IEEE TRAXSACTIONS O N AUTOMATIC
CONTROL
90
January
Example I
Generalized Isocline Method of Plotting
Phase-Plane Trajectories
B. L. DEEKSHATULU ASD I. S. N. AIL-RTHY
Consider the system with nonlinear damping given by
i+&+2=0
Abstracf-By considering simultaneously the N-x (or the N-x),
where N = d b / d x and the k-x planes, second order
non-linear autonomous systems (not easily amendable to the existing methods)
describedby the differentialequation ? = F ( x , k) canbestudied.
in initial conditions,
Changes in system behavior due to changes
nonlineardamplingandrestoringforcescaneasilybestudied.
Simple and general a s i t is, the method presented here is believed
to be novel.
On the phase plane by the existing
isocline method one should
soh-e the quadratic equation in i namely ri-2+~Ti+x=0and both
the roots are valid.By the present method the procedureis as follows:
1 ) On the :\:-.s plane. curl-es of :\- vs. .u are plotted from the
equation
for particular values of 2 , say 2 =0, f 1 , + 2 , . * . . This is a simple
job since it amounts to dra\\-ing straight lines with different slopes
and different shifts from the origin. See Fig. 1 .
-4general second-order nonlinear autonomous system is governed
2 ) A number of horizontal lines corresponding to N= rt: 1 , k2,
b>- thedifferential equation
. . . are then drawn. These are already present on the graph sheet.
= F ( x , +)
(1) See Fig. 1.
3) Consider a horizontal straight line say for N = 1.5. Determine
If a.e denotethe slope of a trajectory on thephaseplaneby
all possible points of intersections (x and L?) of this straight line with
iV( = d i / d x ) , ( 1 ) reduces t o
1). These are (0.5, - O S ) , (0.5, -l.O),
the curves obtained under
LV.f = F ( x , i)
(2) ( - 1.0, 0.5) (-1.0, -2.0), (-2.5, l.O), ( - 2 . 5 , -2.5) and (-7.0, 2.0).
4) These points are plotted on the
i-x plane, joined and the
There are numerous methods to construct indix-idual phase-plane
slope lines marked with slope N = 1.5. The procedure is repeated for
trajectories starting from a given set of initial conditions (like the
isoclines (u-ith the respective
other x-alues of S , and the corresponding
methods of Lienard [7], [SI, Ku [4], and Pel1 [l]. But there are only slope lines marked on them) are obtained on the phase
plane. Joining
a felv techniques which permit the study of the behavior of certain
as shown in Fig. 2.
these slope lines give the entire phase portrait
class of nonlinear systems corresponding to many initial conditions.
The system represents thatif the initial point is interior to the sepaFor example Szego [ 5 ] suggested a procedure. lvhich needs the plotis a n oscillation (not a harmonic motion). The
ratris,themotion
ting of curves of certain equations on a transparent paper which are singular pointis a center.
used later for plotting the trajectories. Brodestky's [6] method
inExample 11
volves the division of the phase plane into regions with positive and
negatix-e d x / d x and d2.+/ds?. Deekshatulu [ 7 ] made use of simple
Consider the equation
transformations to effect rapid plotting of trajectories on the entire
phase plane or in any desired region. The existing isocline method
s+ri+z=o
necessitates plotting the curves of f 1-s. s from ( 2 ) for different conwe have :\:=
-x(2+1)/
constant values of ,??, from \vhich the phase trajectov may be
In this case the curves of :V vs, x for different values of f are
structedfroman>chosen initialcondition,butplotting
of i - x
c u n e s for any given value of A
' is not alx\-a>-spossible; for example, straight lines through the origin. See Fig. 3. Therefore, the present
equations like f + cot is+ sin x=O, and f+f+.++ log x=O do not technique is much simpler than the usual isocline method (which is
applicable for this example) since latter method necessitates plotting
yield to the existing isocline method. In what follon-s. a new, general
of the cur\-es of f vs. x from the equation i = -s/;V+x for different
and simple method of plotting phase-plane trajectories forsecondof S . The complete trajectories are shown
in Fig. 4.
values
order nonlinear systems governed by(1) is presented.
TSTRODITCTIOS
DISCUSSIOS
THEPROPOSED
METHOD
Equation ( 2 ) is rewritten as follows:
S = F ( x , ?)/i
i3)
Then,
1) On the :\'-x(~y-L?) plane, plot curves of :V vs x ( S vs. 2 ) for
different constant x-alues of ?(x). This is always possiblefor any
F ( s , x)
The following important points may be noted:
1) a ) \\-bile thepresentmethod
is generall>- applicabletoany
second-order nonlinear autonomous system, it is particularly useful
for the following equations:
r +j(.i) +
j: +j(& +
2
=0
2 =
0
2 ) Draw lines of constant K say corresponding to S = A',, :\'z,
b ) For equations of the type x + j ( 2 ) + g ( x ) =O one would plot the
LV3,. . . . Thisamountstodrawing
a number of horizontal lines
~ Y v xs curves from the equationX= - f ( f ) + g ( x ) / j for different conparallel to thes(i)
axis.
stant d u e s of i. Instead, a simple procedure would be t o imagine a
3 ) Keepingan>-horizontal
line in viex,saycorrespondingto
scale change of the I' axis from M to LV.?(L?being a constant each
N = :\'I, determine all possible points of intersection of this line with
time) and to shift the cun-e of g(x) with the proper shift j(i)from
the curves of constant i(x) as obtained under 1).
the origin depending on the value of 3E. The scale change of
axis for
4 ) Plot these points on the 2 - x plane and join through a smooth
each .i.is to be taken careof at the timeof reading the pointsof intercurye.Thiscurvethenrepresentsthe
isocline curvefor : \ - = S 1 .
sections of the :\7-s curves and the constantS lines.
Similarly other isocline curves corresponding to :Y= :\'z, 'V3, . . . are
2 ) Effect of variation of nonlinear damping and/or nonlinear restorobtained.
ing forces on system performance can be studied easily by the pro5) Small slope lines with slopes S I . '\'?, :Y3,. . . are then marked
po*d
method since it amounts to only interpretation of the shifts
on each of the respective isocline curves corresponding to :V=-Tl, -Y?,
(from the origin)or of the scale change on the Sasis. For example the
-V3, . . . . From these slope lines, the phase trajectory may be
conS-s curve for the two equations f + h ( i ) + s = O and I+f(.+)+x=O
structed for any given initial conditions or the entire phase portrait
are theSame when the shifts are suitably interpreted.
may be obtained.
To illustrate the procedure thetn.0 follon.ing simple examples are 3) The nonlinear functions can be given either anal>-ticallyor in the
form of curl-es.
gix-en:
4) Sometimes a S - x curve and the constantiViine may run so close
Manuscriut received June 29, 1964: r e v i d Septem,kr 23, 1964.
The authors are xx-ith the Dept. 01 Electrical Engmeermg. Indian Institnte of
Science, Bangalore. India.
to one another (for a certain range of x) as to make their points of
intersection difficult to locate. This is not a disad\.antage but merely
91
l?h5
\
,
/
++,
\
-9
%
8
Fig. 1. F - x curves for constant f for the equation ;+;*+x
Fig. 3. J,'-.x curves for constant .
i
for the equation : + x :
+.x=O
=O.
, , , , , , , , . ,, ,
-7
-5
i:
Fig. 2. Isoclines and phase-plane trajectories for the equation ;+.iZ+x
=O.
Fig. 4. Isoclines and phaw-plane trajectories for the equation
';+xi+x=O.
indicates that on the phase plane that
isocline curve is nearly vertical
in that range of x.
has been well developed, but i t is a trial-and-error procedure. It can
thereforebetediousandinaccurate.Insuch
cases, a n analq.tical
solution is preferable. Fromtheanalyticalsolution,
anaccurate
5 ) The well-known Leinardconstructionforplottingphaseplane
calibrated root locus can be obtained. The analytical procedure for
trajectories is clearly explained in \Yest [2] and Thaler and Pastel
obtaining the root locus of (1) is described in this paper. Anal>-tical
131. The Leinard procedureis a step by step construction for drawing
solutions of the root locus for some practical s>-stems are obtained
anyparticulartrajectorycorrespondingto
a givenset of initial
and discussed. The gain or time constant is computed analytically.
conditions, whereas the present method determines the entire phase
The application of this method to solve higher-order algebraic equaportraitindicatingthesystembehaviorforallinitial
conditions.
tions isdiscussed.
Further, the Leinard construction (even
for drawing a n individual
trajectory) is not applicable to all second-order nonlinear differential
AULYTICALSOLVTIOSS
equations, ahile the present method is applicable to every secondThe
functicn
G
(
s
)
H
( s )in (1) can be writtenin the form
orderautonomoussystem
described bythe
differential equation
P = F ( x , i).
Co~cLusross
A new, simple, general, rapid, and useful method is presented to
determine the behavior of nonlinear second-order autonomous systems for all pertinent initial conditions, and for changes in system
characteristics.
where K is a parameter which is proportional to the system
Substituting (2) into (1)yields
D(s)
The authors are grateful to Prof.H. h
'. Ramachandra Rao, Prof.
S . Ramaseshan, Dr. C. Ramasastry, and Prof. P. Venkata Rao for
their encouragement and discussions during the preparation of this
paper.
R,D(r + j y )
ImD(x jy)
E..A'oxlinear A z r h a f i c Confrol. Kew Yorl;: McGraw-Hill.
[4l Ku. Y. H.. Analysis and CrnJrol of Xonlinear Sysfcms. Kew York: Ronald,
1958. chs 1 and 4.
[SI SzegS, G . P. -4 new procedure for plotting phaseplane trajectories, AZEE
T r a m . , pt 2, Jul 1962, pp 120-125.
161 Davis, H . T., Inlrodudion to Nonlinenr Diserenfial and Integral Equations.
Y, S. Atomic Energy Commission, 1960. pp 25-32.
[il Deekshatulu. B. L.. Techniques ior analysis oi certainnonlinear systems.
I E E E Trans. oa Applicatiom and Induslry. Jul 1964. pp 258-262.
An Analytical Method for Obtaining the Root Locus
with Positive and Negative Gain
s. CHANG,
MEMBER, IEEE
Abstract-An analyticalprocedure for obtaining the root locus
with positive and negative gain is described. Solutions of various
systemsconsisting
of rational polynomial and/ortranscendental
functions of s are obtained. The root locus for a varying timeconstant is investigated. Accurate root locus curves can be plotted from
the analytical solution. The gain or time constant is calculated for
each point on the root locus. Application of the method is particularly suitable for obtainingthe root locus of a characteristic equation
containing a transcendental function or with a varying time constant
for which the graphical method is dficult to apply.
+ G(s)H(s) = 0
(1)
The functionG ( s ) H ( s )in general may bea rational polynomial andior
a transcendentalfunction of s. Thegraphicalmethod
[ l ] , [2], [3]
based on the angle condition
/G(s)H(s) = T
+ ha,
IZ =
0, k l ,
=0
Inserting this value of K into ( 6 ) , the equation of the root locus can
be putin the form
Reference to (4)shows that the 1,s" has a factor y. Therefore ( 8 ) also
has a factor y, revealing that the real axis,y=O, is a part of the root
locus, and the corresponding gain is R = - D ( x ) / X ( x ) , which may
be either positive or negative. If D ( 5 ) and X ( s ) are polynomial in s,
then in view of (4),the factor inside the bracketof (8) is a function of
y? and x. For many practical control systems, the equation
of root
locus can be solved for y 2 in terms of x; then the root locus can be
obtained with ease.
EXAMPLES
1) If
G(s)H(s) =
(s
+
K
u1
fjwd(x
+
UI
-jud(s
+ ud
where UI,W I ,uz are real numbers. The solutions are
y=o
]I;=-,( x
The root locus is defined to be the trajectory in the s plane, followed by a root of the characteristic equation as some parameter of
the corresponding system is varied continuously. The characteristic
equation of any system can be written
in the form
(5)
(6)
=0
where R, indicates the real part, andI , indicates the imaginary part
of a comples function. The gain constant obtained from (5) is given
by
ISTRODKCTION
1
+ KR,:h'(x + j y )
+ + KI,,,:V(x + j y )
1963.
1.
121 West, John C., AnalTticalTechniques for A'rndisear Control Systems, London:
English University Press, 1960, ch 4.
[31 Thaler. G. J., and Pastel, hl. P.. Analysis and Design of Xonlinear Feedback
Control Sptems, New Yorl;: McGraw-Hill. 1962.
CHI
(3)
where x and y are real variables. Then,(3) implies the two conditions
REFEREXES
cn
+ RX(S) = 0
Let s=s+jy. From the binomial theorem, it can be shown that
ACKNOWLEDGMENTS
[ I ] Gil+vm. John
gain.
+ +
4
2
w121(5
+
u2)
and
+
y? = 3x2
j K = yY3x
(&I
+ 2 4 2 + + 2u,u2 + w t
+2Ul+
u12
02)
- [(x
+ +
Ud2
WlP](X
+
U?)
The second set of equations represents the root locus in the complex
portion of the s plane. Using the condition y==O permits evaluation
of x at the break-in and break-away points. The result
is
. .
*
Manuscript received April 2i. 1964: revised September 21. 1964.
The author is with the Dept. oi Electrical Engineering. Air Force Institute of
Technology. Wright-Patterson Air Force
Ohio
Base.
Case I . I f ( U I - U : ) ~ > ~ W ~ there
,
arebreak-inandbreak-away
points. The rootloci are the real asis
plus a pair of hyperbolaas shown
in Fig. l(a).