Aggarwal, J. K.

Infante, E. F.

English

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SOME REMIw(S ON THE STABILITY OF TIME-VARYING SYSTEMS 
J. K. Aggarwal* 
Department qf Electrical Engineering 
The University of Texas 
Austin, Texas 
and 
E. F. Infante** 
Division of Applied Mathematics 
Brown University 
Providence, Rhode Island 
GPO PRICE $ 
CFSTl PRICE(S) $ 
Hard copy (HC) 3..@ 
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ff 653 July 65 
* 
This research was supported by the National Science Foundation under 
Grant GK-1879. 
This research was supported in part by the Air Force Office of 
Scientific Research, Office of Aerospace Research, United States Air 
Force; under - , in part by the National Aeronautics and 
Space Administration under Grmt Eo. 40-002-013 and in part by the 
United States Army under Grant No. DA-31-124-ARO-D-270. 
** 
https://ntrs.nasa.gov/search.jsp?R=19680015896 2020-03-12T10:27:37+00:00Z
. 
SOME REMARKS ON THE STABILITY OF TIME-VARYING SYSTEMS 
A l a rge  number of r e s u l t s  a re  present ly  available f o r  t h e  de- 
termination of s t a b i l i t y  conditions f o r  l i n e a r  time-varying systems; 
among these i s  the  w e l l  known " c i r c l e  c r i te r ion"  [l] and i t s  var ian ts .  
The &,jezt of t h i s  note i s  t o  discuss t'nrough examples some of 'the 
r e s u l t s  which a re  of an essent ia l ly  d i f f e ren t  nature from the  c i r c l e  
c r i t e r ion .  The examples chosen are  of second order but they a re  
r a the r  i l l u s t r a t i v e  of some of t h e  advantages and d i f f i c u l t i e s  of t he  
techniques discussed. 
Example 1: Consider the  system, f irst  discussed by Marcus and 
Yamabe [2] 
- 
1 - a s i n  t cos t 1 2 -1 + a cos t 
2 
-1 - a s i n  t cos t -1 + a s i n  t 
I 
with a > 0. The s t a t e  t rans i t ion  matrix f o r  t h i s  system i s  
X ( t )  = 
2 
eigenvalues of t he  system are  given by 
a - 2 t  P- a - 4  
%,2 = 2 
and are  time invariant;  they l i e  i n  the  left-hand plane f o r  a < 2. 
Example 2: The second order equation 
2 + 26% t [l+f(t) Jx = 0 
t 
with 6 > 0, If(t;)l < 03, and E ( f ( t ) ]  = l i m  - 1. I f(7)d.t = 0 i s  a 
t+ U F O  t 
0 
simple gneraliza,tion of the  damped Mathieu equation. 
Example 3: Consider t h e  simple modification of the  above equa- 
t ion 
j ;  + 2g(t)S + f ( t ) x  = 0 
with 0 < g 5 g ( t )  5 and 0 < a! 5 f ( t )  - 2 i ( t )  I p. -
These th ree  examples const i tute  t he  nucleus of the  following 
discussion. 
3 
S t a b i l i t y  Results 
There seem t o  be two possible viewpoints f o r  t he  determination 
of s t a b i l i t y  conditions for  time-varying systems. One of these 
viewpoints i s  t o  i n s i s t  t h a t  the "frozen" system be s t ab le  f o r  a 
r m g e  n f  var ia t ion  of the time-varying parameters and then t o  fur ther  
r e s t r i c t  t h i s  range and perhaps impose bounds on the r a t e  of change 
of these parameters t o  obtain su f f i c i en t  conditions f o r  s t a b i l i t y .  
This i s  the  viewpoint which leads t o  the  c i r c l e  c r i t e r ion .  It i s  
self-evident  t h a t  such a viewpoint can be, i n  some cases, extremely 
r e s t r i c t i v e .  S t a b i l i t y  i s  a concept which demands observation f o r  
a l l  subsequent times a f t e r  t he  s t a r t i n g  time, and it i s  not d i f f i c u l t  
t o  conceive of a system which is  a l t e rna te ly  frozen time s tab le  and 
frozen time unstable but which i s  asymptotically s t ab le  i n  t h e  sense 
of L i apunov . 
The second possible viewpoint i s  t o  consider t he  time-varying 
system as a perturbation on a n  "average" autonomous system. If t h i s  
average system i s  asymptotically s t ab le  then f o r  s u f f i c i e n t l y  small 
perturbations,  i n  some appropriate sense, it i s  t o  be expected t h a t  
t h e  time-varying system w i l l  be s tab le .  A s t a b i l i t y  r e s u l t  of t h i s  
type i n  which t h e  averaging i s  done over the  time has been obtained 
i n  [ 3 ] :  
Theorem: The time-varying system k = A(t)x, A ( t )  bounded, i s  
almost surely asymptotically s t ab le  i f ,  f o r  some pos i t ive  
d e f i n i t e  matrix 8 and some f > 0 
4 
t 
lim - 1 X,,[A' ( T ) + B A ( T ) B - ~ ] ~ . T  5' -E .  
0 
t +aJ t-to t 
Furthermore, f o r  su f f i c i en t ly  l a rge  times 
xt ( t )Bx( t )  5 X I  ( to)Bx(to)e 4(t-t0) 
This theorem represents  an attempt t o  d r a w  information from the  be- 
havior of t he  eigenvalues of  A ( t ) .  
values of A ( t )  do not ca r ry  a g rea t  dea l  of information; on t h e  
other  hand, t he  above theorem indica tes  t h a t  the  "average" i n  t h e  
time sense of t h e  eigenvalues of A f ( t )  + R A ( t ) B - l  do. Naturally,  
t he  above theorem only gives su f f i c i en t  conditions and i s  not con- 
s t r u c t i v e  i n  t h e  sense t h a t  nothing i s  sa id  about matrix B. It 
should be noted t h a t  t h i s  r e s u l t  includes the  familiar s t a b i l i t y  
c r i t e r i o n  t h a t  t h e  eigenvalues of A ' ( t )  + A ( t )  be negative. 
Example l shows t h a t  t h e  eigen- 
This simple theorem, i n  the case of Example 1, y ie lds  t h e  b e s t  
poss ib le  result by l e t t i n g  the  matrix 
t h e  number E: can then be taken as 2 - 2a and then it becomes 
necessary f o r  s t a b i l i t y  t h a t  a < 1. From the  t r a n s i t i o n  matr ix  it 
i s  c l e a r  t h a t  
B be the  i d e n t i t y  matrix; 
5 
while the  theorem yie lds  x' ( t ) x ( t )  5 x' (0)x(0)e2(a-1)t f o r  suf-  
f i c i e n t l y  la rge  times. 
Perhaps of more i n t e r e s t  is t h e  appl ica t ion  of t h i s  theorem t o  
Example 2 i n  standard phase space. I n  t h i s  case l e t t i n g  
an elementary computation yields  t h a t  E can be taken as 
2 
E ( f  } < .which y ie lds  as a s u f f i c i e n t  condition f o r  s t a b i l i t y  t h a t  
4e2. 
t i o n s  f ( t )  
per iodic  square wave of height 25 - 6, f o r  some s m a l l  pos i t ive  6. 
However, f o r  
s t a b l e  when 
It is  i n t e r e s t i n g  t o  note, i n  t h i s  case, t h a t  among the func- 
f o r  which the  system i s  asymptotically s tab le  i s  an odd 
5 > 3 and such a wave the  system i s  not frozen time 
f ( t )  = -25 + 6. 
It i s  a l so  of i n t e r e s t  t o  compare t h i s  r e s u l t  t o  two recent  ones. 
Michael [4] has considered by the Liapunov method the  equation 
2 P + 3% + (w +p cos 2 t ) x  = 0 
6 
and in this notation, after an appropriate time scaling, the above 
result yields as a sufficient condition for asymptotic stability 
that 
rather than the much more meager and more complicated result obtained 
in [4]. 
made of averaging of the coefficients. 
published a most interesting stability condition which depends only 
on eigenvalue behavior. 
stability for any a > 0 in the case of Example 1 and yields the 
rather restrictive sufficient conditions 
The reason for the improvement is evident: heavy use is 
Davidson [ 5 ]  has recently 
However, his method does not guarantee 
a 2  10E2 -E < 1 + f(t) < - 9 9 
in the case of Example 2. 
This last computation is clearly more in consonance with the 
viewpoint of imposing bounds on the variations of the time-varying 
parameters than on the averaging indicated by the quoted theorem. 
Along this vain, Brockett [6] has considered the system 
j ;  + 2: -t f(t)x = 0. (4) 
7 
Brockett 's  s t a b i l i t y  r e s u l t s  can be extended t o  t h e  system of 
Example 3 considered i n  the  form 
2 = y - 2g( t )x  
i = - [ f ( t ) -2k( t ) ]x .  
It i s  e a s i l y  seen, r e fe r r ing  t o  Figure 1 and t o  the  imposed bounds 
0 < - g 5 g ( t )  5 El 0 < a 5 f(t) - 2 & ( t )  5 By t h a t  t he  worst possible  
t r a j e c t o r i e s  i n  t h e  sense of s t a b i l i t y  t h a t  can be achieved are  
given by the  autonomous equations 
k = y - 2gx 
9 = -w 
i n  region I and i n  region I1 by 
k = y - 2gx 
$ = -px 
and by extensions symmetric about t h e  or ig in .  Considering the  t r a -  
j e c t o r y  AB emerging from (1,O) and t h e  t r a j e c t o r y  ED passing 
through 
s t a b i l i t y  of  t h e  so lu t ions  i s  t h a t  
(-l,O), it i s  c l e a r  t ha t  a s u f f i c i e n t  condition f o r  
8 
and for  asymptotic stability that the strict inequality hold. 
simple but somewhat tedious computation of the solutions of these 
equations yields that the above condition becomes 
A 
For 
This approach is similar to one followed by Aggarwal [7] fo r  non- 
linear second order systems. 
g(t) 5 1 this condition reduces to the one given by Brockett. 
References 
[l] Brockett, R. W. and L. J. Forys, "On the  s t a b i l i t y  of systems 
containing a time-varying gain", Proceedings of 2nd Allerton 
Conference on Ci rcu i t  and System Theory, 1964. 
[2! Marcus, L., Yamabe, H., "Global st.abilit,y c r i t e r i a  f o r  d i f f e r -  
e n t i a l  systems", Osaka Math. J., vol.  12, pp. 305-317, 
1960. 
[ 3 ]  Infante, E.F., "On the  s t a b i l i t y  of' l i n e a r  non-autonomous random 
systems", JournKL of Applied Kechanics, vol. 36, no. 1, 
March 1968, p. 7. 
[4] Michael, G. J., "Expl ici t  s t a b i l i t y  cr i ter ia  f o r  damped Mathieu 
equation", IEEE Transactions on Automatic Control, vol. 
AC-12, no. 3, June 1967. 
[ 5 ]  Davidson, E. J., "The s t a b i l i t y  of an nth order nonlinear t h e -  
varying d i f f e r e n t i a l  system", IEEE Transactions on Automatic 
Control, vol.  AC-13, no. 1, February 1968. 
[ 6 ]  Brockett, R. W., "Variational methods for s t a b i l i t y  of periodic 
equations, Di f fe ren t ia l  Equations and Dynamical Systems", 
(Proceedings of the  Internat ional  Symposium held a t  the 
University of Puerto Rico, Mayaguez). Academic Press, 1967. 
[ 7 ]  Aggarwal, J. K., "Nonlinear osc i l la t ions :  amplitide bounds f o r  
second-order systems", Journal of Franklin Ins t i t u t e ,  
vol. 282, no. 1, July 1966. 
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FIGURE I 


Some remarks on the stability of time-varying systems

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