) depends on that of (4.11). Since f(z) i s of order z f the solutions of (4.15) approach zero as t t provided that the impulse response of the system described by (4«ll) i s bounded; 58 or i f , in particular, B(t) = B 1 + B 2(t) where B 2 0 as t-v<=c and B^ has characteristic roots with negative real parts. It ; i s not possible to obtain an analytical solution of (4.11) for' a general input y'; hence the behaviour for some particular inputs w i l l be discussed. Step Input Let y 1 be a step of magnitude k. Then from (4»9), , - a t u - J- ( 1 - * 0 ) ao Equation (4.11) becomes d_ dt k_ - f* ~^  r -a Q e 1 = U ao + -Ak 0 L oJ o J ro , Ak (. a„ -a„t o 0 0 or.' f f = (B± + B 2 ( t » . (4.16) where B, 0 as t -v o C , i t being assumed that a remains o essentially constant over the convergence time of B > The equation (4.16) i s proved stable in Appendix 3 and i t s solutions approach zero as t oC , provided the equation d z -p dt = B l z (4.17) is asymptotically stable. The characteristic roots of are • given by the equation 2 Ak' A + a A + sa-t.s 0 0 ao 59 or A - - * - 2 ao 0 Since the roots both l i e in the left-half plane, (4.17) i s stable and consequently (4.16) i s stable. In order to estimate the convergence time, the variable e is eliminate from the equations (4.6), (4.8), ( 4 . 9 ) . Differentiating (4.8) Substituting for |~ from (4*9) and ff- from (4.6), d 2 5 dt dt - = - A(ey' - a eu + u 8 y 1 - a eu) dt 2 d 2 b d S or —2-°- + 2a Q + Auy» $ Q = - Aey' (4.18) dt Substituting y 1 k, u = J- (1 - 6 ° v), ao d 2S d^- . . .2 - a t . . .2 - a t ^ + a _ 2 5 . . A e f r 0 + ^ L- S c- 0 (4.19) ,,2 0 dt a„ o a„ o dt o o If the r.h.s. i s omitted, the transient response of the free -a system approaches zero as g. ^ — t ; the actual terms on the r.h.s. approach zero at a much faster rate than this and hence the response of the system i s like the free response. The free response i s _ & ~ _ 0 - t 8 = C €• 2 sin( oat + 0 ) (4.20) 60 where C, 0 are constants and -o It may he noted that the convergence time depends on the actual location of the process pole a Q; i f i t s value i s small, the system converges very slowly and also, the frequency of oscillation co becomes large. Sinusoidal Input In general, the inputs to the system w i l l be very different from a step input and the behaviour of the system response w i l l also be quite different. It i s not possible to obtain an analytical treatment for the case of a general input; however, a sinusoidal input may be examined to detect some of the features of system behaviour i n response to these general inputs, and also to see the technique that can be employed to obtain an approximation to the system transient behaviour. Let y ' = k sin a t. If a>»)6Jt equation (4.6) can be solved by neglecting 8Q etc. fetf- a e = S y ' = & k sin tot dt I O 0*7 o or e l o — o j_ sin ( ot-0) ( u > + a o } where • tan 0 = - — 61 It may be noticed that the error signal is now of the form e = $ Q<i.0t with q Q independent of S Q5 i.e. the system behaves as a Type I system for sufficiently large values of co , and an estimate of the convergence time may be obtained as in Chapter 3, If co i s small, the equation (4.19) may be'considered. d 2 & d c% — + 2a n jTT-Z + Auy' 5 = -Aey' (4.19') air 0 a 0 From ( 4 . 9 ) , u = 6 5: sin ( wt-0) + 0 ( 8 ) (co 2+a 0 2D± 0 Substituting for u and y' and neglecting second order terms in i S Q, (4.19') becomes dS A V 2 — 5 - 0 - + 2&n - T T - ° - + ,x .• sin cot sin ( < o t - 0 ) . £ = -Aey' di; 0 a t V w + a o The above is a linear equation with time-varying coefficients in order to guarantee stability, i t i s necessary that the homogeneous equation d 2 Sn d S A V2 0 - cos(2 ot -0))S = 0 dt +aQ (4.21) be stable. The above equation i s similar to one obtained by Margolis and leondes; 1 i t i s demonstrated by them that i t may be reduced to a Mathieu equation, which i s unstable under certain conditions. The Mathieu equation i s 62 + (a-2q cos,(2c -0))z = 0 where Z = »t a — ° r z = <5 4 <° o and a, q are functions of a Q, A, k, <o . The condition for stability is a Q > , where ^ uis a complicated function of a, q. If co is large, a and q are small and consequently /u- is very small, thus ensuring stability; while i f <o is very small, aQ > /"-oa and the system is again stable. This confirms the conclusions reached earlier that i f w i s large, the system i s stable and i t s behaviour can be approximated by that of a Type I system; while i f co is very small, the behaviour of the system is similar to the response to a step input, which is stable. • The unstable cases occur when to is of the same order as <5 o and a Q is small, so that i t may be expected that a Q < <^.*o . Margolis and Leondes1 have demonstrated some examples of in s t a b i l i t y i General Input An estimate of the transient behaviour of the system in response to a general input is possible under one of the following two assumptions: a* The input varies very slowly, so that u, y' are essentially constant over the convergence time of the system. b. The input varies very rapidly, i.e. I y ' | » | 5 | 63 If assumption (a) holds, the transient behaviour may be approx-r imated by the response to a step input; while i f assumption (b) is true, the time-varying quantities y' etc. can be replaced by their mean values and the transient behaviour approximated by the behaviour of a Type I system. These two assumptions are generally sufficient to give an estimate of the convergence time of the system for a large number of inputs, including stochastic inputs. It may be noted that for slowly varying inputs, the behaviour of the system is like a second order system; while for faster inputs, the system behaves like a f i r s t order system. 4.4. n-Parameter. Type II Systems As in the case of a single parameter system, a large number of configurations may be constructed whose behaviour in the linear range around £^ = 0 corresponds to that of a Type II system. For example, additional parameters can be easily incor-porated into the systems illustrated in Fig. 6(ii) and Fig. 6 ( i i i ) . As observed in the previous section, some general conclusions about the system transient response can be arrived at only when the transient response can be approximated by that of a Type I system ;±n the.linear range, and that this approximation i s valid when the inputs to the system vary rapidly compared to the rate of change of the controlled parameters. For example, a two parameter system may be considered for which the process is defined by the operator 64 Proceeding along the lines discussed in section 4.2, the following differential equations are obtained for e, u , u-, , where u-1 " ^oc± (4.22) (4.23) (4.24) (4.23) and (4.24) may be substituted into (4.22); and, under the assumption that | 8 | , \ S , | << |y"| , e may be obtained as The adjusting equations for $ Q, 8 m a y n o w D e solved as described in section 3.3. The results obtained are the same as in that section, since e is seen to be identical with the error signal considered for the system described in i t . 4 . 5 . Compensation with Lead-Lag Network A common type of compensation network that can be used for control is the lead-lag network. The diagram of a system using this kind of compensation is shown in Fig. 7. = 8 u + § - U -o o 1 1 65 I fp+a.Xp+a,) P+b, P + l>x Process Lead-Lag Network Ref, Model b^ + bg = m^  b l b 2 = mo P + * , p + ^ i . Fig. 7. Compensation with Lead-Lag Network The system is assumed to be of a configuration similar to that of Fig. 6(i) for convenience of analysis; the results obtained may be used to approximate the behaviour of systems with con-figurations similar to Fig 6(ii) and 6 ( i i i ) . The expression for the error in terms of operators is given by f ( P + * n )(p+*o) i I P 2 +m l P +m o W^K^J "12 1 p +m1p+mQ Let v = p +m1p+mQ .(p+a 1) (p+a2) If | a-J , | a 2| << 1 and can be neglected, the expression for the error becomes e = [ ( ^ ^ ^ g - a ^ a g j p + ( 4 ^ 2 " a l a 2 ^ - ' v (4.25) 66 The signals ^ ^ — = u^, — = u 2, can be found by differentiat-ing (4.25). v± = (p + <* 2 ) v (4.26) u2 = ( p + < * i ) v (4.27) The adjusting equations for ^ and °< 2 then become &o( = - AUo^+^-a-^a^p + ( ^ 1 ^ 2 _ a 1 a 2 ) ] v . (p+^ 2)v d « (4.28) ^ j - 2 - = - A[(^ 1+< 2-a 1-a 2)p + (°< 1 ^ 2 - a 1 a 2 ) ] v . ( p W ^ v The assumption i s now made that |v| »|o<^| , so that the functions 2 v etc. may be replaced by their mean values. Remembering ~ ~~2 ~ w = 0, and writing v = g^, v = g 2, d* dt = " A [ ^ 1 + - < 2 - a 1 - a 2 ) « 2 + ^ 2 ( ° < l 0 ' 2 - a l a 2 ) « l ] do. 2 ( * . 2 9 ) dt = - A [ ( ^ i + ^ 2 . a i - a 2 ) g 2 4- ^ 1 ( ^ 1 ^ 2 - a 1 a 2 ) g 1 ] Writing 2 = << 1 ~ a 1 , &2 = ° < r 2-a 2, (4.29) becomes d8 dt - i = - A[( « x+ 3 2 ) g 2 + a 2 g 1 ( a 1 S 2 + a 2 ^ 1 ) ] + 0( 8^, Z = - A[ S 1(g 2+a 2 g l ) + S 2(g 2+a 1a 2g 1)] + 0( 8 ±2, 8 (4.30) ^r-= = -A[ ^ 1(g 2+a 1a 2g 1) + S^g^&^g^ ] + 0( S 2) 67 where a^, a 2 are assumed essentially constant over the convergence time of £ ^ , Equation (4.30) can he written in matrix form as + o( ^ 2 , s22) d_ dt or "V = - A . 2 . dS _ dt (g2+a22gl) ( g 2 + a i a 2 g l ) (g 2+a 1a 2g 1) ( g p + a ^ ) _ (4.3D where f ( 5 ) i s 0( S ). The stability of the singular point at 20 S = 0 depends on the behaviour of the solutions of the equation f f - = BS (4.32) The characteristic roots of B can be obtained from the equation A 2 + A(2g 2 + a 1 2 g 1 f a 2 2 g 1 ) A + (a^a^A^gg. = 0 21^  2 a, +a, or a 2+a 2 A = -A(g 2 i ^ 1 2 ° 2 g l ) - A I [ g 2 i 1 2 & 2 g - J 2 - ( a ; L-a 2 ) 2 g l g 2 The second term under the square root sign i s very small compared to the f i r s t term. Hence, an approximate value of A is given by 2._ 2 /_ _ 2^. > A f _ , a l 2 + a 2 2 N + , al ^ + a 2 ^ v 1(al-a2 )Xg2 A = - A (g 2 H ^ — g ± ) - A(g 2 H g S ±) ^ l - i ~~ 2—T" a l +a2 _ N2 (g 2' X 2 * g ±)'= - A(.2g2+a1^g1+a22g1), - A — i — 2 2g 2+a 1 2g 1+a 2 2g 1 68 Since both the above roots l i e in the l e f t half plane, the system is stable. The solution is of the form. c A 1 * A 2* S 1 = C1 € + C 2 * Since A » A ^ , the convergence time is determined by the smaller root A 2» In. fact, the part of the transient due to A becomes negligible before there i s any appreciable change in the transient due to A g . Thus, i f the system starts with an i n i t i a l value ^ ( 0 ) = + ,C2, i t moves quickly to make the term due to G equal to zero, subsequently, the term due to Cg is made zero much more slowly. The term is proportional to a^ Sg + a 2 5 ^ ; hence the system f i r s t reduces a^ 82 + a 2 8 ^ to zero i.e. i t reaches a point on the curve °^ 2 = a l a 2 * The overall behaviour of the system is comparable to that of a' system with an equality constraint *1*2 " a l a 2 = 0 It may be noticed that i f g^, g 2 are of the same order, the second 2 term in the r.h.s. of (4.29) is approximately a^ times the f i r s t term; hence the second term behaves like a penalty function on the system and the system moves so that this term i s reduced to zero f i r s t . Having once reached the curve <** 2 = a^ a,,, the system i s constrained to move along this curve un t i l oi ^ + oC 2 = a^ + a^, this movement taking place much more slowly since the larger forcing term on the r.h.s, is zero. 4*6. Example of a 2-parameter Type II system 69 A two parameter system was simulated on a PACE 231-R. computer using a stochastic input, with a process function P = Z p2+a1p+aQ and a compensation network P 2 + * 0  M _ _ . p + m-lP+ m with the following parameter values Z = 10 m1 = 10 * 1(0) = 0 a]_ = 4 mQ = 10 °<0(0) = 0 a Q =4 The computer diagram i s shown in Pig, 8. It was observed that an optimum value of gain exists for different levels of input and different values of the para-meters a^ and a Q . This may he expected, since i f the gain A is too small, the tracking system moves very slowly while i f A is very large, instability may occur. In fact, the system was found to be unstable for a gain exceeding 10 for the values of a., , a„ indicated above, 1 o The tracking behaviour of the system was studied by varying a^ and a Q according to a ramp of slope 0.01, 0.05 and 0.07-70 The nature of the response i s indicated in Fig, 9. It may he seen that in each case, the system became unstable when a^ and a Q reached a small enough value, the actual values being depend-ent on the gain used. On reducing the gain, the system resumed tracking the values of a^ and a Q. 4 , 7 . Example of Lead-Lag Network Compensation A two parameter system with a process function P = K (p+a1)(p+a2) was simulated on the PACE Analog Computer with a tandem compensa-tion network (p+^ 1)(p+oC 2) H = -g p +m1p+mQ The following parameter values were used K = 10 m1 = . 20 ^ ( O ) = 0 a1 = 10 mQ = 100 t < 2 ( 0 ) = 0 a 2 = 4 The computer circuit diagram i s shown in Fig, 1 0 . It may be noticed that u-^  i s obtained as = _ J _ . ( P + < * 2 ) ( P + * - l ) . 1 1 P+^ ± p 2 + m ip+m o (p 2+a 1)(p+a 2) and i s not exactly equal to = (p+ o 6 2)v, since the various 71 operators do not commute. However, in the linear range, when <*• i s small, the operators may in fact he treated as i f they were commutable. Since ot^(0) = o ^ ^ ) ' the two tracking loops are identical and w i l l arrive at the point «< ^  = << ^  = *J a]_a2 o n the curve ^ 2 ~ a i a 2 * Since ^\ ~ ^2* ^ e r e l a " ^ 1 0 n q l ' * l + q2 5 2 = 0 can he satisfied i f S = -'$2* 8 0 "kne system may never enter the linear range. However, random differences between the two loops result in small differences between a n d ^ which leads to eventual convergence. The tracking action in the para-meter plane i s shown in Fig. 11. It can be seen that after arriving on the hyperbola ^2.^2 = a l a 2 ' ^ e system continues to move along i t in the manner of riding a constraint. The con-vergence time for the system is much slower than the system described in section 4.6, as may be expected from the discussion on p. 68. Input p i +m ( p + m0 p + rn,p + m0 p"4m,p + m0 X I Pi + a,p+a0 \ P2+m,p+m0 X P +*,P + * e -* P x l X Pig. 8 . Computer Circuit Diagram for Two Parameter Type II System fV> 73 Fig. 9. Response of Two-Parameter Type II System 'nput X Q P + b. p + b 3 X p + b a (P+b.)(p + b J •*x] X P +«<x Fig. 10. Computer Circuit Diagram for Lead-lag Network Compensation Fig. 11. Response of Lead-Lag Network Compensation 76 5. CONTROL WITH IDENTIFICATION 5.1. Configurations of Indirect Controllers The over-all control system shown in Fig. 1 differs from the controllers described in the previous chapter in the fact that an estimate of the process dynamics is f i r s t obtained before proceeding with the adjustment of the parameters in the compensation networks'. Since the adjustment of the networks i s not carried out directly with the error signal obtained from the process and the reference model, these controllers may be called indirect controllers as opposed to those of the previous chapter. The estimation of the process dynamics is carried out with the identification model described in Chapter 3, and these estimated values serve as inputs to the adaptive loop. Since the optimum policy i s to make the over-all system dynamics identically equal to that of the reference model, whose parameters are known, a set of functional relationships may be obtained for the adjust-able parameters ft^ by equating the over-all system dynamics' to the model dynamics. These functional relationships w i l l have the form f ^ ^ , ^ . . . . ^ ) = «(. i = 1, 2 n (5.1) where °(^ are the estimated values of the process parameters. The solution of these functional relationships has already been discussed in section 1.2. In general, there are two methods available:-equations d T J l + 77 (a) The parameters are adjusted according to the A f ^ / ^ , /^2.../2»n) = A * ± i = 1, 2...n (5.2) If the system converges to some point on the parameter plane, d£ ^ — -» 0 as t -s. oc and the functional relations are a l l satis-fied at this point. (h) A performance function i s defined as n p = S [ f i ^ i > £ 2 * - - < V - * i ] 2 ( 5 * 5 ) i=l and the parameters adjusted according to the method of steepest descent. The adjusting equations are = _ A-|^— = - 2A]> (f Ju£i.i = l,2...n (5 .4) The two controllers w i l l he referred to as Controller I and Controller II respectively. In a practical case, Controller I has the advantage of leading to a simpler instrumentation; how-ever, stability cannot always he guaranteed for the method, while Controller II i s always stable. The stability problem becomes more d i f f i c u l t when simultaneous identification and control i s being carried out, as there is some interaction between the identification and adaption loops. It w i l l be recalled that the identification model of Fig. 2 i s of Type I, and i t may be shown that the Controllers I and II also behave like Type I systems 78 by themselves. However, the interaction between the two loops introduces further terms in the expression for the error signal, so that the over-all combined system no longer behaves like a Type I system. 5.2. The functional relations ^ " " ^ n ^ = °* i Let the forward loop network be represented by ^|^y and the feedback loop network by ^| p|. The operator represent-ing the over-all system may now be equaled to the operator representing the model K(p) . I>1(p)H(p) _ M1(p) Mpf ' P2(p)H(p) + G(p)Px(p) ~ MgTpT Stri c t l y speaking, a l l the operators on the l.h.s. are non-commutable; however in the linear range around the optimum point of operation, they may be treated as commutable, by neglect-ing second order effects. The operators P^(p), Pg(p) may now be replaced by their estimates P^'(p), P 2'(p). Then ^ P2'(p)H(p) + (}(p)P1'(p) M 2 t p ) or K(p)M2(p)P1»(p)H(p) = M 1(p)L(p)[P 2'(p)H(p) +G(p)P 1'(p)] (5.5) The coefficients of equal powers of p may now be equated on both sides, yielding the required functional relations between 7 9 and oC ^  for the system and model to he identical. These may be written after some manipulation, as f i ( $ v P 2 . . . . P>J = °<± i = 1 , 2....n (5i.l) It may be noticed that these relations are nonlinear in general since K(p), L(p), G(p), H(p) a l l contain the variable parameters let b-^ , ^ " " ^ n t e ^ e s o l u * i o n o f * n e equations ( 5 . 1 ) , and let $ ± = b ± + € ± . Then f i(h 1,b 2....b n) = oi± and, expanding f i in a Taylor series about the point (b 1,b 2...b n) in parameter space, n ( ^ f 3 .1=1 u J ^ ^ b j J ( 5 . 6 ) A linear range of operation may hence be defined in the neigh-bourhood of the point (b^,b2.,.b^) where f^ may be approximated by the f i r s t two terms of ( 5 . 6 ) . The equation ( 5 . 2 ) for Controller -I hence becomes - + A 2 ^ sr&- + = o ( 5 . 7 ) dt ^ J £ / i Similarly, the equation (5 .4) for Controller II becomes J=l k=l * 80 As indicated in Appendix 3, the stability of (5.7) and (5.8) depends on that of the equations obtained by rejecting the bt second order terms. (It Will be noticed that - -t i s evaluated at ft . = b . and is hence independent of (I '..) J t) J In practice, a large number of alternative configura-tions exist for the forward and feedback loop compensating net-works that can be used to make over-all system performance eaual to the reference model performance, since K(p), L(p), 0(p), H(p) can be freely chosen, with the restriction that ^fp}» H"fpj" must be physically realizable. This gives, the condition that equation (5.1) must be satisfied for positive values of fi> ^. This restricts the range over which the parameters a^ of the process may be tracked. For example, considering P1Cp) = K 1(p+a 3) K(p) = K(p+ P>±) 2 P 2(P) = P +pa-]_+a2 L(p) = p + £ 2 M1(p) = kQ G(p) = M2(p) = p+mo H(p) = 1, the functional relationships must be obtained from the equation ' K • P+fi x K^p+a^) _ K Q P + ^ 2 p 2+a 3 ip+a 2+^ 3E ; i(p +a 3) = P + mo or K v o 81 ^ 1 - ^ K l = a l -m o ^ l " a 3 ^ 3 K l = a 2 a0+a~(m -a.) m o *1 V a , = 2 ' " 3 V o "V o 3 A positive solution for fifi^* (^3 1 8 hence possible under the following conditions m >^ a,, either ni a, 0 - ^ 3 o 1 or a 2 > mo(a1-mo) mo < a 3 ' mo < a l and a 0 <C m (a--m ) ^ 0 l 0 Since mQ i s specified, this restricts the range over which variations in a^, a^, a^ may be compensated by this particular configuration. It is hence necessary to know the complete range over which these parameters are expected to vary before selecting a suitable configuration for the compensating networks. 5 . 3 . Stability of the General System The over-all system in Pig. 1 may be considered to be constructed from two separate control systems, one for identi-fication and one for adaption, each of which behaves like a Type I system by i t s e l f ; however, as stated in section 5.1, the 82 interaction of the two systems produces some additional terms in the error signal e, which can no longer he considered to he a linear function of the o \ and consequently the over-all system behaviour is like that 6f a Type II system. A linear range of operation may be defined for the system when c< ^  is close to the process parameters and the adjustable parameters fi ^  are close to the optimum values b.. If o< = a.+5 . and fi. = b.+ &. J i i i J 0 3 i t w i l l be shown that the interaction between the two systems 2 2 contributes terms of order ( o \ . j €• . ). In order to investigate the behaviour of the systems in the linear range, these terms may be neglected and hence, within this range, the two systems are essentially independent of each other and are stable. Referring' to Fig. 1, the output y i s related to the input V by the differential equation i [P2(p)H(p) + P 1(p)G(p)]i(p)y = P 1(p)H(p)K(p)V i (5.9) where H(p), G(p), L(p), K(p) contain the variables fi^. Hence y = P( fi±, P»2.... fij In the linear range, this may be expanded i n a Taylor series about the point (\>^ f b2....bm) to yield y = y 0 + 0 ( Similarly x = x^ + 0 ( £ . ) o .1 83 The error signal e is given by e = H 2G fy - H-^x (5.10) Hence the adjusting equations for & ^  are ^--v-fcr = _ A e (0 y ) + 0( S * j i The second order terms may be neglected in the linear range; then S^ is independent of & y The stability of the above set of equations may then be established by the ordinary proce-dure applicable to Type I systems. The adjusting equation for 6- . may be found similarly. d £ m n ST1 = ~ A 2 [ £ £ k c . k + 1 8 ±c<.±] + 0 ( * 2) j = 1, 2...m k=l i=l = 1, 2...n (5.11) (5.12) 84 where c ., , c'.. are constants. The equations (5.11) and (5.12) may he written in matrix form as f f = B ( t ) z + f ( z ) (5.13) where z = n m B(t) = - Al»l«n 0 ,-A0c' -A.c 2 nm 2 ml f(a) = 0 ( 2 2 ) If z(0) i s small enough, the solutions of (5.13) approach zero as t — o C , provided that f t • B(t)z; (5.14) 85 has solutions that approach zero as t - ^ o C and further i f the 20 system described by (5.14) has a bounded impulse response. The behaviour of (5.14) can be investigated by considering the f i r s t n equations that involve only the 6^; the solutions of these equations may then be regarded as inputs for obtaining €r .. These solutions can be considered as approximations to the behaviour of 6"V, 6- ^ in the linear range. The analytical solution 1for a one-parameter system is considered in the following section for some particular inputs. As in the case of the Type II systems examined in Chapter 4, i t is not possible to obtain a solution for a general input; however, the methods applied for investigating the one-parameter system can be.used to obtain an insight into the behaviour of systems with a larger number of parameters. 5 . 4 . A One-Parameter Indirect-Oontroiler The diagram of a one-parameter controller with a process described by the operator is shown in Fig. 12. The reference model has a transfer function M = - J -s+m and the system is controlled by an adjustable compensation net-work in the forward loop, The compensation network is represented by an operator block H = - J2± 86 0 p+m H l The process is identified hy a model g- where = 1, Hg = p+»< . The functional equation relating ft to °< i s simply £ = ©< . Hence the adjusting equation for is dt - V P ' and the adjusting equation for i s d«< . d F = " A l e y Let u = p^^j_» then the system i s described by the following set of equations:-e = | f + * y _ x (5.15) = (-a) y ff - - V ff = - A 2 ( P - « ) Writing (i = a + 6, «< = a + £ , y = u + A , the above equations may be rewritten as ££• + a.A = + f u . (5.16) 87 f f = - A 1 £ ( U + ^ ) ; or, writing in matrix form, d 8 dt -a 0 u -0 -k^T 0 0 +A 2 - A 2 6 0 -2A 1 5 4 u-A^a 2 0 p+ tri P P I P + a X At Fig, 12, One-Parameter Indirect Controller Step Input Let u d* dt dP> dt k, y = — k . Then 2,_2 (^k = - A ^ r f - a ) = - A]_ -a) A 2( (5.17) 88 Differentiating the last equation, JjT^ A2 dt + A2 dt or dt a a (5.18) Let = x-j^ , p = Xg, The above equation may be rewritten as dx 2 dt" = X l dx A k 2x 2 2 2 d T = " <V X 2 >X1 " A1 A2 V I + A1 A2 ^ X l (5.19) a a This system of equations has a singular point at x^ = 0, x^ = a. The nature of the singular point may be investigated by consider-ing a small deviation v^, v 2 from i t . Writing - v^, x 2 = a+v2, 2 2 2 dv -(A 2+A 1= ?(a+v 2) 2)v 1 - A A 2 k^(a+v 2) 3 + A A ^ a + v J 2 1 a a a dv 2 , v x The stability of the singular point may be determined by consider-ing only the linear terms.in v^, v,,. Then d V l - ( A 2 + A ! k 2 ) v i - AjAgk^g dv^= v 1 The nature of the singularity i s indicated by the characteristic roots. 89 A = - ^(Ag+A^) I ^ ( A p + A ^ r - 4A 1A 2k^ = - A 2, - A x k 2 The s i n g u l a r i t y i s hence a stable node; as may be expected, the two roots correspond to the roots obtained f o r the i d e n t i f i c a t i o n system and the con t r o l l e r without any interaction present. Sinusoidal Input Let u = k s i n 60 t; then f f + ay = k(ocos6>t + /Ssina>t) If oo > > I (h I , , y ~ — k j ° ^cos(cot-0) + r-£-±&ix.(c»t-$) (5.20) V u> +a sj to +a where , , _1 dD 0 = tan — El and a i s assumed t o be e s s e n t i a l l y constant over the convergence time of oC, ^ . Then |f = - A lk 2(o<-a) ^ 2 + f i 2 + ^ ^ s i n 2(cot-0) (^ 2 l ) co +a Eliminating oC as before, 90 d^ft , /. ,A ,2 A A 2 + ^sin 2(*>t-flhaft TTT + ^ 2 + A l K ,,2_,_ 2 ; d t dt oo +a + A ^ k 2 ^ f t V ^ s i n 2 ( ^ t - i Z l ) ( f i , a ) = 0 ( 5 # 2 2 ) +a U s i n g t h e same change o f v a r i a b l e s as f o r t h e s t e p i n p u t case, d_ dt V A l k -1 A^Agk 0 2 J a s i n 2(cQjt~0) A . "2 a s i n 2( «ot-0) 7 2 2 ^ A g K 2 2 0 0 2 J (5.23) o r = (B + ; C ( t ) ) v The c h a r a c t e r i s t i c r o o t s o f B a r e A = - A 2 , - A.jk so t h a t a l l s o l u t i o n s o f dv:' dt Bv' (5.24) approach z e r o as t - V c*3 . As shown i n Appendix 3, t h e same h o l d s f o r the s o l u t i o n s o f (5.23), p r o v i d e d || C || < c - ^ < b, — b t 2 where || v'|) £ Q 2 4 ~ . I n the p r e s e n t case, II Oil < A^Agk ; hence the system i s s t a b l e i f t h e i n i t i a l c o n d i t i o n s a r e c l o s e enough 91 to the point v-. = 0, v 0 = 0, so that c0 i s smaller than 1 9 . It may he noticed that though the above system is also a Type II system, there is no instability in the linear range indicated as was found for the examples discussed in Chapter 4» The transient behaviour for systems with more than two parameters can therefore be approximated in the linear range by the solu-tions obtained for the identification and controller systems without interaction. The behaviour of the identification system has already been investigated in Chapter 3; some examples of the behaviour of the controller w i l l be discussed in the following sections. 5.5. Performance of Controller I The 4-parameter system described on p.80 was simulated on a PACE 231-R analog computer with a Controller I being used for adjusting the parameters of the compensation network. The process is described by the operator 2 p +a1p+a2 i The reference model has a transfer function K M(s) = — ° -s+m0 The compensating network in the forward loop i s represented by L T P T P+ |£ 2 92 while the feedback loop contains an adjustable gain ft> ^, The functional relationships between £^ and are given on p.8t£; accordingly, the adjustment equations for the parameters are of + A 2 K l ' E = A2 Ko < 5' 2 5 ) d £ 2 + A 0 |S 0 = A 0<X„ (5.26) dt 2 r 2 2 3 There are two possible choices for adjusting ft^ and £ ^  since there are two more functional relationships that involve both these parameters and either one of these may be selected for adjusting either one of the parameters. The choice used in the simulation was d£. dt + A 2 ( m o / 3 1 - c < 3 K 1 ' p 3 ) = A 2 * 2 (5,27) d P_K_ * I F^-=- + A 2 ( E l - ^ 3 - ^ ) = A g d n ^ ^ ) The computer circuit diagram is shown in Fig. 13. In the linear range, the substitution £ . = fi.-b., <5j_ = ^ i ~ a i m a y D e made. Then the equations (5.25), (5.26), (5.27).become E | ^ ( A K ) + A ^ A E = A 2 A K ' + 0 ( ^ K 2 ) 2 + k06 . = A c S , (5.28) dt 2 2 2 3 d 6 i + A 2(m oe 1-a 3K 1e 5) = A g( 5 g+K^ 6^) + 0 ( $ 3 2 ) dt 95 The last two equations may he written as = " A 2 m 0 "a 3 " [ 6 i - + A 2 d dt - 1 1 L- 5i + o ( S 2 ) Ur = B 6 + P(<5 ) + 0 ( < 5 2 ) ( 5 . 2 9 ) The characteristic roots of B can be obtained from the equation A 2 + A 2(l+m o )A + (m0-a3.)'A22 = 0 The system is stable i f mQ > a^. Since S —> 0 as t <*> with a stable identification system, £ also approaches zero as t->«o. The response of the system obtained using the following parameter values mQ = 1 0 a 2 = 5 K Q = 1 0 0 fi±(0) = 0 a i = 6 a^ = 4 Kj^  = 1 0 £ ( 0 ) = 0 i s shown in Pig. 1 4 . The response was also calculated using equations ( 5 . 2 7 ) , assuming the ol to be constant over small intervals of time; the results obtained are also shown in Pig. 14 and the agreement with the experimental results i s good. 5.6. P e r f o r m a n c e o f C o n t r o l l e r I I . 94 The s y s t e m d i s c u s s e d i n s e c t i o n 5.5 was a l s o s i m u l a t e d u s i n g a C o n t r o l l e r I I f o r a d j u s t i n g t h e c o m p e n s a t i o n n e t w o r k s . The c o m p u t e r c i r c u i t d i a g r a m f o r t h i s i s s h o w n i n F i g . 15. The a d j u s t i n g e q u a t i o n s f o r K a n d /^>2 a r e "the same a s i n t h e p r e v i o u s c a s e . The r e m a i n i n g t w o p a r a m e t e r s a r e a d j u s t e d h y s t e e p e s t d e s c e n t , u s i n g t h e p e r f o r m a n c e f u n c t i o n P = [/^-V P 3 - * 1 + m o ] 2 + [ ^ 1 - * 3 V f V * 2 ] 2 ( 5 * 3 0 ) The a d j u s t m e n t e q u a t i o n s a r e t h e n d£> 1 _ dt = - A 2 ( l + m o ^ ) ^ 1 + A 2 ( l + m 0 ^ 5 ) K 1 « P> ? + A 2 ( - m Q + o ^ + m ^ 2 ) - f e C ^ ^ j ) = A 2 ( l + m o ^ 5 ) ^ - ^ ( 1 + * 2 ) X j / ^ (5.31) + A 2 ( m o - o ^ 1 - ^ 3 ^ 2 ) The e q u a t i o n s a r e s e e n t o h e s i m i l a r t o t h o s e o b t a i n e d f o r C o n t r o l l e r I p e r f o r m a n c e . A n e q u a t i o n s i m i l a r t o (5.29) may b e o b t a i n e d f o r o p e r a t i o n i n t h e l i n e a r r a n g e b y w r i t i n g A. = b . +6 . . a n d oi = a . + £ . . T h e e q u a t i o n t h u s o b t a i n e d 2 2 2 i 1 i may b e w r i t t e n a s dj_ dt = B £ + f ( S ) + 0( S2) where B = - A , 1+m - l - m 0 a 3 -1 -m a , o 3 1+a 2 95 The characteristic roots of the matrix are A i , A 2 . . - ( n l ^ L j A j ± . ± A 2 J ( 2 + m 0 2 + a 3 2 ) 2 - 4Cm 0-a 5) 2 The system can he seen to he stable. The transient response of the system obtained with the same parameter values as in the previous section i s shown in Fig. 1 6 . The over-all response is very similar to that for a Controller I, the present system having a slower response for the same value of A 0. 96 Pig. 13. Computer Circuit Diagram for Controller I .97 — Theoretical Response x Experimental Values ( 2 3 4 5 6 7 8 * ? ' 2 3 n 5 6 7 b 1 Pig. 14. Response of Controller I 98 F i g . 15* Computer Circuit Diagram for Controller II 100 6. CONCLUSION It has been shown that systems of the type illustrated in Fig. 1 can be investigated by considering a single error signal (2.3) and the adjustment equations (1.10). A further simplification in the investigations can be effected by defining a Type I system as one in which the functions q^ are independent of the adjustable parameters. A Type I system i s stable for a l l possible inputs and i n i t i a l conditions provided the condition on p. 18 is satisfied guaranteeing the q to be linearly independent. An estimate of the transient behaviour of a Type I system can be obtained. If a system is not of Type I, i t can be designated as Type II. It has been shown that the behaviour of a Type II system can be approximated by that of a Type I system within the linear range defined in terms of S^. It is not possible to guarantee stability for Type II systems uncondi-tionally? but i t has been observed experimentally that the conditions under which these systems become unstable occur infrequently. The tracking systems derived from Fig. 1 are relatively simple to instrument in terms of the operator blocks described in the thesis; these operator blocks are shown to result in adequate approximations for the exact mathematical functions within the linear range of operation. The linear range is of special interest, since i t i s expected that the control system w i l l be operating within this range i.e. close to the optimum point of operation for the most part. However, neither the signal representation (2,3) nor the approximation by a Type I system behaviour i s valid once the error signal becomes large. 101 Further investigations are required to obtain an adequate estimate of the behaviour of the general system outside the linear range of operation. 102 APPENDIX 1 Pole-zero Cancellation for Systems with Time-Varying Parameters The technique of compensating the behaviour of control systems by canceling unwanted poles and zeros of the process transfer function by zeros and poles of the transfer function of a compensating network is well-known and described in the literature of classical oontrol t h e o r y . - ^ , 1 4 y^ie method may be extended to systems with time-varying parameters provided certain restrictions are placed on the behaviour of these time-varying parameters. In particular, i f the system is described hy y = Hx where H i s an operator as described in section 1.2, and ^ and a^ are the time-varying parameters in the numerator and denominator of H respectively, the case of interest is when °^(t) ELa^(t), and i t must be determined i f H = 1 in such a case; also i f **-j_(t) — a ^ ( t ) as t —b. oC , i t must be determined i f y —> x. It is being assumed here that the numerator and denominator of H are of the same order. The parameters may be rewritten as follows:-a i(t) = a 1 Q + a^Ct) i = 1 , 2 . . . . n ( A . 1 . 1 ) ^ ( t ) = a i 0 + S ± ( t ) i = 1 , 2 . . . . n The system equations can then be expressed in the form 103 |f + (A Q+A 1(t))y = ( B O + B 1 ( t ) ) x (A.1.2) where y, x are column vectors and A, A^, B Q , B^ are nxn square matrices. The stability and behaviour of (A.1.2) is connected with the behaviour of |f + (A o+A 1(t))y = 0 (A.1.3) This equation is discussed in Appendix 3; i t is shown that i f A^ has characteristic roots with negative real parts and i f \ JlJ^Ct)."!! .dt < oO , the solutions of (A.1.3) are bounded; o further, they approach zero as t i f IJiA^t)//). —± 0 as t — ^ o < 3 . If Y(t) is the fundamental matrix of solutions for (A.1*3), the condition that the solutions of (A.1.2) be bounded for bounded x requires: 1. Y _ 1 ( t ) exists and ll.YY"1//. is bounded for t >t 2. //.B-^t) || i s - bounded for t > t If the solutions of (A.1.2) are bounded, they may be expanded in a series of the form"'""'" y = y 0 + f'y-L + P 2y 2 + •• where y is the solution for A, = 0 and f = 'II^ JL It can be 11 . ^ A l shown that i f P 1 or i f '.7).^—ill <C«=C(1, the series converges. A f i r s t order approximation may then be made to y. 104 Writing (A.1.2) as g + A Qy = B Qx + B 1(t)x - A 1(t)y (A.1.2') The last two terms on the r.h.s. may he considered as perturbation terms. Substituting y = y Q f Py^, d y o + A y = B x dt o^o o Since A Q has characteristic roots with negative real parts, y — x as t <?C . Let Y be the fundamental matrix of solutions ° o o for the above equation, dy dt or 7i = i + = B 1(t)x - A x ( t ) y o Y ( t ) Y _ 1 ( r )B]_( r)x( t ) d T - j " Y(t)Y _ 1( T )A 1( -c)yo( r ) d r o (A,.1.4) If A = B and B, A. as t-V"< , y = x for t large enough o o 1 1 1 J o e & and y^ — ^ 0 as t -V<^3. Hence y — x to a f i r s t approximation. The f i r s t term on the r.h.s. of (A.l.4) can be written as JYY ^ x d t = B l J T T"" 1xdr - J dB d r 2 , Y Y _ 1 x d T . f d r 2 dB If l/~-^f=l|. <T<^ 1, the second term may be neglected. Then JYY-^xdr ^ B ^ x 105 where ^ B' is the operator defined on p. 5i Then It follows that the solution y of (A.1.2) can he expanded in the form y = Y 0 + B I T B ^ x + A i ^ ; x around the point B, =0, A, =0, as stated on p.-14. 106 APPENDIX 2 Stability when ^ is negative semidefinite It has been established in section 2.3 that i f n 3 i=l is chosen as a Lyapunov function for the system described by equation (2.6), i t s derivative i s obtained as This shows that V i s negative semidefinite and hence i t follows that the system i s stable but not necessarily asymptotically stable. It can be proved, however, that in such a case V — ^ 0 as t — > o o . Since V i s bounded as t -s> 0 0 , V — c ^ for t large enough, where c^ i s some constant. A heuristic argument would be as follows: by the mean value theorem, V ( t 2 ) - V ( t 1 ) = ( t 2 - t 1 ) V ( r ) , t 2 > r. > t 1 If t 2 , t^ are large, the l.h.s. approaches zero; hence V ( T ) - V O as "C — 0 , 0 . The result may be proved more rigorously by stipulating that V satisfies a Lipschitz condition, i.e. < |.v(t2) - v c t ^ i < c ( t 2 - t 1 ) 107 where c i s a p o s i t i v e constant, t 2 > "t^ , Then, since V ( t ) f~ 0, V ( t 2 ) < V^) + cC'tg-^) Considering C 2 • T(t2) - V t ^ ) = \ V ( t ) 4 t Since Z > t ^ r we have *2 V ( t 2 ) - Y(t±)^ j [ V ^ ) + c ( c - t 1 ) ] d r t. J l 2 t 2 ^ v ( t 1 ) ( t 2 - t 1 ) + icCtg-^) 2 V(t ) ' S u b s t i t u t i n g t 2 = t 1 ~ - ( > t 1 since V ( t 1 ) 0) v ( t n ) 2 V ( t ) - Y(h) < - - g j -V ( t n ) 2 or I f now t-^ o O , V(t-^), V ( t 2 ) -> c, and the l . h . s . — V 0; hence V(t- L) 0. This proves the r e s u l t c i t e d i n s e c t i o n 2 . 3 . 108 APPENDIX 3 Stability of Differential Equations with Time-Varying Co-efficients 20 The following theorems cited by Bellman are quoted without proof. Theorem 1 If a l l solutions of | f = Ay ' (A.J.I) where A i s a constant matrix, are bounded as t —> °<3 f the same is true for the solutions of ^ || = (A + B(t))z (A.3.2) provided ^ IIf.B(t) ||. dt < ^ . o Theorem 2 If a l l solutions of (A.3.1) approach zero as t — ^ c < 3 t the same holds for the solutions of (A.3.2), provided that "||i;B(t)'|| ^ for t > t Q , where c-^  i s a constant that depends upon A. Theorem 3 If Z i s the fundamental matrix of solutions for (A.3.2), the equation | f = (A + B(t)>+ 0(t) (A.3.3) has bounded solutions for bounded 0(t) i f O Z Z - 1 ) ) dt <C oo 0 109 This is equivalent to stating that the impulse response of the system described by (A.3.2) i s bounded. Theorem 4 Consider the two equations (A.3.3) and || = (A + B(t))z + f(z,t) (A.3.4) where || f - i | * ^ - - || — > 0 as ' |j z '// 0 . If the solutions of (A.3.3) are bounded for a l l bounded 0(t), the solutions of (A.3.4) are also bounded; further, they approach zero as t-^ < ?°, i f the i n i t i a l values are small enough. 110 REFERENCES 1. Leondes, C.T»,: and Margolis, M», "A Parameter Tracking Servo for Adaptive Control Systems," IRE Wescon Conv. Record, pt. 4, .pp. 104-115; 1959. : 2. Leondes, C.Ti, and Margolis,-M., "On the Theory of Adaptive Control Systems - A learning Model Approach," Proc. IFAC Congress. (Moscow), Vol. II, pp. 556-563 3. Narendra, K , , and McBride, I.E., "Multiparameter Self-' Optimizing Systems Using Correlation Techniques," Trans. IEEE. Vol. AC-9, pp. 31-38; Jan., 1964. 4. McGrath, R.J., Rajaraman, V., and Rideout, V.C., "A Parameter-Perturbation Adaptive Control System," Trans. IRE, Vol. AC-6, pp*'154-161; May, 1961. 5* Rajaraman, V., "Theory of a Two parameter Adaptive Control System," Trans. IRE. Vol. AC-7* pp. 20-26; July, 1962, 6. Puri, N.N.., and Weygandt, C.N., "Transfer-function Tracking and Adaptive Control Systems," Trans. IRE, Vol. AC-6, pp. 162-166; May, 1961. 7» Kitamori, T., "Applications of Orthogonal.Functions to the. Determination of Process Dynamic Characteristics and to the Construction of Self-optimizing Control Systems," Proc. IFAC Congreas (Moscow), Vol. II, pp. 613-618; 1961. 1 8. Bongiorno, J.J., Jr.', "Stability and Convergence Properties of Model-Reference Adaptive Control Systems," Trans. IEE, Vol. AC-7, pp. 30-41; April, 1962. 9. Roberts, J.D., "A Method of Optimizing Adjustable Parameters in a Control System," Proc. IEE. pt. B, Vol. 109, P P * 519-528; Nov., 1962. 10. Eykhoff, P., "Some Fundamental Aspects of Process Parameter Estimation," Trans. IEEE. Vol, AC-8, pp. 347-358; Oct., 1963. 11. Gibson,- J», "•Nonlinear Automatic Control," McGraw H i l l , 1962. 12. Graham, D», and McRuer, D., "Analysis of Nonlinear Control Systems." Wiley, 1961. 13. Bohn, E.V., "The Transform Analysis of Linear Systems," Addison Wesley, 1964. I l l 14. Truxal, J. G., "Automatic Feedback Control System Synthesis," McGraw H i l l , 1 9 5 5 . 15. Mishkin, E., and Braun, L., Jr.., (ed.), "Adaptive Control . Systems." McGraw H i l l , 1 9 6 I . " ' 16. Chang, S.S.L., "Synthesis of Optimum Control Systems," McGraw H i l l , 1962. ' 17. Leitman, G., (ed.) "Optimization Techniques," Acad. Press, 1962. 18. Levine, L., "Methods for Solving Engineering Problems," McGraw H i l l , 1964. ! 19. Beckenbach, E., (ed.), "Modem. Mathematics .for the Engineer," McGraw H i l l , 1956. 20. Bellman, R», "Stability Theory of,Differential Equations," McGraw H i l l , 1953* 21. Minorsky, N., "Introduction to Nonlinear Mechanics," J.W. Edwards, 1947. 22. Cunningham, W.J., "Introduction to Nonlinear Analysis," McGraw H i l l , 1958. 23. Bonn, E.V., Butler, R.E., and Mukerjee, M.R., "Parameter Tracking Models for Adaptive Control Systems," to appear in Proc. IEE. 

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