67
TECHNICAL NOTES AhW CORRESPONDENCE
part of the multiplier, and is hence undesirable. It. turns out, however, that such a rest,riction on 6 appearing in Section I11 of [ l ] is
superfluous.
CORRECTION
+ 11'.
(13)
+
+
For el = 0, (13) yields BZ = [ 11 dj) l ] ? 'v 6.5, as compared
with B = 4 from (4),but still conservat.ive compared with the
optimum result 6 = 11.6 obtained by Brockett's variational technique 121.
Beginning with Sect,ion 111-B on mathematical preliminaries, a
major slip inthe integralsl a , 11,etc., of [ 11 is the employment of u ( t ) ,
+( ~ ( t )for
) funct,ions vanishing outside the interval
0 2 t 2 T.Hence,
the first correction tothe lemmas and resultsis:Replace
u(t),
+ ( u ( t ) ) in the integrals lo, 4 , etc., with u * ( t ) , + * ( u ( t ) ) , which are
equal to t.heir nonstarred counterparts in ( 0 , T )and to zero outside
this interval. Once this is accomplished, note from (29) that m ( T )
is identically zero, which implies that the term (35)
is also identically
zero. This leads immediat,ely to the following correct form of Result
1.
Result 1: Let a,,8 be posit.ive constants and zla(s) be as defined in
[ 11. Then the integral
S,
T
REFERENCES
111 A . C. Tsoi and H. M. Power, "Equivalent predictions of the circle criterion
and an optimum quadratic form for a second-order system,"
I E E E Trans.
Automat. Contr. (Tech. Notes and Corresp.), vol. A C l i . pp. 565-566, Aug.
19i2.
[21 R.
,v. Brockett, .dVariational methods for stability of
equations,v9
in Difeerrntial Equations andDvnarnical Sastems. J. Hale and J . P. La Salle.
Ed. New York: Academic.1967,pp. 299-308.
[3]--,"Thestatus
of stabilitytheory
for deterministicsystems,"
IEBE
Trans. Automat. Contr. (Survey
Papers).
vol. AC-11,
pp.
596-606,1966.
July
laln
=
k(t)+*(u(t))( a u l ( t )
t
+ 19 dc*
+ Zlau*(t)
dt
-
is positive [except for a constant, term dependingonly on
for some const.ant €0 > 0,
dk
(r@i)k(t) 5 -
-+n
dt
1.
ff
2 - R(t) B
EO,
dt
0 1 1 if,
V t E [O,m ).
REJZ~RKS
On the Stability of Nonlinear Time-Varying Systems
Y.
17.
VENKATESH
Abstract-concerning the absolute stability of a nonlinear timevarying feedback system, an important correction to the results of
the author [ l ] is presented.
INTRODUCTION
The author's paper [l]gives some new conditions for the absolute
stability of a feedback system governed by t.he set. of equations
x ( t ) = exp (At)x(O) -
exp (A(t -
[0, ). For an explanation of the notation, terminology, and for a st,atenlent of the problem, definitions, and assumptions, see [ l ].
The multiplier employed in [l] to establish absolutestability
condit.ions, when, for instance, +( . ) E P,w, is of the form
+ 6s +
mi
74s
i=l
+
PiPi)/(S
FROH
O'SHEA[2])
The system is asynlptotically stable for all constant. negative(linear)
feedback, regardless of magnitude. W-ith the choice of
z(s) =
(s
(S
+ 0.0001) (-s + 0.1) (s + 40)
+ 0.005) ( - s + 2)
)+
+ Pi)
m?
+i = l
ESLMPLE(ADAPTED
Consider a system with
7 ) ) b k ( ~ M ( u ( 7 d7
))
on theinterval
4s) = a
1) Byasimilar
argument, the lower bound on /3 appearing in
Resu1t.s2-6
is seen tobe superfluous. Consequently,Theorems
3 and 4, with their corollaries, remain valid wit.hout the restriction
on j3.
2) The upper and lower bounds on coefficients vi, v;', &,and Bi'
of the multiplier employed in Theorem 3 and 4 imply, in essence,
a time domain constraint [of the type appearing in O'Shea, Zames,
and Falb (see [I, references])]on the multiplier. The author has
obtained absolute st.ability conditions for (1) in t,enns
of a multiplier
Gont.aining general causal and noncausal functions with a certain
time domain const.raint . These d l be report,ed elsewhere.
+
+
(2)
where constants 4 , P i L i > 0; l]ityi 2 0; 0 2 vi, & < 1 forall i;
l[w/{i(l - &)I, which dependsonlyon the noncausal
and 6 >
part of (2). This lower bound on swamps the ett'ect of the noncausal
In-
z2
79.6029
-
which satisfies the conditions of the corollary to Theorem 3, it can be
verified t.hat
Re
z ( j ~ ) G c j w )2 0 for all real W. N0t.e that here gS =
0.004439, VP = 0.001617. Therefore, the systemis asymptotically
stablefor all monotonic nonlinearities if, for some (arbit.rarily
small) posit.ive ea,
-0.004439k(t)
Manuscript received July 20, 1971; revised August 31, 1973.
Theauthor is xvith theDepartment of ElectricalEngineering,Indian
stitute of Science, Bangalore 12, India.
(39.89
dk
5 - 6,,
dt
5 0.001617k(t) - EO,
'#t
E [O,
a).
This constraint on k(t) is apparently very trivial. But it appears
68
IEEE TRANSACTIONS O N AUTOM-ITIC CONTROL, FEBRUAFtY
that no improvement, is possible usingmultipliers of the present.
form. It is ynite like11 that the system becomes unstable for larger
variations of (dk!dt); t.his remains to be demonstrated.
1973
should be finite. It can also be shown t.hat this time is the minimal
time of intercept.
Two restraint sets for the controls are examined here, viz.,
ACRNORZEDGJIEKT
11. A. L. Thathacharand
Theauthor
wishes tothankDr.
M. K. Sundaresan for their critical
remarks
and
on [I].
1u11
5
1,
It%/
I 1,
(8)
i.e., the controls restricted to theunit. hypersphere or unit hypercube.
Carrying out the maximization operations in the first case giva
R.EFEREXCES
111 T . V. Venkatesh.“Soncausalmultipliersfornonlinearsystemstability.”
I E E E Trans. Automat. Conlr.. rol. .\C-l5, pp. 195-204, Apr. 19iO.
121 R.P. O‘Shea,‘*.inimprovedfrequencyrimedomainstahilitycriterion
for autonomousconttnuoussystems.”
I E E E T r a n s . Auiomat. Contr.. vol.
A C 1 2 . pp. 72.5-731, Dee. 1967.
( H e( +HepB+e8)p’ ~I f’* t ‘
II 1HP+PBPI,
u = .
t’ =
II yH&Be II
x
(9)
where the prime indicates transpose. Substibuting in(4),
W ( ~ p a , ~ ~ , y=, t y) ( H p O p r 9
- H.+AI)
+
A Linear Minimal-Time Intercept Problem
- ( ‘ ~ H G M Jd)a .
X R . AIUKERJEE
The minimal-time intercept problem for a pair of liuear systems
has been studied [ l ; . [2] asa linear difl’erential game. The ba.4c
system cnnxists of the pursuer and evader systems, the equations
of motion being given by
+
A e ( t ) l e ( t ) + Be(t)v(t),
= A,(t)TP(t)
t(t)
=
IjP(t)l4f),
r p ( t o ) = Z@
(1)
d t o ) = reo,
(2)
respectively, where rp,rcare n p - and n,-dimemional state vectors
and Z I , C are p - and q-dimensional control vectors of the pursuer and
evader, aud A,, Is,,A,, 8,are continuous time-varying matrices of
appropriate dimensions. An intercept occurs when
y( 1’) =
(10)
A sufficient.condition for intercept is t.hat the second term be positive
for all t. Thus a sufficient condition is that thematrix
Absfrocf-SufEcient conditions for intercept are obtained for a
linear minimal-time intercept problem, with the controllers restricted
to the unit hypersphere and unit hypercube. It is shown that an
expression called an interceptibility matrix can be defined whose
positive definiteness is therequired sufficient condition.
&(t)
(il-Jfp+pBpl~
H p ( T).z,( 7’) - He( T)z,( T ) = 0.
X(t,t+)
( H p @ p B p B p ’ @ p ’ Hp f H e + e B e B k @ : f f e ’d)a
=
(11)
be positive definite for all t . This matrix may be termed an output
interceptibility matrix. If H , He are identity matrices (i.e., if the
dimensions of pursuer and evader are the same), a state interceptibility matrix is obtained, which can be recognized as the difference
between the cont.rollability matrices of the pursuer and evader.
The same result, with some modification was obtained in [ 5 ] and
indicates that the pursuer should be “more“ controllable than the
evader in every Euclidean dimension.
In the second case, using (8),
ui
=
sgtl y ( H p + p B p ) ;
vi =
sgn y(H&Be)i
(12)
where the subscript. denotesthe column. The expression (4) becomes
W ( ~ p O , r c o , ~ ,=
t) ~ ( H p + g p ~He+ela~)
nt
(3)
The objective of the pursuer is to effect the intercept in the minimum
possible time, and that of the evader is to prevent a11 intercept. The
case considered earlier [I], [2] is the special case where y is a scalar.
A more general case is investigated here.
Let be all arbitrary unit row vector. Then it can be shown, by a
modification of a result. obtained elsewhere [3] that a necessary and
sufficient condition for an intercept. to be possible is a.s follows. Let
I t is seen by expanding that.
1
~ ( H p + p B p B p ‘ + p f H-p fHe+eBtBe’+e’He’)y da
-,
W ( X F O , X ~ O , Y , ~=) ? ( H p + p ( f , t o ) r p o
+
l+f
m?
- He@e(t,to)~eo)
-
-,,Hp+,(t,a)B,7c d e
l”
nlax -,H,@.(t,a)B,c d a
(4)
Thus, if (11) is positive definite, the second term of (13) is also
positive for all f. Thus, (11) being positive definite is a sufficient.
cundition for intercept in both cases. The actual optimal strategies
can be determined by obtaining the adjoint variables as indicated in
[21.
and let
h ( ~ p ~ , r , ~=, tmin
) T.V(xp~,r~,y,f).
(5)
I
Then thesmallest positive root o f the equation
X(ZpO,P,O,t) =
Manuscript receired July 18. 1972.
The author is withtheDepartment
of Roorkee, Roorkee, India.
0
of ElectricalEngineering.Gnirersity
REFEREWES
[ l ] T . C. Ha and S. Baron, “A minimal time intercept problem,” I E E E Trans.
Automat. Contr. (Corresp.), vo1. AC-10. p. 200,Apr. 1965.
[2] E. J. Davison and J.
&I. Seugnet, “ A minimumtimeinterceptproblem,”
I E E E T r a n s . Automat.Contr. (Corresp.), vol. AC-15,pp.116-118.
Feb.
1 ~ 7 n
(6)
I31
~~
ai. R.
tion.
Mukerjee, “On linear differential games.” Int. J. Confr.,communica-
[. I.] B. h’. Pshenichnyi, “Linear differentialgames.”
I41
eames.”
Automat. Remote Contr.
(USSR). pp. 55-68, Jan. 1968.
[5] T. C. KO. A . E. Bryson, Jr., and S. Baron, “Difierential games and optimal
I E E E Trans. Aufomaf. Contr.. vol. A C 1 5 ,
pursuit-evasionstrategies,”
pp. 385-389, Oct. 1965.