278
IEEE TBAWSXCTlONS ON AUTOhL4TIC COSTROL, APRIL
1970
Comments on “On the Optimal Angular Velocity
Control of Asymmetrical Space Vehicles”
Abstract-The purpose of this correspondence is to explain
the surprisingly simple feedback laws derived by Debs and Athans
in their paper,l and to indicate a broader class of systems that
can be solved by this approach.
then 1; el exp (ai)11 < m and e, exp (at) 11 < m .
Two useful lemmas which can be easily derived are stated first.
I. S ~ X L ROF
Y THE RESULTS OF DEBSA X D ATEL~SS~
The state equations of the system are
Lemma 1
Let. P be any positive (strongly positive) operator; let T be a
time-varying gain whose derivative exists almost everywhere. Then
rP is positive (strongly positive) if T is posit.ive and nonincreasing
dmost everywhere.
21 = alZ?za
Lemma 2
Let Q be a positive (strongly positive) operator; let r be a timevarying gain. Then rQr is posit.ive (strongly positive).
PToof: Let x E IT,,,. Referring to Fig. 3, let 11f = z(j w )
~ j w
where conditions 1) and 2) hold. Now, the inputs to the
system are in
L.H I k strongly positive as per (3) and Lemma 1. X * ( j ~ =)
M (j w
a) and hence, MI, also belongs to the class of allowable
multipliers for AT,,,[SI. Hence, H? is positive from Lemma 2. Thus,
from a well-knom theorem of Zames [l], and €2 are in Ls. From (2)
it follows that el and e2 are in L a .
The result for N E N o , can be established similarly using t,he corresponding mult.iplier. These resdts can be improved by extending
the interval of definition of the mult,iplier from [0, m ) to ( - Q , m )
~41.
+
+
= a223X1
53
= afl12s
+
JI
=
+ + X?) + (l/q)(ul2+ US’ +
2
(~(zI’ 5 2 ’
l[
where p
[fa2)
1 dt
1 ) H ( s - j3) is strictly positive real (SPR);
2) H ( s ) [ G ( s - a) 1 / K ] is SPR;
3) f I
2Bf(l - f/K).
up = -px2,
u1 = -qx1,
u3
=
(3)
-qx3.
If the performance index is
JZ =
[
IpCf~(xd+A(m)Cf3(23)1
fk(0)
u1 =
-qh1(21),
I21
I31
u2 = -qhr(Zd,
h t ( 0 ) = 0, h,t-l(.) exists,
> 0.
(5)
u3 = -ph3(xa)
> 0,
hk(~l;)~l;
fk(2d =
h’
hk(2)
n’[(-ut/p)hk-l(-Uk/Q)
x1
#
0, X.
=
1,2,3
ax
- fk(hk-’(--lLk/Q))l.
11. EXPLANATION
OF THE RESULTS
The results of the optimization problem considered in Section I
become obvious if the corresponding optimization problem of a n
equivalent system is considered. Thesystem equivalent to (1)
for the performance indices considered is the simple decoupled
linear system giwn by
x 1 = u1
x 2
[41 - “Stability. conditions for svstems with- monotone and slope
resJr&Jed nonlinearities,” S I A M ‘ J . Control, vol. 6, no. 1, pp.891U8, 1 Y 6 8 .
[5] M . Gruber and J. L. W i e m s , “On a generalizationof the circle
crit,erion,” PTOC.4th Ann. Allerton Conf. Circuzt and System Theory,
196% nn
_=. 837--RA!?
161 M . I. Freedmanand
G. Zames,“Logarithmicvariationcriteria
S I A M J.
forthestability
of systemswithtimevaryinggains,’’
Controt, vol. 6, pp. 48‘7-505, August 1968.
Q
where hl;(- ) are continuous and differentiable scalar valued functions of a single variable such that
H. S. RXYGANATH
Dept. of Elec. Engrg. and
Indian Institute of Science
gl;!ut) =
Bangalore 12, India
1
.11. G. Zames. “On the inDut-outoutst,ahilit.v of time-xrarrinenonlinear feedback s y s t e m , p t . I < c o n d i t i ~ n s ~ d e r i ~ . e d uii&ip:s
sin~
of loop
gain,
conicit,y,
and
positivity
IEEETrans.
Automatic
Control. vol. AC-11, pp. 228-238, April i966.
- “On theinput-output
stabllay oftime-varyin*nonlinear
feeddack systems,pt. 11: con$tionsinvolvingcirclesoin
the freauencv Dlane andsectornonhnearit.ies.”
IEEE Trans. Automatic
k = 1,2,3,
gt(0) = 0,
= 0,
The control law which minimizes ( 4 ) is
M. A . L. THATHACHAR
REFERENCES
(4)
where ft(-) and g l t ( - ) , k = 1,2,3, are positive definite functions
of a single variable such that
+
PTooj: Fig. 3 reduces t.0 a linear system G(s - a) f 1;‘K with a
time-varying feedback gain ( f ( t ) ) / ( l - f ( t ) / K ) when N E N t .
The Theorem follows from earlier results [5] recast in a functional
setting. Some recent results of Freedman and Zames [6] can also be
used here. These involve a v e r e e logarkhmic variation criteria on
j ( t ) in contrast t o condition 3).
(2)
> 0, the optimal feedback control law is
+ (l/dcg*(Ul)+ gn(u2) + 93(~”)11dt
The system (1) with 11’ E Nt is exFonentially bounded with order
if there e.nists an H ( s ) such that
-_--. --. ---.
(1)
a3 = 0.
t CYIhf a2
Corresponding to a performance index which is to be minimized,
given by
~
Themem 2
a
x2
+ u1
+ US
+ u3
=
u2
53 = ua.
Manuscript received October 9, 1969.
1-4. S. Debs and M. bthans, I E E E Trans. Automatic Control (Short
Papers), vol. AC-14, pp. 80-83, February 1969.
279
CORRESPONDENCE
The nonlinear terms have no effect on either the optimal control
or the minimum cost of the system.
Considering the system ( 6 ) with performance indices ( 2 ) and
( 4 ) , the corresponding Hamilton-Jacobi equations can be easily
solved. In both the cases the gradient of the optimal cost function
V , (z) is given by
v z (2) = (51,zZ,z3)
(7)
has characteristicroots
the equation
-3 and -4,
A’P
5 =f(z)
+u
(8)
where z 4 (zl,zq,za)’and u
(u1,Uz,?&)’, and if i t were to have
the same optimalcost function V ( s )and the samefeedback control
law, the term due to f ( z ) in the Hamilton-Jacobi equation should
vanish. This requires
(f(z),Vz’)
= 0.
(9)
Due to a lemma on inner products
f(z)
[a],
(8) implies
p
QVz
0
.-I
= 1,
+ P A + 2uQ = 0
=
I[
26 -81
21
-8
11
from which it is easily seen that Q - P is not positive definite.
The error in Man’s proof lies in the incorrect statement that since
A
UZ is asymptoticallystable, ( A
uZ)‘Q &(A
ul) is
negative definite for any symmet.ric positive definite matrix Q.
+
+
+
+
S. BARNETT
School of Mathematics
University of Bradford
Bradford, Yorkshire, England
(10)
Theauthor is indebted to Dr. Barnett for e-xhibiting by a
counter example a flaw in the proof of t.he Theorem.’ It had been
overlooked that the symmet.ric matrix
S
-azz1
Q
Aulhor’s Reply2
where Q is a skew-symmetric matrix. It can be easily verified that
for the system (1)
L-a122
u = 2,
has solution
’.
If the nonlinear system (1) is represented by
andtaking
=
(A
+ uI)‘Q + Q ( A + uZ)
isnot necessarily negative definite for any symmetric posit.ive
definite matrix Q. However, S can be shown t o be negative definite
forsome subset of Q. This set is nonempty since, according .to
Lyapunov’stheorem, ( A
uZ) is a stable matrix if and only if
there exists a symmetric positive definite mat.rix Q such that S < 0.
If, for example, in Barnett’s counterexample Q is taken as
+
Lz3J
+ +
as al a4 a3 = 0. Systems (1) and ( 6 ) thus belong to an equivalence class as defined by Liu and hake.*
By choosing several skew-symmetric matrices whose elements
are functions of z, a whole class of nonlinear systems for which the
given control laws are optimal with respectto thegiven performance
indices can be constructed.
Q
=![
-34
- 341
1
S is computed to be
111. CONCLUSIOXS
It is shorn that the optimal control of the system of nonlinear
equations of an asymmetrical space body is equivalent to thatof a
system of decoupled linear equations, and this explains the simple
feedback laws obtained by Debs and Athans.
”.
which is negative definite. Furthermore, P is given by
68
-50
-50
SARhPA
from which P - Q is seen to be negative definite as specifled by
Dept. Of
the Theorem.’ Thus, t.he Theorem’ is still t.rue if the phrase “where
IndianInstitute Of Science Q is any symmetric posit,ive definit.e matrix” is amended to “where
Bangdore 12, India Q is some symmetric posit.ive definite matrix.”
It is desirable, however, to obtain a result which holdsforall
2.R. W. Liu and- R. J. Leake. ‘‘ExhaFtire equivalenceclasses of
optlmal
systems
mlth separable cont,rols, S I A M J. Control,
4,
symmetlic positive matrices Q . The f o l l o ~ n gproposition thus
no. 4, pp. 678-685, 1966.
a.rises.
1701.
Proposition 1
There exists a symmetricpositive
Comments on “A Theorem on the Lyapunov
Matrix Equation”
The sufficiency part of a recent t.heorem by &Ian1 does not hold,
as the following counterexample illustrates.
The matrix
A
=
[-:I:]
Manuscript received October 23, 1969.
1 V. T. Man. I E E E
Trans. Automatic Control (Correspondence).
vol. AC-14, p. 306, June 1969.
A’P
definite mat.rix P satisfying
+ P A + 2uQ = 0
where Q is any symmetric positive definite matrix and u is some
positive scalar, for which
P < Q
if and only if
+
+
a) the symmetricmatrix ( A
A‘
2 ~ 1 iq) negative definite,
and
b) the real parts of all the eigenvalues of the matrix ,4 are less
than -u.
Manuscript received Sovember 10. 1969. This work mas support.ed
by the National Research Council of Canada under Grant 8-4396.
f