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Design studies of a class of multivariable feedback control systems Baird, Charles Robert 1962

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DESIGN STUDIES OF A CLASS OF MULTIVARIABLE FEEDBACK CONTROL SYSTEMS by CHARLES ROBERT BAIRD B.E., Nova Scotia Technical College, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard Members of the Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia September 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allox^ed without my written permission. Department of -JEle, e tui cal Engineering The University of British Columbia, Vancouver 8, Canada. Date Oct. 4, 1962  ABSTRACT Methods of designing mult ivar iable feedback control systems based on system eigenvalues and matrix d iagonal izat ion are discussed. It i s shown that these methods allow s ing le -var iab le graphical analysis and design techniques to be applied to mult ivar iable systems. The experimental determination of system eigenvalues i s shown to be f eas ib l e . The s u i t a b i l i t y of these methods i n conjunction with simulation studies for inves t igat ion and design purposes i s also shown. A simulated two-axis tracking system i s used to compare the eigenvalue method and the diagonalized method. The eigenvalue method i s applied to a system of four para l l e l -operated synchronous machines and graphical methods of s t a b i l i t y inves t igat ion are discussed. ACKNOWLEDGEMENT The author would l i k e to thank the supervising professor of th i s projec t , Dr. E . V . Bohn, for his help and guidance during the course of th i s research. The author i s indebted to the B r i t i s h Columbia E l e c t r i c Company Limited for a scholarship awarded i n 1960 and to the National Research Council for addi t ional support from the Block Term Grant (BT-68) i n the form of a Research Ass i s tantsh ip . i i i ? TABLE OP CONTENTS page XjXS*b O f I 1 1XIS ~t I* £l"fc IL O JtTS * t m 9 9 * o * Q 9 » a » o Q m o * 6 0 9 * » n a e 9 v v L i S "fc O f T £lt) 1 © S • « » « « « « « * 0 9 » O 0 a o » o 0000000 • • o 0 » 0 » # • « V l l l Acknowledgement . . . . . ix 1. Introduction . 1 2. The Eigenvalue Method of Analysis and Design . 7 2.1 I l l u s t r a t i v e Example 2.1 . . . . . . . . . a . . . . . . 13 2.2 I l l u s t r a t i v e Example 2.2 16 2.3 I l l u s t r a t i v e Example 2.3 18 2.4 Test Example 2.1 . . . . . . . . . . . 21 2.5 Test Example 2.2 27 3. Noninteraction 41 3.1 Diagonal izat ion of Open-Loop Transfer Matrix 42 3.2 I l l u s t r a t i v e Example 48 4. The Two-Axis Tracking System A Comparative Study of the Eigenvalue Method and the Diagonal izat ion Method 55 4.1 Open-Loop Two-Axis Tracking System 55 4.2 Closed-Loop Two-Axis Tracking System . . . . 58 4.3 Simulation Study of Two-Axis Tracking System . 60 4.4 The S t a b i l i t y of the Open-Loop System . . . 64 4.5 The S t a b i l i t y of the Closed-Loop System . 67 4.6 Diagonal izat ion of Open-Loop Transfer Matrix . 69 iv page 4.7 S t a b i l i t y Comparison ..................... 72 4.8 Mean-Square E r r o r C o n s i d e r a t i o n s .....»<.». 75 4.9 Comparative Comments .....a................. 78 5. Eigenvalue Method A p p l i e d to a S t a b i l i t y Study of Pour P a r a l l e l - C o n n e c t e d Synchronous Machines 83 5.1 Nyquist S t a b i l i t y I n v e s t i g a t i o n .......<,.<, 88 5.2 Root-Loci S t a b i l i t y I n v e s t i g a t i o n o . . . . . . . 90 6 a COITClUSlOn 0 « 0 « 0 0 o « « 0 « e 0 0 0 « 0 O 0 0 0 0 0 O 9 0 3 0 O o o 0 e i o o o 94 AppGIlCLiX. X aatfevtfoesootfooeooeeeoooeoeeoooeeoeooeoos 93 Ap P 6 n d 1 X XX a e o o 0 O 9 0 0 0 » 0 0 * 0 0 o a 0 o « 0 0 0 0 o a 0 o o o a 0 e o e 0 d t i 97 R© f G J T G I 1 C G S 9 0 9 0 0 0 O 9 O 0 0 9 9 0 0 9 0 9 9 0 0 0 0 a 0 0 0 0 0 C 0 0 0 0 0 0 O 0 0 0 99 V LIST OP ILLUSTRATIONS Fi g u r e page 1- 1 A L i n e a r M u l t i v a r i a b l e Feedback Co n t r o l Sy S ~fc 6 ITl 0 O 0 » 0 0 O 0 0 6 0 e 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 ' 0 » « 0 e O « 2* 2- 1 E q u i v a l e n t Single-Loop System . . . . . » . . » . . « . 9 2-2a Gain Margin m = 20 Log 1 0|(x k(3« 1)G c( . 11 2-2b Phase Margin 0 = arg G ^ j c ^ ) - arg x~(ja> ) 1 1 k c 2-3 N i c h o l s P l o t by Bode P l o t Technique ....... 12 2-4 I l l u s t r a t i v e Example 2.1 ............. 13 2-5 Matrix Representation of Fi g u r e 2-4 ....... 14 2-6 Nyquist P l o t of j^) a n d G c(jw) ......... 17 k - i 2-7 N i c h o l s P l o t of ^ and Gc(j<o) ......... 17 k 2-8 Nyquist P l o t of a n d G c ( j 6 > ) ......... 19 k 2-9 N i c h o l s P l o t of x~(j<o) a n d G C ^ W ^ 1 9 k 2-10 Nyquist P l o t of - X f c ( j t t ) and G 20 2-11 Matrix Form of Test Example 2.1 21 2-12 C i r c u i t Form of Test Example 2.1 21 2-13 Computer S i m u l a t i o n of Test Example 2.1 ... 22 2-14 Nyquist P l o t f o r Test Example 2.1 ......... 26 2-15 Matrix Form of Test Example 2.2 ........... 27 2-16 C i r c u i t Form of Test Example 2.2 .......... 27 v i page 2-17 Computer Simulation of Test Example 2.2 . . 29 2-18 Nyquist Plot of Table 2.1 31 2-19 Nyquist Plot of x "( 3 4 2-20 Nyquist Plot of Xk(jfl>) and G "( j t t ) 3 6 2-21 X-Locus from Table 2.3 38 2-22 X-Locus from Table 2.3 38 2- 23 X-Locus from Table 2.3 39 3- 1 , Open-Loop System 42 3-2 Diagonal izat ion of Open-Loop Transfer Matrix 44 3- 3 Two-Axis Tracking System 49 4- 1 Two-Axis Tracking System 56 4-2 Nyquist Plot of Rjjjtt) = - X k ( j « ) + F ( j » ) . 59' 4-3 Two-Axis Tracking System 61 4-4 Arrangement for Simulation of Two-Axis Tracking System 62 4-5 Simulation C i r c u i t for Two-Axis Tracking System 63 4-6 Nyquist Plot for Test 1 66 4-7 Nyquist Plot for Test 3 70 4-8 Nyquist Plot for Test 4 70 4-9 Block Diagram of Diagonalized Two-Axis Tracking System 73 4-10 Network Analog of the Diagonalized Two-Axis Tracking System 74 v i i page 4-11 Eigenvalue System Response to Single Rectangular ¥ a v e Pulse (Very l i t t l e 4-12 Diagonalized System Response to Single Rectangular ¥ a v e Pulse (Adequate Damping) 76 4-13 System ¥ i t h Noise i n Input . . 75 4-14 Experimental Determination of e 78 4-15 Eigenvalue System Mean-Square Error . . . . . . 79 4-16 Diagonalized System Mean-Square Error . . . . 79 4-17 Eigenvalue System Compensation 81 4-18 Diagonalized System Compensation . . . . . . . . . 81 1! 4- 19 Approximate T Network 82 5- 1 Single-Loop Parallel—Operated Synchronous 5-2 Equivalent Mult i -Loop Single Variable Sy S "t 6 HI O 0 0 0 0 0 0 0 0 0 0 0 e « O e 0 0 0 0 0 » 0 O 0 0 0 « « 0 « * 0 * « 89 5-3 Nyquist Plot of R k ( j « ) = - y - + G(j<o) 91 5-4 Root-Locus Plot of the Function X, K 92 v i i i LIST OP TABLES Table page 2.1 Experimental Results of Test Example 2.1 . . 31 2.2 Experimental Results of Test Example 2.2 . . 32 2.3 Experimental Results of Test Example 2.2 . . 37 3*1 E X ^ G I T V ct X U 6 S « o e » o « « 0 0 « « a o o « 0 0 a 0 0 * 0 « « 0 » o 6 0 0 « 38 DESIGN STUDIES OF A GLASS OF MULTIVARIABLE FEEDBACK CONTROL SYSTEMS 1 . INTRODUCTION In the design of conventional s ing le -yar iab le l inear feedback control systems Nyquist and root- locus diagrams are of considerable p r a c t i c a l use. These graphical methods are based on the complex frequency response and enable the designer to choose a suitable system configurat ion and to study the effect of parameter v a r i a t i o n s . A suitable choice of system configurat ion and parameter values can then be made based on engineering experience or a simulated study pf the system. An analyt ic design approach to a complete system synthesis i s possible i f suitable c r i t e r i a for optimum response are formulated a n a l y t i c a l l y . The two most suitable and therefore most often used c r i t e r i a are the minimization of mean-square drror and the spec i f i ca t ion of closed-loop response. Both the a n a l y t i c a l and the graphical methods have been applied to the design and synthesis of mult ivar iable control systems. A mult ivar iable control system is one with n independent inputs and m dependent outputs where n and m 2 are integers and n > 1 m ^ 1 Consider Figure 1 which represents a l inear mult ivariable feedback control system and l e t ri = m. Figure 1 - 1 . A Linear Mul t ivar iab le Feedback Control System X and Y are column matrices whose elements are the Laplace transforms of the input and output signals respect ive ly . X = I x ( s ) and G G n ( s ) K l ( s ) / A n ( s ) \ A nl< s > / C n ( s ) • Gln< s> • • G ( s) , nn / • c • o • « A l n ( s ) A (s) nn c m ( s ) C n l < s ) • • • Cnn< s )< are transfer matrices. Analys is of the system of Figure 1-1 gives the fol lowing equations: AGE = Y . . - . . (1-1) E = X - Z . . . (1-2) = X - CY . . . (1 -3 ) Thus AG(X - CY) = Y . . . ( l - 4 ) or AGX = (AGC + l )Y . . . ( 1 - 5 ) where I i s the unit matrix. Also Y = (I + AGC) _ 1 AGX provided (I + AGC) is nonsingular. The transfer matrix AGC w i l l be defined as the open-loop transfer matrix and . . (1-6) H = (AGC + I) "-""AG . . . ( 1 - 7 ) 4 w i l l be d e f i n e d as the c l o s e d - l o o p t r a n s f e r matrix. The poles of the c l o s e d - l o o p t r a n s f e r matrix are determined by the c o n d i t i o n s X = 0 Y ^ 0 ...(1-8) which, from equation 1-5, can only be s a t i s f i e d i f the determinant |AGC + l | = 0 . . . (1-9) Equation 1-9 i s known as the c h a r a c t e r i s t i c equation. I f the system i s to be s t a b l e , the values of s which s a t i s f y equation 1-9 should a l l l i e i n the l e f t - h a l f s-plane. P o v i s i l and F u c h s ^ ^ have developed a s y n t h e s i s method which co n s i d e r s the c o e f f i c i e n t s of the c h a r a c t e r i s t i c equation as f i x e d . T h i s , of course, s p e c i f i e s the poles of the c l o s e d - l o o p t r a n s f e r matrix and the syn t h e s i s deals with the r e a l i z a t i o n of the poles by s u i t a b l e c r o s s - c o u p l i n g and feedback connections. Kavanagh^^ (3) and Freeman both assume that a s p e c i f i e d H i s given and then r e a l i z e t h i s H by means of a p h y s i c a l l y r e a l i z a b l e ( 4 ) compensating matrix. Horowitz c o n s i d e r s the design problem from much the same p o i n t of view as the above two (2 3) authors, ' i:..e., the d e s i r e d system response i s known. He a l s o d i s c u s s e s at some le n g t h the response v a r i a t i o n s due to changes i n the system c o n f i g u r a t i o n and al s o due to 5 system parameter v a r i a t i o n s . However, i n most p r a c t i c a l appl icat ions H i s not a p r i o r i known and these methods are then not app l i cab le . (5) Hsiefa. and Leondes ' discuss the appl i ca t ion of the mean-square error c r i t e r i o n , an extension of the Wiener method of synthesis , to mult ivariable systems. However, a n a l y t i c a l methods based on minimizing the mean-square error are not appl icable i f the s t a t i s t i c a l properties of the input and the disturbance signals are inadequately known. Even i f such information is ava i lab le , the problem may very well prove to be mathematically i n t r a c t a b l e . Under these conditions a graphical approach may be more su i tab le . K r a s o v s k i i ^ ^ and Newman^^ discuss the appl icat ion: of the conventional Nyquist c r i t e r i o n to a two-dimensional system. B o h n ^ ' 9 ^ introduces a procedure for applying known s ing le -var iab le feedback control system s t a b i l i z a t i o n techniques to a special class of mult ivar iable systems. In "actual design, s imulation of the system i s , i n general , e s s e n t i a l . Optimization i s then performed on the simulated system by experimental evaluation of the minimum mean-square e r r o r , system s e n s i t i v i t y to parameter var ia t ions and non^-linear e f fec ts , or any other suitable c r i t e r i o n . This study w i l l deal with graphical analysis and design methods and t h e i r experimental v e r i f i c a t i o n . The 6 mean-square error w i l l be invest igated experimentally. The systems studied w i l l be r e s t r i c t e d to a configurat ion such as that shown i n Figure 1 - 1 . The study b a s i c a l l y deals with l inear two-variable feedback control systems for two main reasons: ( 1 ) Many such systems occur i n p r a c t i c e . ( 2 ) Relat ive ease of simulation to afford experimental v e r i f i c a t i o n of t h e o r e t i c a l l y predicted r e s u l t s . 7 2. THE EIGENVALUE METHOD OF ANALYSIS AND DESIGN Consider once more the system shown i n Figure 1-1„ Suppose now that the elements of the G matrix have the common factor G c ( s ) . A new matrix can then be defined by G = G c ( s ) G ' . . . (2 -1) Subst i tut ing th i s into equation (l-9) and replacing G c ( s ) by y i e lds AG'C - XI = 0 . . . (2 -2 ) The values of X which sat i s fy th is equation are the eigenvalues of the transfer matrix AG'C. G (s) , or represents the common factor which can be considered as a var iab le element while a l l other elements and the system configurat ion are e s sent ia l l y f i x e d . The s t a b i l i t y of the system can be determined by an app l i ca t ion of the Nyquist c r i t e r i o n . Let X^(s); k = 1, 2 , . . . , n be the eigenvalues of the AG'C matrix. It follows from equation (1-5) that the roots of equation (2-2) are the poles of the closed-loop transfer matrix. The system i s stable i f there are no values of s i n the r ight ha l f s-plane which are roots of equation (2-2). To determine i f th i s i s the case, consider the function ^ ( s ) defined by R k ( s ) = A - f e ) + G c ( s ) . - . (2-3) From equations (2-2) and (2-3) i t i s eas i ly seen that the roots of equation (2-2), when X = X^, are the zeros of R^Xs), 8 The condit ion for s t a b i l i t y i s that the zeros of ^ ( s ) must a l l l i e i n the l e f t - h a l f s-plane, i . e . , they must a l l have negative rea l par t s . This can be determined by considering the Nyquist p lot of R^(s) and appl i ca t ion of the general condit ion for s t a b i l i t y N = Z - P where P i s the number of poles and Z the number of zeros of R^(s) i n the r i g h t - h a l f s-plane respect ive ly and N i s the number of pos i t ive revolutions of the radius vector R^(ja>) . If Z = 0, the system i s s table . If Z > 0, there are values of s i n the r i g h t - h a l f s-plane which are roots of equation (2-2) and the system i s unstable. Equation (2-2) can be expressed as a polynomial i n A and may be wri t ten i n the factored form . . . U - A) (A 1 - A) (A 2 - A) n 0 or B 1 A 1 R 2 A 2 R A n n G n = I n " T T V k = ° (2-4) (2-5) c k = 1 The mult ivar iable system can be considered to be reduced to n equivalent s ing le -var iab le systems represented by the factors of equation (2-4). The eigenvalue method e s sent ia l ly considers the system in terac t ion as an ent i ty d i s t i n c t from the var iable element. A^ may be ca l l ed an in terac t ion parameter and represents the effect that the system has on the equivalent s ingle- loop system shown i n Figure 2-1. 7\ 9 F i g u r e 2-1. E q u i v a l e n t Single-Loop System A n a l y s i s of the system of Figure 2-1 y i e l d s G v v = u, ...(2 - 6 ) K 1 + G A. K c k or G v i , = ; — ~ — • . . . (2-7) »A In a conventional s i n g l e - v a r i a b l e system (A^ = l ) we consider a s t a b i l i t y v e c t o r R = 1 + G c and the c o n d i t i o n N = Z - P I f P = 0 , N = Z = 0 f o r s t a b i l i t y (see page 8). In the eigenvalue system of F i g u r e 2-1 we consider the s t a b i l i t y v e c t o r as d e f i n e d by equation (2-3). R, (s) = — + G (s) ...(2-8) I f ^k^ s^ i s a constant, — i s a c r i t i c a l p o i n t and i s A k ( s ) considered the same as the -1 p o i n t i n a conventional 10 s ing le -var iab le case. However, i f A, (s) i s a funct ion of s, i s a c r i t i c a l locus and considerable care must be exercised to determine N. The above discuss ion i l l u s t r a t e s the app l i ca t ion of the Nyquist c r i t e r i o n to mult ivar iable systems where eigenvalues can be introduced. Another method of inves t igat ing system s t a b i l i t y i s to consider the phase and gain margins by using the Bode Plot Technique to obtain the Nichols plot of the functions and G ( j t t ) . The condit ion for o s c i l l a t i o n i s = o (1«>) + — — = 0 This requires the magnitude condit ion | G Ai*)\ = * k(j<e) and the phase condit ion arg. G (j_«o) = arg . — — A k ( j a ) The gain margin i s i l l u s t r a t e d i n Figure 2-2a and the phase margin is i l l u s t r a t e d i n Figure 2-2b. Figure 2-3 i l l u s t r a t e s the phase and gain margins i n a Nichols p lot of the functions — — and G (jtt) . The above discuss ion shows how gain and phase margin c r i t e r i o n may be applied to mult ivar iable systems where eigenvalues can be used. 1.1 12 Using s imi lar reasoning, the eigenvalue method may be used to make other s ing le -var iab le techniques, such as root- locus p l o t s , applicable to mult ivariable systems. angle Figure 2-3. Nichols Plot by Bode Plot Technique 13 2.1 I l l u s t r a t i v e Example 2.1 As an i l l u s t r a t i v e example, consider the system shown i n Figure 2-4. Figure 2-4. I l l u s t r a t i v e Example 2 i l A n a l y s i s of the system of Figure 2-4 y i e l d s G11< X1 - C l l V - A 2 2 ( X 2 - C 2 2 T 2 ) = T 1 G 2 2 ( X 2 - C 2 2 Y 2 ) + A n ( X 1 - C ^ ) =J2 In matrix form equations (2-8) become ...(2-8) G l . l A22 L l l G 22 G11 C11 A 1 1 C 1 1 — A r 22 22 G22 C22 The c h a r a c t e r i s t i c equation i s G11 C11 A 1 1 C 1 1 A 2 2 C 2 2 G22 C22 1 0 0 1 1 0 = 0 "2, ...(2 - 9 ) ...(2-10) The system shown i n Figure 2-4 can be put i n t o the form shown i n Figure 2-5. X + B y c y F i g u r e 2-5. Matrix Representation of Figure 2-4 To determine B and C consider the equation BX = (BC + I)Y Comparing equations (2-9) and (2-11) y i e lds / B = 11 11 A 2 2 \ 22 C = C l l 0 0 c 22/ As an example, consider now the case where C l l = C22 = 1  G l l = G22 = G c ( s )  A l l = A22 = a G c ( s ) G„(s) K i 1 + s T l ) ( 1 + - S T 2) Therefore, from equation (2-12) B = G (s)B' = G (s) 1 -a \ a 1 f Subst i tut ing th i s into equation (2-11) y ie lds 1 B' + G c ( s ) = 0 or, i f oc(.) i s replaced by - A , B 1 - Al | = 0 This i s the eigenvalue equation and i t has the form Solving th is equation for .the eigenvalues y i e lds A l , 2 . = 1 ± j a T — and T — represent the c r i t i c a l l o c i . To discuss system 1. 2 s t a b i l i t y , we can sketch the Nyquist p lot of the functions R, (s) = — - + G (s) RAs) = — + G (s) 2 A 2 ( s ) as shown i n Figure 2-6. I f the gain K of the G c element i s increased u n t i l the G (ja>) locus intersects the j^- po int , c A 2 i . e . , R2(jfl>) = 0, the system becomes unstable. Figure 2-7 i l l u s t r a t e s the Nichols p lot of — and G (jte). If the gain K of the G element i s increased c c u n t i l the gain and phase margins are zero, the system becomes unstable. 2.2 I l l u s t r a t i v e Example 2.2 As a further example, suppose that G , , = G__ = G 11 22 c 1 + s T 3 where G c i s considered to be the var iable element. Solving the eigenvalue equation (2-16) y i e lds F i g u r e 2-7. N i c h o l s P l o t o f — a n d G ( j c o ) . X k(jw) ° 18 1 + sT, + j a x 2 _ 1 + sT - Ja The Nyquist plots of R (^jte) and R^Cjw) are shown i n Figure 2-8. If the gain K of the G £ element i s increased u n t i l the radius vector R (^jto) = 0 (or R2(jtti) = 0)^the system becomes unstable. Figure 2-9 i l l u s t r a t e s the Nichols p lot of the functions -1 \ ( j < a ) and G (jo). If the gain K of the G element i s increased u n t i l the phase and gain margins become zero, the system becomes unstable. 2.3 I l l u s t r a t i v e Example 2.3 As a t h i r d example, suppose that i n equations (2-12) and (2-13) ', G = G = G (s) = ^ 1 1 22 c ( 1 + s T i ) ( ] _ + A l l = A22 = a  C l l = C22 = 1 It follows from equation (2-11) that the charac ter i s t i c equation i s G + 1 - a c a G + 1 c = 0 I m . \ i \ / V / 'V CO= oo \ yo/a-co OJBO J UJaO / -/ / \ 1 \ \ 1 \ / >» ^' ' CO s oo Figure 2-8. Nyquist Plot of — and G (jco). X k(jto) C 20 Log. Magnitude 4 Figure 2-9. Nichols Plot of — and G (303). X k (jw) ° The eigenvalue notation can be introduced by replac ing G c with A. Carrying out th i s subst i tut ion and solving for the eigenvalues y i e lds (A + l ) 2 + a 2 = 0 X l , 2 = - 1 ± J a To determine s t a b i l i t y , we consider the Nyquist p lot of the function Bj j s ) = - A k ( s ) + G c ( s ) . . . (2-17) which i s shown i n Figure 2-10. If the radius vector = 0, the system becomes unstable. Figure 2-10. Nyquist Plot of -A (j<e) and G (ja) The f o l l o w i n g two t e s t examples compare the r e s u l t s of s i m u l a t i o n s t u d i e s on an analog computer wi the t h e o r e t i c a l l y p r e d i c t e d r e s u l t s . 2.4 Test Example 2.1 v A + G x' A 9 F i g u r e 2-11. Matrix Form of Test Example 2.1 y F i g u r e 2-12. C i r c u i t Form of Test Example 2.1 Figure 2 -13 . Computer Simulation of Test Example 2 . 1 23 In F i g u r e 2-13, c o n s i d e r i n g the s e c t i o n from X to Y, the f o l l o w i n g equations are v a l i d : X-. - Z-, X 0 - Zy Y, - Z-, _ i 1 + _ i 1 + _l 1 = 0 11 12 f l i i X 2 " Z2 + X l " Z2 + Y 2 " Z2 = 0 r22 r21 r f 2 In matrix form these equations become 1 1 ijl 11 1 12 1 / r21 r22 / r l l r12 r f l f l 0 0 f 2 0 0 1 + 1 \ Y l Y^ I 2 / . 1 , \ r22 r21 f l / \ Z, = 0 Equation (2-18) may be w r i t t e n i n the form (2-18) or FX - QZ + CY = 0 C 1 F X C 1QZ + Y = 0 ...(2-19) Now since and we can set Therefore, Now from F i g u r e 2-11 Z1 « 0 z 2 « 0 Z = 0 C - 1FX' = -Y AX* = Y 24 therefore -A = C F . . . (2-20) A l s o , from Figure 2-11, AGX = (AG + I)Y . . . (2-21) Now from Figures 2-12 and 2-13 we can see that G i s diagonal and that G l l = G22 = G c ( s ) provided potentiometers 1 and 2 have ident i ca l se t t ings . Therefore we can write equation (2-21) as AX = (A +• — I)Y G c ( s ) The charac ter i s t i c equation is 1 A + = 0 . . . (2-22) Replacing t>y ~X y ie lds the eigenvalue equation A - Al = 0 . .(2-23) From equations (2-18) and (2-19) we have J L l r l l r 12 r 21 r 22 C = f l 0 0 •f 2 / Subst i tut ing these values i n equation (2-20) y i e lds 25 A = f l ' l l L f2 f i *12 f2 '22 . . . (2-24) The eigenvalue equation (2-23) becomes f l *11 v - X f 2 L,21 f l T2. \ f2 '22 = 0 A / / . . . (2-25) Figure 2-13 shows that r l l _ r 12 - r 21 - r 22 " r f l " r f 2 ~ 2 M Equation (2-25) becomes 1 - X 1 1 1: - X 0 Solving th i s determinant for the X values y i e lds X± = 0 X 2 = 2 For the s t a b i l i t y inves t igat ion we consider the Nyquist p lot of V s ) = 7 7 , * G c ( s ) Figure 2-13 shows that 26 G l l = G22 = G c ( s > = ~ " J - ± ^ c i + S T with T = 1, provided oc^  = c^. Figure 2-14 shows the Nyquist p lot of th is example. . .m Re Figure 2-14. Nyquist Plot For Test Example 2-1 i In the simulation tes t s , when k = 0 . 5 the system became unstable. Thus experimental resul t s v e r i f i e d the pred ic t ion that the system would be unstable i f t the gain k of the G (s) element reaches 0.5. 27 2.5 Test Example 2.2 As a second example, consider the system shown i n Figures 2-15, 2-16 and 2-17. X G C y Figure 2-15. Matrix Form of Test Example 2.2 Figure 2-16. C i r c u i t Form of Test Example 2.2 28 Analysis of the system of Figure 2-15 gives GX = (GC + I ) Y Analysis of the system of Figure 2-16 gives . . . (2-26) G 11 0 G 0 22 11 0 G 0 22 C l l C12 C21 C22 1 0 0 1 I Y 2 . . . (2-27) By inspect ion i t i s apparent that equations (2-26) and (2-27) are i d e n t i c a l . Now suppose that G 1 1 = n G c ( s ) G 2 2 = mG c(s) Then G = G (s) n 0 0 m Equation (2-26) becomes G'X = (G'C + G c ( s ) G (s)G c The eigenvalue equation G'C - X I = 0 i s obtained by replac ing —-— with - A . Equation (2-28) . . . (2-28) then takes the form o c ( . ) n C 1 ; L - A nC 12 mC 21 mC 22 = 0 . . .(2-29) 29 Figure 2-17. Computer Simulation Test Example 2.2 30 F i g u r e 2-17 shows t h e a n a l o g c o m p u t e r s i m u l a t i o n o f t h e s y s t e m where -<x-G ( s ) = c v ' r . 1 -K 1 + s r f c f 1 + s T V i t h t h e s w i t c h e s s^ and s 2 i n p o s i t i o n 2 C l l - C22 ~ 1 "C12 - C21 ~ 1 Now s u p p o s e we s e t oc^ = a.^ = cx T h e r e f o r e n = m and e q u a t i o n (2-29) becomes 1 - X -1 1 1 - X w h i c h y i e l d s t h e e i g e n v a l u e s X 1 = 1 + j 0 1 - j Now s t a b i l i t y i s d e t e r m i n e d b y a N y q u i s t p l o t o f R ^ s ) = + G c ( s ) A k ( s ) Now and -1 h -1 A„ -1 1 .+ J -1 1 " 3 Thus i n s t a b i l i t y occurs when A 1 + j 1 + sT which y i e lds K = 1 and »T = 1 i Parameter Values Result ing Componen- ; Values i n an Unstable System C f r . I T = r f c f a r f K = a — r . l f r eq . 65 T 10 5 .1 1 1.0 1.0 1 5 0.1 0.2 1 1 .158 .315 1 1 Table 2.1 Experimental Results of Test Example 2.1 Figure 2-18. Nyquist Plot of Table 2.1 The experimental resul t s agree with the predicted r e s u l t s . Suppose now that n = 2 and m = 1. Equation (2-29) becomes 2 - X -2 1 1 - X = 0 which y i e l d s the eigenvalues Therefore, -1 h -1 1.5 + j l . 3 2 1.5 - j l . 3 2 -0.67 1 + jO.88 -0.67 1 - jO.88 Now i n s t a b i l i t y occurs when -1 -0.67 -K which y i e l d s X, 1 + jO.88 1 + jcoT K = 0.67 and coT = 0.88 Parameter Values R e s u l t i n g Component Va! .ues i n an Unstabl Le S;\ astern r f 2 C f 0.1 r . l 1 T 0.2 a l .68 a 2 .34 K .68 f .67 wt .86 Table 2.2 Experimental R e s u l t s of Test Example 2.2 The experimental r e s u l t s v e r i f y the p r e d i c t e d r e s u l t s . Thus we see that i t i s p o s s i b l e to determine a X-value by choosing a G c element and v a r y i n g i t s parameters 33 u n t i l the system becomes unstable. As a somewhat more in teres t ing case, suppose we determine a A-locus by the same procedure. Suppose switches S-^  and °^ Figure 2-17 are i n pos i t i on 1. This gives 1 C l l ~ C22 1 + s " C12 - C21 ~ 1 Now l e t t i n g n = m, i . e . , oc^  = a^, the eigenvalue equation (2-29) bee omes 1 + s 1 - X -1 1 + s - X = 0 which y i e lds A (jtt) .= — i + j 1 + j» A 2(j») = 1 + jw. As before, from equation (2-3), the system i s unstable when R^JM) = 0. Figure 2-19 i l l u s t r a t e s the Nyquist p lot -1 of * k ( j « ) 34 Im Figure 2-19. Nyquist Plot of i — xk(d») In th i s p a r t i c u l a r case i t i s easier to plot X k ( j » ) and ——- rather than — - — • and G ( j<o). This i s shown i n G c ( j co ) X k ( j c o ) c Figure 2-20. If the locus intersects the c r i t i c a l X- locus , at <* c(j») a c r i t i c a l frequency (0 c, the system i s unstable. Thus, i t has been shown possible to determine the A-locus experimentally by choosing a convenient G £ . The transfer funct ion G £ can be sui tably var ied by se lect ing various t T s, i . e . , r^ and c^, and adjusting i t s gain, i . e . , and a^t u n t i l the system i s on the verge of i n s t a b i l i t y and then measuring the frequency of free o s c i l l a t i o n of the system. Now -1 _ 1 + jcoT Gc(j<o) K Figure 2-20 shows that for each value of T selected, two points of the c r i t i c a l locus may be determined. They are G K K c r.i. and -1 1 , j<o T where — tt + n G K K c <D"T O>'T . , " / ' -—— = f- but K p K K K The experimental resul t s given i n Table 2.3 and Figures 2-21, 2-22, and 2-23 v e r i f y the predicted resu l t s F i g u r e 2 - 2 0 . N y q u i s t P l o t o f A, (jco) a n d k G c ( j « ) r i 1.0 10.0 0 .5 0.105 0.11 0.12 0.13 0.14 0.15 K (0 1.05 1.10 1.20 1.30 1.40 1.50 0.1751 1.75 0.20 ! 2 . 0 0.30 i 3 .0 , 0 .06 i 0 .07 I 0.08 ! 0.09 I o . i o 0.15 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 1.0.60 ! 0 .70 i 0 .80 I 0 .90 ! 1.0 ; 1.50 i I 0 .10 I 0 .20 0 .30 0 .40 0.50 0.60 0.70 0.80 0 .90 1.0 1.50 1.70 1.58 1.45 1.37 1.32 1.27 1.20 1.15 1.08 1.43 1.31 1.24 1.20 1.16 1.09 0.705 0.640 0.605 0. 580 0.565 0.555 0. 545 0. 540 0.538 0.530 0.523 0.805 0.73 0.642 0. 582 0.535 0.495 0.433 0.376 0.267 1.25 1.06 0.906 0.83 0.75 0.53 0.647 0.532 0.465 0.412 0.383 0.341 0.322 0.298 0.279 0.265 0.205 K (0 T t K 0.588 0.633 0.69 0.73 0.757 0.787 0.83 0.87 0.927 0.70 0.763 0.497 0. 506 0.531 10.553 JO.567 10.585 0.632 0.652 0.742 0.525 0.567 K 0.80510.585 0 .834 |0 .622 0.862 0.918 1.42 1.56 1.65 1.72 1.77 1.81 1.84 1.85 1.86 1.89 1.91 0.650 0.732 10.092 10.166 JO.231 10.248 iO.338 J0.368 |0.416 10.441 10.468 JO.50 10.59 2. 58 2.90 3.50 4.03 4.60 5.13 6.55 8.00 15.50 3.54 4.62 5.69 6.85 8.00 (0 1.22 1.34 1.53 1.70 1.85 1.93 2.27 2.53 3.56 3.05 3.60 4.03 4.65 5.02 K 15 .52 .12 .6 1.73 2.29 2.85 3.46 4.07 4.74 5.44 6.16 6.93 7.70 12.66 1.54 1.86 2.09 2.36 2.60 2.78 2.88 3.24 3.34 3.65 4.63 0.388 0.345 0.286 0.248 0.218 0.195 0.153 0.125 0.065 0.283 0.217 0.176 0.146 0.125 0.065 0.578 0.437 0.3 50 0.289 0.246 0.211 0.184 0.162 0.144 0.130 0.079 n (0 T n K 0.497 0.508 j 0.524 i 0.548 0.565 0.565 0.607 0.632 0.69 ) P i g . 2-21 0.517 0.546 0.568 0.612 0J628 0.705 1 )~F±g. 2-22 I i J 0.089 ^ 0.162 0.220 0.273 0.320 0.3 52 Wig. 2-23 0.370 ' 0.420 0.434 0.473 0.550 j Table 2.3 Experimental Results for Test Example 2.2 - H — i — I — i — I — i — I ' M ) A " l o c u s (Theoretical) X-Locus (E xpen mental) A - rift, + i Re. Figure 2-21 X-Locus from Table 2.3 lm, X~locus {Tneoret/ca/J \~Locus (Expertmen td/J J Re. Figure 2-22 X-Locus from Table 2.3 Im. .1 . A .6 .8 /.0 AZ /.4 />& to Figure 2-23 A-Locus from Table 2.3 40 In general , for a complex system, the ca l cu la t ion of the c r i t i c a l A - l o c i i s very complicated even with the a id of d i g i t a l computers. Using a convenient var iable element, G (s) , i n a simulated system, i t i s possible to f ind the c r i t i c a l A-locus r e l a t i v e l y e a s i l y . The var iable element i s adjusted u n t i l the system i s on the verge of i n s t a b i l i t y . This determines a. point on the c r i t i c a l A- locus . After the c r i t i c a l A-locus i s completely determined i t can be used to design a suitable var iab le element for the closed-loop system. 41 3 . NONINTERACTION In the analysis of mult ivariable feedback control systems much e f fort has been spent developing methods for a t ta in ing noninteract ion, that i s for a t ta in ing a system whose input v a r i a b l e , say X^, controls only i t s corresponding output var iable Y >^ Stated mathematically, the condit ion for noninteract ion i s that the closed-loop transfer matrix H i s d iagonal . Once noninteract ion has been accomplished, the system can be treated as cons is t ing of n s ing le -var iab le systems. Boksenbom and H o o d ^ ^ did some of the e a r l i e s t work on achieving noninteract ion i n mult ivar iable systems. Their major interes t was i n obtaining the conditions for noninteract ion i n l i n e a r mult ivar iable systems with (11) p a r t i c u l a r emphasis on an engine-type problem. Meerov discussed a specia l c lass of systems which became noninteract ing i f the gains of the system are increased without l i m i t . In general th is means increasing loop gains u n t i l the effect of the cross-coupl ing elements become n e g l i g i b l e . Another method of achieving noninteract ion consists of a t r i a l and error se lec t ion of elementary transformations which resu l t i n the d iagonal izat ion of the open-loop matrix. There i s no unique method of d iagonal iz ing the open-loop rbransfer matrix. A convenient method, which usual ly resul t s i n simple phys i ca l ly rea l i zab le elements, w i l l now be discussed. 3.1 Diagonal izat ion of Open—Loop Transfer Matrix To i l l u s t r a t e th i s method, consider Figure 3-1 Figure 3-1 Open-Loop System where E and Y are column matrices where elements are the input and output s ignals respect ive ly and G, F and D - 1 are n x n transfer matrices . G i s taken to be diagonal and i s c a l l e d the ampl i f i ca t ion or gain matrix, F i s the forc ing matrix and D _ 1 i s ca l l ed the dynamical matrix. Analysis of Figure 3-1 y i e lds . . . (3 -1 ) D 1 FGE = Y or , premult ip ly ing both sides by D, FGE = DY . . . (3 -2 ) One method of diagonal iz ing a 2 x 2 system is shown i n Figure 3-2 where ' Y l !  T 2 , ' G l l 0 K i D i 2 ' D21 D 2 2 | and T 2 , C 2 and represent transformation and compensation matrices which are to be determined. Analys is of Figure 3-2 y i e lds D ~ 1 F G C 1 C 2 T 2 E = T . . . (3 -3 ) or D " 1 F G C 1 C 2 T 2 ( X - T) = T . . . (3 -4 ) According to the design philosophy to be discussed, the open-loop transfer matrix D - ^ F G C ^ C 2 T 2 i s to be diagonalized and the diagonal elements are to be sui tably chosen to rea l i z e good dynamic performance. For the p a r t i c u l a r configurat ion shown i n Figure 3-2, th i s method resul t s i n a diagonal H (closed-loop transfer matr ix) . Suppose we write the D matrix i n the factored form F Figure 3-2 Diagonal izat ion of Open-Loop Transfer Matrix D = T 1 D * T 2 . . . (3 -5 ) The inverse dynamical matrix i s D 1 = T 2 1 ( D ' ) " 1 T 1 1 . . . (3 -6 ) Now choose T^ so that T 2D" 1FGC 1 = (D*) 1 T 1 1FGC 1 . . . (3 -7 ) = ( D ' ) _ 1 G . . . (3 -8 ) This requires that T^G = FGC X . . . ( 3 -9 ) Thus T 2 " 1 ( D I ) " 1 G C 2 T 2 X = ( T 2 ~ 1 ( D ' ) " 1 G C 2 T 2 + l ) T . . . (3-10) t The matrices T^, T 2 , C^, C 2 and D w i l l now be determined. Equation (3-9) i s s a t i s f i e d i f a new matrix c'^  i s defined by the matrix commutation and C-^  i s chosen to be GC1 = C1 G . . . (3-11) = F 1 T 1 . . . (3-12) Matrix T, i s chosen to have the form 46 T 21 t 12 1 1 . . . (3-13) where F i s a suitable common factor and T, ~ and T 0 1 are c 12 21 transfer functions which are to be determined. Pos tmul t ip l i ca t ion of equation (3-5) with T0 ^ y i e lds DT, T 1 D . . . (3-14) The design philosophy i s to diagonalize the open-loop transfer matrix and sui tably choose the diagonal elements to rea l i z e both good dynamic performance and simple, phys i ca l ly r e a l i z a b l e , compensating elements. This objective i can be achieved i f D has the form D D 11 0 0 22 . . . (3-15) and i f -1 1 a 12 \a22- 1 . . . (3-16) where and o c ^ are constants. The inverse matrix i s 1 - a 1 2 a 2 1 -a 12 -a 21 . . . (3-17) where 1 " a 12 a 21 ^ 0 Subst i tut ing equations (3-13), (3-15) and (3-16) into equation (3—14) gives D l l D12 D21 D22 a a 21 12 1 21 or •12" 1 D r i o 0 D 22; D l l + D 12 a 21 D l l a 1 2 + D12 D21 + D 22 a 21 D 2 1 a 1 2 + D 2 2 D 11 D T 11X21 i i D T 22 x12 D 22 . . . (3-18) Equating elements y i e lds D-,, .= Dn -, + L\ o^c '11 11 12 21! D 2 2 = D 2 2 + D 2 1 a 1 2 . . . (3-19) . . . (3-20) 12 •21 D l l a 1 2 •+ D12 22 D 22 a 21 + ^21 11 . . . (3-21) . . .(3-22) Combining equations (3-19), (3-20), (3-21) and (3-22) gives 12 D12 + a 1 2 D l l D22 + a 12 D 21 . . . (3-23) 48 :21 D 2 1 + a 2 1 D 2 2 D l l + a 21 D 12 . . . (3-24) Thus T-j^  has now been determined i n terms of the D matrix and the two constants oc^2 and oc,^ . T 2 i s given by equation (3-17). C-^  and, hence, C-^  can be determined from T^ (see equation (3-12)). C 2 i s selected to be a diagonal compensating matrix. Therefore, a l l elements of the diagonal ized system have been determined. There i s a r e s t r i c t i o n on a^ 2 and o c ^ i n that the transfer functions given by equations (3-23) and (3-24) must be phys i ca l l y r e a l i z a b l e . It i s evidently desirable to have simple phys i ca l l y rea l i zab le elements for T^ and T 2 which allow high loop gains and y i e l d a stable system. Once suitable transformations are found, more quanti tat ive root- locus methods may be appl ied to determine gain parameters, e t c . , for the var iable elements. The s u i t a b i l i t y of the design may then be tested by simulation studies . 3.2 I l l u s t r a t i v e Example As an example, l e t us apply th i s procedure to the system shown i n Figure 3-3. Analysis of th i s system y i e lds 11 0 F 0 22 11 0 G 22 E, F 22 J 22 F 11 J 11 . . . (3-25) 4 9 4" 7 / f ) 3", F i g u r e 3 -3 Two-Axis T r a c k i n g System -2 A comparison of e q u a t i o n s (3-25) and (3-2) shows t h a t F n G = G , , 0 G '11 0 F = 22 P 1 1 J 1 1 11 0 0 22 . .(3-26) F 2 2 J 2 2 1 I f i s assumed d i a g o n a l , we have 50 c 2 = 0 '11 0 "22 . . . (3 -27) Equation (3-17) gives T -1 " a 12 a 21 - a 12 -a 21 1 . . . (3-19) If G i s diagonal and G-^ = G 2 2 > i t follows from the matrix commutation QC1 = C^G . . . (3-13) and equation (3-12) that C l = C l = P F 11 22 P 2 2 0 0 F 11 T, = 11 0 1 0 F 22 . . . (3-28) Equation (3-13) gives T, = F 1 c 1 T •21 12 1 . . . (3-15) Therefore / _1 F 11 0 I H 22' 1 T 12\ T 2 1 1 . . . (3-29) 51 Now i f F,-, = F~„ = F , we obtain I i 22 c •21 12 1 . . . (3-30) Subst i tut ing the values of the D matrix elements into equations (3-23) and (3-24) gives L12 a 12 + J 11 F 11  1 + a 12 J 22 F 22 . . . (3-31) 21 a 21 + J 22 F 22 1 + a 2 1 J n - P 1 1 . . . (3-32) From equation (3-3) the open-loop transfer matrix B i s B = D •FGC 1 C 2 T 2 . . . (3-33) The design philosophy i s to diagonalize B and sui tably choose elements B ^ and B 2 2 . Let us consider the case where • J l l = J 22 = J ( s ) F l l = F 22 = F ( s ) G l l = G22 = G ( s ) . . . (3-34) )/j — C~ = Cj ( s ) ^11 22 ^ Subst i tut ing these values into the matrix B and cance l l ing where possible we obtain B = C 2 PG (1 + (J F r ) ( l - a 1 2 a21 ) ' l + J F T 2 1 - « 2 1 ( T - [ 2 + JF) T ^ 2 + JF - a 1 2 ( l + J F T 2 1 ) T21 " J F " a21 ( l " J F T i 2 } 1 " J F T12 a 1 2 ( T 2 1 - JF) B = C 2 FG (1 + (J F r ) ( l - a 1 2 a 2 1 ) B 11 B 1 i 12 B 21 B 22 ..(3-35) where we have set J = J ( s ) , F = F ( s ) , etc.,vto s impl i fy the notat ion. Now for noninteract ion we set B12 = - a12-?12 J F T21 + T12 + J F = 0 •••(3-36) B 2 1 = - « 2 1 + « 2 1 J F T ; 2 + T 2 1 - JF = 0 ...(3-37) These two equations are s a t i s f i e d i f we set a12 = -a21 = a t i i T = - T — T 12 21 ~ ...(3-38) ...(3-39) These re la t ionships and equations (3-31) and (3-32) y i e l d a - JF 1 + ocJF ...(3-40) S o l v i n g e q u a t i o n (3-35) f o r B^^ and B 2 2 us^-nS e q u a t i o n s (3-38) , ;v(3-39) and (3-40) y i e l d s B l l = B 2 2 C 2 F G 1 + aJF The f i n a l open-loop t r a n s f e r m a t r i x i s ' C 2 P G B = 1 + ocJF 0 C 2FG 1 + aJF/ Thus by means of t h i s c h o i c e of T^ and T 2 the open-loop t r a n s f e r m a t r i x i s d i a g o n a l i z e d and the system d e s i g n reduces t o a d i s c u s s i o n of the two n o n i n t e r a c t i n g systems each h a v i n g an open-loop t r a n s f e r f u n c t i o n of the form B C 2FG k k 1 + ocJF The c h o i c e of compensating networks and g a i n c o n s t a n t s f o r these systems can be found by c o n v e n t i o n a l g r a p h i c a l methods. T h i s procedure does n ot n e c e s s a r i l y l e a d t o the optimum d e s i g n because i t r e q u i r e s the open-loop t r a n s f e r m a t r i x t o be d i a g o n a l i z e d . T h i s may not always be d e s i r a b l e s i n c e the o f f - d i a g o n a l elements o f t e n improve system performance. The method does, however, a l l o w the d e s i g n e r t o i n v e s t i g a t e t h e p o s s i b i l i t i e s w i t h r e l a t i v e l y f e w c o n s t r a i n t s o n t h e s e l e c t i o n o f t h e c o m p e n s a t i n g e l e m e n t s a n d a v o i d s t h e p r a c t i c a l d i f f i c u l t i e s o f a p u r e l y t h e o r e t i c a l a p p r o a c h . 4. THE T¥0-AXIS TRACKING SYSTEM A COMPARATIVE STUDY OP THE EIGENVALUE METHOD AND THE DIAGONALIZATI0N METHOD This chapter deals with a treatment of the two-axis tracking system shown i n Figure 4-1. This p a r t i c u l a r configurat ion has been discussed by a number of authors. (6) Krasovski i has dealt with i t as an antisymmetric system, i . e . , the cross-coupl ing transfer functions are the negative of each other J l l = ~J22 Krasovski i defines the input and output signals as complex X == X x + j X 2 Y = T 1 + j Y 2 This procedure enables him to reduce the system to an equivalent s ingle var iable system which he then treats i n ( 7 ) the conventional Nyquist manner. Newman'sv paper deals with the analysis of a s i m i l i a r system.. . The methods of both these authors are r e s t r i c t e d to two var iable systems with the necessary symmetry. 4.1 Open-Loop Two-Axis Tracking System Consider now the two-axis tracking system shown i n block diagram form i n Figure 4-1. Analysis of the open-loop system (C-,, = C00 = 0) y i e lds 56 Figure 4-1 Two-Axis Tracking System 57 F 1 1 G 1 1 E 1 " F 1 1 J 1 1 Y 2 = Y l F G E - F J Y - Y * 22 22*2 *22 22 1 ~ 2 ...(4-1) . . . (4-2) In matrix form these equations become F 0 *11 u 0 F 22 G l l 0 0 G 22 F 1 1 J 1 1 F 2 2 J 2 2 From equation (4-3) the c h a r a c t e r i s t i c equation F 1 1 J 1 1 F 2 2 J 2 2 = 0 l Y 2 ...(4-3) ...(4-4) i s obtained. Consider now the case where F ^ = F 2 2 ~ F * I f the s t a b i l i t y of open-loop i s to be i n v e s t i g a t e d , the eigenvalue method can be used. F i s then r e p l a c e d by X. S u b s t i t u t i n g i n t o equation (4-4) and s o l v i n g f o r the eigenvalues y i e l d s + 1 •1,2 ...(4-5) The eigenvalue equation has thus been f a c t o r e d i n the form or (X - X 1 ) ( X - A 2 ) = 0 (F - A 1)(P. - X 2 ) = 0 ...(4-6) S t a b i l i t y i s then determined by a Nyquist p l o t of the f u n c t i o n (see Figure 4-2)= = - X k ( s ) + F ( s ) ...(4-7) Rj^jtt) may be c a l l e d a s t a b i l i t y v e c t o r . The system w i l l o s c i l l a t e at the frequency ( 0 c i f \(i<*c) = 0 4.2 Closed-Loop Two Axis T r a c k i n g System Let us now consider the cl o s e d - l o o p system of Figure 4-1. A n a l y s i s y i e l d s the f o l l o w i n g matrix equation F 1 1 G 1 1 0 0 F G *22 u22 1 + C 1 1 G 1 1 F 1 1 J 2 2 F 2 2 J 1 1 F 1 1 1 + C 2 2 G 2 2 F 2 2 ...(4-8) Now suppose we l e t J l l = J22 = J -p — P — F r l l - 22 ~ G l l = G22 = G C l l - C22 - C Equation (4-8) y i e l d s the c h a r a c t e r i s t i c equation 1 + CGF J F JF 1 + CGF 0 ...(4-9) The eigenvalue equation i s obtained i f J i s r e p l a c e d by X. This equation then gives the eigenvalues 59 Figure 4-2 Nyquist Plot of Rj^jw) 60 X1 2 = + + CG . . . (4-10) In the case where = -J22 ~ ^ 9 ^ e e igenv?i lues are \ 2 = + j | + CG . . . (4-11) 4 . 3 Simulation Study of Two-Axis Tracking Systems In the two-axis tracking system shown i n Figure 4 - 3 , we have the fol lowing transfer funct ions: G l l = G22 = ^G Power Ampl i f ier Gain K F F-i = F 0 0 = Transfer Function of an 1 1 2 2 s ( l + BT )(1 + sT b ) Ampl i f ier ^11 = ^02 ~ a s Transfer Function due to Gyroscopic Torque P r a c t i c a l d i f f i c u l t i e s ar ise i f one attempts to simulate the der ivat ive elements as. The configuration shown i n Figure 4 - 4 , which has the same response as that of Figure 4 - 3 , includes no derivat ive elements and was used for analog t computer s imulat ion. To rea l i z e the transfer function F (see Figure 4 - 4 ) with one operational ampl i f ier the network discussed i n Appendix I was used. The network analog of the system i s shown i n Figure 4 - 5 . From Appendix I we have Figure 4 - 3 Two-Axis Tracking System 62 A , 5 a "5. c" IT" I S F i g u r e 4 - 4 A r r a n g m e n t F o r S i m u l a t i o n o f T w o - A x i s T r a c k i n g S y s t e m Figure 4 - 5 Simulation C i r c u i t for Two-Axis Tracking System For the simulation the values R = r 4 = 1 Meg C = C-j^  = 0.1 [if c 2 = 0 are chosen. Thus p' = 100/3 (s + 5.45)(s + 24.55) From Figure 4-5 we have 4.4: The S t a b i l i t y of the Open-Loop System The s t a b i l i t y of the open-loop system w i l l be discussed f i r s t , i . e . , with switches s-^  and s 2 of Figure 4-5 open. Test 1 - Open-Loop System with Symmetric Cross-Coupling Switches and are i n pos i t i on 1. t i Thus = = a From equation (4-5) i n s t a b i l i t y occurs when 65 or when P = + J 11 J 22 100/3 (s + 5.45)(s + 24.55) + 1 a . . . (4 -12) i . e . , | -i F = ± J The lef t -hand side of equation (4-12) must be e n t i r e l y +' 1 rea l to equal — since a i s en t i re ly r e a l . Therefore s = j 0 . Now P(s=0) = ° ' 2 4 8 Therefore, t h e o r e t i c a l l y the system i s unstable when 1 a = .248 4.03 Experimental Values | System Condit ion 4.10 stable but highly underdamped 4.125 steady o s c i l l a t i o n 4.15 unstable Comparative Results Theoret ica l a 4.03 Experimental 4.12 io Difference 2.0 66 The Nyquist p lo t of F (j<o) i s shown i n Figure 4-6. '.m unstsb/e -"r 1 \_ /. ., A / L i m i t oT Figure 4-6 Nyquist Plot for Test 1 Test 2 - Open-Loop System with Antisymmetric Cross-Coupling Switch S^ i s i n pos i t i on 1 and switch S^ i s i n p o s i t i o n 2. Thus t i J l l = ~ J22 = a From equation (4-5) we know that for i n s t a b i l i t y F (j«>) = 100/3 ± j . . . (4-13) 134 -<o + j 3 0 » a Now F (jtt>) must be e n t i r e l y imaginary to sa t i s fy equation (4-13), therefore 0 ) = Vl34 =11 .6 rad/sec Thus P ( d « c ) = J0.096 67 Th e r e f o r e the l i m i t of s t a b i l i t y o ccurs a t a = — = 10.4 0.096 E x p e r i m e n t a l l y i n s t a b i l i t y o c c u r r e d a t a = 10.4 and co . = 2TX x 1.82 = 11.5 c Comparative R e s u l t s T h e o r e t i c a l E x p e r i m e n t a l fo D i f f e r e n c e a 10.4 10.4 0 f 1.84 1.82 1 4.-5 The S t a b i l i t y of the Closed-Loop System The s t a b i l i t y of the c l o s e d - l o o p system w i l l now be d i s c u s s e d , i . e . , w i t h s w i t c h e s S-^  and S 2 i n F i g u r e 4-5 c l o s e d . Test 3 - Closed-Loop System w i t h Symmetric C r o s s - C o u p l i n g S witches S^ and S^ are i n p o s i t i o n 1. Thus , , J l l = J 2 2 = a T h e o r e t i c a l l y i n s t a b i l i t y o c c u r s when (see e q u a t i o n (4-10)) as = ± s ( s + 5.45)(s + 24.55) + 1 ...(4-14) 100/3 E x p e r i m e n t a l l y the l i m i t of s t a b i l i t y was found a t a = 4 and f = .167cps (co = 1.05). S u b s t i t u t i n g these v a l u e s i n e q u a t i o n (4-14) g i v e s 68 L.H.S. •= J4.2 E.H.S. = j'4.18 + .006 Thus the experimental r e s u l t s v e r i f y equation (4-14). F i g u r e 4-7 shows a Nyquist type p l o t of the f u n c t i o n s of equation (4-14). Test 4 - Closed-Loop System with Antisymmetric Cross-Coupling Switch S^ i n p o s i t i o n 1} switch S^ i n p o s i t i o n 2. Thus , , J 2 2 = " J l l = a From equation (4—11), t h e o r e t i c a l i n s t a b i l i t y occurs when as = + j s(s + 5 . 4 5 ) ( s + 24.55) + ± . . . ( 4 - 1 5 ) 100/3 E x p e r i m e n t a l l y the l i m i t of s t a b i l i t y was found at a = 10.4 and f = 1.82cps (ft) = 11.5). S u b s t i t u t i n g these values i n t o equation (4-15) giv e s L.H.S. = J119.5 R.H.S. = J118.2 + 0.518 Thus the experimental r e s u l t s v e r i f y equation (4-23). Figure 4-8 shows a Nyquist type p l o t of the f u n c t i o n s of equation (4-15). When making a Nyquist type p l o t of the f u n c t i o n s of equations (4-14) and (4-15) i t i s e a s i e r to p l o t the inve r s e f u n c t i o n s . Thus f o r t e s t 3 we p l o t 1 ± 100/3 and as s(s + 30s + 134) + 100/3 69 as shown i n Figure 4-7 and for test 4 we p lot ± j 100/3 and as s(s + 30s + 134) + 100/3 as shown i n Figure 4-8. Two other tests were conducted, one with F l l = 5 a n d J 22 = a and the other with J-, -i = — and J o n = —a 11 s 22 In both these tests the experimental resul t s v e r i f i e d the predicted r e s u l t s . 4.6 Diagonal izat ion of Open-Loop Transfer Matrix The two-axis tracking system w i l l now be considered using the d iagonal izat ion method discussed i n chapter 3. The diagonal ized system has the form shown i n Figure 3-2. From equations (3-19), (3-30) and (3-26a), the desired transfer matrices of the system with antisymmetric cross-coupl ing i . e . , J^-^ = -^22' w e r e found to be D 1 FJ G = - F J 1 F = F 0 *11 U *11 0 G 0 22 C, = 0 ] 1 T T' 1 22 70 Figure 4 - 8 Nyquist Plot of Test 4 71 0 C „ = (1 + o T ) '11 T„ = 1 + a' 0 1 a "22 - a •1 From the f i r s t eigenvalue t e s t , equation (4-12), we can obtain JF = a 100/3 (s + 5.45)(s + 24.55) Subst i tut ing th i s value for JF i n equation (3-4l) y i e lds . . . (4-16) cx(s 2 + 30s + 134 - a - i g ° - ) ( s 2 + 30s + 134 + oca 100/3) . . . (4-17) This transfer funct ion can be conveniently r e a l i z e d by means of an act ive network. This r e a l i z a t i o n i s discussed i n Appendix I I . In order for T to be rea l i zab le i n the form given i n Appendix I I , a must be chosen such that 134 -or a a 100 3a a 100 0 (3)(134) . . . (4-18) From the analysis of the system based on the eigenvalue method, we know that the l i m i t of s t a b i l i t y was reached at a = 10.4 (see Test 4, page 68 ) . At a•= 10.6 the system was stable but h ighly underdamped. Suppose that a = 10.0 72 and that the system is to be designed using the d iagonal izat ion method. I f a = 10.0, we must have was r e a l i z e d by the network shown i n Appendix I I . Figure 4-9 shows a block diagram of the diagonalized system and Figure 4-10 shows the network analog of the diagonalized two-axis tracking system. To test the degree of noninteract ion achieved by the d iagona l i za t ion , a f ixed s inusoidal s ignal was applied to an input terminal (say X^) of both the diagonalized and the undiagonalized system and the amplitude of the outputs was measured. For the undiagonalized case both outputs Y^ and Y 2 had equal amplitude. For the diagonal system the amplitude of the output was approximately 15$ of the amplitude of Y^. Two design criteria:. ' were used for comparative tests of the eigenvalue system and the diagonalized system. The c r i t e r i a are s t a b i l i t y and mean-square e r r o r . S t a b i l i t y w i l l be discussed f i r s t . 4.7 S t a b i l i t y Comparison S t a b i l i t y was tested by examining the system response to an input wave cons is t ing of a rectangular pulse a > 1000 402 a > 2.485 For the s imulation tests a was set at 3.0 and T -v I °i 21 5*. Figure 4 - 9 Block Diagram of Diagonalized Two-Axis Tracking System Figure 4-10 Network Analog of the Diagonalized Two-Axis Tracking System applied manually by a switch. The undiagonalized system was highly underdamped, ind ica t ive of i t s nearly unstable s tate . The diagonalized system response had approximately a 10% to 15% overshoot and then the o s c i l l a t i o n died out very rap id ly ind ica t ing a stable system. These resul ts are shown i n Figures 4-11 and 4-12., 4.8 Mean-Square E r r o r Considerations In addi t ion to various closed-loop response criteria>u; one of the most important feedback control systems design c r i t e r i o n s i s the minimization of the mean-square e r r o r . This was the second c r i t e r i o n used for comparative purposes. Consider Figure 4-13 A/to X(sy 4- + ) H(s) 1 Figure 4-13 System With Noise In Input Analys is of the system of Figure 4-13 y i e lds Y = H(X + N) For a s ing le -var iab le system the t o t a l mean-square error can be found by the equation A Figure 4-12 Diagonalized System Response to Single Rectangul Wave Pulse (Adequate Damping) 77 ° 2 = 2 7 " J { l H ( J t t ) | 2 W ^ + | H d ( J w ) " H ( 3 « ) | 2 $ x x ( 3 " ) } <*» . . . ( 4 - 1 9 ) w h e r e H ( j » ) = s y s t e m t r a n s f e r f u n c t i o n $ ^ ( ; j <6 ) = N o i s e a u t o - c o r r e l a t i o n f u n c t i o n $-JQ^(J<O) - S i g n a l a u t o - c o r r e l a t i o n f u n c t i o n H^(jco) = d e s i r e d s y s t e m t r a n s f e r f u n c t i o n I f we a r e c o n c e r n e d w i t h a n i n p u t s i g n a l w h i c h i s e n t i r e l y n o i s e , e q u a t i o n ( 4 - 1 9 ) r e d u c e s t o e 2 = 2 ^ - J " | H ( j < o ) | 2 $ M ( j < o ) d < o . . . ( 4 - 2 0 ) — C O I t i s a p p a r e n t t h a t , f o r a m u l t i v a r i a b l e s y s t e m , t o a p p l y t h i s e q u a t i o n i n m a t r i x f o r m a n d s o l v e f o r e 2 = e , 2 = e , 2 + e 0 2 + e 2 . . . ( 4 - 2 l ) 1 1 2 n w o u l d i n v o l v e a g r e a t d e a l o f l a b o u r . I f a s y s t e m i s a t a l l c o m p l i c a t e d , t h e m o s t r e a l i s t i c m e t h o d f o r d e t e r m i n g t h e o p t i m u m c h o i c e o f p a r a m e t e r s f o r a f i x e d c o n f i g u r a t i o n i s b y 2 e x p e r i m e n t a l d e t e r m i n a t i o n o f e . T h i s i s d o n e q u i t e s i m p l y i n t h e m a n n e r s h o w n i n F i g u r e 4 - 1 4 . T h i s t y p e o f p r o c e d u r e was a p p l i e d t o b o t h t h e d i a g o n a l i z e d a n d u n d i a g o n a l i z e d s y s t e m s . A r a n d o m s i g n a l Feed back A/. e * to System ,2. e Figure 4-14 Experimental Determination of e with a white noise output i n the range 0.04 cps to 10 cps 2 2 was introduced at and e^  and e 2 were measured and recorded. This was also done for the s ignal introduced at X 2 « The parameter a representing the gain of the cross-coupling elements was v a r i e d . The resul ts of these tests are shown i n Figures 4-15 and 4-16. From Figures 4-15 and 4-16 we can see the mean-square error remains r e l a t i v e l y f l a t (within the accuracy of the experiment) as the degree of cross-coupl ing i s var ied for both the diagonalized and the undiagonalized systems. Consequently i n th i s p a r t i c u l a r case a mathematical analysis to determine e would be f r u i t l e s s since i t i s evident that the minimum mean-square error (with respect to a) c r i t e r i o n does not have any s ign i f i cance . 4.9 Comparative Comments It i s seen that i n the case considered the z.o. — n -h >*- —;—* Signal Input at Xf or X% /2 .8. .4 Stability timtf H 1 I- H 1 1-6 F i g u r e 4 - 1 5 E i g e n v a l u e S y s t e m M e a n - S q u a r e E r r o r Signal Input st At . ' / .8.. Signal Input at X% Stability > limit H h" 8 Tjj ana/ J^z F i g u r e 4 - 1 6 D i a g o n a l i z e d S y s t e m M e a n - S q u a r e E r r o r 80 diagonal ized system i s more stable than the eigenvalue system, since with a = 10.0 the l a t t e r was near the l i m i t of s t a b i l i t y while the s t a b i l i t y margin of the former was adequate. The mean-square error tests d id not show one system to be superior to the other. In both systems, i . e . , the eigenvalue and the diagonal ized, no compensating networks have been used to improve system performance. If such networks were des irab le , i t would be simpler to deal with the diagonalized system. Consider the s-plane Figures 4-17 and 4-18,. It i s quite apparent that to compensate Gc(jco) i n Figure 4-17 so that the G (jto) locus avoids the X,(JG>) locus would be considerably C n. more d i f f i c u l t than to compensate B^Cjco) i n Figure 4-18 so that the 8^ (3(0) locus does not enc irc le the -1 po in t . Considerably more elements are required i n a diagonal ized system (approximately twice as many). To reduce the number of components required i t may be possible to achieve approximate noninteract ion by approximating the transfer functions using simple RC networks. For example T 1 1 = s 2 + 30s + 23 s 2 + 30s + 1134 can be very crudely approximated by the network shown i n Figure 4-19. 81 F i g u r e 4 - 1 7 E i g e n v a l u e S y s t e m C o m p e n s a t i o n F i g u r e 4 - 1 8 D i a g o n a l i z e d S y s t e m C o m p e n s a t i o n ft* o. 5~n it F i g u r e 4 - 1 9 A p p r o x i m a t e T N e t w o r k U s i n g t h i s p a r t i c u l a r n e t w o r k i n t h e s y s t e m t h e same t e s t s a s b e f o r e w e r e c a r r i e d o u t . The r e s u l t s o f t h e s e t e s t s w e r e : ( 1 ) D i a g o n a l i z a t i o n was n e a r l y n o n e x i s t e n t , i . e . , i n t e r a c t i o n was v e r y s t r o n g . ( 2 ) R e s p o n s e t o a s i n g l e s q u a r e wave p u l s e was s o m e w h a t b e t t e r t h a n t h e e i g e n v a l u e s y s t e m . O v e r s h o o t was a p p r o x i m a t e l y t h e same a s w i t h t h e a c t i v e n e t w o r k ( 1 0 - 1 5 % ) b u t t h e o s c i l l a t i o n s t o o k a p p r o x i m a t e l y 5 t i m e s l o n g e r t o damp o u t . ( 3 ) The m e a n - s q u a r e e r r o r s t a y e d r e l a t i v e l y f l a t a s t h e c r o s s - c o u p l i n g was v a r i e d , h o w e v e r , ~~2 t h e v a l u e o f e was a p p r o x i m a t e l y t w i c e t h a t o f t h e " a c t i v e n e t w o r k " s y s t e m . The u n s a t i s f a c t o r y r e s u l t s f r o m t h i s t r a n s f e r f u n c t i o n a p p r o x i m a t i o n i s p r o b a b l y due t o t h e f a c t t h a t t h e a p p r o x i m a t i o n i s i n a d e q u a t e o v e r a s u f f i c i e n t b a n d w i d t h . 83 5 . E I G E N V A L U E METHOD A P P L I E D TO A S T A B I L I T Y STUDY OF FOUR P A R A L L E L CONNECTED SYNCHRONOUS M A C H I N E S The f o l l o w i n g i s a n e x a m p l e o f t h e e i g e n v a l u e m e t h o d a s a p p l i e d t o a s y s t e m o f f o u r p a r a l l e l c o n n e c t e d s y n c h r o n o u s m a c h i n e s . The b l o c k d i a g r a m f o r i n c r e m e n t a l o p e r a t i o n i s s h o w n i n F i g u r e 5-1 w h e r e Gp = s ( M ^ s + l ) = t r a n s f e r f u n c t i o n o f t h e p r i m e m o v e r o 1 G T = = t r a n s f e r f u n c t i o n o f t o r q u e T (1 + s T ^ U + s T 2 ) p r o d u c i n g e l e m e n t G = = c o m p e n s a t i n g n e t w o r k C (1 + sT,) s 1 sD^ = L a p l a c e t r a n s f o r m o f i n c r e m e n t a l s p e e d d e v i a t i o n T ^ = s y n c h r o n i z i n g t o r q u e c o e f f i c i e n t o f k^* 1 a n d i ^ * 1 l i n e s L ^ = l o c a l l o a d d i s t u r b a n c e T h i s i s a f o u r g e n e r a t o r s y s t e m a n d a l l t h e g e n e r a t o r s a n d c o n t r o l e l e m e n t s a r e a s s u m e d t o h a v e i d e n t i c a l t r a n s f e r f u n c t i o n s . T h i s i s n o t a r e s t r i c t i o n ; i t m e r e l y s i m p l i f i e s t h e p r o c e d u r e f o r i l l u s t r a t i n g t h e p r i n c i p l e i n v o l v e d . T h i s s y s t e m c a n be a n a l y z e d . u s i n g t h e e i g e n v a l u e m e t h o d . A n a l y s i s o f t h e s i n g l e l o o p s h o w n i n F i g u r e 5-1 y i e l d s Figure 5-1 Single-Loop of P a r a l l e l Operated Synchronous Machines 0 0 n ( G ^ + s G T ( K l l G c + K 1 2 ) + U + G c G T ) Y T l k ) 1 f k = 2 n - (1 + G C G T } y T i k = L I k = 2 Nov setting ( T " + S G T < K I I G C + K12> x _ I _ _E G ~ 1 + G CG T and -writing equation .(5-1) i n general form gives n n ( - X + 2 ] T i k ) A i - ^ T i k A k = 1 1 1 + S o G I where i = 1,2 ....n k — 1,2 ....n i £ k Let n A = \ T mm / mk 1 m: ^  k A - -T ink ~ mk be the elements of the matrix A. 86 System s t a b i l i t y can be determined from a Nyquist p l o t of the f u n c t i o n X k ( s ) + G ...(5-5) where the X^ s are the eigenvalues of the matrix A. .(12) Such a system has been d e a l t with by Crary A network an a l y z e r was used to determine the s y n c h r o n i z i n g c o e f f i c i e n t s and the swing curves evaluated by numerical i n t e g r a t i o n . The f o l l o w i n g d i s c u s s i o n w i l l deal w i t h the s t a b i l i t y of the l i n e a r i z e d system under v a r i o u s p o s s i b l e network c o n d i t i o n s . From equations (5-4) we o b t a i n the eigenvalue equation -X + T 11 -T 21 -T 31 -T 41 where -T 12 X + T 22 -T 32 -T 42 -T 13 -T L23 -X + T 33 -T 43 -T 14 -T 24 " T34 -X + T 44 = 0 ...(5 - 6 ) kk 3=1 To i l l u s t r a t e the a p p l i c a b i l i t y of the eigenvalue method, l e t us use the t i e - l i n e coef f ic ients given by Crary Since we are dealing with the s t a b i l i t y for the l i n e a r i z e d system, we sha l l neglect the i n t i a l incremental deviations i n angular displacement and the power angles. It i s apparent from equation (5-7) that one X value i s going to be zero. This i s seen by adding the second, t h i r d and fourth columns to the f i r s t . Using C r a r y 1 s values for the synchronizing coef f ic ients the fol lowing determinants were solved on the Alwac III E d i g i t a l computer: (1) Fault On -X + .7931 -.7480 -.0231 -.0220 -.7480 -X + 1.0440 -.1510 -.1450 -.0231 -.1510 -X + .2378 -.0638 -.0220 -.1450 -.0638 -X + .2307 (2) Fault P a r t i a l l y Cleared -X + .863 -.782 -.038 -.043 -.782 -X + 1.311 -.250 -.279 -.038 -.250 -X + .413 -.0638 -.043 -.279 -.125 -X + .447 = 0 (3) Fault Cleared -X + 1.205 -.960 -.116 -.129 -.960 -X + 2.567 -.760 -.847 -.116 -.760 -X .+ 1.247 -.371 = 0 -.129 -.847 -.371 -X + 1.347 The eigenvalues for these determinants are given i n table 5.1. Eigenvalues System Condit ion X-^  X 2 X-j X^ Fault On 0 1.689 .318 .29 Fault P a r t i a l l y Cleared 0 1.936 .559 .513 Faul t Cleared 0 3.424 1.69 1.251 Table 5.1 Eigenvalues E s s e n t i a l l y we have reduced the mult ivar iable system of Figure 5-1 to four equivalent single var iable systems as shown i n Figure 5-2. 5.1 Nyquist S t a b i l i t y Invest igat ion Now to graphica l ly study the s t a b i l i t y of the system, we construct a Nyquist p lot of the function R ^ s ) = + G(s) Figure 5-2 Equivalent Multi-Loop Single Variable System CO 90 Substitution of the proper transfer functions into equation (5-2) yields G(s) = [s(l+ T l S ) ( l + T 2 s ) ( l + T 3s)+ l ] ( l + M l S) (1 + Tjs) (1 + T2-s) (1 + T 3s) + s ( l + Mjs) (Ki;j_ + K 1 2 s ( l + T 3s)) To plot the function G(s), we consider the behaviour of the function as s —*- 0 and ¥e f i n d that Knowing the necessary time constants and gains we can then plot the G(s) locus. as shown i n Figure 5-3. From Figure 5-3 we know that the system w i l l be unstable i f the G(j<o) locus intersects or encloses the c r i t i c a l points - r — (k = 1,2,3,4). 5.1 Root-Loci S t a b i l i t y Investigation. Another way to graphically study the s t a b i l i t y of the system i s to determine a root-locus plot of the function G(s) H3» ( S —*- oo ) G(s) • (s — 0) 1 In the case considered, suppose the G(s) locus i s Suppose we consider the root-locus plot of the F i g u r e 5 - 4 R o o t - L o c u s P l o t o f t h e F u n c t i o n K function KG where K i s var iable from 0 to o o A root- locus p lot of KG can then take the form shown i n Figure 5-4. The system w i l l remain stable provided that the r o o t - l o c i remain i n the l e f t - h a l f s-plane. One such diagram is su f f i c i en t for a complete: root- locus analysis of the system. The roots are found by locat ing the value of K = A^., f ° r example the condit ion K = 0 gives the roots for A = 0 as shown i n Figure 5-4. t For d i f f erent network conditions the A^ s change. The corresponding locat ions of the charac ter i s t i c roots i n the s-plane can be determined from Figure 5-4. I f a root- locus analysis of the charac ter i s t i c equation of the system i s attempted by conventional methods, a polynomial with 20 zeros would have to be considered and considerably more labour would be involved. Thus i t i s apparent that the eigenvalue method provides a simpler root- locus analysis for th i s system. 6. CONCLUSION Si n g l e v a r i a b l e g r a p h i c a l a n a l y s i s and design techniques have been shown to be a p p l i c a b l e to c e r t a i n types of m u l t i v a r i a b l e systems when the eigenvalue method and the d i a g o n a l i z e d method are used. The experimental determination of system eigenvalues has been i n v e s t i g a t e d and shown to be f e a s i b l e . The s u i t a b i l i t y of s i m u l a t i o n s t u d i e s to v e r i f y design, to i n v e s t i g a t e the i n f l u e n c e of parameter v a r i a t i o n s and to evaluate the mean-square e r r o r has-been shown; A comparison of the eigenvalue method and the more complicated d i a g o n a l i z a t i o n method has been made. I t was seen t h a t the d e sign methods are very u s e f u l to determine i n i t i a l system c o n f i g u r a t i o n f o r s i m u l a t i o n s t u d i e s . The simulated system can then be used to perform o p t i m i z a t i o n and f u r t h e r improvements. The eigenvalue method has been a p p l i e d to a system of f o u r p a r a l l e l - c o n n e c t e d synchronous generators and g r a p h i c a l methods of s t a b i l i t y i n v e s t i g a t i o n have been d i s c u s s e d . A p p e n d i x I C o n s i d e r t h e n e t w o r k s h o w n i n F i g u r e A I - 1 a -A/VWV *3 -VWW - A A A A / V -F i g u r e A I - 1 ( 1 3 ) Wadhwa h a s s h o w n t h i s n e t w o r k t o h a v e t h e t r a n s f e r f u n c t i o n V 0 " b 0 y — = 3 2 • • . ( A I-1 ) 1 a - j S + a 2 s + a^s + 1 w h e r e b o = r ~ (1 + 3 a ) = 2 RC + (1 + a ) R C 6 1 (1 + 3 a ) 1 = a d ±-2a) R 2 C 6 ( C 2 + ^ ^ (1 + 3 a ) 96 2 o oc 3 3 (1 + 3a) 2 4 6 R-^  = = R^ ~ R Ry — Rg = ocR a - ( l + 2a) &± > —— a 2 ( l + 3a) Now s e t t i n g = 0 makes a^ = 0 I f we set a = 1 c 2 = c 6 = c equation AI-1 becomes ^ 3 B V < . • * $ ) ( . • * $ ) ...(AI-2) 97 A p p e n d i x I I C o n s i d e r t h e n e t w o r k c o n f i g u r a t i o n s h o w n i n F i g u r e A I I - 1 V 2. Z, 4 Z. z . F i g u r e A I I - 1 The t r a n s f e r f u n c t i o n h a s t h e f o r m Z. 0 1 + — Z l Z 4 I + I + i + I Z l Z 2 Z 3 S + 1 + 1 + 1 Z 5 Z 6 \ Z 1 Z 2 Z 3 Z 5 ( A I I - 1 ) Now i f we s e t 98 I = 1 x Z l R l Z = C 2 S - = - - • = C Z 3 : R 3 Z 4 : 2 3 I _ 1 1 _ P Q Z 5 - R 5 Z , = C 6 S S u b s t i t u t i n g t h e s e v a l u e s i n t o e q u a t i o n A I I - 1 y i e l d s V 0 s 2 ( C 2 R 3 C 4 ) + s R 3 C 4 ( | i + J 3 + + ^ V . I s 2 ( C 2 R 3 C 6 ) • + s R 3 C 6 ( | + | + i ) + | . 1 J 5 5 Now i f = we o b t a i n s + S p ( ^ + n IP- - ° 2 R l R 3 R 5 R 1 R 3 C 2 C 4 , A T T , v V . - 2 , 1 / 1 , 1 , l w 1 W» 2 JX-^  Jtt-j Itf^ I t ^ I ^ l ^ l s g 99 R E F E R E N C E S 1 . P o v e j a i l , D . J . a n d F u c h s , A . M . , " A M e t h o d f o r t h e P r e l i m i n a r y S y n t h e s i s o f C o m p l e x M u l t i - L o o p C o n t r o l S y s t e m s " , T r a n s . A I E E , P t . 2 , V o l . 7 4 , p p . 1 2 9 - 1 3 4 , J u l y , 1 9 5 5 . 2 i K a v a n a g h , R . J . , " M u l t i v a r i a b l e C o n t r o l S y s t e m s S y n t h e s i s " , T r a n s . A I E E , P t . 2, V o l . 7 7 , p p . . 4 2 5 - 4 2 9 , N o v e m b e r , 1 9 5 8 . 3 . F r e e m a n , H . , " A S y n t h e s i s M e t h o d o f M u l t i p o l e C o n t r o l S y s t e m s " , T r a n s . A I E E , P t . 2 , V o l . 7 6 , p p . 2 8 - 3 1 , M a r c h , 1 9 5 7 . 4 . H o r o w i t z , I . M . , " S y n t h e s i s o f L i n e a r M u l t i v a r i a b l e F e e d b a c k C o n t r o l S y s t e m s " , I R E T r a n s . , P G A C , V o l . A C - 5 , p p . 9 4 - 1 0 5 , J u n e , 1 9 6 0 . 5 . H s i e h , H . C * , a n d L e o n d e s , C . T . , " O n t h e O p t i m u m S y n t h e s i s o f M u l t i p o l e C o n t r o l S y s t e m s i n t h e W i e n e r S e n s e " , I R E N a t i o n a l C o n v e n t i o n , M a r c h , 19~W. ' ~~ 6 . K r a s o v s k i i , A . A . , "Two C h a n n e l A u t o m a t i c R e g u l a t i o n S y s t e m s w i t h A n t i s y m m e t r i d : Crosse? C o n n e c t i o n s " , A u t o m a t i o n a n d R e m o t e  C o n t r o l . P t . 2 , V o l . 1 8 , p p . 1 3 9 - 1 4 9 , F e b r u a r y , 1 9 5 7 . 7 . Newman, D . B . , " T h e A n a l y s i s o f C r o s s - C o u p l i n g E f f e c t s o n t h e S t a b i l i t y o f T w o - D i m e n s i o n a l , O r t h o g o n a l , F e e d b a c k C o n t r o l S y s t e m s " , I R E T r a n s . . P G A C , V o l . A C - 5 , p p . 3 1 4 - 3 2 0 , S e p t e m b e r , 1 9 6 0 . 8 . B o h n , E . V . y " D e s i g n a n d S y n t h e s i s M e t h o d s f o r a C l a s s o f M u l t i v a r i a b l e F e e d b a c k C o n t r o l S y s t e m s B a s e d o n S i n g l e - V a r i a b l e M e t h o d s " , T r a n s . A I E E , p a p e r N o . 6 2 - 7 5 , D e c e m b e r , 1 9 6 1 . 9. B o h n , E . V . * " S t a b i l i z a t i o n o f L i n e a r M u l t i v a r i a b l e F e e d b a c k C o n t r o l S y s t e m s " , T r a n s ; , I R E , S e p t e m b e r , I 9 6 0 , p p . 3 2 1 - 3 2 2 . ' 100 1 0 . Boksenbom, A* , and Hood, R . , "General Algebraic Method Appl ied to Control Analysis of Complex Engine Types", N a t l . Advisory Committee  for Aeronautics. Washington, D . C , Rept. No. 9 8 0 , A p r i l , 1 9 5 9 . 1 1 . Meerov, M . V . , "The Autonomy of Mult i -Loop Systems Which are Stable When Their Steady-State Prec i s ion i s Increased Without L i m i t " , Automation and Remote Contro l , No. 5 , pp. 4 1 1 - 4 2 4 , 1 9 5 6 . 1 2 . Crary , J * B . , Power System S t a b i l i t y , John Wiley and Sons, New York, V o l . I I , Chapter 5 , Sections 18 and 1 9 , 1 9 4 5 . 1 3 . Wadhwa, L . K . , "Simulation of Third-Order System with One Operational Ampl i f i er" , Proc . IRE, pp. 2 0 1 - 2 0 2 , February, I 9 6 0 . 1 4 . Kavanagh, R . J . , "Noninteracting Controls i n Linear Mul t ivar iab le Systems", Trans. AIEE, Pt . 2 , V o l . 7 6 , pp. 9 5 - 9 9 , May, 1 9 5 7 . 15« Freeman* H,y " S t a b i l i t y and Physical R e a l i z a b i l i t y Consideration i n the Synthesis of Mult ipole Control Systems", Trans. AIEE, P t . 2 , V o l . 7 7 , Appl icat ions Industry, pp. 1 - 5 , March, 1 9 5 8 . 1 6 . Bohn, E . V * r and Kasvand, T . , "The Use of Matrix Transformations and System-Eigenvalues i n the Design of Linear Mul t ivar iab le Control Systems", (to be Publ ished) . 1 7 . T r u x a l f J*G*? "Control Systems Synthesis . McGraw-H i l l , New York, pp. 4 5 4 - 4 5 7 , 1 9 5 5 . I 8 v Kirchmayer, L . K . , "Di f ferent ia l Analyzer Aids Dispatching System Design", Trans« AIEE, P t . 2 , V o l . 7 5 , pp. 5 7 2 - 5 7 9 , January, 1 9 5 9 . 1 9 . Mesarovic, M . D . , The Control of Mul t ivar iab le Systems, The Technology Press, Cambridge, Massachusetts and New York, Wiley and Sons, 1 9 6 0 . 

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