GRAPHICAL. TESTS FOR THE ABSOLUTE STABILITY OF MULTIVARIABLE NONLINEAR SYSTEMS by JACKSON LAU B.S.E.E. Oregon State University, Corvallis, Oregon 1969 B.Sc. (Math) Oregon State University, Corvallis, Oregon 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE r We accept this thesis as conforming to the required standard Research Supervisor . Members of Committee Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA November, 1971 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for ah advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Three graphical methods, similar to Popov's approach, are derived to test a s t a b i l i t y criterion for a general class of multivariable nonlinear feedback systems. Various simplifications of these methods are then derived for four particular classes of system: the series system, the parallel system, the internal feedback system, and the symmetric system. These s t a b i l i t y tests may be applied to active networks that consist of passive linear time-invariant elements and many nonlinear controlled sources. It is further shown that certain special network structures may be represented as one of the four particular classes of system. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT v i i 1. INTRODUCTION 1 1.1 Description of System 1 1.2 S t a b i l i t y of Single Variable Systems 5 1.2..1 Popov .Is . S t a b i l i t y .Criterion. , 5 1.2.2 Graphical Test f or a Single Variable System 7 1.3 S t a b i l i t y of M u l t i v a r i a b l e Systems. 7 1.4 Relations Between Networks: and Systems • •••« » 8 2. GRAPHICAL STABILITY TESTS FOR MULTIVARIABLE SYSTEMS 9 2.1 Introduction 9 2.2 Previous Works on Graphical Tests 9 2.3 Graphical Tests f o r the General System 10 2.4 Remarks 27 2.5 Graphical Tests f o r P a r t i c u l a r Systems 28 2.5.1 The Series System.... •.. 28 2.5.2 The P a r a l l e l System. 38 2.5.3 The In t e r n a l Feedback System 42 2.5.4 The Symmetric System 53 3. SYSTEM REPRESENTATION OF NETWORKS CONTAINING MANY NONLINEAR CONTROLLED SOURCES .... .... 58 3.1 Introduction 5 8 3.2 System Representation of the General Network 59 3.3 Networks with Series System Representation 63 3.4 Networks with P a r a l l e l System Representation 66 3.5 Networks with Symmetric System Representation 69 i i i Page 4. CONCLUSIONS • 7 3 REFERENCES 7 4 c LIST OF ILLUSTRATIONS Figure Page 1.1 General time-invariant nonlinear multivariable system 2 1.2 Time-invariant characteristics of nonlinear elements 2 1.3 The series system with n nonlinearities ... 3 1.4 The paral l e l system with n nonlinearities 3 1.5 The internal feedback system with n nonlinearities 3 1.6 Graphical test of Popov's st a b i l i t y criterion for a single variable system 6 2.1 Graphical interpretation of (2.9) ... 12 2.2a Plot of (2.16a) for (j u) with a pair of Popov lines 12 2.2b Plot of (2.16a) with just one Popov line 12 2.3 Plots for the graphical method of Theorem 2.1 14 2.4 Graphical interpretation of (2.22) 19 2.5 Plots for the graphical method of Theorem 2.2 20 2.6 Plots of Re {g i ; L(j co)} for 1=1,2,3 16 2.7 The rotated curves for g 1 2 ( s ) f o r E x a m P l e 2 - l • 2.8 Plots of w^(ju) for off-diagonal elements of matrix (2.19) 16 2.9a Curves (2.29) for Example 2.3 with N=l 24 2.9b Curves (2.29) for Example 2.3 with N=2.25 25 2.9c Curves (2.29) for Example 2.3 with N=2.5 26 2.10 Rotated curves of g(s) of Example 2.4 30 2.11 Curves of (2.29) for Example 2.4 31 2.12 A three dimensional series system with linear blocks of different transfer functions 33 2.13 Rotated curves for ( s) of Example 2.5.. • 35 2.14 Curves of (2.29) for the case of i=l in Example 2.5. 36 2.15 Curves of (2.29) for i=2/i=3 of Example 2.5 37 v Figure Page 2.16 Curve's (2.23) of g 1 2 ( s ) i n Example 2.6 39 2.17 Curves (2.29) for i = l of Example 2.6 40 2.18 Curves (2.29) for i=2 of Example 2.6 41 2.19 An i n t e r n a l feedback system with n=3 47 2.20 Rotated curves f o r g^(s) of Example 2.7 48 .2.21 .Curves (.2..9) of Example 2.7 49 9 2.22 The curves w_.(jcu) for Example 2.7 50 2.23 Curves (2.29) for i=l/i=3 of Example 2.7 51 2.24 Curves (2.29) for i=2 of Example 2.7 52 2.25 The rotated curves of ? ^ — of Example 2.8... 56 s +s+l 2.26 The Popov p l o t of —x of Example 2.8 56 s +0.1s+0.1 2.'27 Curves (2.29) for i=l/i=3 of Example 2.8.. 57 2.28 Curves (2.29) f o r i=2 of Example of 2.8. 57 3.1 Network n 60 3.2a A modified RC o s c i l l a t o r 62 3.2b An ac-equivalent c i r c u i t 62 3.2c Equivalent c i r c u i t of o s c i l l a t o r i n form of network n 62 3.3 The network n 64 s 3.4 Network for Example 3.1 64 3.5 The cascade configuration of network n g which continas only c o n t r o l l e d current sources 65 3.6 Network v 68 P 3.7 Network for Example 3.2...... 68 3.8. Network n 7Q sym 3.9 Network for Example 3.3 70 v i ACKNOWLEDGEMENT I wish to express my sincere gratitutde to Dr. M. S. Davies, the supervisor of this project, for his invaluable guidance, and Dr. H. Chinn for reading the manuscript. Grateful acknowledgement i s given to the National Research Council for the financial support received under NRC Grant A-4148. I would also like to thank Miss Linda Morris for typing the manuscript. v i i i 1 1. INTRODUCTION 1.1 Description of System This thesis considers the absolute s t a b i l i t y of a class of multi-variable feedback systems which consists of a memoryless, nonlinear part N, and a linear time-invariant plant G. Such a system i s depicted in Fig. 1.1. Here cr, and y are a l l n-vectors. The nonlinear part, N, consists of a set of n time-invariant memory-less nonlinear gain elements defined by C. = f.Co-J ( i = 1, n) (1.1) It i s further assumed that ( A 1 ) 0 $-a. f . ( c r . ) S k.a 2 for a. 4 0 i I I x I l f.(0) = 0 such that each of these elements i s bounded within a sector in the f i r s t and third quadrant, as i s shown in Fig. 1.2. The linear plant G of the system is characterized by an n x n linear time-invariant transfer function matrix G(s) = (g^j(s)}. It i s assumed that (A2) g^j(s) is rational (A3) g^j(s) has a l l poles in the l e f t half of the complex plane. A wide variety of systems may be treated by choosing certain specific structure for G(s). This thesis considers the following particular classes of systems. (1) The Series System. The forward path consists of linear time-invariant blocks separated by amnesic nonlinearities. This i s shown in Fig. 1.3. The linear transfer function matrix is of the form 2 Fig. 1.2 Time-invariant characteristics of nonlinear elements _ 9 — i '9,(3) N; 9j(s) N, n Fig. 1.3 The series system with n nonlinearities 9 — ^ y9n(s) *> Fig. 1.4 The parallel system with n nonlinearities N, Nn 'n Fig. 1.5 The internal feedback system with n nonlinearities i 4 G(s) = 8 l ( s ) 0. 0 -g n(s) o 0 (1.2) (2) The parallel system. The forward path consists of n-parallel branches, each containing a nonlinear element in series with a linear time invariant transfer function. This i s shown in Fig. 1.4, with transfer matrix g-j/s) g 1(s) ...... g 1(s) g 2(s) g 2(s) g 2(s) G(s) = g n(s) g n(s) g n(s) (1.3) (3) The internal feedback system. The forward path consists of n-single nonlinear feedback loops. This is shown in Fig. 1.5. G(s) = n n n g x(s) Eg.(s) 1 1 Eg.(s) 1 -Eg,(s) -E g,( s) -Eg.(s) g 2(s) Eg.(s) 2 n -Eg.(s) 1 1 g 3(s) n • • • • -Eg.(s) 1 1 n " -Eg.(s) 1 x Eg.(s) 1 Eg.(s) 2 1 where n Eg.(s). 3 n n •'• 2 g. (s) g (s) i l n n-1 (1.4) g t(s) = H.(s)/[1 + rr H^s)] (1.5) (4) The symmetric system. This class of system has a block diagram 5 s i m i l a r to the general system of F i g . 1.1. But i t s l i n e a r t r a n s f e r function has the s p e c i a l property of g..(s) = g..(s), such that when wr i t t e n out i n f u l l , L g u ( s ) g 1 2 ( s ) g 1 3 ( s ) .. g l n ( s ) g 1 2 ( s ) g 2 2 ( s ) g 2 3 ( s ) .. 8 2 n ( s ) g 1 3 ( s ) § 2 3 ( S ) • -g 3 3 ( s ) .. » .. g 3 n ( s ) • (1.6) • • g 2 n ( s ) • g 3 n ( s ) .. • nn 1.2 S t a b i l i t y of Single Variable Systems Most of the e a r l i e r work on the s t a b i l i t y of nonlinear feedback systems was done for s i n g l e v a r i a b l e systems, that i s , systems as d e t a i l e d i n Section 1.1 with n=l. L u r i e [1] took the c l a s s i c a l approach based on Lyapunov's Second Method. However, the s t a b i l i t y c r i t e r i o n obtained was dependent on the choice of a s u i t a b l e Lyapunov function, and r e s u l t s were possible only a f t e r a great deal of computation. Popov [2] took a d i f f e r e n t approach and obtained powerful r e s u l t s which are s i m i l a r i n many respects to the Nyquist t e s t f o r l i n e a r feedback systems. The s t a b i l i t y c r i t e r i o n obtained i s dependent only on the frequency c h a r a c t e r i s t i c s of the l i n e a r part of the system and i t has a convenient graphical i n t e r p r e t a t i o n to f a c i l i t a t e a p p l i c a t i o n . 1.2.1 Popov's S t a b i l i t y C r i t e r i o n A nonlinear feedback system as depicted i n F i g . 1.1, with the assumptions of (Al) to (A3) holding and n=l, i s absolutely stable i f there e x i s t s a r e a l constant q such that k _ 1 + Re (l+jwq)G(jto) > 0 - (1.7) 6 Fig. 1.6 Graphical test of Popov's s t a b i l i t y criterion for a single variable system 7 1.2.2 Graphical Test for a Single Variable System The s t a b i l i t y criterion expressed in (1.7) may be rewritten as k" 1 + X(w) - qY(w) > 0 (1.8) where X(w) = Re G(jw) (1.9a) Y(o)) = wlm G(jw) (1.9b) Now, (1.9) describes a curve on the X-Y plane with a) as the parameter, and the limiting case of (1.8) may also be .represented on .the X-Y plane as a straight line with slope of l/q, and passing through point (-k 1 , 0). (This straight line w i l l be referred to as the Popov l i n e ) . Hence, Popov's st a b i l i t y criterion i s satisfied when a real constant q can be chosen such that a Popov line may be drawn to l i e completely on the l e f t of the curve (1.9). (See Fig. 1.6). 1.3 Stability of Multivariable Systems Popov's s t a b i l i t y criterion has been extended by Jury & Lee [3] to the case of multiple nonlinearities. Since their work is the basis for the following chapters of this thesis, their result i s stated in the following theorem. Theorem 1.1 Under the assumptions (Al) to (A3), the system of Fig. 1.1 is absolutely stable, i f there exists a real diagonal matrix Q such that P(u) > 0 for a l l to * 0 (1.10) where P(u) = W(jw) + W*(JOJ) + 2k"1 (1.11) W(jto) = (l+jwQ)G(j(o) (1.12) K = diag{k i sk., .... k } (1.13) and W*(j(u) is the complex conjugate transpose of W(jto). 8 Working independently, similar results were obtained by Anderson [4], who used a multiplier of a more general form, (A + jwB), in place of (I + jwQ) in (1.12). In a paper'by Narendra and Goldwyn [5], i t is shown that the criterion (1.10) is a necessary and sufficient condition for Lyapunov functions of a particular form to exist. Sandberg [6] and Zames [7], used functional analysis techniques, and obtained results that bear close comparison with those of theorem 1.1. 1.4 Relation Between Networks and Systems The problem of network sta b i l i t y follows directly from that of system s t a b i l i t y , since many classes of networks may be shown to have an equivalent systems representation. It was Zames [7] who f i r s t pointed out the close connection between the single variable nonlinear system and a passive network terminated by a nonlinear element. Since then, papers by Sandberg [8], Trick [9], Goldstein and Frank [10] have applied the s t a b i l i t y results of nonlinear systems to classes of networks that contain passive and nonlinearly controlled elements. In a recent paper [11] , Ho and Figueiredo considered the s t a b i l i t y of a broader class of networks that may contain controlled sources and active elements. As an extension of this latter work, Chapter 3 of this thesis w i l l show how a network containing many, nonlinear controlled sources may be represented as a multivariable nonlinear feedback system, and how certain classes of this network are related to the particular classes of systems under special investigation in Chapter 2. 9 2. GRAPHICAL STABILITY TESTS FOR MULTIVARIABLE SYSTEMS 2.1 Introduction The ava i l a b i l i t y of a graphical test for the single Variable non-linear feedback system has greatly facilitated the application of Popov's st a b i l i t y criterion. Following the same approach, we develop in this chapter graphical tests for the s t a b i l i t y of multivariable nonlinear feedback systems. As discussed in Section 1.3, a sufficient condition for a multi-variable nonlinear feedback system to be absolutely stable i s that criterion (1.10) of Theorem 1.1 be satisfied. This requires that an (n x n) matrix, p = {p£-}> be tested for positive definiteness. There are two ways to achieve this. One is Sylvester's test, which requires a l l principal minors of P be positive. The other test is that a row sum dominance condition [12] n P i ± > Z |P ±J (i=l,n) (2.1) i=l J i£j be satisfied. Sylvester's test is both necessary and sufficient but the condition (2.1) is only sufficient for positive definiteness. However, although the latter test yields more conservative results, the amount of computation work w i l l be less since each row of P may be tested separately. 2.2 Previous Works' on Graphical Tests The procedure for testing absolute s t a b i l i t y may be greatly simplified when either of the two mentioned tests for positive definiteness of the matrix P can be related to the frequency plots of the matrix elements. Successful work in this direction has been done by Davies [13] and Chang [14] who obtained results for the series system and a special case of the parallel system. But no such graphical test has yet been developed for the general m u l t i v a r i a b l e system f or which only the numerical approach has been used [15,16]. But th i s involves a great amount of computation and the use of a d i g i t a l computer. In the following sections, we develop three graphical techniques to test the s t a b i l i t y of the general nonlinear systems and give various s i m p l i f i c a t i o n s of these tests when they are applied to several p a r t i c u l a r classes of systems. 2.3 Graphical Tests for the General System We f i r s t express the s t a b i l i t y c r i t e r i o n (1.10) i n terms of the; elements of the l i n e a r t r a n s f e r function matrix. We l e t G(jto) =' {g^Cjw)}; W(ja>) = {w^ja))} (2.2) and Q = d i a g t q ^ q 2 > q^} (2.3) Then P(.w) has the form 2k 1 1+w 1 1(jo))+w 1 1(-j(o) w12(jo))+w21(-j(»)) w l n(joj)+w n l(-jaj) w 2 1(j<jj)+w 1 2(-jco) 2k 2 1+w 2 2(j(o)+w 2 2(-jcj) w 2 n(ja))+w n 2(-ja)) . % l U u ) + w l n ( - J o j ) w n 2(j(o)4v 2 n(-ja J) .... 2k; 1-h. n n(jco)4v n n(-ja )) (2.4) And by (2.1), a s u f f i c i e n t condition f o r P(w) to be p o s i t i v e d e f i n i t e i s 1 n 2k" x + 2Re{w (ju)} > E |w (ju)+w,.(-ja>)|; (i=l,n) (2.5) j = l s u b s t i t u t i n g for-Iw^Cju)! •+ I W j . H u ) ! * Iw^Cjco) + w j l(-jo))| (2.6) 11 and |w..(jco)| = |w..(-juO| (2.7) we have as a sufficient condition, 2k"1 + 2Re{w (jo)} > Z |w (jco)| + E |w (jco) | ; 1 1 j=i X J j=i J j ^ i j ^ i ( i=l,n) (2.8) Now, i f there exists constants b., y . such that (2.9) n n Z |w..(jco)| + E |w..(jco) |; (i=l,n) j=l 1 J j=l J 1 j5^i j ^ i (2.10) Then, in place of (2.8), we may consider the more conservative condition Then, the graphical approach developed by Popov may be applied in a modified form to obtain the constants k^, b^, and y ^ . In fact, there are two approaches to obtain y ^ such that we can derive two graphical methods for the s t a b i l i t y condition of (2.11). A third graphical method is derived by applying Popov's technique in a modified form to the s t a b i l i t y condition of (2.8). We now present these three graphical tests for the s t a b i l i t y of the general multi-variable system. Then, in the next section, we shall remark on their advantages and differences. We f i r s t note the similarity of (2.9) to (1.7). Hence, the constants b^ may be determined by choosing a certain constant q such that Popov lines may be drawn completely to the l e f t of curves defined by 2k"1 - 2b ± > Y. ; (1=1,n) (2.11) X(co) = Re{g1.(jco)} Y(co) = wlm{gli(ja))} (2.12a) (2.12b) This is shown in Fig. 2.1. Fig. 2.1 Grahpical interpretation of (2.9) Y X Fig. 2.2 (a) Plot of (2.16a) f o r w ^ (jw) with a pair of Popov lines Y X Fig. 2.2 (b) Plot of (2.16a) with just one Popov line 13 In order to determine Y^J we consider the i n e q u a l i t y |Re{w,.(jw)}|2 + |lm{w,,(jco)}|2 = |w..(jco)! 2 (2.13) and /2 a. . > |w (ju>) | (2.14) where the constants a., are to s a t i s f y both i n e q u a l i t i e s and Expanding (2.15) a ^ * |Re{w i j(jo))}| (2.15a). a., >, |lm{w. .(jco)} | (2.15b) l j i j a^j ^ Re{w ij(jco)} 5 - a ^ (2.16a) a.. * Im{w..(j(o)} 5 -a.. (2.16b) Now, (2.16a) i s s a t i s f i e d when the curve defined by X(w) = Relg^Qio)} (2.17a) Y(w) = coIm{ g i j(ju))} (2.17b) i s bounded i n s i d e a p a i r of p a r a l l e l s t r a i g h t l i n e s of slope l/q^.and passing through points (-a.,, 0) and (a.., 0), as shown i n F i g . 2.2a. To take advantage of symmetry, we rotate the curve (2.17) 180° about the o r i g i n , and superimpose t h i s onto the o r i g i n a l curve of (2.17). We s h a l l r e f e r to t h i s combination as the rotated curve of (2.17). Then, (2.15a) i s s a t i s f i e d by the drawing of j u s t a s i n g l e Popov l i n e to the l e f t of t h i s rotated curve of (2.17). This i s shown i n F i g . 2.2b. Equation (2.15b) may s i m i l a r l y be s a t i s i f e d by considering the rotated curve of X(fo) = Im{g (jco)} . (2.18a) Y(to) = -coRefg^juj)} (2.18b) 1=1 Y / y x A Y -n / A 1 k¥ x i=2 Y X X ; \ X -°nz Y X -C25\ ( X Y X -a2f\ 1\S \ 1% Y Fig. 2.3 Plots for the graphical method of Theorem 2.1 15 The constant a ^ of (2.14) i s then determined by p l o t t i n g both rotated curves of (2.17) and (2.18) on the same axes i n the X-Y plane and drawing a si n g l e Popov l i n e to the l e f t of'both curves. Thus, with t h i s f i r s t g r aphical method, we s a t i s f y the s t a b i l i t y conditions of (2.11) by p l o t t i n g one group of curves for each row of the matrix G(jio). Each group consists of a plo t of (2.12) f o r g ^ ( j a ) ) , and a plot containing both rotated curves of (2.17) and (2.18) f o r each g^j(jw) when i ^ j , and j = l , n. Then, f o r each of these groups of p l o t s , a constant i s chosen such that Popov l i n e s , a l l with the same slope of l / q ^ , may be drawn to the l e f t of each of these p l o t s . With the constants b. and a.. thus obtained, v, may be found as a sum of the a,, and a,, terms. Then i i j j i can be determined with (2.11). This method i s i l l u s t r a t e d i n F i g . 2.3, and the r e s u l t i s formulated i n the following theorem. Theorem 2.1 Under the assumptions (Al) to (A3), the system of F i g . 1.1 i s absolutely stable i f there e x i s t s _ r e a l constants such that -1 ; 72 k ^ > ~ [ I (a.,+a,.)] + b.; (i=l,n) j = l (2.19) where a ^ , and b^ are defined In (2.14) and (2.9) r e s p e c t i v e l y . Example 2.1 Consider a three-dimensional general system with N 1 1 G(s) = s+6 1 s+8 N s+10 s+12 s+3 s+1 s+4 1 N s+9 s+2 1 1 (2.20) where N i s a p o s i t i v e constant. 16 - 0 . 5 --10 Fig. 2.6 Plots of Re{g . (jw)} for i - 1,2,3. Fig. 2.8 Plots of h7..(jw)| for off diagonal elements of matrix (2.19) s-f4 0.3 Fig. 2.7 The rotated curves for g^Cs) of Example 2.1 18 Taking the case of N=l for this example, i t can be shown that the linearized system i s stable for a l l positive k., (1=1, 2, 3) and Rae and MacLellan [17] < < have established s t a b i l i t y for the nonlinear system with k^ = 5, k 2 ~ 2, k^ ^.1. Using numerical and analytical methods, Dewey [16] obtained the result k^ = = k^ = 10 with q=0. Chen [15] improved that result to k l = k2 = k3 ^ 1 8 W i t h q = ° a n d k l = k2 = k3 ^ 3 2 3 W i t h q = °' 2' To apply theorem 2.1, we show the curves for (2.9) in Fig. 2.6 and rotated curves of (2.17) and (2.18) for g-^s) in Fig. 2.7. Similar plots fo r the other off-diagonal elements are made. With a choice of = q 2 q^ = 0, we have Hence, b l = b2 = b 3 = o a 1 2 = 0.250 a 1 3 = 0.167 a 2 3 = 0.125 a 2 1 = 0.110 a 3 1 = 0.100 a 3 2 = 0.083 k± = 2.30, k 2 = 2.48, k 3 =-2.97 We now develop another graphical method to obtain of (2.10). This approach makes use of the inequality c.. >, |w..(jo))| for a l l w (2.21) the right hand term of which may be expanded with 2 i ,_ . , v . i2 |w1;)(ja))r - la+j^g^cjw)!-= (l+qV)|gij(jco)|2 • |-g±j(jw) I2 + q l^g^Cju) such that we have for (2.21) c 2^ >, X(&>) - q 2Y(u) (2.22) 19 Fig. 2.4 Graphical interpretation of (2.22) 21 where . X(to) = Re 2Cg. ,(jco)} + Im2{g .(jco)} (2.23a) Y(u>) = -co2X(co) (2.23b) Now, (2.22) i s s a t i s f i e d g r a p h i c a l l y by drawing a s t r a i g h t l i n e , having 2 2 a slope of l / q . and passing through point (c.,, 0), which l i e s completely i i j to the r i g h t of curve (2.23). This i n t e r p r e t a t i o n of (2.22) i s shown i n F i g . 2.4. Hence, i n t h i s second method, we s a t i s f y the s t a b i l i t y condition of (2.11) by p l o t t i n g one group of curves f o r each row of the matrix G(jco). Each group consists of a p l o t (2.12) f o r g^^(jco) and a p l o t of (2.23) for each g..(jco) f o r j = l , n and i ^ j . Then, a constant q. i s chosen f o r each group such that a Popov l i n e may be drawn to the l e f t of the curve (2.12) 2 while a s t r a i g h t l i n e with slope l / q ^ may be drawn to the r i g h t of each curve C-2-.-23-). With the constants b^ and c^. thus obtained, may be found as a sum of the c . and c , terms. Then constants k. can be determined with (2.11). This method i s i l l u s t r a t e d i n F i g . 2.5, and i t i s stated as the following theorem. Theorem 2.2 Under the assumptions (Al) to (A3), the system of F i g . 1.1 i s absolutely stable i f there e x i s t s r e a l c o n s t a n t s s u c h that _ i i n k. > E ( c + c ) + b.; (1=1, n) (2.24) j s = l X J J 1 1 where c.,, and b. are defined by (2.22) and (2.9) r e s p e c t i v e l y . Example 2.2 We apply Theorem 2.2 to the system of Example 2.1 with equation (2.20) and N = 1. The curves f o r (2.23) and (2.9) are shown In F i g s . 2.8 and 2.6 r e s p e c t i v e l y . With a choice of q^ = q^ = q^ = 0, we have 22 Hence, b1 = b 2 - b 3 - 0 c 1 2 = 0.250 c 1 3 = 0.167 c 2 3 = 0.125 c 2 1 = 0.111 c 3 1 = 0.100 c 3 2 = 0.083 ^ $ 3.18, k 2 $ 3.51, k 3 $ 4.20. In.the third graphical approach, we consider the less conservative sta b i l i t y conditions of (2.8). Substituting the inequality I Re{ g . (jco) } | + | coq Im{g. (jco) } | + | Im{ g (jco) } | + | coq Re{g (jco) } | •1-1 -^.1 I I i j 1 ' i i j ' I J 1 I J * | Refw^ (jco) } | + | Imfw^ (jco) } | * |w (jco) | (2.25) into (2.8), we have 2k i 1 + 2Re{gi:L(jco)} - 2coqiIm{gii(jco)} n (2.26) > E [|Re{g. .(jco)}| + |coq Im{g (jco)} | + | Im{g. . (jco) } | + | coq Re{g,.(jco)}| ] n + E [|Re{g ..(jco)} l + l coq Im{g . . (jco) } | + | Im{g , (jco) } | + fcoq Re{g (jco)}|] j ^ i Now, i f a l l q^ are chosen to have the same value, by taking q = q ( i = 1, n) (2.27) then, we may separate a l l the q terms to arrive at a form similar to Popov's criterion, k" 1 + X(co) - qY(co) > 0 (2.28) where, 1 n X(co) = Re{g. .(jco)}- | Z [|Re{gij(jco)}| + |lm{g.j(jco)}| + |Re{gji(jco)}| + |lm{gj.(jco)}|] (2.29a) n Y(co) = co(Im{gi:.(jco)}+ | E t|Re{gij(jco)}| + |lm{g.j(jco)}| + |Re{gj.(jco)}.| + |lm{gj.(jco)}|]) (2.29b) Since X(io) and Y(u)) may be evaluated f o r each point i n w by taking the numerical sum of the absolute value functions, equation (2.29) describes a set of two curves i n the X-Y plane for each case of i , one for each choice of sign i n (2.29b). The s a t i s f a c t i o n of the s u f f i c i e n t condition (2.28) then, requires that these two curves of (2.29) l i e completely to the r i g h t of a s i n g l e s t r a i g h t l i n e passing through point (-k^, 0) and having slope l / q . This r e s u l t may be stated simply i n the following theorem. Theorem 2.3 Under the assumptions (Al) to (A3), the system of F i g . 1.1 i s absolutely stable i f there e x i s t s a r e a l constant q such that the s u f f i c i e n t conditions of (2.28) and (2.29) are s a t i s f i e d . Example 2.3 We apply Theorem 2.3 to the system considered i n example 2.1. For the case of N = 1, the curves of (2.29) are shown i n F i g . 2.9a. With a choice of q = 0, we have the r e s u l t s k x J? 3.3, k 2 $ 4.2, k 3 £ 5.1 Hence, for the system of t h i s p a r t i c u l a r example, the method of Theorem 2.3 y i e l d s the large s t nonlinear sectors that are comparable with those obtained by Dewey [16]. However, i n extending the example to cover other values of N, r e s u l t s are obtained with a choice of q that i s non zero. F i g . 2.9b shows the curves for N = 2.25, and with q = 0.04, ~~ 3*3} ^2 A • 3 j ~~ 3*2* And, when a higher value of N i s considered, ( i . e . N = 2.5), i t i s found that a l l the curves l i e below the X-axis, as i n F i g . 2.9c. Then choosing q to be a large constant, q = 2, we have 24 F i g . 2.9(a) Curve of (2.29) f o r Example 2.3 with N=l 2.4 Remarks i) As in the single variable case discussed in Section 1.2, the graphical methods presented by the three theorems here provide no means to obtain an optimum choice of the constants q^. This is done mostly by inspection of the plots, or by t r i a l and error. i i ) The method used in Theorem 2.1 follows closely the approach taken by Popov, in that modified Nyquist plots are to be drawn for each element of G(jco), except that the plot of each off-diagonal element requires 2 a combination of a pair of rotated curves. A total of 2n -n curves must be plotted. i i i ) The use of method of Theorem 2.2 requires more work in the preparation of the plots, which are expressed in terms of the square of the magnitude of the off-diagonal elements of G(jto), and which are to be 2 bounded by straight lines of slopes l/q^,. With q^ close, to zero, this method have minimum values for constants c.. and hence may yield results that are les conservative than those of Theorem 2.1. The total number of curves to be 2 plotted is n . iv) The method of Theorem 2.3 is also a direct extension of Popov's approach, in that each curve is in the form of Popov's modified Nyquist plot, and that k^ are found directly from the plots without the use of auxiliary constants e.g. a.., b., and c. ,. However, the directness of i j i i j the method is offset by the greater amount of computation required in combining the elements for each curve. v) It should be noted that whereas the amount of computation in preparing the plots is greatest for Theorem 2.3 and least for Theorem 2.1, the results from the theorems are only potentially different with 28 this increased amount of complication. There i s , however not a direct trade-off, since the differences are more dependent on the system that is under consideration than on the approach of the methods. As we shall show in the next section, there are particular systems for which just one of the three theorems has a simplication, and i t is not always the same one. Hence, the results of these three methods should only be compared in reference to a •certain system, and no general statement can be made as to which method is preferred until the class of system under investigation is known. 2.5 Graphical Tests for Particular Systems In this section, graphical methods are presented to test the absolute s t a b i l i t y of the four particular classes of systems mentioned in Section 1.1. It is found that the general theorems of the previous section, when applied to these particular systems, may be simplified in two steps. The f i r s t step is based on the fact that these particular systems are described by the transfer function matrix G(jto), whose elements may have the special property of being identical to one another or of Vanishing altogether. Hence, the total number of test curves for these theorems may be reduced. The second step arises when the general s t a b i l i t y criterion (2.5) may be further simplified and expressed in a more convenient form for graphical tests. The results of this second step are presented as corollaries to the main theorems of the previous section. Examples are given for each particular class of system to i l l u s t r a t e these points of simplification. 2.5.1 The Series System We consider the system of Fig. 1.4, whose linear transfer function matrix is given by (1.2). Due to the number of zero elements in this matrix, the number of test curves required for the application of Theorems 2.1, 2.2, 29 are reduced to 2n, n respectively. In the following corollary, we formulate a further simplification of Theorem 2.2. Since there are no diagonal entries in G(jco), constants of (2.9) are a l l identically zero, and hence only (2.22) need be tested. Since X(co) of (2.23a) is always positive, test curves (2.22) l i e completely inside the fourth quadrant. And since the constants c^ are determined by •straight lines with positive slope of l/q^, -they have their minimum values 2 2 when q^ is chosen to be. zero. With this choice of q^, c^ ^ |g^(jw)| . Thus for the series system, the constants c^ may be determined either by finding the maximum value of the magnitude of g^(jco), or by enclosing the Nyquist plot of g^(jco) within a c i r c l e of radius c^. Corollary 2.1 Under the assumptions (Al) to (A3), the series system of Fig. 1.4, whose linear part is described by (1.2), is absolutely stable i f where k i X " i ^ i - i + ci> 5 ( i = 1 » n ) c 2 * |gn(jw) | 2, and c? $ |g 1(jw)| 2. (2.30) (2.31) Example 2.4 Consider the system of Fig. 1.4, with n = 3 and a l l identical linear transfer functions. (1.2) has the form 0 0 -g(s) G(s) = g(s) 0 0 0 g(s) 0 where 100 g ( s ) = 8 i ( s ) = (s+l)(s+5)(s+20) > <i-1.2,3) F i g . 2.10 Rotated curves of g(s) of Example 2.4 o Fig. 2.11 Curves of (2.29) for Example 2.4 We f i r s t apply Theorem 2.1, with the p a i r of rotated curves of g(joj) shown i n F i g . 2.10. We choise q^ = q 2 = = 0, then a ^ 3 = a ^ = a 3 2 = 1» and a ^ = a 2 3 = a 3 i = ®' ^ o r a ^ s e r * e s systems, = 0 f o r a l l i . Hence, we have kx = k 2 = k 3 = 0.707. Corollary 2.1 i s applied instead of Theorem 2.2. From F i g . 2.10 i t i s obvious that the maximum magnitude of g(jco) i s bounded by unity. Thus, C o = C l = C2 = c 3 = 1 * and hence, k 2 = k 2 = k 3 = 1. The a p p l i c a t i o n of Theorem 2.3 requires the p l o t t i n g of a set of two curves f o r each i . Since a l l g^(jto) are the same, we show only one set of curves i n F i g . 2.11. With q = 0, we have k l = k2 = k 3 " ° ' 8 0 -The following points can be noted: i ) The method of Corollary 2.1 required the l e a s t amount of work i n computation and p l o t t i n g , and for t h i s example also y i e l d e d the largest nonlinear sectors, i i ) " The r e s u l t s obtained above by a l l three methods remain v a l i d when the system i s extended f o r higher dimensions, ( i . e . n -> °°) . i i i ) Chang [14] obtained b e t t e r r e s u l t s by the use of a l e s s conservative graphical method. For n - 3, and q = 0.788, he had k^ = k 2 = k 3 = 1.40 For higher orders of n, h i s r e s u l t s converge to that of the more conservative r e s u l t s of C o r o l l a r y 2.1. F i g . 2.12 A three dimensional s e r i e s system with l i n e a r blocks of d i f f e r e n t t r a n s f e r functions 34 Example 2.5 Consider a ser i e s system with n = 3 but with d i f f e r e n t t r a n s f e r functions for the l i n e a r blocks, as i n P i g . 2.12. G(s) = 0 1 s+4 0 0 s+4 0 sN-0.1 s+0.1 For ser i e s systems, b^ = 0 for a l l i . Thus, Theorem 2.1 may be applied by p l o t t i n g a p a i r of rotated curves for each g^(jw). The pl o t s for g^Cjw) and g^(jco) are i d e n t i c a l and are shown i n F i g . 2.7, while the plo t s f or g 3 2(jco) a r e shown i n F i g . 2.13. Choosing q • = = q^ = 0, w e have k x = 2.82, k 2 = 0.14, k 3 = 0.14. Corollary 2.1 may be applied with the curves of F i g . 2.7 also. The maximum magnitudes of g^(ju)) are obtained by drawing c i r c l e s of radius c^ to enclose the respective curves. Hence, we have c = c. = c_ = 0.25; c„ = 10.0; o 1 3 2 and ^ s 4.0, k 2 = 0.195, k 3 = 0.195. To apply Theorem 2.3, we must p l o t a t o t a l of 6 curves. F i g . 2.14 gives the curves for the case of i = 1. The cases i = 2 and i = 3 are i d e n t i c a l , and the set of curves for these are p l o t t e d on F i g . 2.15. With q = 0, we obtain kx = 3.33, k 2 = 0.162, k 3 = 0.162. Thus, for seri e s systems, Corollary 2.1 gives r e s u l t s that aire the l e a s t F i g . 2.13 Rotated curves f o r g 3 2 ( s ) of Example 2.5 Fig. 2.14 Curves of (2.29) for the case of 1=1 in example 2.5 ON F i g . 2.15 Curves of (2.29) f o r 1=2/1-3 of Example 2.5 conservative with a minimum of p l o t s . 2.5.2 The p a r a l l e l System This class of system i s depicted i n F i g . 1.4 and i t s l i n e a r transfer function matrix i s given by (1.3). This matrix has the s p e c i a l property of i row, may be determined with j u s t one p a i r of rotated curves f o r g^(jco), while the constants b^ may be obtained also from the unrotated portion of these same curves. Thus a t o t a l of only 2n curves need be p l o t t e d . Theorems 2.2, and 2.3 may both be applied d i r e c t l y with the p l o t t i n g of 2n curves also. The following c o r o l l a r y modifies the condition (2.24) of Theorem 2.2. I t requires the same curves to be p l o t t e d as for Theorem 2.2, but with the introduction of a constant S, the amount of summation work to compute y^ i s reduced. C o r o l l a r y 2.2 Under the assumptions of (Al) to (A3), the p a r a l l e l system of F i g . 1.4 whose l i n e a r part i s described by (1.3) i s absolutely stable i f there e x i s t s r e a l constants q. such that (jto) = g, (jto) ; ( i = 1, n) (2.32) For the a p p l i c a t i o n of Theorem 2.1, the constants a., for each k" 1 > |[(n-2)d.+S] + b± ; (i=l,n) (2.33) where n S = (2.34) o d 2 = Iw.Cjco)^ = {| g j(jco)| Z} - q j { V | g j ( j U ) r Z } 2r 2| (2.35) Example 2.6 Consider a parallel system with the transfer function matrix 1 1 G(s) s+4 s+4 s2+0.1 s+0.1 s2+0.1 s+0.1 Figs. 2.7 and 2.13 give the two pairs of rotated curves of g^(jw) that are necessary for the application of Theorem 2.1. Choosing q.j=q2=0 we have Hence, b 1 = t>2 = 0, a 1 2 = 0.25, and a 2 1 = 10.00. k 2 = k 2 s 0.137 For Corollary 2.2, the constants are found from Figs. 2.7 and 2.13., d^ from Fig. 2.8, and d 2 from Fig. 2.16. With q-^j^ chosen to be zero, we have b l = b2 = °» d l = ° * 2 5 , d2 = 1 0 * 0 0 ' * Hence, we have S = 10.25, and k x = k 2 § 0.195 For Theorem 2.3, k^ and k 2 are each determined by the set of curves shown in Figs. 2.17 and 2.18 respectively. With q = 0, the nonlinear sectors are found to be ^ $ 0.83 , k 2 £ 0.17. 2.5.3 The Internal Feedback System This class of system is shown in Fig. 1.5, with the linear part CM 3 N «-T-' S C W TH C W M C w CO CM 1-1 I CM •I-) I I I I ,—\ 3 3 I-I '—s~ -H & c W -i 1 r—• •—\ 3 3 I-I s • E 6 M M RO W < ( O H H <N Csl •N 1-1 1 CM N I CU pi CM I CO AS CM Pi CM TH CM I «-Y-< W CO CM C W CO I 3 . CU *-»-> I-I Pi »—' CM 3 •H I-I & + E & M 1 CM TH E CM W II CM H •H CO W CM CM CM I - 1 1 N + •H C W CM CM •—s *—N 3 3 I-I !-•-> I-) •H 3 & I-I & c c W CO 3 TH 1 1 N & r*-\ r*~i v ' s •H . CU 3 3 & Pi I-I N CM TH •H S M + & & . C W TH •N TH . E . E "' CM 1 TH H M + A5 CM W TH CO W TH CM CM CM 44 characterized by (1.4). Since the special property of this class of system is not evident in the matrix elements, the general Theorems 2.1, 2.2, and 2.3 do not reduce directly to simpler methods. However, when g^(jco), the components of these matrix elements, are considered, the following corollaries are derived. Let wj,(jto) = (l+jioq^Goj) (2.36) where g^(jco) is given in (1.5). Then, for this particular class of system, (2.4) has the form (2.37) as shown on page 43. Taking the same approach as that of Theorem 2.1, we expand the |w (jco)| terms with i /2 a > |wi(jw) | ; ( i = l,n) (2.38) where the constant a satisfies both inequalities a = |Re{wi(jw)}|; ( i = l,n) (2.39a) a= Im{ W i(ju)}|; ( i = l,n) (2.39b) Substituting (2.38), (2.39) into (2.37), we have for (2.8) _1 n-i+1 i n-2 1n-i-1 2k.1 + 2Re{w. . (jco)} > 2a( Z j+ Z i+SZ{ Z - — Z j - j I j}) (2.40) 1 1 1 j=2 j=2 j=l V l J=l for ( i = 1, n) Expanding the sums on the right hand side, we may denote n-i+1 i n-2 n-i-1 i-2 T = I j + Z j + /2{ Z j - -f Z j - j Z j} j=2 j=2 j=l Z j = l Z j = l (2.41) = |(n+4)(n-l)+i(i-n-l)+l+ V^{(n-4)(n-l)-2i(i-n-l)-2} Hence, with (2.40), we have the following corollary. Corollary 2.3a Under the assumptions (Al) to (A3), the internal feedback system 45 of Fig. 1.5, described by (1.4) and (1.5), i s absolutely stable i f there exists a real constant q such that k" 1 > Ta + b ± ; . ( i * 1, n) (2.42) where the constant a satisfies inequalities (2.39), T is defined in (2.41) and b_^ satisfy (2.9). A graphical method of satisfying (2.39) is to plot a l l 2n rotated curves of g/(jco) on the same X-Y plane. Then, i f a constant q may be chosen such that a l l these curves l i e completely to the right of a Popov line, the constant a is determined. The constants b^ may be obtained also from the unrotated portions of these curves, thus, the total number of curves required by Corollary 2.3a is only 2n. We now derive a corresponding corollary of Theorem 2.2 by sub-stituting into (2.37) the inequality |wk(ja))U|lm wk(jco)}| (2.43) Then for (2.8), we have 2n+l-i 2n-l-i n+1 2k:1 + 2Re{w (ju>)}: > E |w (jco) | + E |w (jco) | -2 £ |w, (jco) | (2.44) 1 1 k=l R k=l k k=l k Expanding the sums in the right hand side we have the following. Corollary 2.3b Under the assumptions of (Al) to (A3), the internal feedback system of Fig. 1.5, described by (1.4) and (1.5), i s absolutely stable i f there exists a real constant q such that k" 1 > [-|(n+l)(3n-2) + -|i(i-n-l) ]c + b ±; (i=l,n) (2.45) 2 |2 where c :>|w. (jco)| ; (j=l,n) (2.46) and satisfy (2.9). The constant c of (2.46) is determined graphically by drawing , 2 2 a straight line, having slope 1/q , and passing through point (c> 0), to the right of a l l n curves, each defined by (2.23). The total number of curves required by Corollary 2.3b is then also 2n. Example 2.7 Fig. 2.19 depicts an internal feedback system with n=3, with G(s) = gj.Cs)-g 1(s)+g 2(s) -g 1(s)-g 2(s)-g 3(s) g 2 ( 8 ) g 1(s)+g 2(s)+g 3(s) g 2(s)+g 3(s) -g 1(s)-g 2(s)-g 3(s) -g 1(s)-g 2(s)-g 3(s) g 3(s) where and H.(s) = ~ 2 s +S+1 1=1,2,3 H, (s) *±™~—r—. l+TH k(s) s7+3s^-H5s"^+7s^+6s3+3s2+s = s8+4s7+10s6+19s5+22s4+19s3+10s2+4s+l ; for i=l,2,3. We f i r s t apply Corollary 2.3a by plotting the pair of rotated curves of g^(jco) for i=l,2,3. These are shown in Fig. 2.2D. Choosing q=0, we have a = 1.35, b^=b2=b3=0, and thus k^=k3= 0.130, and k2=0.137. For the application of Corollary 2.3b, we show the Popov plot of g^(jco) in Fig. 2.21, from which the constants b_^ are determined. The curve of (2.46) i s plotted in Fig. 2.22 to obtain the constant c. Hence, < < with q=0, we have b^=b2=b3=0, c=1.35, arid k^=k3=0.135, and k2=0.148. F i g . 2.29 An i n t e r n a l feedback system with n=3 53 1 Theorem 2.3 is applied directly with two curves plotted for each i . These are shown in Figs. 2.23, 2.24 for i=l/i=3, i=2 respectively. With q=0, we have k ^ k ^ . 1 1 8 , and k 2 =0.129. Thus, in this example, the same number of plots i s required for a l l three methods. And i t is noted that for this special example of the internal feedback system, the method of corollary 2.3b gives the least conservative results. 2.5.4 The Symmetrical -System This class of system has linear transfer function matrix (1.7), with the special property given by g i : j(jw) = g.^Jto). (2.47) The equality of these elements of the matrix reduces the total number of 2 curves required for the application of Theorems 2.1 and 2.2 to n , and '"'^2+1^ respectively. But the number of curves for Theorem 2.3 remains unchanged at 2n. We now show that a further simplfication i s possible. By substituting the property (2.47) into (2.4), we have Re w12(jco) k^+Re w^jio) Re w13(jco) ... Re wln(jco) Re W 1 2 ( J L O ) Re w13(jco) Re wln(jco) -1 k 2 +Re w22(ju>) Re w 2 3(jio) ... Re w2n(jco) -1 Re w 2 3(jto) +Re w^Cjco) ... Re w^(jco) 33 3n' -1, Re w2n(jco) Re w3n(jco) ... k R +Re w n n(jw) (2.48) where a l l In^w^ (jco)} terms have vanished. Hence (2.5) reduces to n k~ +Re{w .(jco)} > E | Re{w. . (juO} | J (i=l,n) j = l ^ (2.49) 54 Thus, in the development of graphical methods, we may disregard a l l Im{w^j(ju)} terms. Both Theorems' 2.1, and 2.2 reduce to the following corollary. Corollary 2.4a Under the assumptions (Al) to (A3), the system of Fig. 1.1 with the symmetrical property (2.47) is absolutely stable i f there exists real constants such that -1 n k / > E a. . + b. ; (i=l,n) (2.50) 1 J - l l j 1 where constants a.. = a,., and satisfies a t j |Re{wi;j (jw)}| (2.51) and b i satisfies (2.9). Similarly, by setting a l l ImCw^(jw)} terms to be zero, we restate Theorem 2.3 with the following corollary. Corollary 2.4b Under the assumptions (Al) to (A3), the system of Fig. 1.1 with the symmetric property of (2.47) is absolutely stable i f there exists a real constant q such that k" 1 + X(to) - qY(co) > 0 ; (1=1,n) (2.52) where n X(oi) * Re{g,,(ju)} - I |Re{g..(ja))}| (2.53a) j=l J n Y(to) * a>(lm{g.,(jw)} 4- I | Im{g.. (jco) } |) (2.53b) ~" j =1 3~H 55 Hence, the t o t a l number of curves required f o r the a p p l i c a t i o n of Corollary 2.4a i s j n ( n + l ) , and that of Corollary 2.4b i s 2n. Example 2.8 Consider a symmetric system x^ith G(s) = s +0.1 s+0.1 1 s 2+s+l s s 2+0.1 s+0.1 s 2+s+l s 2+0.1 s+0.1 s 2+0.1 s+0.1 s 2+s+l s 2+s+l s2+0.1s+0.1 In applying Corollary 2.4a, we need to p l o t the rotated curve (2.51) f o r the off-diagonal elements of G(s). These are found i n Figs. 2.2 5 and 2.13. The Popov pl o t of the diagonal elements (which are a l l i d e n t i c a l ) i s shown i n F i g . 2.26, from which b^ are obtained. With a choice of q=0, we get b^=b2=b3=13.8, a^2 =l> and a^3=a3^=10.0. Thus from (2.50) we have k1=k3<: 0.040, k 2 * 0.0625. With. q=3, we get b 1=b 2=b 3=7.0, a 1 2 = a 2 1 = a 2 3 =a 3 2~3.5, and a 1 3=a 3 ]=12.0. And we have k ^ k ^ J? 0.045, k 2 $ 0.072. Corollary 2.4b i s applied with the p l o t t i n g of a set of two curves for each 1. But the cases i = l and i=3 are i d e n t i c a l , such that a t o t a l of two sets of curves need be pl o t t e d . These are shown i n Figs. 2.27, and < < 2.28. With q=0, ^=^=0.0475,^2=0.0645. And with a choice of q=3, we < < improve the res u l t s to k^=k3=0.080, k2=0.l4l. Fig. 2.25 The rotated curve of —z s +s+l of Example 2.8 57 Fig. 2.28 Curves (2.29) for i=2 of Example 2.8 58 3. SYSTEM REPRESENTATION OF NETWORKS CONTAINING MANY NONLINEAR CONTROLLED SOURCES 3.1 Introduction The s t a b i l i t y theorems of the previous chapters may be used to study the s t a b i l i t y of nonlinear networks, provided that these networks can be represented by the nonlinear feedback system described i n Section 1.1. This l a s t condition imposes r e s t r i c t i o n s on the types of network elements and the manner by which these elements are connected together. The work of Zames [6], Sandberg [8], and T r i c k [9] showed that passive networks which contain one nonlinear element may be represented as a s i n g l e v a r i a b l e nonlinear feedback system. In extending t h i s approach to the m u l t i v a r i a b l e case, Sandberg [8], and T r i c k [18] considered passive networks that contain many nonlinear capacitors, while Goldstein and Frank [10] considered an even broader c l a s s of passive networks that contain both r e s i s t i v e and reac t i v e nonlinear elements. However, i t was the work of Ho and Figueiredo [11] which, with the use of nonlinear c o n t r o l l e d sources, opened the way for the study of act i v e networks. The following sections are a d i r e c t extension of t h i s l a t t e r work. We s h a l l f i r s t give a general system representation of a network which contains passive l i n e a r elements and many nonlinear c o n t r o l l e d sources. This network may include most active devices since these are modelled with c o n t r o l l e d sources. We s h a l l then show that with some s p e c i a l c h a r a c t e r i s t i c s , c e r t a i n classes of t h i s network may be represented as the p a r t i c u l a r systems that were considered i n Chapter 2. 59 3.2 System Representation of the General Network Fi r s t , we briefly summarize the approach developed by Ho and Figueiredo for representing a general nonlinear network as a system. Consider the network n which consists of linear time invariant elements and many nonlinear controlled sources, as shown in Fig. 3.1. These nonlinear sources include the following types: voltage-controlled voltage sources, v=h(v c), ..voltage-controlled-current sources, i=g(v c), current-controlled voltage sources, v=h(i c), and current-controlled current sources i= g ( i c ) . The characteristics of each of these functions i s assumed to l i e in a sector, ^ as shown in Fig. 1.2. It is further assumed that the linear network obtained by deleting the nonlinear sources (opening a l l current sources and shorting a l l voltage sources) is asymptotically stable, i.e. that the assumptions (Al) to (A3) are satisfied. Then, regarding a l l the nonlinear controlled sources as inputs to the remaining linear network, we may write the state variable equations for n as x = Ax + Bf(y) , («=d/dt) (3.1) y = Cx + Df(y) (3.2) where -x=col. (x^,^, x ) is the state Vector of the network, the output vector y-eol. ( i ^ . i ^ , • • • . i c l , i C B l . 1 , . •. . i c k . v c l , . . . . v ^ . v ^ , • • ; ,v c p) contains a l l the controlling Variables, and the input vector f(y)=col. ( g ^ ( i c ^ ) , g^Ci^^ »••• ' ^ i ^ v c i ^ »•••»•••) contains a l l the controlled variables, and A, B, C, D are constant real matrices. Using the state Variable equations of (3.1), (3.2), we can derive the transfer function matrix relating the input to the output vectors, GCjw) = C^.jul-A)"^ + D, (3.3) c l C £ cm+1 A ck o— V , o-c l o— + cm v c l + l ^ o— + cp IV =h (v ) ra m cm ' Vmfl = hm+l ( ic m f l> ' V h k ( i c k > G 0 ^ - f l ^ ^ l ^ c ^ ^ kJ P P CP Fig. 3.1 Network n 61 such that we have a systems representation of the network y = -G(j(o)f(y) (3.4) Thus, the s t a b i l i t y of t h i s network n i s determined by the t r a n s f e r function matrix G(_j u), which r e l a t e s the c o n t r o l l e d v a r i a b l e s to the c o n t r o l l i n g v a r i a b l e s of the nonlinear sources of the network. Example 3.1 We apply the above approach to f i n d the conditions under which no o s c i l l a t i o n s would occur i n the modified RC o s c i l l a t o r c i r c u i t as depicted i n F i g . 3.2a. The nonlinear r e s i s t o r R has the c h a r a c t e r i s t i c s of F i g . 3.1b, and i t i s modelled as a curre n t - c o n t r o l l e d Voltage source, V 2 = ^ 2 ^ c 2 ^ " ^ nonlinear current source i s used i n the hybrid it t r a n s i s t o r model, and F i g . 3.2c shows the ac-equivalent of t h i s c i r c u i t i n the form of network.n. Choosing the voltages of the capacitors as state v a r i a b l e s , we can write -a -a -a -(a+b) 0 0 ,*-3 . x. 1 • *2 • 'cl Lc2 -a d 0 •(a+b) -1 d 0 0 x l X2 X 3 + d -a d -(a+b) h -b 0 b V2 — b=0.10 x 10 3 , and d=10^, and 1 0 1 , -(a+2b) Hence, with (3.3) we derive, 0 G(s) = 1 - ds+b x_ 2(a+b)s+ab -(a+2b) + (2a 2+2ab+b 2)s+(a+b)ab ^ b 2/d ds 2+2(a+b)s+ab d d s 2 + 2(a+b)s + ab S + b 62 F i g . 3.2a A modified RC o s c i l l a t o r •Oluf .Olyf .Olyf hybrid IT model for t r a n s i s t o r F i g . 3.2b An ac-equivalent c i r c u i t F i g . 3.2c Equivalent c i r c u i t of o s c i l l a t o r i n form of network n. 63 Theorem 2.1 is applied to yield the results with q ^ l . <12=0> k x » 1.414 , k 2 » 1.40 Thus, the circuit w i l l not oscillate as long as the nonlinear resistor i s bounded in the sector (0,1.400,), and the gain factor of the voltage controlled current source i s bounded in the sector (0, 1.414). 3.3 Networks with Series System Representation Consider the network n in ••Fig. 3.3. This network is a subclass s of n and is characterized by the special property that every controlled source is connected to the passive linear tnultiport through an LTI one-port which contains a controlling variable of one of the sources. More speci-f i c a l l y , every nonlinear voltage (current) source i s connected in shunt (series) With an LTI one-port which contains a voltage (current) as a controlling variable. For this network n , we can apply KVL and KCL, and s express the controlling vector in terms of the controlled vector with 0 z± 0 0 - 0 s • • 0 - - - 0 c l 'c2 cm cm+1 cn-1 cn 0 0 0 0 0 0 0 m 0 y m + l ° , 0 0 -y n 0 'n-1 f l ( v c l > V Vc2> f (v ) m cm ^ l ^ c m + l ^ n-1 cn-1 / f ( i ) >— n cn (3.5) where z = 1 -y, = i -ai z ,+z, . ai b i J b i z .+z, . ai b i ai y a i + y b i y a i + y b i (i=l,n) (i=l,n) (3.6) (3.7) 64. LTI n-plot network an ^bru cn %1 a l "bm~l 'am-1 bm am am+1 ybm+l an-1 'bn-1 v -c l « + cra-1 - o f v -o-f cm cm+1 F i g . 3.3 The network n © •F-O V =f (v ) m m cm o Vm+l _ fm+l ( icm+l) © 1m+2r=fm+2<-:Lcm+2^ Lcn-1 =f ( i ) V_y n n cn 0 V 2 " £ 2 ( i C 2 F i g . 3.4 Network f o r Example 3.1 passive l i n e a r time-invariant multiport 1 1' 2 2 ' n n V I L c l l al f 2 © " b 2 a2 I I n (V, A cn "bn" an I 1 F i g . 3.5 The cascade configuration of network n which contains only c o n t r o l l e d current sourc s r 66 The network equation (3.5) i s i n the form of (1.2), hence, network n may be representated by the s e r i e s system depicted i n F i g . 1.3, and i t s s t a b i l i t y may be studied with C o r o l l a r y 2.L The s e r i e s c h a r a c t e r i s t i c s of network ft i s s better i l l u s t r a t e d i n F i g . 3.5 where the network i s shown as a cascade of co n t r o l l e d sources connected to an n-port. It i s noted here that the s t a b i l i t y of n i s dependent only on the one-ports z ,, z, ., y . and y, .-, and i s not s r J • r ax b i •'ai •'bi affected by the n-port. Example 3.2 The system equations f o r the network of F i g . 3.4 i s R 1 '•cl Lc2 0 -(1 2 sL V l T c o V o l ' £2< 1c2 ) Thus, applying C o r o l l a r y .2...1., _k^ = k^ S 1.0. 3.4 Networks With P a r a l l e l System Representation In F i g . 3.6 we show another c l a s s of network, n . For t h i s subclass P of n» a l l the nonlinear c o n t r o l l e d sources are current (voltage) sources. These are connected i n shunt (series) and terminated at an LTI one-port which contains the c o n t r o l l i n g currents and voltages. In order to express the c o n t r o l l i n g v a r i a b l e s i n terms of the con t r o l l e d v a r i a b l e s , we f i r s t pick out a l l the n c o n t r o l l i n g v a r i a b l e s as port terminals, so that we see the LTI one-port as a (n+l)-port with_one port terminated with the combination of the nonlinear sources, and the n ports e i t h e r open- or s h o r t - c i r c u i t e d depending on whether the c o n t r o l l i n g v a r i a b l e i s a voltage or a current. Then, the equations of the (n+l)-port are 67 where u 0 A B y V n+1 0 C D (3.8) u = [ i - , . . . , i ,v .......v ] 1' 5 m' m+1' n T y = [v .. v , i . 1 ] 17 1 c l cm cm+1' cn A is a (nxn) constant matrix T B , C is a ( l x n) vector and D i s constant. Expanding the f i r s t n rows of (3.8) we have 0 = Ay + B i (3.9) with which the controlling Vector y may be expressed in terms of i n +^» which is the sum of the controlled variables y = -A _ 1B i n+1 -1 n -A B E f. (•) i=l 1 (3.10) or, rewritten in f u l l , we c l i i i v cm 1cm+l cn have h l i i i h m h n m - h„ m m+1 m+1 m+1 n h n f (v ) m cm fm+l ( ±cm+l ) f (1 ) n cn (3.11) Now, the network equations of (3.11) are in the form of (1.3), and hence the network i s representated by the parallel system of Fig. 1.4, and i t s s t a b i l i t y may be studied with Corollary 2.2. 68 F i g . 3.6 Network n o— 1/s ' c l © © © Ha a CM U CM CM CO u r o C O F i g . 3.7 Network f o r Example 3.2 69 Example 3.2 Consider the network of Fig. 3.7, s 2+ s+l s +2s+l s+1 s+1 s+1 s +2s+1 s +2s+1 Hence, we have h l = ~2 h2 " 2 s +2s+1 s+1 s +2s+l s +2s+1 s+1 s2+2s+l s 2+s+l s2+2s+l B = , s 2+s+l h3 = 2 s +2s+l And the network can be represented by the system equation V c l y h l h l _ " fl<*d> Vc2 = - h2 h2 h2 f2<vc2> Vc3 h3 h3 h3 f3< Vc3 ) 3.5 Networks with Symmetric System Representation The network nsytn of Fig. 3.8 is a subclass of n. It consists of a passive LTI n-port with a l l the port Voltages (currents) as controlling variables. Each of these h ports i s terminated with a LTI impedance (admittance) and a voltage controlled current source (current controlled Voltage source) in series (shunt). : 170 1 =f (v ) n n cn I 1 LTI n-port network N Fig. 3.8 Network nsym. H r | z =0.1 v c i ! Ar + v 'c2 0 1 2 = f 2 ( v c 2 ) 'c3 0 i 3 - f 3 ( v e 3 ) L. J F i g . 3.9 Network f o r Example 3.3 71 We define the n-port N to include a l l the linear passive elements of n , such that there i s a symmetric matrix Z, which i s the short-circuited sym' . impedance matrix of N. r— — • Z • V n V n where Z = {z . .} i s symmetric (3.12) 'We then take <K-VL around loop 1^, for a l l i=l, n to obtain v l " • _ V c l • 0 * • = - • + • • • V n V cn 0 z n i n (3.13) Hence we can describe network n with sym 'cl cn z i r z i 'In '12 Z13 ' V ' Z l n Z — z « • • • z ~~ z 2n 3n nn n f l ^ c l > f (v ) n cn (3.14) Thus, the equation (3.14) is in the form of (1.6), and the s t a b i l i t y of net-work n g m may be studied as a symmetric system with the Corollaries 2.4a, and 2.4b. Example 3.3 -Consider the network of Fig. 3.9, where the short-circuit impedance matrix of N is given as 1 Z = s2+4s+3 s2+3s+l s2+2s+l s2+2s+l s2+2s+l s2+2s+l 1 s2+2s+l s2+3s+l With z 1 = z 2 = = 0.1, we have 'cl 'c2 0.9s +2.6s+0.7 2 s +2s+l s +4s+3 v c3 s +2s+l 1 0.9s2+1.6s+0.7 s2+2s+l s +2s+l 0.9s +2.6s+0.7 f l ( v c l ) j f2<Vc2>l f3<Vc3>l 4. CONCLUSIONS Three graphical methods,are derived to t e s t the absolute s t a b i l i t y of a general rcultivafiable nonlinear feedback system. The graphical approach of these methods i s a d i r e c t extension of Popov's, i n that i t consists of drawing s t r a i g h t l i n e s as bounds to the modified Nyquist p l o t s of the elements of the t r a n s f e r function matrix of the l i n e a r part of the system. Due to the d i f f e r e n t approximations taken, these three methods d i f f e r from one another i n three aspects: the amount of computation i n preparing a p l o t , the t o t a l number of plo t s required for t e s t i n g a system, and the conservativeness of t h e r e s u l t s . However, these factors are so dependent on the cl a s s of system under i n v e s t i g a t i o n that no one method can be preferred above the others for t h e general system. S i m p l i f i c a t i o n s of these methods are derived f or four p a r t i c u l a r classes of systems:the s e r i e s system, the p a r a l l e l system, the i n t e r n a l feedback system, and the symmetric system. While the r e s u l t s of these graphical methods are often more conservative than those obtained by the numerical approach [15, 16], they are f a r simpler i n a p p l i c a t i o n and more s u i t a b l e f o r system design work. F i n a l l y , these s t a b i l i t y tests are applied to.an.active network t h a t consists of passive LTI elements and many nonlinear c o n t r o l l e d sources, b y f i r s t representing i t as a nonlinear feedback system. I t i s further shown t h a t c e r t a i n s p e c i a l structures of t h i s network correspond to the three •particular classes of system under consideration. 74 REFERENCES 1. A.J. Lurie, "Some Nonlinear Problems i n the Theory of Automatic Control", Translation: Her Majesty's Stationery Office, London, 1957. 2. V.M. Popov, "Absolute Stability of Nonlinear Systems of Automatic Control", Automation and Remote Control, Vol. 22, No. 8, pp. 85 7-875, August, 1961. 3. E..J-. Jury and B.W. Lee,, "The .Absolute Stability of Systems with Many Nonlinearities", Automation and Remote Control, Vol. 26, No. 6, pp. 945-965, July, 1965. 4. B.D.O. Anderson, "Stability of Distributed-Parameter Dynamical Systems", International Journal of Control, Vol. 3, p. 535, 1966. 5. K.S. Narendra and R.M. Goldwyn, "Existence of Quadratic Type Liapunov Functions for a Class of Nonlinear Systems", Int. J. Engng. Sc., Vol. 2, p. 367, 1964. 6. I.W. Sandberg, "On the L-Boundedness of Solutions of Noninear Functional Equations", B e l l System Int. Jounral, Vol. 143, No. 4, 1964. 7. G. Zames, "On the Input-Output Stability of Time Varying Nonlinear Feedback Systems", Parts I, II, IEEE Trans. Automatic Control, Vol. AC-11, pp. 228-238, 465-476, 1966. 8. I.W. Sandberg, "A Stability Criterion for Linear Networks Containing Time-Varying Capacitors", IEEE Trans. Circuit Theory, Vol. CT-12, pp. 2-11, 1965. 9. T. Trick, "Asympotic Stability of a Class of Network Containing a Nonlinear Time-Varying Capacitor and Periodic Inputs", IEEE Trans. Circuit Theory, Vol. CT-16, p. 217, May, 1969. 10. M. Goldstein and H. Frank, "Stability Criteria for Nonlinear RLC Networks", Journal of Franklin Inst., Vol. 288, No. 5, p. 351, 1969. 75 11. C.Y. Ho and R.J.P. Figueiredo, "Global Asymptotic S t a b i l i t y of Networks Containing Several Nonlinear Controlled Sources", SWIEECO, A p r i l 22-24, at Dall a s . 12. P.A. G r a y b i l l , Introduction to Matrices with Applications i n S t a t i s t i c s , Chap. 12, p. 333, Wadsworth Publishing Co. Inc., Belmont, C a l i f . , 1969. 13. M.S. Davies, " S t a b i l i t y of a Class of Nonlinear Systems", E l e c t r o n i c s L e t t e r s , V o l . 4, No. 5, p. 322, July, 1968. 14. T.L. Chang, "The Absolute S t a b i l i t y of Nonlinear Systems", M.A.Sc. Thesis, U.B.C., February, 1970. 15. C.F. Chen, "On The Absolute S t a b i l i t y of Nonlinear Control Systems", M.A.Sc. Thesis, U.B.C., February, 1970. 16. A.G. Dewey, "Frequency Domain S t a b i l i t y C r i t e r i a f o r Nonlinear M u l t i -v a r i a b l e Systems", In t e r n a t i o n a l Journal of Control, V o l . 5, No. 1, pp. 77-84, 1967. 17. W.C. Rae and G.D.S. MacLellan, " S t a b i l i t y of Two Classes of Nonlinear M u l t i v a r i a b l e Control Systems", Paper 12c, 3rd IFAC Congress, London, 1966. 18. T.N. T r i c k , "Bound Input-Bounded Output S t a b i l i t y of RC Networks Containing N Time-Varying Nonlinear Capacitors", 11th Midwest Symposium, p. 352, 1968.
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Graphical tests for the absolute stability of multivariable nonlinear systems. Lau, Jackson 1971
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Title | Graphical tests for the absolute stability of multivariable nonlinear systems. |
Creator |
Lau, Jackson |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | Three graphical methods, similar to Popov's approach, are derived to test a stability criterion for a general class of multivariable nonlinear feedback systems. Various simplifications of these methods are then derived for four particular classes of system: the series system, the parallel system, the internal feedback system, and the symmetric system. These stability tests may be applied to active networks that consist of passive linear time-invariant elements and many nonlinear controlled sources. It is further shown that certain special network structures may be represented as one of the four particular classes of system. |
Subject |
Non-linear theories. Multivariate analysis. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0101644 |
URI | http://hdl.handle.net/2429/33498 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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