Stability of periodic solutions of a nonlinear differential equation arising in servomechanism theory

C ^ ^ REPORT AERO No. 191

TECHNISCHE HOGESCHOOL D E l H

VUEGTUIGBOUW KUNDE

BI&UOTÜEEK

THE COLLEGE OF AERONAUTICS

CRANFIELD

'T'ABILITY OF PERIODIC SOLUTIONS OF A NONLINEAR DIFFERENTIAL '

EQUATION ARISING IN SERVOMECHANISM THEORY

by

1

P. A, T. C h r i s t o p h e r

m

THE COLLEGE OF AERONAUTICS CRANFIELD

Stability of Periodic Solutions of a Nonlinear Differential Equation Arising in Servomechanism Theory

b y

-P . A. T, Christopher, D . C . A e . , A . F . I . M . A .

SUMMARY

The stability of the periodic solutions of a particular nonlinear differential equation of third order is discussed in t e r m s of the asymptotic stability of the corresponding variational equation and, thereby, in t e r m s of the characteristic exponents of this equation. A method, due to C e s a r i , is used to evaluate the characteristic exponents and thus the boundaries of asymptotic stability. Taking the solution to have, in the first approximation, an amplitude F and frequency w , it is shown that one of the asymptotic stability boundaries corresponds to the locus of vertical tangents of the frequency response curves in the u , F plane. This result is similar in character to, and a modest generalization of, the well known stability criterion associated with Duffing's equation.

CONTENTS

Page Summary

1. Introduction 1 2. A nonlinear control systemi 1

3 . The frequency response 4 4. Asymptotic stability 6 5. A numerical example 14

where dots indicate differentiation with respect to t. In the case

b > 0 , c > 0 , c > 0 , it is well known that the asymptotic solution of this equation as t •• » i s , in the first approximation, a sinusoid

X = F Sin (<jt + 4>), Eq. 1.2 where the amplitude F is given by

(c^ -u* + | C j F M ' + «"b" = (Q/F)^ Eq. 1.3 See (1), Chapter 4; (2), Chapter 14, Theorems 3 . 1 , 3.2 and 3.3; (3), Chapters 7 and 8; (6).

The graphs of u , F , described by Eq. 1. 3 and known as the frequency response curves, a r e markedly different from the linear case, c^ = 0 , in that they may possess points of vertical tangency. In an e a r l i e r paper (4) the author has discussed how the loci of vertical tangents of the response curves correspond with the boundaries of asymptotic stability of the associated periodic solutions. In particular, it was shown how the method due to C e s a r i , (5) Chapter 8, for evaluating the characteristic exponents of the related variational equation, could be used to obtain the stability boundaries. Because the variational equation of Eq. 1.1 can be reduced to Mathieu's form then the well known stability theory of this equation may be used, and C e s a r i ' s m.ethod offers no advantage. F o r nonlinear

equations of higher order, the method provides a uniform approach to the problem of determining the characteristic exponents and it is the primary purpose of this paper to show how it may be used to determine the asymptotic stability boundaries of a third order equation, arising in servomechanism theory.

2. A nonlinear control system

The control system to be considered is shown in Fig. 1. Being nonlinear it is not possible to use the 'transfer function' form of representation. However, the functional relationships between the variables of the system have been denoted by N N , some of which a r e linear, and others nonlinear, differential equations. These a r e given by the following equations:

N T? + T ^ r ) = A : e , T > 0 , K > 0 Eq. 2.1 N x + (b + 3b x*)x + e x + e x ' = ?7, b > 0, c > 0 Eq. 2.2 N g,x + g^x' =5 Eq. 2.3 N T3(4 + T^^) = T^(6 + T , é ) , T^ > T3 > 0 Eq. 2.4 4 y - 6 = e , E q . 2 . 5

2

-w h e r e , f o r t h e p r e s e n t p u r p o s e , y h a s a s i n u s o i d a l f o r m -which -will be specified l a t e r .

When t h e n o n l i n e a r t e r m s Sb^ x x , c x and g^x a r e a b s e n t t h e s y s t e m r e d u c e s t o a l i n e a r one having component t r a n s f e r functions Y^ , Y^ which a r e a s s o c i a t e d in an obvious way with N , N^ and a r e given by

Y, (p) = J7_(p) = /c/(l + T^p) E q . 2 . 6 e Y^ (p) = 2 L ( P ) = 1/(P* + b^P + c j E q . 2 . 7 and \ (P) = _1<P) = g , E q . 2 . 8 X Y, (p) = _6_(p) = T , ( l + T ^ p ) / p r ^ d + T, p ) " | E q . 2 . 9

e L J

In o r d e r t o r e l a t e t h e s e e q u a t i o n s to the e n g i n e e r i n g a s p e c t s of t h e s y s t e m , Y^ m a y b e looked upon a s r e p r e s e n t i n g an a c t u a t i n g e l e m e n t combined with an a m p l i f i e r of gain /c, Y^ a ' p l a n t ' capable of r e s o n a n c e , Y^ a ' p h a s e a d v a n c e n e t w o r k ' included to i m p r o v e t h e s t a b i l i t y and Y, a feedback gain c o m p e n s a t o r . T h e o v e r a l l t r a n s f e r function i s X (p) = (a p + a o ) / ( b p * + b p ' + b p * + b p + b ) E q . 2 . 1 0 ' 4 3 2 1 ° y w h e r e a^ = «T^T^ . a^ = /cT , \ = T T T , b^ = T^(T, + T3 + b^T, T ), b^ = T r 1 + b , (T, + T , ) + c< T , T, 1 h = b T + c T ( T + T ) + KB T T , b„ = c T + K g , T , 1 1 2 1 2 * 1 3 S , 2 3 ' ° 1 2 ° 1 3

T h e p r i n c i p a l n o n l i n e a r i t i e s o c c u r in t h e plant and a r e defined by N^ . T h e s e n o n l i n e a r i t i e s a r e looked upon a s being i n c a p a b l e of modification in t h e plant

itself, and t h e n o n l i n e a r i t y in t h e feedback p a t h , i . e . in N ^ , i s i n t r o d u c e d with t h e a i m of i m p r o v i n g t h e o v e r a l l c h a r a c t e r i s t i c s .

F r o m E q . 2 . 3 ,

4 = ( g , + S g ^ x ^ ) ^ E q . 2 . 1 1 F r o m E q s , 2 . 1 and 2 . 5

6 = y - e = y - {n+T,h)lic. E q . 2 . 1 2 which upon differentiation gives

5 = y - ê = y - ('7+ T / J 7 ) / « . E q . 2 . 1 3 Substituting f o r g , ^ , 6 and é in E q . 2 . 4 gives

T , f g , x + g ^ x ' + T j g , + 3g3X»)x j = T^ j ^ y + T ^ y - f n + (T, + T,)i7 + T, T , S j /K j E q . 2.14

Differentiating Eq. 2.2 with respect to t gives li as a function of x, x, x and'x, and a second differentiation gives i} as a function of x, x, x, x and ii,

Substituting for rj, ij and V in Eq. 2.14 gives, after some r e - a r r a n g e m e n t , the nonlinear equation

f X + f (x)x'+ f (x, x)x + f (x, x)x* + f (x) = K(y + T, y), Eq. 2.15

^ 3 2 1 n ^ where f = T T , 4 1 3 ' f (x) = T (1 + b T ) + T + 3 T T b x ' 3 1 1 3 3 1 3 3 f^(x, x) = 1 + b^(T^ + T^) + c J ^ T + 3 fbjCT^ + T^) + c ^T j l x* + 18b T T XX 3 1 3 2 X f^(x, x) = b , + c,(T, + T j ) + Kg^T, + 3 P b , + c^iT^ + T , ) + Kg,T 1

+ 6 r b ( T + T ) + c T T n x x + 6 b T T (x)"

| _ 3 1 3 3 1 S_J 3 1 3 f (x) = (c + Kg T / T )x + (c + Kg T / T )x ' o 1 1 3 2 3 3 3 2 A Special Case

A desirable feature of many control systems is that they have linear characteristics in the 'steady s t a t e ' . F o r the present system the steady state equation is

(c, + f g . T , / T ) x + ( c + K g T / T ) x ' =(cy Eq. 2.16

1 ° 1 3 ' 2 ' S . S . ' » 3 3 ' 2 ' S . S , • ' S . B . ,

which, in general, is nonlinear. If, however, g is chosen to be

g = -c T / ( K T ), Eq. 2.17

3 3 2 3 then Eq. 2.16 reduces to

(c + K g T / T ) x = (cy E q , 2.18

1 ^ 1 3 ' a ' S . S . • ' s . B . ^

which is a linear relation in x , y Choosing, for convenience

B i S t S > S »

g = T / T , Eq. 2.19

4

-t h e n

Xg g / y g g = ' ' / ( c ^ +K) E q . 2 . 2 0 If t h e s y s t e m i s s u c h that t h e gain K m a y b e chosen so that K » c , t h e n

X «» y , i m p l y i n g good s t e a d y s t a t e a c c u r a c y .

A s i m p l e choice of t h e coefficient g , h a s r e m o v e d t h e n o n l i n e a r i t y f r o m t h e s t e a d y s t a t e c h a r a c t e r i s t i c . Unfortunately t h e effect of t h e t e r m gjX i s not r e s t r i c t e d t o t h e s t e a d y s t a t e . In p a r t i c u l a r , e x a m i n a t i o n of f , ( x , x) s h o w s t h a t t h e t e r m 3Kg T x , which i s n e g a t i v e , can h a v e a c o n s i d e r a b l e influence on t h e m a g n i t u d e of f, (x, x) at h i g h e r v a l u e s of x a n d , t h e r e b y , modify t h e s y s t e m s t a b i l i t y . It will be s e e n l a t e r in Section 5 that t h i s effect i s p a r t i c u l a r l y m a r k e d in s y s t e m s having l i m i t e d loop gain, i . e . for s o m e r e a s o n , s u c h a s a n o i s y input s i g n a l y, t h e gain /c h a s t o be kept s m a l l . It i s d e s i r e d to d e m o n s t r a t e t h e effect of t h e t e r m 3KgjTjX^ in f^ (x, x) on t h e s t a b i l i t y . In o r d e r t o do t h i s c o n v e n i e n t l y , and t h e r e b y r e d u c e t h e b u r d e n s o m e m a t r i x m a n i p u l a t i o n r e q u i r e d in Section 4 , it will be a s s u m e d that T^ = 0 and b^ = 0. T o g e t h e r with E q s . 2 . 1 7 and 2 . 1 9 , t h i s a s s u m p t i o n r e d u c e s E q . 2 . 1 5 to x"+f^x + f^(x).x + f . x = *:(y + T y ) / T , E q , 2 . 2 1 w h e r e f = (1 + b T ) / T , f (x) = m + m x " , f = (c + K ) / T , 2 1 3 3 1 1 2 o 1 3 m = (b + c T + K T ) / T . and m = 3c (T - T ) / T 1 1 1 3 2 ' 3 ' 2 3 * 3 2 3 T a k i n g y t o b e s i n u s o i d a l , t h e r e s t of t h e p a p e r will be devoted to a n a n a l y s i s of t h e p e r i o d i c s o l u t i o n s and a s s o c i a t e d s t a b i l i t y of E q . 2 . 2 1 . 3 . T h e f r e q u e n c y r e s p o n s e T a k i n g y in t h e f o r m y = Q S i n w t , t h e n E q . 2 . 2 1 b e c o m e s x"+ f .X + f ( x ) x + f .X - p Sin (ojt + #•), E q . 3 . 1 2 1 o w h e r e 1 P = Q K U + w*T* ) ^ / T and wT = T a n ^. ^ 3 3 3 ' C l e a r l y P i s f r e q u e n c y d e p e n d e n t . In o r d e r to simplify t h e p r o b l e m it will be a s s u m e d that t h e v a l u e of T i s s m a l l such t h a t o v e r t h e f r e q u e n c y r a n g e of i n t e r e s t (uiT )* << 1; i m p l y i n g that P m a y b e t a k e n to be 3 P = Q K / T

independent of f r e q u e n c y . T h i s a s s u m p t i o n i s a valid one in m a n y e n g i n e e r i n g s i t u a t i o n s a s m a y be s e e n f r o m Section 5.

elements of the system, may readily be obtained by substituting P = i" in the appropriate transfer function. The modulus of the resulting complex number is then the amplitude-ratio between the output and input sinusoids, and its magnitude

is independent of the input amplitude. A typical frequency response curve for the plant alone, described by Eq. 2 . 7 , is shown in Fig. 2, Curve 1. In order to

achieve a flatter frequency response and a transient response with less 'overshoot', a negative feedback through a phase-advance network is introduced. The effect of t h i s , in a typical case, is shown by Fig. 2, Curve 2.

When the nonlinearities a r e present, as in Eq, 3 . 1 , a periodic solution for X can only be guaranteed if the nonlinearities a r e small, i. e. if m^x « m^. It should, perhaps, be added that periodic solutions for large nonlinearities a r e often observed on analogue computers, see (8) pp. 392-4, but a rigorous mathe-matical proof is absent except in the case of equations of second order; see (6) and (7). In the present problem particular interest will centre on what happens to the solution when u is close to the frequency of the free oscillations of the

equation

x"+ f X + m X + f X = 0 Eq. 3.2

2 1 o

If free oscillations of Eq. 3. 2 a r e to exist at all, the associated characteristic equation must have a pair of imaginary roots. Since Eq. 3.2 is the variational equation of Eq. 3.1 with m = 0, then these conditions a r e precisely those required for the application of the existence theorem of (2), Chapter 14, Theorem 3 . 1 . F u r t h e r , since Eq. 3.1 is a r e a l , analytic expression, its periodic solution will be analytic in m^ for m,, sufficiently small. This means that the solution may be obtained by the well known analytic perturbation procedure. In the present problem a first approximation is sought, akin to Eq. 1. 2 for Duffing's equation. F o r this solution the perturbation procedure reduces to taking a solution in the form

X = F Sin (ut + ^ + 9I) = p(t), Eq. 3.3 substituting it into Eq. 3.1 and obtaining the 'bifurcation' or 'determining' equations, (2), Chapter 14, Equation 3.27, by equating the coefficients of Sinwt and Cos ut, respectively. This yields the equations

w F ( u ^ - m - i m F*) Sin ( ^ + é ) + F(f - u'i ) Cos (f + ^)

1 2 0 2

= P Cos f Eq. 3.4

and

F(f - u^f ) Sin (^ + ^) - wF(w' - m - i m F*) Cos (•• + ^)

o at 1 2

= P Sin * Eq. 3.5 Squaring E q s . 3.4 and 3.5 and adding gives

(f - u*f f + u^{<^' -m - i m F ) = ( P / F ) , Eq, 3.6

0 2 1 2

6

-Alternatively Eq. 3.6 may be written (a.')' + l^f^ - 2(m, + i m ^ F ' ) ] {u>'Y

+ r(m^ + i m F*)' - 2f f n oj* + f' - (P/F)* =0, Eq. 3. 7

| _ ' 2 . O Z { O

a cubic in u in which only the real positive values of w and F a r e relevant, F o r frequencies away from the resonance 2v/u will differ considerably from the period of the solutions of Eq. 3.2 and the appropriate existence theorem is (2), Chapter 14, Theorem 1 . 1 . The application of the relevant perturbation procedure (2), pp. 350-1, to this case then yields the same bifurcation equations as above and, therefore, E q s . 3.6 and 3 . 7 . These latter equations, then, define the frequency response curves associated with Eq. 3 . 1 , in the first approximation, over the whole frequency range.

Later, the locus of vertical tangents of the frequency response curves, given by Eq, 3,6, will be required. These may be obtained by differentiating Eq. 3.6 implicitly with respect to F and imposing the condition dw/dF = 0. Thus

(f - u«f )''+ u'(a)* - m - i m F * ) ( w * - m - | m F ' ) = 0 Eq. 3.8

o 2 1 * 2 ' ' 1 * 2 ^ o r alternatively, 3 u^m*(F'')* - ü)^(u^ - m )m F* + ( / ( " " - m )* + Ï6 * (f - u^f f = 0, Eq. 3.9 o *

a quadratic in F in which the real positive values of « and F a r e relevant. 4. Asymptotic stability

It is required to determine the regions in the u, F plane for which the solution, Eq. 3 . 3 , of Eq. 3 . 1 , is asymptotically stable. F r o m (2), Chapter 13, Theorem 2 . 1 , it is known that Eq. 3,3 will be asymptotically stable as t •• " provided the trivial solution of the variational equation associated with Eq. 3 . 1 , rrij / 0, is asymptotically stable. The variational equation will be derived and C e s a r i ' s method employed to determine the characteristic exponents of this equation. The boundaries of asymptotic stability of the variational equation, and hence of the solution, Eq. 3 . 3 , of Eq. 3 . 1 , then follow.

The scalar Eq. 3.1 may alternatively be written as the 3-vector equation X = J(t, x), Eq. 4.1 where

T h e v a r i a t i o n a l equation of E q . 4 . 1 with r e s p e c t to E q , 3 . 3 i s

i= ) - ^ (t. p ( t ) ) . 5 . = J ^ ( t . p ( t ) ) , 5 . J = l '

E q , 4 . 2

w h e r e J i s t h e J a c o b i a n m a t r i x of J with r e s p e c t to x . On e v a l u a t i n g t h e p a r t i a l d i f f e r e n t i a l coefficients 9 J^ /3x^ a j ^ / d x ^ and s u b s t i t u t i n g into E q . 4 . 2 , t h e v a r i a t i o n a l equation b e c o m e s 0 0 -f - 2m .X, X. - m ^ - i^gX^ 0 1 -f. 5 , E q . 4 . 3 E q . 4 . 4 E q . 4 , 5 o r in s c a l a r f o r m 'i'+ f^i + (m^ + m ^ x * ) 4 + (f^ + 2m^x^x^)5 = 0 Writing z = wt + f + ^, t h e n d z / d t = u , and d^/dt = u Ê ' , w h e r e ^ " d é / d z . In t e r m s of z, E q , 4 . 4 b e c o m e s e ' + h 5* + (h + e C o s 2 z ) 5 ' + (h - 2e Sin 2z)g = 0, 2 1 o w h e r e h = f / u , h = (m + i m F * ) / « ^ e = - ^ m F * / " * . h = f / w ' , 2 2 1 1 2 2 e o

and e i s looked upon a s a ' s m a l l p a r a m e t e r ' .

In o r d e r to put E q . 4 . 3 o r E q . 4 . 6 in t h e a p p r o p r i a t e s t a n d a r d f o r m (5), C h a p t e r 8, E q , 8 . 1 , f i r s t w r i t e E q . 4 . 7 E q . 4 . 6 t h e n h 2 h 1 h e E q . 5 ' = = = 4 + K + 1 1 + K 1 a e a 1 . 6 b e c o m e s K g ' + € ' + K — 1 5 = + C o s 2 z ) t ' - 2 S i n 2 z 4 j E q . 4 . 8

8

-It will b e o b s e r v e d t h a t t h e a s s o c i a t e d c h a r a c t e r i s t i c equation f o r t h e c a s e e = 0 i s {\' + 1)CK+ K^) = 0,

h a v i n g a p a i r of i m a g i n a r y r o o t s and one r e a l r o o t . T h u s E q , 4 , 8 i s now in t h e f o r m of a p e r t u r b a t i o n p r o b l e m in t h e neighbourhood of t h e f r e e o s c i l l a t i o n of e * + K eT + 5 ' + « 5 = 0, 1 1 Now E q . 4 . 8 m a y b e w r i t t e n a s t h e s y s t e m

e' = eg +e*(z).5.

E q , 4 , 9 w h e r e 0 0 1 0 -1 0 1 E q . 4 , 1 0 and • ( z ) = 0 0 2 Sin 2z T h e c h a r a c t e r i s t i c equation of C i s det (C - XE) = 0, which h a s t h e r o o t s X = +i, - i , -K 1 I 2 f 3 1 0 0 (or + Cos 2z) 1 0 0 E q . 4 . 1 1 E q , 4 . 1 2

R e d u c t i o n of C to diagonal f o r m i s then a c h i e v e d by the s i m i l a r i t y t r a n s f o r m a t i o n

T C T diag (+i, - i , - K ) = D , E q . 4 . 1 3 w h e r e T i s a 3 x 3 m a t r i x whose coefficients m a y b e d e t e r m i n e d f r o m t h e m a t r i x equation

T C = D T

E x p a n d i n g E q , 4 , 1 4 and c o m p a r i n g coefficients then gives -i i 1 0 K E q , 4 . 1 4 T = 1/K , 1 1 - K i 1 1 + K i 1 E q , 4 , 1 5

T h e d e t e r m i n a n t of t h i s m a t r i x i s det T = 21(1 + K^)IK' 1 1 F o r m i n g t h e adjugate m a t r i x and s u b s t i t u t i n g in t h e r e l a t i o n T ' * = (det T ) ' ' . adj T E q , 4 , 1 6 g i v e s

"' = KJ I 21(1 + '^*) J .

1 + «c i 1 -K + i 1 -1 - K i 1 - 1 + K i 1 1 - *c i 1 T r a n s f o r m i n g E q . 4 . 9 by t h e s i m i l a r i t y t r a n s f o r m a t i o n g i v e s X ' = Dx + e l ' ( z ) , X w h e r e • (z) = T * ( Z ) T ' ' F r o m E q s . 4 . 1 5 , 4 . 1 7 and 4 , 1 9 21/K -21 2K 1 1 • ( z ) = 1/

[21(1 + K*)j

w h e r e P i 1 1 1 -/9 i 2 ^ i 2 ^ ^ -/9 i 3

e

1

3 1 3 E q . 4 . 1 7 E q . 4 , 1 8 E q . 4 . 1 9 E q , 4 . 2 0 ^ = 2(1 + jc i ) Sin 2z - (-(f + i ) ( a + C o s 2z) + cr (1 + K 1) 1 1 1 1 « 1 P = 2 ( - l + K i) Sin 2z - (ic + i ) ( a + C o s 2z) - a (1 - <c i) I 1 1 1 2 1 and /9 = (4i Sin 2z)/(c + 2i (cr + C o s 2z) - 210" K 1 1 1 2 1 E q . 4 , 1 8 i s now in t h e a p p r o p r i a t e s t a n d a r d f o r m . E q . 4 , 2 1 E q . 4 . 2 2 E q , 4 , 2 3 T h e next s t e p i s t h e f u r t h e r r e d u c t i o n of E q , 4 , 1 8 t o t h e f o r m given by (5), C h a p t e r 8, E q , 8 , 5 , F o r t h i s p u r p o s e w r i t e x = col (u, v ) , w h e r e u i s a 2 v e c t o r and V a one v e c t o r , and c o n s i d e r t h e t r a n s f o r m a t i o n

u = exp (X + e y ) z , exp F d i a g (+1, -i) - X E H Z , W -\

E q , 4 , 2 4 V = exp (X + e y ) z . r

10

-w h e r e -w i s a t-wo v e c t o r , r a one v e c t o r and Y any finite complex n u m b e r . No-w d i a g (+1, -i) - X E = d i a g (i - i, - i - i) = diag ( 0 . -21), and E q , 4 . 2 4 b e c o m e s u = exp (X + e y ) z , e x p . d i a g ( 0 , -21) z, w

]

E q , 4 , 2 5 E q . 4 . 2 6 v = exp (X + e y ) z , r . D i f f e r e n t i a t i n g E q . 4 . 2 6 with r e s p e c t t o z gives u' = (X + ey) exp ( X + e y ) z , e x p . d i a g (0, -21) z . w + exp (X + € y ) z , diag ( 0 , -21). e x p , diag (0, - 2 i ) z , w + exp (X + e y ) z , e x p , diag (0, - 2 i ) z . » v ' ,

and

v* = (X + cy) exp ( X + ey) z , r + exp (X + ey) z . r ' 1 1 1 E q . 4 , 1 8 m a y b e w r i t t e n in t e r m s of u and v a s 1— — u'

v

L J =

r+i ! 1

! 0 - i ! 0 I -K _ 1 _ ~ "1 u V — -1

[id +

K*)J

r-/9i

1

1

^'

L^^

- ^ 1 2

pi

? icp 1 a -^'~

Pi

, j _ _ K p 1 3 u V + Ul 1(1 + K* E q . 4 . 2 7 and p a r t i t i o n e d a s shown by d a s h e d l i n e s . E x p a n d i n g and s u b s t i t u t i n g f o r u, v , u ' and v' g i v e s , a f t e r c a n c e l l a t i o n of t h e c o m m o n f a c t o r exp (X + e y ) ,

e x p , d i a g ( 0 , -2i)z,«r' = diag (+i, -1) e x p . diag (0, - 2 i ) z , w - (X + ey) e x p , diag (0, - 2 i ) z . w - diag (0, -21) e x p . diag (0, - 2 i ) z . w + e"* (z) e x p , d i a g (0, - 2 i ) z , w 11 + e * (z) e x p . diag ( 0 , -21) z . r , 1 2 = d i a g ( - c y , - e y ) e x p . diag (0, - 2 i ) z . w + e * (z) e x p . diag (0, - 2 i ) z , w + e * (z) e x p , diag (0, - 2 i ) z . r 1 2

and r ' = - K^r - (X + ey)r + e* (z) e x p . diag (0, - 2 i ) z . w + e * ( z ) . r , w h e r e 1 1

1/

[i(

« (z) = i / i d + K') -pi -pi pi pi 1 2

.« ^(z) = i / [ i ( l + (C») j .

-p 1 3 T h u s JZ) = i/ [ i d . K|) ] . [ . , P^ . / J . •„(Z) = iV [ i d . K^) ] . [ K ^ ^ ^ ] .

w = -cyw + e exp | - d i a g ( 0 , -21) z * ( z ) . e x p . diag ( 0 , - 2 i ) z . w

+ e exp r - d i a g ( 0 , -2i) z j * ( z ) . e x p . diag ( 0 . - 2 i ) z , r

and

E q . 4 . 2 8

r ' = (-K - i - ey)r + e* ( z ) . e x p . d i a g (0, - 2 i ) z . w + e* ( z ) . r , E q . 4 . 29 which a r e now in t h e r e q u i r e d f o r m .

If y can be d e t e r m i n e d in s u c h a way t h a t t h e s y s t e m , E q s . 4 . 2 8 , 4 , 2 9 , h a s a p e r i o d i c solution of p e r i o d T = 2tr/2 =ir ( i . e . the p e r i o d of the t e r m s Sin 2z and C o s 2z i n ^ ^ , p^ and p^, and h e n c e of *(z) ), then t h i s solution g i v e s , f r o m E q . 4 . 2 4 , a solution of E q . 4 . 1 8 of the f o r m

X = exp (X + ey)z, p(z), p(z + ir) = p(z). E q . 4 . 3 0 T h i s i m p l i e s t h a t X, + ey i s a c h a r a c t e r i s t i c exponent of E q . 4 . 1 8 . T h e c h a r a c t e r -i s t -i c exponent -i s unchanged u n d e r a s -i m -i l a r -i t y t r a n s f o r m a t -i o n so that X, + ey -i s a c h a r a c t e r i s t i c exponent of E q . 4 . 8 a s w e l l .

T h e solution for y, in t h e f i r s t approxinnation, c o m e s r e a d i l y f r o m C e s a r i ' s g e n e r a l i t e r a t i v e m e t h o d for t h e p e r i o d i c s o l u t i o n s of d i f f e r e n t i a l e q u a t i o n s c o n -t a i n i n g a sn-tiall p a r a m e -t e r . T h e p r e s e n -t p r o b l e m is a p a r -t i c u l a r c a s e of -t h i s m e t h o d and t h e r e s u l t i s s u m m a r i z e d in (5), C h a p t e r 8, T h e o r e m 8 . 1 , In o r d e r t o u s e t h i s t h e o r e m , define t h e m a t r i x G(y, e = 0) = J_ f r< T J e z)dz, E q . 4 . 3 1 w h e r e

12

then the theorem states that y may be evaluated from the determinant

d e t r G ( y , e = 0) - y E T = 0 Eq, 4,33 Substituting for * (z) in Eq, 4. 32 gives

r ( z ) = i / d + K'). pe 2iz -P e 2 P -2iz Eq. 4.34 + o- ) - i( ( K ) \ .

Substituting for r(z) and the/9'B in Eq. 4.33 gives the following coefficients for the 2 x 2 matrix G: ir

G,, = 1/ [ir (1 + Kp 1 . I -p^dz = i / d + '^^ ). [ -(<r^<f, + a ) + 1(0-^ - <r^«,)J .

O G,, = i / [IT (1 + KM j . ƒ /9^ dz = i / d + «M. [ -(cr^K^ O TT G ^ = i / [rr (1 + KM ] . ƒ - /S^e'^^'^dz = i / ( K^ - i), O

^21 = 2/['^<l + ' ' ' ) ] • ƒ ^^e'^^^dz = -i/(K^ +1).

o

Substituting for G G in Eq. 4. 33 then gives

or G - y 11 ' <\. G - y 22 ' = 0 y* - (G + G )/ + (G G - G G ) = 0, ' 11 2 2 ' 1 1 2 2 12 21 '

which has the roots

y = i ( G + G ) ± i l ( G - G ) * + 4 G G !

1 . 2 11 22 (_ 11 22 i a 2 l j

Eq. 4.35

Eq, 4,36 Two of the characteristic exponents of Eq. 4 . 8 a r e , therefore,

X + e y = i + i e ( G + G ) ± i € l (G - G )* + 4G G E q , 4,37 1 11 xa L ' ^ ** 1 2 2 1 J

and, a s explained by Hale (5), p. 71, a r e the critical ones from stability con-siderations, The third characteristic exponent, which is K when e = 0, is not critical since K^ dominates any change in the r e a l part arising from the t e r m s on

the right hand side of Eq. 4 . 8 . It i s , of course, the real part of the characteristic exponent which determines the stability of Eq. 4 . 8 . Two cases a r i s e as follows.

a

Case 1 (G - G ) + 4G G < 0

11 22 i a 21

F o r asymptotic stability the real part of the characteristic exponent must be negative, which in the present case implies that

e ( G + G ) < 0 11 xa

o r

-e(<r K + a- ) / ( l + K*) < 0

1 1 2 1

Since K is finite and r e a l , then

1

- eio^ K + a ) < 0

1 1 2

or upon substitution frona Eq. 4 . 7 , h (2 - h ) - h <0,

o ' 1 2 '

o r , after multiplication throughout by ui',

f^ 2 - (m^ + i m ^ F ' ) / u ' ' - u ' f < O Eq. 4,38 The asymptotic stability boundary corresponds to the vanishing of the left hand side of Eq. 4 . 3 8 , which gives

F * = 2/m^. I u' (2 - u^f^/f^) - m^ , Eq. 4.39 The real positive values of F satisfying Eq, 4. 39 describe a boundary in u, F ; the region for which F < F is then one of asymptotic stability, for the solution, Eq, 3 , 3 , of Eq, 3 . 1 , and the region for which F > F. is one of asymptotic in stability. It will be observed that in Eq. 4. 39, « may be a multi valued function of F* .

Case 2 ( G , , - G, j ' + 4G ^^G^^ > 0

In this case the condition for asymptotic stability is k

r z ^

e(G_{' i i xa'} + G ) ± e

[<Gii - G . a ) ' + 4 G , . G „ J ' <0

l e ( G + G )| > c (G - G ) * + 4G G ' 1 1 82 ' I. 11 xa IX XI J

14 -or e*(G + G 1 1 2 or ) ' > e* (G - G ) ' + 4 G G a L l 1 22 1 2 2 1 _ 4 e * (G G - G G ) > 0 E q , 4 , 4 0 11 22 ia a i

Substituting for G G and multiplying throughout by (1 + if)' gives

e'((T K + o- )* + e ' ( a - o" JC ) ' - i e* (1 + K* ) > 0,

1 1 a 1 2 1 1

which upon substitution from Eq. 4 . 7 , becomes

r ( h - l)h +(h - h ) ~ ] ' + r ( h - l ) - ( h - h )h ~1 * - i e * ( l +h*) > 0

| _ 1 o ^ ' ' J L _ ' X o o _j o

o r after sonie r e - a r r a n g e m e n t

p h ^ - h/

+

(h

-if - i e*^ d + h^) > 0

Provided h^ = K ^ i, which was postulated, then the region of asymptotic stability is defined by

(h - h )' + ( h , - D ' - i e S 0,

2 O 1

which upon substitution for h , h , h , e and multiplication throughout by u , gives

o 1 2

(f - u'^ff +uMm. + i m F * - w*)» - ( i ) ' m ! < ^ F * > 0,

O 2 1 Z 2

which upon r e - a r r a n g e m e n t becomes

(f - u^f )* + u*(u* - m - i m F * ) ( w » - m - | m F*) > 0 Eq. 4.41 The asymptotic stability boundary corresponds to the vanishing of the left hand side of Eq. 4 . 4 1 , and agrees exactly with Eq. 3 . 8 . Thus, one of the asymptotic stability boundaries, of the solution,Eq. 3. 3,of Eq. 3 . 1 , corresponds to the locus of vertical tangents of the response curves in the u, F plane, defined by Eq. 3.6. As far as the author can determine, this is the first time that the locus of vertical tangents of the response curves has been established a s an asymptotic stability boundary for systems of higher o r d e r than the second,

5, A numerical example

In o r d e r to illustrate some of the previous results a numerical example has been evaluated. F o r this purpose a gain limited system has been considered. The coefficients associated with the plant, N , were taken to be

b = 0 . 1 , c = 1 , 0 and c = 0 . 2 .

1 ' 1 3

The unit linear frequency response for the plant alone is shown in Fig. 2, Curve 1 and p o s s e s s e s a very marked resonance at 1 radian/second. Following the usual methods of system synthesis, K was taken to be its maximum permissible value,

chosen in this case to be 10, T^ was neglected, and T^ , Tj were chosen such that the resonance was effectively suppressed. Simultaneously, checks were made on the transient response to a step function of y in order to ensure a small r i s e time and modest overshoot. The final values chosen were

T = 0 . 2 , T = 0 , 0 4

a ' 3

and the associated unit frequency response curve is Curve 2 of Fig. 2. These values a r e not optimum in t e r m s of the above criteria but a r e sufficiently close to optimum in order to give a realistic illustration of the effects of the nonlinearity in f on the frequency response.

With these values the coefficients of Eq. 3.1 become

f^ = 1.004/0.04, m^ = 2 . 1 4 / 0 . 0 4 , m^ = - 0 . 0 9 6 / 0 . 0 4 , f^ = 1 1 / 0 . 0 4 . The value of P is

1

P = lOQ (1 + 0.00166)^)^/0.04,

which, in the frequency range 0 to 6 radians/second, may be approximated to by P = 250Q,

and, thereby, b e a r s out the argument at the beginning of Section 3.

Substituting these values into Eq. 3.7, the frequency response curves for various Q have been obtained. This was done by choosing various F for a given Q (and thus P) and solving Eq. 3.7 by extracting, by iteration, the one real root which must exist; the remaining roots followed readily. The response curves a r e shown in F i g . 3, on which a r e shown the stability boundaries given by Eqs. 4.39 and 3 . 8 , It can be seen that for Q = 1 the response curve is little different from Curve 2 of F i g . 2. Increasing Q to 2 produces the dramatic change shown in Fig, 3, where upon the response curve meeting the locus of vertical tangents, which encloses an asymptotically unstable region, gives r i s e to 'jumps' in amplitude, akin to those exhibited by Duffing's equation. In the present case these jumps c a r r y the amplitude to values above the stability boundary defined by Eq, 4. 39, and thereby into a region of asymptotic instability. The overall effect of the instability associated with the locus of vertical tangents i s , therefore, to reduce the maximum stable amplitude, in the region of resonance, below that given by Eq. 4 , 3 9 .

Although this example is p r i m a r i l y concerned with the limited gain system described, an approximate assessment of the effect of high « on the position of the stability boundary given by Eq. 4.39 can be obtained. By assuming T^ « T^ , which is r e a l i s t i c , and that K> 10, then

m^ /m^ « - i ^ / C j ,

16

-Thus for c = 0 . 2 and « = 100, F = 18, and clearly this permits a very much larger operating amplitude than that of the gain limited system.

References

1. Stoker, J . J . Nonlinear Vibrations.

Interscience Publishers, N.Y. (1950). 2. Coddington, E . A . and

Levinson, N.

Theory of Ordinary Differential Equations. McGraw-Hill (1955).

3. Struble, R.A. Nonlinear Differential Equations. McGraw-Hill (1962),

4. Christopher, P . A . T . College of Aeronautics, Cranfield, Report Aero. No. 180.

5, Hale, J . K . Oscillations in Nonlinear Systems, McGraw-Hill (1963),

6, Yoshizawa, T. Memoirs of the College of Science, University of Kyoto, Series A, Vol. XXXIH, Mathematics No. 2, 1960, pp. 301-8,

7, C e s a r i , L . Contributions to Differential Equations, Vol. 1, No. 2, pp. 149-187.

8. Davis, H . T , Introduction to Nonlinear Differential and Integral Equations.

N.(<.«) { _NsC^.O

FIG.I. A NONLINEAR CONTROL SYSTEM.

IL

)

I

<

FIG.2. UNEAR FREQUENCY RESPONSE.

8 - _{Stability boundary} defined by equation ^ 3 9 )

3

< 4

2 3 w - radiant per second