Absolute stability notion extended to include parameter variations of linear part of syste

Siljak, D.

English

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ABSOMlTE STABILITY AND PARAMETm SENSITIVITY 
D. h J A K  
University of Santa Clara 
Santa C l a r a ,  California i t  
ABSTRACT 
E\ 
This paper extends t h e  notion of absolute s t a b i l i t y  t o  include 
the  parameter variations of t he  l inear  part of the  system. 
analytic procedure is proposed t o  calculate the  regions of absolute 
A simple 
3 s t a b i l i t y  i n  the  parameter space. Then, a parallelpiped of max'imum 
volume is  imbedded i n  t h e  region t o  interpret  Its boundardes and 
obtain readily t h e  information about parameter variations which do 
not affect  t he  system s t ab i l i t y .  
"c 
i 
INTRODUCTION z09 mod AlllI3Vd 
Stabi l i ty  and sei .si t i \-by are two essential properties of dynamic 
control systems. Whiie s t ab i l i t y  assures a proper .'unctioning of the 
system, the  sen is t iv i ty  indicates t h e  a t i l i t y  of t h e  system t o  re ta in  
required performance characterist ics despite chang s **n the  operating 
conditions. These changes may occur due t o  the  fac t  t ha t  the 
*The research reported herein was supported by the National 
and Space Administration under the grant no. IVGR 05-017-010. 
https://ntrs.nasa.gov/search.jsp?R=19680012971 2020-03-12T09:01:01+00:00Z
2 
of physical systems deviate from t h e i r  nominal VRlues ei ther  because 
of inaccuracies i n  the  system cmponents (timebinvariant case) , or 
because the system parameters vary i n  t i m e  (time-varying case). 
fore,  a simultaneous consideration of s t a b i l i t y  and parameter sens i t iv i ty  
There- 
I '  
i n  system analysis is desired. 
The L u r ' e  absolute s t ab i l i t y  concept [l] and t h e  related cr i te r ion  
of Popov [21 are significant contributions t o  s t a b i l i t y  analysis of 
dynamic systems. 
i s  meaningful i n  a large c lass  of closed-loop control systems, and the 
Popov cr i te r ion  provides a simple procedure t o  conclude tha t  kind of 
This is mostly because the absolute s t a b i l i t y  concept 
j 
s t ab i l i t y .  
In the  absolute s t a b i l i t y  analysis, the  nonlinear character is t ic  
is not completely specified and it should only belong t o  a cer ta in  de- 
fined c lass  of f'unctions. On the other hand, t h e  parameters of the  
l inear  part  are specified numerically. This paper proposes an absolute 
s t a b i l i t y  definit ion which w i l l  relax the  conditions on-the l i n e a r  part 
and allow system parameters t o  deviate from t h e i r  nominal values. 
Then, a simple analytical  procedure based upon the Popov cr i te r ion  is 
presented t o  determine i n  t h e  parameter space the  region of parameter 
deviations which do not violate t he  absolute s t ab i l i t y .  
A graphical procedure fo r  evaluation of t he  absolute s t a b i l i t y  
regions i n  the parameter plane was given i n  reference [SI. Under 
cer ta in  conditions tha t  technique which is based upon the  envelope 
. 3 
cr i ter ion can also be extended t o  considerations i n  the parameter space. 
ABSOLUTE STABILIm I N  THE P m  SPACE
The Problem of W ' e  [l] is formulated far a class of closed-loop 
control systems described by t he  equations 
(1) T x = PX i q + ( u ) ,  u = r x 
where x, q, r are real n-vectors, P i s  a r ea l  n x n matrix, the  pair  
(P,q) is  completely controllable, and + ( a )  is  a real continuous scalar 
function of the real scalar u such that it belongs t o  the  class  
AK : + ( O )  = 0, 0 u+(u) K U ~  . One asks: Is the equilibrium 
state of the system (1) asymptotically stable i n  t h e  large for  any 
+(a) € AK , i.e., is the  system absolutely stable. 
The most important solution of the Problem of Lur'e was given by 
Popov [2] i n  terms of t h e  frequency character is t ic  
which is t h e  transfer function of the  l inear  part of t h e  system (1) 
from t h e  input 4 t o  the output (-a), and X = 6+ Jw i s  the  complex 
variable. 
t h a t  if +(u) E AK and a l l  the roots of 
half-plane Re X < 6 <Os and if there  is a real number u such 
t h a t  a Popov type inequality 
Yakobovich [4 ]  generalized the results of Popov and proved 
IP - X I 1  = 0 are i n  the  
- 
is satisfied, then there ex is t  positive constants p and E such that, * 
fo r  any solution x ( t )  of (1) and any t t one has 
Ix( t ) l  2 plx( to) l  
il - 0' 
expE(6 - c ) ( t  - to)] 
I 
Yakubovich 141 also treated the forced system 
2 where +(a)  e A,' Cb(0) = 0 ,  0 a+(a) 
f ( t )  is a bounded function on the interval  (a, +-), and shoved tha t  
a modification of ( 3 ) ,  
KO , 0 a+' (a) < #co2 , 
assures tha t  there is a unique bounded solution 
(--, +.PI and t h a t  for  any x ( t )  and t - > to, one h a  
I x ( t )  - x o ( t ) J  5 p Ix(to) - xo(to)l 
paper [ 41, Yakubovich t reated t h e  discontinuous functions $(a)  and 
showed t h a t  the absolute s t ab i l i t y  is based upon the  s&e inequalit ies 
( 3 )  or ( 5 ) .  
xo( t )  of (1) on 
exp[(d - e ) ( t  - t o ) ] .  In the  same 
I n  application of the system (11, the  l inear  part of the system 
contains parameters which may deviate from t h e i r  nominal values. 
it is necessary t o  relax t h e  conditions on the  l inear  part of the system 
and allow these parameters t o  vary i n  some neighborhood of t h e i r  nominal 
values while preserving the absolute s t a b i l i t y  of the system. 
Then, 
Let  UB assume tha t  t he  transfer f'unction x(X,  p1,p2, , pI is 
,/' 
-- 
I 
- 
. A  
. .- " - .  . . . . 
5 
a function of X and L parameters (pl, p2, ..., pk), and l e t  us 
suppose tha t  the  solution x ( t  , pl, p2, . . . , p,) , of (1) 
[SI for parameter values i n  a certain region R 
euclidian space 
s t a b i l i t y  for  system (1) can be reformulated t o  include the parameter 
is  well-defined 
of t h e  L-dimensional 
. I  
(pl,p2, ,pL) . Then, the  def ini t iawof absolute 
variations. 
The equilibrium s t a t e  x = 0 of t h e  system (1) is said t o  be 
absolutely stable i f  it is asymptotically stable i n  the  large fo r  any 
When the  system (1) is specified, one ' i s  interested eo find: (a) 
The greatest value of K and the largest  region R;  (b) A value of IC 
is given and the  largest  region R i s  t o  be determined. A graphical 
solution of these problems was given i n  [3] where the region 
determined by the  envelope cr i ter ion as the  largest  se t  { ( p1 ,p2 , ,pQ) d 
R w a s  
RI -H > 0 ,  vw - > 0 1. 
I n  t h i s  paper, a simple analytical solution is  presented which 
first yields the  region R i n  terms of a set of algebraic inequalit ies 
involving parameters. Then, a rectangular parallelepiped of maximum 
volume i s  imbedded i n  the  region t o  yield a convenient interpretation 
of the absolute s t a b i l i t y  region i n  the parameter space ( t h i s  i n t e r -  
pretation technique w a s  proposed by George [6,7] for approximation of 
f i n i t e  regions of asymptotic s t ab i l i t y  and l i n e a r  system analysis) ,  
Assume the t ransfer  function of t he  l inear  part t o  be a rat ional  
function of the complex variable X , 
7 Ck ak 
. k=O 
bk Ab 
k=O 
n > m  
il 
i n  which the  coefficients bk and ck are real functions of t he  
parameters pi( i  = 1, 2, ..., 11) . men, l e t  us express 
a k = K +  j rk 
where A = 6+ j w  , and 
Functions 9 and Yk can be easily calculated using the recurrence 
- 2X1Yk + formulas : \+l - 2X1% + (xl + y1 = 0, yk+l 2 2 
+ (xl 2 + Y1 2 )Yk-l,= 0 ,  xo = 1, x1 - = 5 ,  Yo = 0, Y1 = w 0 
When 6 is specified i n  an absolute s tab i l i ty  problem, and (71, (8) 
are substituted in ( 3 )  or ( 5 ) ,  one obtains 
2n 
k=O 
where the  coefficients 
the parameters. For convenience, i n  (91, 1 / ~  and u of ( 3 )  are 
considered as parameters. Note that  u is not a physical parameter 
and only i ts  existence is required such that 
ak = $(pl, p2, . , p,) a re  real f'unctions of 
n > 0, v w 2 0 . 
From (g), one can readily conclude tha t  the system (11, or ( 3 ) ,  is  
absolutely stable if  t h e  corresponding polynomial w has no posit ive 
7 
real roots. For t h i s  t o  take place, it is suff ic ient  that t h e  following 
set  of algebraic inequalit ies 
ir 
a > O ,  %LO, ( k = 1 ,  2, ..., 2n) 
0 
(10) 
J 
is satisfied. 
For example, if t h e  t ransfer  function 
2 
3 + P2A + P 
X b ,  P1,P2’P3) = pl( X + 1 )  ( X+2 ) ( X+3 I’ 
IC = 1, and 6 = 0 ( A  = J w )  are specified, one obtains (9) as 
Inequalit ies (10) are  
(11) 
B O  6p1 + ?3 - 
49p1 + 1lp2 - 6p3 - 6 - > O 
p2 + 6 > o 14p1 - - 
PI ’ 0 
which determine the  boundaries of E . 
--  
Inequalit ies (10) specify a region R(R c R) of absolute s t a b i l i t y  
i n  the  parameter space which may appear t o  be an overly s t r i c t  region since 
(10) are only suff ic ient  conditions for  
(101, however, lead t o  a convenient interpretation of the s t a b i l i t y  regions. 
f > 0,’v w - > 0 . Conditions 
. 
INTERPRETATION PROCEDURE 
0 
After the inequalities (10) are specified, athe problem of using 
them in practical problems is essentially one of interpretation. Since 
the practical problems may involve more than two parameters, an I 
interpretation procedure for multi-parameter analysis is desired. 
In general, to interpret the absolute stability region, let us 
imbed a parallelepiped II into the convex region R determined by 
inequalities (10) which has sides perpendicular to the coordinate axes 
of the parameter space (pl, p2, . . , p,) 
stable point M(p 1, p2, ... , p,) . Let the volume v of II be 
defined as 
- 
and center at the known 
- -- - 
low, 
(10) 
the function v 
separately considered as a constraint. Thus, a constraint 
should be maximized with respect to each inequality 
may be represented as 
Substituting (16) into (14) and extremizing, a necessary condition for 
(p2 , p2 , ..., p O )  to occur at a maximum of v is that it be a 
solution to 
0 0 
1c 
av 
- I  0 ,  ( i  = 2, 3, 0 . 0 ,  L) 
C I .  
9 
Standard suff ic ient  conditions f o r  t h i s  solution t o  be maximal are 
given in E8l. 
0 0  0 '  L e t  the  solutions (p, , p2 , ..., pg , ' (k = 0, 1, ..., 2n) 
occur at maximum value of v subject t o  constraints (lo), then the 
desired parallelepiped ll is given as 
J 
(i = 1, 2, ..., 11) (18) 
Since each vertex point of fl is located i n  R containing the  point 
M(pl, pz, ..., pg),  it follows that the  parallelepiped 11 is . 
completely imbedded i n  E, i.e., JI c TI . 
--  - - 
In case of t h e  above specific example, l e t  us choose the stable 
point g(0.2 ; 0 ; 0 )  . "he volume t o  be maximized is 
Maximization of v with respect t o  the  constraint 
4 g ~ ,  + l l p 2  - 6p3 - 6 = o 
= 0.226 . According t o  p3 = 0.178, p20 = -0.123, P1 yields : 
these values of parameters, the parallelepiped ll is  determined by 
- 0.21 2 0.022, lp21 2 0.123, I pgl 2 0.226 (21) 1% 
One can readily check that a l l  the vertex points of 
of the  constraints of (13). 
interpretat ion problem under consideration. 
n sa t i s fy  the  rest 
Therefore, (21) is t he  solution of the  
It should be noted tha t  some of the constraintsin (10) may not 
contain a l l  the  parameters, as it is c lear  from inequal i t ies  (13) 
Then, some of the  parameters i n  incomplete inequal i t ies  are arb i t ra ry  
and t o  make the  maximization of v meaningful, one should consider the 
I 
arb i t ra ry  parameters as constants. 
For example, the  optimization of v i n  (19) with respect t o  the  
+ 6 > 0 of (13) should be performed .with constraint  14p1 - p2 - 
p3 = c 
p1 = -0.122, p2 = 4.292. In applying equation (18) t o  determine the  
parallelepiped ll, these values are discarded and ll is  given by (21). 
(c # 0) Then, t he  maximization of v = 8c(p1 - 0 . 2 ) ~ ~  gives 
0 
In  case of the-varying parameters, by the  arguments of reference [I+ ] 
one can use the  inequality ( 5 )  and prove the  s t a b i l i t y  of e i the r  system 
(1) or  ( 3 ) .  
determined region R (or n) the  system is absolutely s table .  
Then, as long as the  parameters are varied inside the  
- 
- - .  
REF'ERENCES 
11 
L u r ' e ,  A. I . ,  "Some Nonlinear Problems i n  t h e  Theory of Automatic 
Control," ( i n  Russian), Gostehizdat, Moscow'~ 1951. 
t rans la t ion :  
(English 
Her Majesty's Stationery Office, London 1957). 
Popov, V.M., "The Solution of a New S t a b i l i t y  Problem f o r  Controlled 
Systems," ( i n  Russian), Automatika i Telemekhanika, Vol. 24, No. 1, 
January 1963, pp 7-26. 
,.* 
h j a k ,  D. ,  "Popov Inequality via Parameter Plane," Proceedings of 
t h e  F i r s t  Princeton Conference on Information Sciences and Systems, 
Princeton University, Princeton 1967, pp 183-187. 
Yakubovich, V.A., "The Method of Matrix Inequal i t ies  i n  the  S t a b i l i t y  
Theory of Nonlinear Control Systems," Part I - The Absolute.Stabil i ty 
of Forced Oscillations,  ( i n  Russian), Automatika i Telemekhanika 
Val. 25, NO. 7, July 1964, pp 1017-1029. 
Petrovski, I .G. ,  "Ordinary Different ia l  Equations ," Printice-Hall 
Inc . , Ehglewood C l i f f s ,  New Jersey 1966, pp 62-66. 
George, J . H . ,  "A Method f o r  t h e  Interpretat ion of Results Obtained 
by Liapunov's Method," Proceedings of the  1966 JACC Conference, 
University of Washington, Sea t t le ,  1966, pp 841-847. 
George, J .H . , .  "On Parameter S t a b i l i t y  Regions f o r  Several Parameters 
Using Frequency Response," IEEE Transactions on Automatic Control, 
vol. AC-12, No. 2, April 1967, pp 197-200. 
Goffman, C., "Calculus of Several Variables ," Harpey and Row, 
New York, 1965. 
h l j a k ,  D., "Nonlinear Systems: 
John Wiley, New York 1968. 
The Parameter Analysis and Design," 
,./ 
i 


Absolute Stability and Parameter Sensitivity

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