311
CORRESPONDENCE
1966
Upon substituting (3) and (5) into ( 6 ) , one
obtains D . R (in expanded form) evaluated
on S
Optimal Control with Bounded
Control Effort
The solution of the problem of optimum
controlwith
saturationconstraint on the
control effort proposed by Letovl was found
invalidin
general.z A necessarycondition
for which the linearoptimal control lawmust
satisfy wasgiven by ( 3 5 ) inRekasius and
Hsia.2 The purpose of this correspondence is
topresenta
differentmethod
by which
(35) can be obtained.
Let the boundaries of the linear control
region defined by (12) in Rekasius and
Hsia2 be denotedby S. Rewriting (12) in
the form
U*(Z) =
* 1,
(1)
then S can be represented by, in view of
(32) and (33)z
+
U . S = (-l)n(-/l*/n-l - amyl
+ (-- l)n(Sz-,’n-l- a,-;2 - + u 2 h
y1
x = Fx - GQ(d)
d =
x1 = 7 1
Z3
=
For D.N = 0 everywhere on S, it is necessary
that the coefficients of all 7 ] ’ s in (7) are zero.
Thus, one obtains the condition
G-1
X,
=
- GYI
qtZYn-1
- U*YZ - *;I
axYn-1-
UI
=0
+ uz 0
+ an-1 = 0
Yn-2
P a ,
(-+
- - 1).
kn
1 n-1
2 --
YiTi.
(3)
b
i-1
I t has been statedz that all optimal trajectories inside the linear region should ever
be tangent to the boundary S. This implies
that the scalar product of the direction D of
thetrajectoriesandthe
normal N tothe
boundaries S should be nonzero everywhere
on S , i.e.:
D.,?: # 0 on S.
(4)
By use of (3), the optimal system [eqs. (1)
and ( 3 0 ) 2 ]on S c a n be written in the folloaing form:
kn
IiarendraandGoldwynhave
condition
derived the
for some diagonal B 2 0 and
(9)
S(S)
= R ’ ( S 1 - F)-’G,
as the condition for absolute stability.
Thisnoteindicates
how bya
simple
transformationthe extension tothe finite
range case can be obtained directlyfrom the
Popovcriterion (P) andtheextended
criterion (P,). In both cases the conditions turn
out to be identical with previous results.
PROBLEM
LVITH NOXINEARITSI N
A
(10)
However,thiscondition
is automatically
satisfied by the optimal control law because
the optimal system is asymptotically stable.
The case D . N = 0 would mean that the optimal trajectories always lie on the hyperplane
S other than go t o the origin.
T. C. HSIA
Dept. of Elec. Engrg.
University of California
Davis, Calif.
FIXITERAKGE
In terms of transfer functions the finite
2) can be statedas
rangeproblem(Fig.
follom-5.
Given the linear system G,(s) with a nonlinear gain G1(ul)/ul in the feedback path,
R1< ‘l(ul)
-< K1+ Ki,
(2)
U1
find conditions for absolute stability. This
problem is convertedtothe
infiniterange
case as follows.
Let
d u ) =-‘l(Ul)
r2
- RlU1
and
I? = ra
Stability with Nonlinearity
in a Sector
Recent investigations of the stability of
a class of nonlinear systems have extended
the results of Popov.1 The blockdiagram
representation of the problem is shownin
Fig. 1where the linear part havinga transfer
function G(s) (satisfying certain conditions)
is followed by a nonlinear gain K ( u ) ranging
from 0 to oc.Then thecondition for absolute
stability of the system is given by the Popov
criterion:
Re [(l +jr&)G(jw)]
Manuscript received October 28. 1965; revised
February 11. 1966.
1 A. Id. Letov. “Analytical controller design 11,”
Automatiox and Remote Control. vol. 21. pp. 561-568.
May 1960.
2 Z. V. Rekasius and T. C. Hsia. “On an inverse
problem in optimal control,* I E E E Tram. on Aulomalic Control. vol. AC-9, pp. 370-355, October 1964.
3 R. C . Buck Adsanced
Calculus.
New York:
hlcCraw-Hill. 1956.
(1)
Ui
(8)
kn
l a ,
-+--1zo.
4~~‘~’)
< a, R diagonal.
+
Again it is required that D/N#O. This yields
the condition that
= 7.-1
O<-
Re [(I ioB)s(iw)2
] 0 for all real o (Pa)
which is identical to eq. (39.2 Consequently, D . N becomes
T2
ka
=
+
1r1,t.-In-1
.
D . N = (-1)-(21)
s: .
H‘x
@(d) = K ( d ) d
Yn2-1-
Furthermore, S is expressed in the following
parametric form3
U1)Tl
This result
has
been extended
to
the
case of thenonlinearity in a finiterange,
0 < K ( u ) <K by several researchers?-5 Extensions to systemswith multiple nonlinearities have been carried out by Narendra and
Goldwvyn4and by Ibrahim and Rekasius!
For the system,
2 0 for all real w
for some p 2 0. (P)
Manuscript received September 27. 1965; revised
Kovember 4. 1965.
1 Y. H. Ku and H. T. Chieb. “Extension of Popov’s
theorems for stability of nonlinear control systems,”
J . Franklin Imst.. vol. 259, pp. 401416. June 1965.
2 R. E. Kalman,“Lyapunov
func$ns forthe
problem of Lur’e in automaticcontrol, Proc. Sat’l
Acad. Sci.. vol. 49. PP. 201-205. February 1963.
Then
so that
a Z. V. Rekasius =A stability criterion fy f e e d back systemswithone
nonlinear element. I E E E
Trans. on AutomaticControl. vol. AC-9, pp. 46-50.
January 1964.
4 K. S. Narendra and R. hl. GoldaTn, “Existence
of quadratic type Liapunov functions for a class
ai
nonlinear vstems.” Internaf’l J . Engrg. Sci.. rol. 2 ,
pp. 367-377. October 1964.
6
A. Aizerman and F. R.Gantrnacher, Absolufe
Sfability of Regulator Sysfems.San Francisco: HoldenDay, 1964.
6 E. S. Ibrahim and Z. V. Rekasius, A
‘ stability
criterion for nonlinear feedback systems.” I E E E
Trans. on Aufoma!ic Coa!ml. ool. AC-9. pp. 154-159.
April 1964.
M
.
IEEE TRANSACTIONS
AUTOMATIC
CONTROL
ON
912
APRIL
Application of Liapunov’s Direct
Method for Determining Stable
Switching Boundaries in the
Phase Plane
Fig. 1.
Fig. 2.
T h e infiniterangeproblem
!
In order to determine stable switching
boundaries in the phase plane for a secondordernonlinear differential equation,Liapunov’s directmethod is used. The state
model for a thermal process is derived b y
considering thefirst and second laws of thermodynamics and thesign of the controlforce
is selected t o satisfyLiapunov’stheorem
after an appropriate Liapunov function
is
selected. The Liapunovfunctioncanbe
easily generated with analog or digital computers and these in turn may be
used to
control a relay which switches on the correct
boundary in the phaseplane.
T h e finite range problem
‘r’
STATE
MODEL
I n the conduction of heat in the unsteady
state, the first and second laws of thermodynamics can be applied to derive the mathematical model for the case of heat being
a
supplied byaninternalsourcewithin
b0dy.l
e-ac=-
dc
at
where
e= heatgeneratedbytheinternal
source during dt
ac= heat transferred from the body t o
surroundings during dt
dc/dt =rate of heat being stored within the
body.
Fig. 3. (a)
Transformation
of the
finite
range
problem to (b) the infinite range problem.
Thus the nonlinear gain &(ul)/ul in a finite
range can be replaced by the nonlinear gain
by setting K1=0, K2=K, we get thecondition as
d(u)/u in the infinite range with a feedback
in parallel with a gain
-(l/K?) around it,
(1
Kl [Fig. 3(a)].
If now we set
+ iwB)(Gl(iw) +- R-)]
1
2 0.
(Pb‘)
Use of a similartransformation in the
case of multiple nonlinearities leads to
the
condition
the system is equivalent to a linear transfer
function G(S) with nonlinear feedback gain
d u ) / u [Fig. 3(b) I
d G)
o<-<
for all real w
for
K
The control
“force”
be
to
applied
If further regulation of the temperature
of the body tosome constant temperatureR
is desired, and if R-c=rl is the temperature error, me may derive the statemodel as
follows.
The changein internal energy must equal
the net heat flow from the body to i t s surroundings or for no heat source within the
body
(PC)
diagonal and
E.
ni
Re [(l
+ iob) G(iw)] 2 0.
Substituting for G ( i w ) from (j),m-e get the
condition for stability for the finite range
case as
1
Re (1
+L B )
-dc = a(c - r-1.
at
U
Since the Popov condition
is now directly
applicable, sufficient conditionforasymptotic stability is given by
is
f (t)=d%/dt.
A s thetransformation
( 3 ) is validfor
monotone increasing nonlinearities,extensions to the finite rangefromthe
infinite
range can be obtained in a similar manner.’
I t should be noted that transformation ( 3 )
does not postulate the existence
of an inverse for the nonlinear function
11. -1.L. THATHACH.4R
M. D. SRIXATH
G. KRISHXA
Dept. of Electrical Engineering
Indian Institute of Science
Bangalore, India
With a heat sourcee within the body and
the surrounding temperature, c- arbitrarily
taken as zero, we ha\.e
Differentiating the above equation with
respect to time, n-e have
d2C
adc
_
-- - - +
-.
at 2
dt
dB
dt
Letting the control force be f ( t ) = d e / d t
and the temperatureerror
XI fromsome
reference temperature R be x, = R -c,
If
7 R. W. Brockett and J. L. WUems. ‘Frequency
domain stability criteria. pts. I and 11.”I E E E Tram.
OR Afrfomalic Control. vol. AC-IO,pp. 253-261 and
407413, July and October 1965.
Manuxript received November 12. 1965; revised
Kovember 29. 1965.
1 F. Kreith. Primiples of Heal Tramfer. Scranton.
Pa.: International Textbook Co.. 1958,ch. 4.