Popov criteria for asymptotic stability of Lurie system of differential equations and Liapunov function

Meyer, K. R.

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LIAPUIVOV FlINCTIONS FOR 
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THE PROBLEM OF LURIE ? y 
I- - A - (PAGES1 
b 
* $  2 
By K. R .  Meyer 
(NASA CR OR TMX OR AD NUMBER) 
(CODE1 
ff653 July65 I 
The search for Liapunov functions fo r  t he  system of d i f f e r e n t i a l  
equations (1) known as t h e  Lurie system has been extensive i n  recent  years. 
A s igni f icant  change i n  the  course of t h i s  research occurred a f t e r  the  appear- 
ance of t h e  1961 paper of  Popov [l]. Popov obtained a c r i t e r i a  f o r  t h e  
asymptotic s t a b i l i t y  of the  Lurie system without t he  use of Liapunov func- 
t ions .  Moreover, he proved t h a t  h i s  c r i t e r i a  i s  s a t i s f i e d  i f  t he re  e x i s t s  
a pos i t ive  d e f i n i t e  Liapunov function of the  type quadratic form plus  the  
i n t e g r a l  of t h e  nonlinearity;  t h a t  is, a function o f  the form (3) whose 
der iva t ive  along the  t r a j e c t o r i e s  of (1) i s  negative def in i te .  
Several authors have proved p a r t i a l  converses t o  Popov' s theorem on 
t h e  existence of Liapmov functions. Two noteworthy papers a re  those of 
Yacubcrvich [ 2 ]  tad K a l m a s  [ 3 ] .  YazuEouich was i b l e  to prove a p a r t i d  con- 
verse by strengthening the  Popov c r i t e r i a  ( see  ( 2 ) )  below from 
F ( b )  > 0 
P(ic0) L 0 t o  
and Kalman w a s  able  t o  e s t ab l i sh  a p a r t i a l  converse by requir ing 
t h a t  t he  system be completely control lable  and completely observable (defined 
below). This note i s  t o  announce t h e  r e s u l t  t h a t  the  Popov c r i t e r i a  implies 
t h e  existence of a pos i t ive  de f in i t e  Liapunov function of the  type quadratic 
form plus i n t e g r a l  of the  nonlinearity t h a t  can be used t o  prove asymptotic 
s t a b i l i t y  i n  the  la rge  without the above r e s t r i c t ions .  
which/used t o  prove t h i s  r e s u l t  can/be used t o  es tab l i sh  fur ther  r e s u l t s  such 
Also the lemma 1 
i 8  a lso  
a n  Theorem 2. 
%is research was supported i n  part bv -Aeronautics and Space 
Administration under C o n t r a c t & ~ - 4 0 - 0 0 2 - 0 1 5 > n  pa r t  by the  United S ta tes  
A i r  Force Office of Sc ien t i f i c  Research under Contract l o .  AFOSR-693-64, and 
i n  p a r t  by t h e  United S ta tes  Army Research Office, Durham, under Contract No. I 
\, 
https://ntrs.nasa.gov/search.jsp?R=19660009065 2020-03-16T23:29:50+00:00Z
-2- 
Let  E" denote Euclidean n-space and I t h e  n x n i d e n t i t y  matrix. 
Let A be a real n x n matrix and b a r e a l  n-vector. The vector b 
will be considered as a column vector and b' w i l l  be i t s  transpose. The 
cyclic subspace generated by b r e l a t i v e  t o  A w i l l  be denoted by [A, b]; 
t h a t  is [A, b] I (x E e: a. b t al& + ... + a 
me reed nmbt r s )  . Let [A, b]' denote t h e  orthogonal complement of [A, b] 
i n  I?. The p a i r  (A, b) is said t o  be completely cont ro l lab le  If [A, b] t En 
and (A, b' ) i s  said t o  be c q l e t e l y  observable provided ( A ' ,  b) i s  com- 
p l e t e l y  controllable.  
An-b where t h e  a ' e  n- 1 3 
The main r e s u l t  is t h e  following lemma: 
Lemma 1: Let A be a real  n x n matrix all of whose cha rac t e r i s t i c  roo t s  
have negative real parte;  l e t  'I be a real nonnegative number and l e t  b, k 
be two real n-vectors. If 
- --- --- 
- - - --- --
--- -
T + 2 Re kt(iuI - A)-% Z 0 
for a l l  real  cb then the re  ex i s t s  two r ea l  n x n symmetric matrices B 
-- -- --
and a real n-vector q such t h a t  D, --- --
A'B + BA - ~ q '  - D 
Bb - k = J T q  
D i s  p o r i t i v e  semi de f in i t e  - and B is pos i t i ve  d e f i n i t e  - -
(x  E En: X'DX 6 01 n [ A I ,  qp= (01 
9 f EA, bIO 
if iC0, 4) r e d ,  is a root of -9' (21 - A)'% + JT then it i s  a 
-- root of b ' (z1  - A)'l  D(z1 - A)-%. 
- - -e-- --- 
-3- 
The beginning of the  proof of t h i s  1- i s  similar t o  the  proof glrm 
by Kn7mm i n  131. A recond lemma which i s  easy t o  e s t ab l i sh  shows a generali-  . 
cation of Popov’r theorem t o  the  c r i t i c a l  canes. It i s  
W 2: Let A be a r e a l  n x n matrix all of whose cha rac t e r i s t i c  roots  - --- --- 
are r b ~ l e ,  d i r t i n c t  and have zero r e a l  g a t s .  
arc all pos i t i ve  then there  ex is t s  a pos i t i ve  d e f i n i t e  matrix 
-- If t h e  residues - of k’ (21 - A)’% - ---- 
B -- such t h a t  -- ---- 
A ’ B + B A = O  and B b - k z O .  
The t w o  l e m m a s  above can be used i n  the  analysis-of several  d i f f e r e n t i a l  
8 7 S t e ~ 6 i  t h a t  occur in control  theory. For ersmple consider 
%here x, b and c are r e d  n-vectors; A is a r e a l  n x n matrix and 
@(u) I n  a r e d  continuow scalar f’unction of the r e a l  scalar u such t h a t  ’ 
&(u) > 0 for u 0. Both x and u are functions of the  r e a l  var iable  
t snd k I E dx . Let  (1) be such t h a t  through each xo E E” t h e r e  e x i s t s  
a unique t r a j ec to ry  of (1). 
lable and c a q l e t e l y  observable provided tha t  (A, b) and (A, c ’ )  a r e  re-  
SpeetiralJ caap1etel.y control lable  and completely observable. 
trp, the 1- 1 and 2 can be used t o  prove: 
The system (1) i s  said t o  be completely control-  
For t h i s  sys- 
Theorem 1. Let there exist two constants a 2 0, B L 0, a + p > 0 
e 
pole of P(z) at bo I8 s l q l e  md ha8 p e i t i v e  residues. Then there e r i a t r  - - -- --
a Llwpuuo~ function V of t he  form - --- 
(3) 
where B is a s p m e t r i c  matrix such t h a t  V is pos i t i ve  d e f i n i t e  for  all - -- -- - --
x and V = (grad V)-(Ax - b+(u) )  S 0 -- f o r  all x. I f  the  linerr Systea! -
f t (A - pbc')x I s  aeymptotically s t ab le  for  all p > 0 then no non zero - -- ----
t r a j c c t o r r  of (1) remains i n  t he  se t  where v = O. 
Moreover) i f  when a = 0 - and A - i s singular 
- ---- 
C 
J0 +(?)d? -- tends t o  + - -- 
w i t h  1.1 then V tends t o  + - w i t h  1 x 11. Thus under the  above con- - 7 -- - ----  
dIt lonsJ  the  solut ion x P 0 of (1) is  aeyppptotically s t ab le  --- i n  t h e  large. - - - .. 
The lema8 1 and 2 can equally w e l l  be used t o  consider the  system (1) 
when +(u) is  r t 8 t r i c t e d  t o  0 < d ( u )  < ku2 and a sharper r e s u l t  can be 
obtained. Another example of the conrequences of lemma 1 is the  following: 
Re c' ( ZI - A)-% i& 0 
-- for all z = I& + X, Q) real, -- then there  e x i s t s  - a nonnegative function K 
defined on [0, 0)  such t h a t  - --
for a l l  t P 0 where x ( t )  i s  the solut ion of (1) such t h a t  x(0) = xo. -- - A- - --
f '  c -5- 
c , 
I11 Popov, V. X., Absolute s t a b i l i t y  of nonlinear system8 of 
automatic control,  Automatika i Telemekh, Vol. 22, 196L 
I 21 Yacubovich, V. A., "he solution of ce r t a in  matrix inequal i t ies  
i n  qutamatic cont ro l  theory, Dokl. &ad. Nauk, SSKR, Vol. 143, 
1w. 
Kalma.n, R. E., Lyapunov functions f o r  t h e  problem of Luric i n  
automatic control, €'roc. Hat. Acad. Sci., 1963. 


Liapunov functions for the problem of Lurie

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