Means for determining stability of feedback systems described by either linear or nonlinear differential equation

Krall, A. M.

English

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A General S t a b i l i t y  C r i t e r i o n  f o r  Feedback Systems -_ 
by 
* 
A l l a n  M. Krall 
Hard COPY (HC),- O N  
Microfiche (MF) a 
ff 653 July 65 
1. I n t r o d u c t i o n .  I f  a feedback system i s  descr ibed by a system 
of l i n e a r  d i f f e r e n t i a l  equat ions with cons t an t  c o e f f i c i e n t s ,  t h e r e  are 
a g r e a t  many techniques f o r  determining whether o r  not  t h e  system i s  
s t a b l e  [l], [3],  [6]. I f  t he  c o e f f i c i e n t s  are dependent on t i m e ,  t hen  
s t a b i l i t y  can be determined by comparing t h e  system t o  one wi th  cons t an t  
c o e f f i c i e n t s  [SI. 
equa t ions ,  s t a b i l i t y  may b e  determined by v a r i o u s  approximations,  
d e s c r i b i n g  func t ions  o r  by techniques f i t t i n g  c e r t a i n  s p e c i a l  s i t u a t i o n s .  
I n  some of t h e s e  cases v a l i d  mathematical proof of t h e  resu l t s  i s  l ack ing  
I f  t h e  system can only be descr ibed by non l inea r  
It i s  t h e  a i m  of t h i s  paper t o  p re sen t  two gene ra l  s t a b i l i t y  
theorems, w i t h  p r o o f s ,  which cover a g r e a t  many s i t u a t i o n s  and which, 
even i n  t h e  l i n e a r  c a s e  wi th  constant  c o e f f i c i e n t s ,  g ives  useab le  new 
r e s u l t s .  
2 .  Exis t ence ,  Uniqueness and S t a b i l i t y .  We w i l l  p r e s e n t  t h e  
fo l lowing  theorems i n  a r a t h e r  general  s e t t i n g ,  t hen  show t h e  a p p l i -  
c a t i o n s .  
L e t  X denote  a Banach space wi th  norm denoted by I I I I .  Let 
F and G be o p e r a t o r s  on X. 
* McAll is ter  Bu i ld ing ,  The Pennsylvania S t a t e  Un ive r s i ty ,  Un ive r s i ty  
Park,  Pennsylvania.  
Grant 39-009-0414 jy b b 1 y 1 6  6 The research was su  ported i n  p a r t  by N.A.S.A. 
a 
* (ACCESSIUN UMBER) 
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https://ntrs.nasa.gov/search.jsp?R=19660003879 2020-03-16T21:34:32+00:00Z
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0 
-2-  
Def in i t i on .  An ope ra to r  F i s  sa id  t o  b e  bounded i f  t h e s e  e x i s t s  a 
number K such t h a t  IlFxll f K 11x11 f o r  a l l  x X. 
The smallest of a l l  such numbers K i s  c a l l e d  t h e  norm of  t h e  
o p e r a t o r ,  All such numbers K a r e  boclnds f o ~  P. I n  what fo l lows  
only  bounds w i l l  be  needed. The norm i t s e l f  w i l l  no t  be necessary .  
It i s  known t h a t  i f  FG i s  a composi.te o p e r a t o r ,  F i s  bounded by K., 
G i s  bounded by L ,  then  FG is bo.unded by KL- 
D e f i n i t i o n .  An ope ra to r  F i s  sa id  t o  s a t i s f y  a L i p s c h i t z  cond i t ion  
i f  t h e r e  e x i s t s  a number IC such t h a t  IIPx - Fyll 5 K IIx - yII  
f o r  a l l  x and y X 
We are now ready t o  consider  t h e  feedback system 3 i l l u s t r a t e d  
by f i g u r e  1. f i s  t h e  "forcing functi.on" or  " input" ,  e i s  t h e  
"er ror" ,  r i s  the "response" o r  "output".  F and G a r e  ope ra to r s  
on X. 
Mathematical ly  3 can be descr ibed  by t h e  fo l lowing  equat ions .  
f - G ( r )  = e , F(e)  = r 9  
o r  by 
P ( f  - G(r))  = r. 
Figure  1. The feedback system if. 
L 
- 3- 
E 
Theorem 1. If F - and G s a t i s f y  a Lipschitz c o n d i t i o n  wi th  c o n s t a n t s  
K and L r e s p e c t i v e l y  and KL < 1, t h e n  F(f - G ( r ) )  = r has a 
- -  
un.ique solut. i .on r - i .n X. (See r.71)" 
Proof .  We develop an  i t e r a t i v e  procedure t o  g ive  a sequence of 
elements i n  X which converge to r. L e t  
r = F ( f ) ,  1 
2 r = F ( f  - G ( r 1 ) ) ?  
and i n  gene ra l  
Eac,h r i s  i n  X and 
n 
We see by inducti .on t h a t  
which approaches zero as n -.) m s i n c e  W, e 1, Since X i s  complete, 
t h e  Cauchy sequence Ern] converges t o  a n  element P i n  X. Fur ther  
so 
-4- 
r = F(f - G(r ) ) .  
I f  r and r' are  both  so lu t ions  then  
I which i s  a c o n t r a d i c t i o n  un le s s  r = r . 
As wi th  o rd ina ry  d i f f e r e n t i a l  equat ions  t h i s  method of succes-  
sive approximations g ives  a procedure f o r  c a l c u l a t i n g  r as w e l l  as 
a n  estimate of t h e  e r r o r  involved. 
It is  a l s o  important  t o  note t h a t  t h i s  theorem guarantees  
uniqueness  only  i n  X.  It is  e n t i r e l y  p o s s i b l e  t o  have o t h e r  s o l u t i o n s  
n o t  i n  X and i n  so doing des t roy  " s t a b i l i t y " .  
Theorem 2. If F G a re  bounded r e s p e c t i v e l y  by  K L ,  
KL < 1 and F(f  - G(r ) )  = r has  a s o l u t i o n  r X ,  t hen  
(See r71). 
Proof .  
and t h e  r e s u l t  fo l lows .  
-5- 
Corol la ry .  If F and G s a t i s f y  a L i p s c h i t z  cond i t ion  wi th  cons t an t s  
K g& L r e s p e c t i v e l y ,  KL < 1 - and P(o)  = 0 ,  G ( o )  = 0,  then 
F(f  - G ( r ) )  = r has  a unique s o l u t i o n  r X ,  - and 
(See P I > .  
Proof .  S ince  F(o) = 0 ,  
I IFx  - FY 
w i t h  y = 0 shows t h a t  K i s  
bound f o r  G .  The r e s u l t s  f o l  
the L i p s c h i t z  c o n d i t i o n  
I < Kllx-YII 
a bound f o r  F .  S i m i l a r l y  L i s  a 
ow from Theorems 1 and 2 .  
These theorems are v a l i d  i n  any Banach space X .  Those most 
commonly considered are t h e  spaces Lp, 1 5 p 5 w 3  C ,  t h e  space of 
bounded continuous func t ions ,  and B t h e  space  of bounded func t ions .  
- -  
(See [ 4 ] > .  I n  p r a c t i c a l  app l i ca t ions  t h e  fvnct. ions i n  t h e s e  spaces  are  
func t ions  of t i m e ,  which i s  permitted t o  va ry  from 0 to m. I f  
nega t ive  t i m e s  are considered t h e  func t ions  are u s u a l l y  t o  be zero  
f o r  t < 0.  
An imp0rtan.t concept assoc ia ted  wi th  fee.dback syste.ms i s  t h e  
concept of s t ab i1 i t . y .  There are many ways of  d e f i n i n g  s t a b i l i t y .  The. 
one fo l lowing  i s  a convenient. choice.  
D e f i n i t i o n .  The system i3 i s  sa id  t o  be s t a b l e  i f  f o r  a l l  f o r c i n g  
func t ions  f 2 X, t h e  response r i s  always an  element o f  X. 
It i s  e n t i r e l y  p o s s i b l e  for f t o  be i n  X w i t h  r n o t .  Such 
syste.ms are uns t ab le .  
P 
L e t  X(O,m> be one of t he  Banach spaces  L (O,m), 1 < p 5 m, C(0,m) - -   
-6 -  
o r  B ( 0 , m )  w i th  norm I I - I I .  For any N > 0 w e  denote  t h e  norm of 
X(0,N) by I I - 1 1 , .  We then  have t h e  fol lowing 
m. Let f be i n  X ( 0 , w )  and l e t  II be i n  X(0 ,N)  
f,or a l l  N > 0. Suppose f o r  a l l  f and g in X(0,N) f o r  a l l  N > 0 ,  
Fg, Fh, Gg, and Gh are de f ined  and s a t i s f y  
IIFg - F h l I  N 2 118 h l I  N 
- w i t h  KL < 1. Then r i s  i n  X ( 0 , m )  and i s  i d e n t i c a l  w i t h  t h a t  of 
theorem 1. 
The proof i s  i d e n t i c a l  w i th  the uniqueness p a r t  of t h a t  of theorem 1. 
S t a b i l i t y  Theorem 2 .  Let f ,  be i n  X ( O , w ) ,  and l e t  r be i n  X(0,N) 
f o r  a19 N > 0 .  Suppose f o r  a l l  g & X(0,N) f o r  a l l  N > 0, Fg 
- and Gg are def ined and s a t i s f y  
w i t h  IU < 1. 
Then r i s  i n  X ( 0 , w )  and 
-
- -
11rH 5 l l f l l .  
- 7 -  
and 
L e t t i n g  N + comp1ete.s t h e  prouf ,  
We f i n a l l y  remark t h a t  t hese  theorems can be gene ra l i zed  t o  
mul t i l oop  syst.ems w i t h  similar r e s u l t s .  
3 .  Appl i ca t ions  
I n v e r s e  D i f f e r e n t i a l  Operators.  The most commonly encountered 
feedback systems are  u s u a l l y  expressed i n  terms of l i n e a r  d i f f e r e n t i a l  
equa t ions  w i t h  cons t an t  c o e f f i c i e n t s .  I n  t h i s  ca se  t h e  o p e r a t o r s  F 
and G are expressed as inve r se  d i f f e r e n t i a l  o p e r a t o r s ,  F = f l (D)/f2(D),  
G = g1 (D) /g2 (D) where f l '  f* '  g1 and g2 are polynomials i n  
deg f < deg f 2 9  deg g1 5 deg g2-  d d t 9  1 =  - These o p e r a t o r s  are uniquely D = -  
def ined  by r e q u i r i n g  t h a t  F f ( 0 )  and Gf(0) be zero a l o n g  w i t h  a n  
a p p r o p r i a t e  number of d e r i v a t i v e s  as long as f ( t )  i s  a s u i t a b l e  f u n c t i o n .  
Another more s u i t a b l e  represe.ntat ion o f  F and G i s  by t h e  USE 
of one s ided  Green 's  func t ions  [5]. It i.s known t h a t  t h e r e  e x i s t  unique 
funct. ions @ ( t )  and r( t)  so t h a t  
Using t h i s  l a s t  r e p r e s e n t a t i o n ,  i f  a l l  of t h e  ze ros  of f 2 ( z )  and 
1 
g (2 )  l i e  i n  t h e  l e f t  h a l f  plane i.t can be shown t h a t  when X i s  L , 2 
L2, La, B o r  C, t h e  norms of F and G are ( O ( t ) l d t  and 
0 
(r(t)  Idt r e s p e c t i v e l y .  
0 
-8- 
a > 0. Then i f  f i s  s u i t a b l e ,  1 A s  an example 1e.t F = - Dt-a ' 
1 2  -1 
F f ( t )  = f e -a(t-'u) f (u )du .  The norm of F i s  a 
Loo, B o r  C .  
i n  L , L ,  
0 
1 2  i s  a bound f o r  F i n  L , L , Lm, B o r  C .  
A s  a c o r o l l a r y  t o  t h e  preceeding d i s c u s s i o n  w e  have t h e  fol lowing.  
1 2 a  Theorem. Let X = L 3 L  iL ,B,C, _Let i3 be a s i m p l e  feP.dback 
system w i t h  F = K/(D+al) . - .  (Wan) ,  ai > 0, G = 1. - Then 3 i s  s t a b l e  
i f  
- -  
1.1 < ala 2 . . . a  . 
n -
G = 1 and 1 Dt-a ' If w e  cons ide r  t h e  feedback system where F = -
determine s t a b i l i t y  by any o the r  method, w e  f i n d  t h a t  t h e  system i s  
s t a b l e  pr0vide.d IKI < a and poss ib ly  u n s t a b l e  i f  1.1 > a .  Thus t h e  
r e s u l t  i s  t h e  "best  poss ib l e" .  Ac tua l ly  t h i s  system i s  s t a b l e  f o r  
K > -a and u n s t a b l e  otherwise.  
S imi l a r  r e s u l t s  may be found f o r  l i n e a r  d i f f e r e n t i a l  systems wi th  
v a r i a b l e  c o e f f i c i e n t s .  I n  t h i s  ca se  t h e  func t ions  0 and r a r e  
f u n c t i o n s  of two v a r i a b l e s  t and T~ 
t t 
0 0 
F f ( t )  = g ( t , ~ ) f ( ~ ) d ~ ,  and G f ( t )  = r ( t ,T ) f (T)dT .  (See [5]). 
T i m e  Lags. A t i m e  l a g  ope ra to r  L def ined by L f ( t )  = f ( t - T )  
7 7 
- TD can  be r ep resen ted  by e . It has norm 1 i n  a l l  t hose  Banach spaces 
mentioned. The r e s u l t s  of t h e  previous s e c t i o n  may be r e s t a t e d  wi th  
t i m e  l a g s  wi th  no change i n  t h e  r e s u l t s .  
-9- 
C e r t a i n  Nonlinear Operators.  There are a g r e a t  many non l inea r  
o p e r a t o r s  such as s a t u r a t i o n ,  dead zone, h y s t e r e s i s ,  backlash which 
f r e q u e n t l y  ocsur a l o n e  o r  i n  any p o s s i b l e  combination. 
l i n e a r  elements are considered s e p a r a t e l y  and a graph of ou tpu t  v e r s u s  
i n p u t  i s  made, t h e r e  u s u a l l y  i s  a p o i n t  on t h e  graph s o  t h a t  a l i n e  
j o i n i n g  i t  t o  t h e  o r i g i n  has maximum s l o p e  IC, It is  easy t o  sez t h a t  
K i s  a bound f o r  t h a t  o p e r a t o r .  (See [3] ) .  
I f  t h e s e  non- 
We n o t e  i n  pas s ing  t h a t  t h e  "desc r ib ing  funct ions ' '  used t o  d e s c r i b e  
t h e s e  phenomena have IC as a maximum v a l u e .  
-10- 
REFERENCES 
W. Kaplan, "Operational Methods for Linear Systems", Addison- 
Wesley Publishing C o . ,  Inc., Reading, Mass., 1962. 
Ja. Kudrevic, "Stability of Nonlinear Systems with Feedback", 
Automat. i Telemek., vol. 25, 1964, pp. 1145-1155. 
B. C. Kuo, "Automatic Control Systems", Prentice. Ball., Inc., 
Englewood Cliffs, N.J., 1962, 
4 .  E. J. McShane and T. A. Botts, "Real Analysis", D. Van Nostrand 
Company, Inc, Princeton, N. J., 1959. 
5.  A. R. Stubberud, "Analysis and Synthesis of Linear Time-Variable 
Systems", Univ. of California Press, Berkeley, Cal., 1964. 
6. J. C. Truxal, "Automatic Feedback Control System Synthesis", 
McGraw-Hill Book Co.,  Inc. New York., 1955. 
7 .  G. Zames, "On the Stability of Nonlinear Time Varying Feedback 
Systems", Proc. Nat Elect Conf I ,  Inc., Chicago, Ill. , 1964 
pp. '725-730. 


A general stability criterion for feedback systems

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