A method was developed for determining the stability of a scalar neutral equation with constant coefficients and constant time delays. A neutral equation is basically a differential equation in which the highest derivative appears both with and without a time delay. Time delays may appear also in the lower derivatives or the independent variable itself. The method is easily implemented, and an illustrative example is presented

Barker, L. K.

Whitesides, J. L.

English

NASA Technical Reports Server

STABILITY OF NEUTRAL EQUATIm 
WITH COWTANT TlME DEIAYS 
L. Keith Barker 
NASA Langley Research Center 
John L. Whitesides ’ 
Joint  I n s t i t u t e  for  Advancement of Flight Sciences 
The George Washington University 
A method has been developed for  determining t h e  s t a b i l i t y  of a scalar 
neutral  equation w i t h  constant coefficients and constant time delays. 
neutral  equation i s  basical ly  a d i f fe ren t ia l  equation i n  which the  highest 
derivative appears both w i t h  and without a time delay. 
also i n  the lower derivatives or the independent variable i t s e l f .  
i s  easi ly  implemented and an i l l u s t r a t i v e  example i s  presented. 
A 
Time delays may appear 
The method 
Ordinary d i f f e ren t i a l  equations with time delays a r e  cal led d i f fe ren t ia l -  
difference equations (ref. 1). Two basic types of differential-difference 
equations a r e  retarded and neutral  equations. The s t a b i l i t y  of the  solutions of 
these equations is  re la ted  t o  t h e  roots of a character is t ic  equation. Generally 
t h i s  character is t ic  equation i s  transcendental and thus has an in f in i t e  number 
of roots. 
A convenient method i s  developed i n  reference 2 for  exadning the  
s t a b i l i t y  of retarded equations with many time delays (not necessarily d i s t i n c t )  
and a scalar neut ra l  equation with one delay. 
is  t o  develop the’bas ic  method of reference 2 for  neut ra l  equations with rzany 
t i m e  delays. 
The purpose of t h e  present paper 
SYMBOIS 
a b c , d  r e a l  constants 
%CS 1 function of s i n  equation (11) 
j’ jy 
i imaginary unit ,  fi 
function of s i n  equation (1.2) 
integer 
757 
https://ntrs.nasa.gov/search.jsp?R=19770003358 2020-03-22T12:10:06+00:00Z
K refers t o  zK 
L(s )  = 0 
L o b )  resulting polynomial with zero delays in  L(s )  
N 
character i s t i c  equation 
highest derivative i n  neutral equation 
p(s 1 function of s i n  equation (20) 
P integer 
Q ( S )  function of s i n  equation (21) 
S 
Isl, 
t 
Q 
*, 
K z 
Lu 
u) m 
complex variable, e + icu 
an upper bound on magnitude of s which sat isf ies  
s = -!J + 5x1 4 L ( s )  = 0, where 
time 
testing function defined in  equation (17) 
scalar flmction of time 
r e a l  numbers 
small positive number 
positive r e a l  number 
specified value of !J 
r e a l  gain constant 
r e a l  part of s 
asymptote of r e a l  part of large modulus roots 
constant r e a l  time delays 
f i n a l  desired value of -rGK 
yaw angle, radians 
imaginary part of s 
an upper bound on u) i n  L(s) = 0, where s = -!J + irU A 
758 
Mathematical notations: 
I I  absolute vab-e or magnitude 
argument 
O+ a rb i t r a r i l y  small posit ive values 
Dots over a symbol denote derivatives with respect t o  time. 
ANALYSIS 
A method is  developed herein for determining the  s t a b i l i t y  of t heneu t r a l  
equation 
N 
c [a .  x(’)(t) + b .  x ( j ) ( t  - z.)]  = 0 
J . j = O  J J 
where 
and x ( j ) ( t )  denotes the  
an # 0, % + 0, O <  2. s~~~ for j = 0,. 17.@. , N - 1, J -  
j t h  derivative of x(t ) .  
The character is t ic  equation associated with equation (1) i s  
j= 0 J J 
It has been shown (ref. 3) that i f  a l l  the roots  
iquation (2)  satisf‘y the  property 
s = Q -f- icu of 
lrhere 
?xponential order as t 4 t ha t  is  
P is  8 posit ive constant, then the  solution of equation (1) is of 
rhere d > 0 i s  a constant r e a l  number and c i s  a rb i t ra ry  on the  
7 59 
in te rva l  (0, 1). 
parts and are  not asymptotic t o  the  imaginary axis, then x ( t )  3 0 
(asymptotically s table) .  
Hence, i f  a l l  the  character is t ic  roots have negative real  
as t 3 m 
If there is  a root of L(s) with positive r e a l  part, then equation (1) 
has a divergent mode and i s  said t o  be unstable. 
Relative S tab i l i t y  
If it can be determined tha t  there a re  no roots of the  character is t ic  
equation with r e a l  parts greater than a specified negative r e a l  number, then 
the solution t o  the  neutral  equation i s  asymptotically stable. 
Relative s t a b i l i t y  for a specified value lu^ of p i n  equation (3)  i s  
indicated herein by the  number of roots of the  character is t ic  equation with 
Q > -!?. 
w2en a l l  the  roots satisfy q < -M < 0, than when there is a root w i t h  
-1-1 < q < 0. Relative s t a b i l i t y  boundaries i n  t‘ne plane of two system pa;nameters 
a re  boundaries corresponding t o  a root w i t h  a = -0. 
For example, the neutralnsystem i s  said t o  be re la t ive ly  more s table  
The s t a b i l i t y  method t o  be presented is  based on determining the  number of 
roots of the  character is t ic  equation w i t h  r e a l  parts greater than a specified 
negative r e a l  number -1-1. The method i s  convenient for determining the  number 
of roots of the character is t ic  equation w i t h  r e a l  parts located between specifie 
negative r e a l  numbers. The approach consists of  separately examining the  
a rb i t r a r i l y  large modulus roots and the f i n i t e  roots. 
are examined by using a simple expression for  t he i r  asymptote; whereas, the 
f i n i t e  roots a re  examined by computing the  magnitude of a complex-valued f’unctio 
on a f i n i t e  interval.  
A 
The large modulus roots 
Large MQdulus Roots 
All roots of equation (2)  must s a t i s fy  the  inequality 
obtained from equation (2). 
the  roots have bounded e. 
It can be shown that since % # 0 and bN # 0, 
Hence, i n  order for the large modulus roots 
(Is I -, m) t o  s a t i s fy  equation (5) 
760 
From equation ( 6 ) ,  ~p becomes a r b i t r a r i l y  close t o  
m bN 
A A 
bN 
A 1  z < T N N = B l n  N 
There are  then no in f in i t e ly  large modulus roots with 
system. It remains t o  examine the number of f i n i t e  roots with Q > -P. 
> -; i n  thenneutral 
N a -
bN 
F in i te  Roots 
+ For zj + 0 L ( s )  has N roots arbitrarily close t o  the  N roots of 
t he  polynomial equation 
and the  remaining roots have a r b i t r a r i l y  large moduli ( ref ,  2).  For 
h zN 4 0’ and 1 ~ 1  > 1 i n  equation (7),  Q, -+ -a < -LL. Therefore, L ( s )  and 
L ( s )  have the  same number of roots wi-th g > -t ( i n i k i a l  r e l a t ive  s tab i l i ty ) .  
Since the  complex roots occur i n  complex conjugate pairs, only roots w i t h  non- 
negative imaginary parts (cu 5 0) are  considered. 
9 
A s  one of the time delays, say z , i s  increased i n  a continuous manner 
with the remaining delays held fixed, fhe f i n i t e  roots of L ( s )  move i n  some 
continuous manner (ref. 2),  generating root  locus curves i n  the  complex root 
plane (s-plane). 
761 
Intersection Points s = -3 + iol, and 
Corresponding Delays 
A root locus curve must intersect  the  -$-line (clashed l ine)  i n  figure 2 
The 
i n  order for  the  number, of roots of L ( s )  with Q > -b t o  change. These 
intersection points (-by w) and the  corresponding values of the  delay Z~ which result  i n  these intersection points a re  discussed i n  t h i s  section. 
change i n  the r e l a t ive  s t a b i l i t y  as a root locus curve crosses an intersection 
point i s  presented i n  the next section. 
For a specific time delay rKY equation (2) can be written as 
where 
3 N j -3 N %(s) = C a .  s + c b .  s e 
J j= 0 j= 0 J 
and 
(12) 
K JK(s) = -bK s 
A At an intersection point s = - ~ . l  + iw, equation (10) is  equivalent, to 
where %(-E, a) = E$(-; + b)., JK(-$y u)) = J,(-G + io))y and 
A 
JK('U> u)) - 7c < arg 57c - (15) 
E$(-& CUI 
It i s  assumed that u) # 0 and HK(-Cy u)) # 0. To handle these special  cases, 
the approach used i n  reference 2 may be followed. Only non-negative values of 
the integer p i n  equation (14) a re  of in te res t  because -rK 2 0 and w > 0. 
762 
Equation (13) gives the points ($, cu) where the root locus curves 
intersect the -$-line i n  figure 2, and equation (14) gives the correspon&hg 
values of T~ which result  i n  these intersection points. In general, the 
values of cu a t  an intersection point must be found by an i teration process. 
The values of cu which may satisfy equation (14) are  restricted to some f in i t e  
interval (0, cyn], where 
equation ( 3 ) .  Also, a use ul bound on the integer p i n  equation (14) is  
obtained as 
i s  an upper bound on 03 determined f r om 9 
where T~ < = 't and cu 5 cum. 
Change in  Number oz Roots 
With Q > -M 
Let N('tK, -$) denote the number of roots of L ( s )  w i t h  g > 0; a t  T~ 
for fixed zj, j # K; and define the testing function 
Then, the following theorem can be used to determine the change in  the number 
of roots of L ( s )  with Q > -0 as zK varies. 
Theorem: 
Let al < LD and a2 > LD be r e a l  numbers for which WK($, al) and WK(+, a2) 
are defined, and such that  there are no other intersection points with 
imaginary parts which l i e  on the interval [al, u2]. Then, for 
arbi t rar i ly  small positive number 
Let ($, cu) be an intersection point with cocresponding delay T~~ 
A 
an 
763 
lWK(-,i?, al) 1 and lWK(-j?, a2 1 a r e  greater than 1 or both less than 1. 
This theorem i s  developed i n  reference 3 by extending the 
method, as refined by Lee and Hsu (ref. 4). 
The theorzm i s  interpreted as follows: 
point, where Y i s  specified and cu is  a root of equation (13). If t h i s  i s  
the  only value of w on the  in te rva l  al 5 w 5 a*, which satisfies 
equation (l3),  then the  change i n  the r%lative s t a b i l i t y  %t the  intersection 
point is  determined by computing ~WK(-U,  ax) !  and lW~(-p, %)io For example, 
from condition 1 of t h e  theorem, if 1 and ~WK( -8, a2) <AI, 
then the  system gains exactly one root with 0 7 -Y; t ha t  is ,  N ( z K  + e, -U)= 
N(ZK, -a) + 1. 
z-decomposition 
kt (-2, w) be an intersection 
Jw,(-~, al) I 
The values of T~ a t  a l l  the intersection points a r e  ordered by 
increasing magnitude to obtain the  change i n  the  r e l a t ive  s t a b i l i t y  as 
increases to i t s  f i n a l  desired value 
delay becomes T~ i n  the  theorem. 
T~ 
7,. A s  each delay i s  varied, t ha t  
Intersection points (-cy w) s a t i s @  equation (131, or IW (-0, w ) /  = 1. K 
In  choosing al and a2 i n  t he  theorem, it is expedient t o  note that 
IWK(-k, w)1 increases as 
A 
p increases for each value of w e ( 0 ,  wm]. 
APPLICATION 
The r e l a t ive  s t a b i l i t y  of the neutral  equation 
where 7 i s  a system gain constant and zK > 0 is  a constant t i m e  delay may 
now be determined. 
damper control system for  an airplane with rudder deflection made proportional 
to the yawing acceleration. 
This equation was used i n  reference 5 i n  examining a yaw 
The character is t ic  equation associated with equation (18) can be writ ten 
as 
where 
764 
With s = -2 + h, equation (19) can be used t o  write 
A 
Now, with 1-1 specified, equations (22) and (23) can be used t o  partition the 
plane of 5 and 'tK in to  different regions as CD > 0 i s  allowed t o  vary. 
The solid lines i n  figure 3 were generated i n  this  manner. Any point on a 
partitioning l ine  or boundary corresponds to a ro&' locus curve intersecting 
the +-line i n  figure 2. 
To examine the s tab i l i ty  condition (stable or unstable) or the number of 
roots with (p > -2 i n  the regions of figure 3, it is usef i l  t o  write 
equation (19) i n  the form 
where 
and 
B 
The i n i t i a l  s tab i l i ty  of equation (24) along the ?-axis (zK 4 0') is 
evaluated by using equations (7) and (91, which became 
and 
L0(s) = (.01024 -I- .l635)s2 f .00704s -I- .250 0 (28 1 
For zK 4 O+ and ? = .04, there is one root with t~ = -.21 and + - CO. 
A s  zX increaszs from 0' w i t h  !f = .04 i n  figure 3, the relative s tab i l i ty  
boundary for -P = 0.5 is intersected. For t h i s  intersection point, it can 
be shown that C L ~  and a i n  the theorem can be chosen as al = 3 and 
a2 = 4. Then, since lWKT-. 'j, 3) f < 1 and IWK(-.5, 4)1 > 1, condition 2 of 
the theorem applies. Thus, the neutral system loses one root w i t h  Q 7 a.5. 
(This  i s  the root which ortginally had Q = -.2L) 
-1-1 = 0.5, there are  no roots with Q > =.3. This  same procedure is  used to 
determine which side of the curves in  figure 3 should be hatched. 
Inside the closed region for 
h 
765 
The hatching convention is  as follows: Passing from the hatched (unhatched) 
side of a boundary l ine corresponding to a particular value of M to the 
unhatched (hatched) side of the boundary results in the gain (loss) of exactly 
one root with Q > -0. 
A t  the point ( 5 ,  T ~ )  = (.04, .2), the system has no roots with Q > -.7. 
The value of Q, i n  equation (27) a t  th i s  point is Q = -2.257. 
W 
COJ!TCLUDING REMARKS 
A method has been developed for determining the s tab i l i ty  and relative 
s tab i l i ty  of scalar neutral equations, with constant coefficients and constant 
time delays. The approach was t o  determine the nlwiber of roots of the 
characteristic equation with r e a l  parts greater than specified negative r e a l  
numbers. The method consists o f  separately examining the large modulus roots 
and f in i t e  roots. 
expression for their  asymptote; the f in i t e  roots are examined by computing 
the magnitude of a complex-valued f'unction on a f in i t e  interval. 
The large modulus roots are examined by using a simple 
The s tab i l i ty  method i s  convenient for determining the number of roots of 
the characteristic equation with r e a l  parts located between specified negative 
r e a l  numbers. An example which has occurred in  practical application has been 
provided to i l lus t ra te  the method. 
REFERENCES 
1. Bellman, Richard; and Cooke, Kenneth L. : Differential-Difference Equations. 
Academic Press, Inc., 1963. 
2. Barker, L. Keith: Stabi l i ty  and Relative Stabi l i ty  of Linear Systems With 
D. Sc. Thesis, George Washington Univ., 1976. Many Constant Time Delays. 
3. Miranker, W. L.: Existence, Uniqueness, and Stabi l i ty  of Solutions of 
Systems of Nonlinem Difference-Differential Equations. 
Mech., vol. 11, 1962, pp. 101~108. 
J. Math. 
4. Lee, M. S.; and Hsu, C. S.: On the z-Decomposition Method of Stabi l i ty  
Analysis for Retarded Dynamical Systems. SIAM J. Contr., vol. 7, no. 2, 
May 1969, pp. 242-250. 
5. Beckhardt, Arnold R.: A Thearetfcal Investigation of the Effect on Lateral 
Oscillations of an Airplane of an Automatic Control Sensitive to Yawing 
Accelerations. NACA TN 2006, 1950. 
766 
‘N 
Figure 1.- R e a l  par t  of large modulus roots.’ 
io 
# 
S-PLANE 
A 
-P 0- cT 
Figure 2.- I l l u s t r a t ion  of intersect ion point. 
767 
A 
$, 
rad 
rad / sec2 
Figure 3. - Relative s t a b i l i t y  boundaries. 
768 I 


Stability of neutral equations with constant time delays

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