ON THE CONSTRUCTION OF LIAPUNOV FUNCTIONS by .

ON THE  CONSTRUCTION  OF LIAPUNOV FUNCTIONS by .
ON THE CONSTRUCTION OF LIAPUNOV FUNCTIONS
FOR THIRD ORDER CONTROL SYSTEMS WITH LIMIT CYCLES
by
.
Michael John Wozny
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF ELECTRICAL ENGINEERING
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
19
6 5
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
I hereby recommend that this dissertation prepared under my
direction by
entitled
Michael John Wozny_______________________________
On The Construction of Liapunov Functions for_____
Third Order Control Systems with Limit Cycles
be accepted as fulfilling the dissertation requirement of the
degree of _________________________________________________________
After inspection of the dissertation, the following members
of the Final Examination Committee concur in its approval and
recommend its acceptance:*
4 (/m utf0 5 ^
H,
17
*This approval and acceptance is contingent on the candidate's
adequate performance and defense of this dissertation at the
final oral examination. The inclusion of this sheet bound into
the library copy of the dissertation is evidence of satisfactory
performance at the final examination.
PLEASE NOTE:
Not original copy. Pages tend to
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STATEMENT BY AUTHOR
This dissertation has been submitted in partial
fulfillment of requirements for an advanced degree at
The University of Arizona and is deposited in the
University Library to be made available to borrowers
under rules of the Library,
Brief quotations from this dissertation are allowable
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when in his judgment the proposed use of the material is
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SIGNED
/C
ACKNOWLEDGMENTS
The author wishes to thank Professor Donald G 0 Schultz
for his guidance» continued Interest, and encouragement
shown In the preparation of this thesis.
Thanks are also due to the author’s wife, Nancy, for
her support throughout this study and for her excellent
typing of the manuscript0
This research was supported by the National Science
Foundation under Grant GP-2237°
ill
TABLE OF CONTENTS
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LIST OF ILLUSTRATIONS
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INTRODUCTION AND ORGANIZATION
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1 ©2 Organization of the Work © © © © © © © © . © , ,
I ©3 Notation © © © @ © © © © © © © @ © @ © © © © < ,
CHAPTER 2s
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2©3
2.4
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CHAPTER 3s
3 ©1
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3.4
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CONSTRUCTION OF LIAPUNOV FUNCTIONS FOR
SECOND ORDER LIMIT CYCLE PROBLEMS © © © © <
LIAPUNOV FUNCTIONS FOR
THIRD ORDER LIMIT CYCLE PROBLEMS © ©
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SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH,
Summary o o o o e e o e o o o o o
Suggestions for Future Research
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47
47
48
55
65
84
Introduction o e o o o e o o o o o o o o o o
Previous Work © ©
Reduction of Third Order Problem e o © o o
Liapunov Formulation of the General Problem
CHAPTER 5s
5©1
5.2
6
Introduction © © © © ©* © © © © © © © © © © © <
Estimation of the Region of
Asymptotic Stability © . © , © © . © © . . « «
Application of the Polncare-Bendixson Theorem ,
Liapunov Functions Definite with
Respect to the Limit Cycle e o e o o e o e e c
CHAPTER 4s
4.1
4.2
4.3
4.4
.
GENERAL BACKGROUND ©
Introduction © © © © © © © © © © © © © © © © (
State Space Representation © © . © © © © . ©
Singularities and Limit Cycles
in Nonlinear Systems .......... . . . » . © ,
Stability and the Second Method of Liapunov ©
Methods of Constructing Liapunov Functions ©
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3
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84
85
94
11.9
134
134
137
v
TABLE OF CONTENTS - - Continued
Page
APPENDIX
A al
A»2
As
SECOND ORDER SYSTEMS WHICH EXHIBIT
LIMIT CYCLES EXPRESSIBLE IN CLOSED FORM „ . 139
Introduction o o o e o o o o o o o © © © © © © 139
A Synthesis Method © « . © . © © © © © « © © © 140
LIST OF REFERENCES © © © © © © © © © © © © © © © © o © ©
1^*9
LIST OF ILLUSTRATIONS
Figure
Page
2.1
Various Representations of the System Solution
2.2
Classification of Singular Points for
Second Order Systems . . . . . . . . . . . . .
.
14
Classification of Singular Points
for Third Order Systems . . . . . . . . . . . .
.
15
Stability Behavior of Limit Cycles
in Second Order Systems . . . . . . . . . . . .
.
19
Geometric Interpretation of the
Polncare-Bendixson Theorem . . . . . . . . . .
.
22
Geometric Interpretation
of the Liapunov Stability Theorem . . . . . . .
31
36
2.3
2.4
2.5
2.6
9
2.7
Geometric Interpretation of the Zubov Equation
.
.
2.8
Approximate Region of Asymptotic Stability
Determined by Variable Gradient Method , . . .
.
45
Inner Bound for van der Pol Equation
via Variable Gradient Method . . . . . . . . .
.
54
.
58
Bound for Second Order Limit Cycle
Using Contact Curves . . . . . . . . . . . . .
.
60
Bound for van der Pol Example of Section 3«3
.
66
Liapunov Function Positive Definite
with Respect to the Closed Curve W(jc) = 1 . . .
.
69
Annular Region Determined by
Proposed Liapunov Function . . . . . . . . . .
.
78
V = Constant Curves for the
Liapunov Function given in Equation (3.102) . .
.
81
Block Diagram of System Considered
in Szego and Geiss (1963) . . . . . . . . . . .
.
90
3.1
3.2
3.3
3.4
3.5
3.6
3.7
4.1
Contact Curve when a(x) is a Linear Form
vi
. . .
vii
LIST OP ILLUSTRATIONS— Continued
Page
Figure
4,2
Typical Trajectories for System (4,1) 9 0 0 , , , ,
4,3
Limit Cycle for System (4,88) , , . , 0 , 0
, , o
4,4
Typical Behavior for System (4,146) , 0 , 0
0 0 0 , 1 31
A,1
Block Diagram of Nonlinear Feedback System
0 , O 0 14 2
A,2
Inner Bound for Limit Cycle of System (A,8 ) S O S O
\
o
93
118
144
ABSTRACT
In this work the Second Method of Liapunov is used
in conjunction with the Poincare-Bendlxson Theorem to ex­
amine methods of determining or specifying limit cycles in
J
second and third order systems.
For second order systems the variable gradient method
is used to find contact curves in the construction of the
annular region required by the Poincare-Bendlxson Theorem.
In this manner it is possible to bound the limit cycle to a
region of the state plane.
With regard to finding the exact limit cycle a method
is formulated by defining a Liapunov function referred di­
rectly to the limit cycle.
Since the limit cycle is unknown
a priori. the Liapunov function is defined as a known function
of an unspecified closed curve.
The form of the Liapunov
function is chosen according to the geometrical requirements
of the Poincare-Bendixson Theorem.
Thus, for the two dimen­
sional case, the Liapunov function represents two concentric
closed curves and, in three dimensions, a toroid.
Making use
of the hypothesis of the Poincare-Bendlxson Theorem that all
the trajectories cross the region bounding the limit cycle in
an inward direction, the time derivative of the Liapunov
function is forced to be negative definite with respect to
viii
ix
the limit cycle.
closed
This condition forces the unspecified
curve to be the limit cycle.
It is shown
that the above formulation is very general
for both second and third order systems. Unfortunately, it
is difficult to establish a routine method of solution in the
third order case.
However# a method of solution is developed
for the special case of systems where the trajectories tend
to a stable surface containing the limit cycle and exhibit a
certain symmetry.
In summary,
this work.
there are three major contributions of
First, a scheme is developed
for applyingthe
variable gradient method to the problem of finding contact
curves for limit cycles in second order systems.
Second,
the special class of third order systems Which have a limit
cycle contained in a stable surface is Investigated and a
method developed for its solution.
Third, the general
problem of enclosing the limit cycle in a toroid is formu­
lated and its properties investigated.
I
CHAPTER 1
INTRODUCTION AND ORGANIZATION
1.1
Introduction
As modern control systems become more sophisticated
and include nonlinear elements, various phenomena occur
which cannot be explained by linear theory.
One such
phenomenon is the existence of sustained periodic oscilla­
tions which are independent of the initial conditions ap­
plied to the system.
This type of oscillation exists in
engineering practice (in relay control systems as well as
in nuclear reactors and guidance systems), and in most cases
it is difficult to eliminate from the system because of the
lack of understanding of its basic mechanism.
In other
cases the sustained oscillation is desired, as in electronic
oscillator circuits.
The investigation of the fundamental
properties of sustained oscillations (limit cycles) is the
subject of this work.
To understand the properties of limit cycles, it is
necessary to formulate the problem in terms of nonlinear
differential equations, and then study the equations by
general mathematical techniques.
The mathematical techniques can be divided into two
classes %
(1 ) the perturbation or analytic methods which are
basically time domain approximation schemes and
(2 ) the
topological methods consisting of the geometrical or state
space approach to the limit cycle problem*
concerned only with the latter approach*
This study is
The Poincare-
Bendixson theory is basic to the topological approach and is
very general in scope, but unfortunately, very difficult to
apply*
This difficulty stems from the problem of specify­
ing the solution curves of the differential equation without
actually finding the complete solution*
In
this work the Second Method of Liapunov Is used
to specify the solution
curves of the systems*
The Second
Method is
a state space technique for determining the sta­
bility of
a solution by examining the solution curves with
respect to that given solution*
The principal feature of
the Second Method is that it abstracts only the stability
information from the system equations and does not require
the entire solution*
Unfortunately, the method represents
only a sufficient condition and requires the existence of
a testing function (Liapunov function) for which only some­
what restricted techniques of construction are available*
The purpose of this work is to investigate methods
of constructing Liapunov functions which can be used in the
study of systems having limit cycles,
A general formulation
is proposed in which a Liapunov function is found displaying
the properties of the Poincare-Bendlxson Theorem,
It is
shown that the formulation provides a consistent approach
to both second order and third order systems*
3
1,2
Organization of the Work
This work is organized around the investigation of
limit cycles by Liapunov methods, and contributions are made
both for second order and third order systems, with empha­
sis on the latter.
Approximately half of the work deals
with background material and second order systems.
Chapter 2
presents the basic properties of singular points and limit
cycles in second and third order systems, A summary of the
basic concepts of the Second Method of Liapunov useful for
studying limit cycle problems is also included in Chapter 2,
In Chapter 3» Liapunov techniques are used to investigate
limit cycles in second order systems.
After the known
methods are discussed, a new Liapunov function is proposed
which is defined in terms of the limit cycle.
The Liapunov
function is shown to be a formulation of the PoincareBendixson Theorem,
The second half of this work, consisting of Chapter 4,
is concerned with the third order problem.
First, it is
shown that the existing methods considered in Chapter 3 are
applicable to the limit cycle problem only because of the
simple structure of the two dimensional plane and cannot be
extended to three dimensions.
-third order problem are given.
Then two formulations for the
The first approach, based on
a paper of Szego and Geiss (1963)» is valid for the class of
systems which have a limit cycle contained in a stable sur­
face,
A method is developed for finding the required surface
4
for a special class of systems.
The second approach is
an extension to three dimensions of the Liapunov function
formulation proposed in Chapter 3•
In this case, the
Liapunov function represents a toroid enclosing the limit
cycle.
This formulation is also shown to be valid for
systems whose trajectories spiral about the limit cycle as
they approach It.
Such a behavior cannot be handled by the
first method.
Chapter 5 summarizes the results obtained, and de­
fines areas for further research.
Examples are presented
throughout the work whenever they can serve to better Illus­
trate a point.
1.3
Notation
Wherever possible, vectors are represented by under­
lined lower case Roman letters, as x = (x^,x2 ,....x^) or as
5 = c o K x ^ . x ^ , ... .x^).
W.
An exception is the gradient symbol
The transpose of a vector is designated by a prime,
as 2**
rule.
Upper case Roman letters designate scalars as a
The n dimensional state space Is designated by En ,
and its Euclidean norm by ||x || .
of class C
A function f (x) is said to be
if it is continuous and has continuous first par­
tial derivatives.
In the notation G : g(x) = 0, G represents
the state space orbit or curve of the equation g(x) = 0.
In general, theorems and definitions are titled and
Indented so as to set them off from the text.
Where many
definitions are given, only the most significant are set
off„ while the rest are simply underlined.
The phrase "second order limit cycle" means "limit
cycle of a second order system."
A similar condition is
used for third order systems.
In Section 2.5 a. lower case v is used in defining
the scalar function associated with the Zubov equation.
This is done for convenience only in Section 2.5®
Elsewhere,
scalar functions associated with the Zubov equation are
designated by upper ease letters.
CHAPTER 2
GENERAL BACKGROUND
2.1
Introduction
In this chapter the background necessary for the
understanding of this work is presented.
Section 2.2 is
concerned with the general representation of a control
system in terms of a set of first order nonlinear differen­
tial equations.
The behavior of the system solutions are
examined in Section 2.3 with emphasis bn the limit cycle
concept for second and third order systems.
In Section 2.4
the notion of stability in the sense of Liapunov is defined
and then discussed with reference to the stability of limit
cycles.
The fundamental concepts of the Second Method of
Liapunov are also included in this section.
Section 2.5
concludes the chapter by presenting two general methods for
generating Liapunov functions which are of importance in
this work.
The topics discussed in Chapter 2 present background
material for this study.
No attempt is made at completeness,
and only those topics which are needed in later chapters
are included.
6
7
2,2
State Space Representation
The control systems considered In this work are
assumed to satisfy the following conditions t
1)
The state of the system, at any time t , Is
completely defined by the variables
x1 (t ), x2(t)
2)
xn (t).
(2.1)
The variables given above are related by a set
of n first order differential equations of the
form
x
= f1 (x1 .x2
X >.
1 = 1,2,...,n,
(2.2)
where the dot Indicates differentiation with
respect to the Independent variable t , and the
f^ are sufficiently smooth to guarantee a unique
continuous solution through every Initial
condition.
The variables
,...,x^ are called state variables, and
the n-tuple (x^,x2 ,...,x^) represents a point In the n
dimensional Euclidean space called the state space. En .
f^ — x ^
,
1 « l,2,...,n - 1 ,
the state space Is called a phase space.
If
(2.3)
The system of
differential equations of the form (2.2), written In vector
notation as
Z = Liz).
(2.4)
8
Is called an autonomous system.
Physically, this means
that the control system Is neither time-varying nor forced.
Thus, only the characteristic behavior of the unforced system
Is Investigated in this work.
The solutions of Equation (2.4) are given various
names depending on the space and form in which they are
represented.
If the solution is expressed in the form
g(x, t) = C,
(2.5)
and represented in the (x, t)-space, it is called an
integral curve.
If solution (2.5) is parametrized by t in
the (jc)-space (i.e., in the state space, En ), it is called
a trajectory or a motion.
The direction of increasing time
is called the direction of the trajectory.
For the case
where solution (2.5) is represented in En as a locus of
points without reference to a parametrization, it is called
an orbit or oath.
Figure 2.1 illustrates the various repre­
sentations of the solution of the system equations.
In addition to the conditions given above, the
systems considered in this work are also assumed to display
an oscillatory equilibrium state, e.g., a limit cycle.
The
properties of limit cycles are taken up in the next section.
2.3
Singularities and Limit Cycles in Nonlinear Systems
A physical system is said to be in a state of equi­
librium if its response assumes either a stationary value or
9
X(0)
INTEGRAL CURVE
TRAJECTORY (ORBIT)
Figure 2,1
Various Representations
of the System Solution
10
a fixed, periodic oscillation.
Furthermore „ nonlinear
systems often have several equilibrium states with related
stability properties.
Thus„ in order to examine systems
with limit cycles„ it is necessary to understand the prop­
erties of equilibrium states in general,
Such an under­
standing is obtained by considering certain aspects of the
mathematical theory of nonlinear differential equations.
Only those topics are considered here which contribute
directly to the understanding of the class of systems of
interest in this work.
Firste properties of stationary equilibrium states
are examined in terms of the mathematical notion of a sin­
gular point,
The local behavior of the system trajectories
is discussed for the various types of singular points in
second and third order systems.
Then limit cycles are dis­
cussed In terms of the mathematical concept of a limit set,
culminating in the Poineare-Bendixson Theorem,
A stationary equilibrium state of a system is charac­
terized by the fact that the solution remains constant and
does not vary with time.
Such a condition is represented as
a fixed point in En and is'called a singular point.
This
concept is expressed more formally in the following def­
initions
Definition 2.1
Singular Point
A point x is called a singular (critical9 stationary)
point of Equation (2.4) if
11
f(xo ) = 0.
(2.6 )
A point which Is not a singular point is called a regular
point.
It should be recalled from Chapter 1 that only the
principal definitions are set off from the text and numbered.
Other definitions are simply underlined.
For the purposes of this discussion it is assumed
that the singular points are isolated and that a singular
point occurs at the origin of the state space.
Any singular
point can be moved to the origin by a simple translation of
coordinates.
Thus, the nonlinear system of interest in this
work is given by
x = f(x),
f(0) = 0.
(2.7)
The general behavior of trajectories near singular
points for second and third order systems is considered
next.
Coddington and Levinson (1955) and Hartman (1964)
consider second order systems of the form
x = Ax + h(x),
(2.8)
where A is a 2 x 2 matrix, and h(x) represents a power
series satisfying
(2.9)
and show
point
the
at the
behavior of the trajectoriesnear the singular
origin is characterized bytheeigenvalues.
12
except for the special cases given below, of the linearized
system,
x = A^.
(2.10)
Hayashi (1964) considers the above conditions for third
order systems.
Singular points having eigenvalues which are either
zero, imaginary, or nondistinct are called singular points
of higher order, and their investigation requires the con­
sideration of the nonlinear terms of the system equations
(Coddington and Levinson 1955# Hayashi 1964).
For the pur­
poses of this study this class of problems can be handled by
The Second Method of Liapunov and will not be considered
any farther.
A singular point x0 is stable if in some
neighborhood of xQ all the trajectories tend to x0 as t -»- *>.
Also the sign of the real part of the eigenvalues determines
the stability of the singular point.
The various types of singular points for Equation
(2.8), as characterized.by the eigenvalues of the linearized
system (2.10), are given below for second and third order
systems.
For second order systems the classification of
singular points is given as follows (Coddington and Levinson
1955# Hartman 1964):
1)
If the eigenvalues are real and of the same sign,
the singular point is a node.
2)
If the eigenvalues are real and of opposite signs,
13
the singular point is a saddle point.
3)
If the
eigenvalues are complexconjugates„
the
singularity is a focus.
4)
If the
eigenvalues are imaginary, the singu­
larity is either a center or a focus.
The various classifications defined above are shown in
Figure 2,2.
For third order systems the classification
becomes (Hayashi 1964)s
1)
If all the eigenvalues are real and of the same
sign, the singularity is a node.
2)
If all the roots are real and not of the same
sign, the singularity is a saddle point,
3)
If one root is real and two are complex conju­
gates such that the real part
of the conjugate
roots is of the same sign as thereal root, the
singularity is a focus.
4)
If one root is real and two are complex conju­
gates such that the real part
of the conjugate
roots is of the opposite sign
of the real root,
the singularity is a saddle-focus.
5)
If the two conjugate roots are imaginary, the
singularity is either a center, focus or a
saddle-focus.
This classification is shown in Figure 2.3°
14
(a)
NODE (STABLE)
(b) FOCUS (STABLE)
(c) SADDLE POINT
Figure 2.2
Classification of Singular Points
for Second Order Systems
15
(b)
(a) NODE
SADDLE
(c) FOCUS
k±>
(d) SADDLE FOCUS
Figure 2,3
(e) CENTER
Classification of Singular Points
for Third Order Systems
16
In addition to the stationary states described above
a nonlinear system may also have oscillatory equilibrium
states.
Oscillatory states which are dependent upon the
Initial conditions of a system, as In conservative systems,
are always associated with centers; hence, they represent a
property of a particular type of singularity, and not of a
phenomenon directly attributed to the nonlinearity of the
system.
Such oscillatory states do not comprise the main
effort in this study.
Isolated periodic oscillations which
are independent of the initial conditions of the system repre­
sent the main topic of interest in this work.
Oscillatory
states of this type are called limit cycles (defined more
precisely below) and are unique to nonlinear systems.
Limit cycles, as contrasted to singular points, do
not appear explicitly in the form of the system equations.
Consequently, their properties can only be determined by
examining the limiting behavior of the system trajectories
as t
The limiting behavior of a trajectory is given
precise meaning through the mathematical notion of a limit
set (Cesari 1963# Coddington and Levinson 1955* Hartman
1964).
Definition 2.2
Positive Limit Set
A positive limit (or limiting) set L(C+ ) of a
solution C+ !2 = x(t), t ^ 0, of Equation (2.4)
is a set of points xQ for which there exists a
sequence t
o
< t, < ... , where t — ► » as n
1
n
17
such that
£( tn ) - ^ x 0 as n - * ™ .
(2.11)
For example, the solution
C+ :x(t) = e atslntit,
has thelimit set L(C+ ) = 0.
a > 0,
Hence, for
(2.12)
the casewhere the
limit set Is a closed curve, Definition 2.2 leads directly
to the general definition of a limit cycle.
Definition 2.3
(+) Limit Cycle
A periodic orbit C0 ijc ■ x^(t) , -<» < t < 00 which is
contained in some limit set L(C+ ), C* c£ C0 , Is called
a (+) limit cycle.
Several statements concerning Definition 2.3 are in order.
It should be noted that this definition Is not restricted to
second order systems, therefore, the limit cycle concept is
valid for
third order systems.
Also, thecondition
C+
C0
implies that the oscillation is Isolated; hence, not every
closed orbit is a limit cycle (e.g., the closed orbits
associated with a center).
Since the limit cycle is a property of the system
trajectories and does not appear explicitly in the form of
the system equations, its existence and stability properties
are difficult to determine.
For second order systems the
motion is sufficiently restricted so that very general re­
sults about the limit cycle behavior are possible.
This is
18
not true In the third order case.
In the remainder of this
section the classical existence and stability properties of
limit cycles In second and third order systems are given.
The body of knowledge concerning the topological
behavior of trajectories of nonlinear second order systems
of the form
x1 = P (x1 ,x2 )
x2 = Q(x1 ,x2 ),
(2.13)
with suitable restrictions (Coddlngton and Levinson 1955)•
Is called the Polncare-Bendlxson Theory.
Only second order
systems which possess limit cycles are considered.
A funda­
mental property of this class of systems Is the division by
the limit cycle of the state plane Into two disjoint re­
gions so that trajectories which originate In one region
cannot cross the limit cycle, and consequently, must remain
In that region for all t ^ 0 (Hartman 1964).
If the tra­
jectories In some neighborhood of the limit cycle tend to
the limit cycle as t -*• 00, the limit cycle is said to be
stable (more precisely, orbitally stable, see Definition 2.5
in the next section).
In addition to the stable limit cycle
other types of two dimensional behavior are unstable and
semistable limit cycles (Pontryagln 1962).
The three types
of behavior are illustrated In Figure 2.4.
Pontryagln
(1962) also states that a limit cycle which is stable for
t > 0 Is unstable for t < 0; hence, the stability of a
(a) STABLE
(b) UNSTABLE
(c) & (d ) SEMISTABLE
Figure 2.4
Stability Behavior of Limit Cycles
in Second Order Systems
20
limit cycle and of the other equilibrium states is reversed
by running time backwards.
This reversal of stability with
a change in the direction of time is also a property of
higher order systems.
In many systems only one singularity
is enclosed by the limit cycle.
In this case the singu­
larity must be either a node or focus whose stability is
opposite to the stability of the interior of the limit
cycle (Coddington and Levinson 1955)•
The Polncare-Bendixson Theorem represents a very
general hypothesis for the existence of a stable limit cycle
in second order systems (Cesari 1963, Coddington and
Levinson 1955,
form is due
Hartman 1964, Struble 1962). The
following
to Struble (1962)%
Theorem 2.1
Polncare-Bendixson Theorem
If C+ is bounded and L(C+ ) consists only of regular
points, then L(C+ ) is a periodic orbit and either
1)
C+ = L(C+ ), or
2)
C+ "spirals" to L(C+ ) on one side
of L(C+ ).
Although the theorem is general, it is not in a form which
can be easily applied.
The following geometric interpre -
tation, due to Poincare, reveals a more convenient form
(Hayashl 1964, Stoker 1950)i
let R be a ring-shaped region
such that R does not contain any singular points.
If all
the system trajectories cross the boundary of R in a
direction toward its interior, then the above hypothesis is
satisfied, and R contains a stable limit cycle
21
(see Figure 2.5) •
It should "be mentioned that even with
the above geometrical Interpretation It Is difficult to
construct a suitable region R.
This theorem Is considered
again in Chapter 3, where Liapunov functions are used to
construct R e
Very few general results are available for the exis­
tence of limit cycles in third order systems.
Fuller (1952)
showed by means of a counter example that the PoincareBendixson Theorem cannot be extended directly to third order
systems by making R a toroid.
The existence of limit cycles
in the several third order systems considered in the litera­
ture were determined by showing, in addition to crossing the
toroid in an inward direction, that the system trajectories
also circulate inside the toroid and converge to a single
curve (Friedrichs 19^6, Rauch 1950» Perello 1965),
In this work limit cycles are examined via the
Second Method of Liapunov,
Although this approach restricts
the class of problems considered, it is suitable for a large
class of practical problems,
The next section presents the
basic concepts of the Second Method of Liapunov,
2,4
Stability and the Second Method of Liapunov
The Second Method of Liapunov is a method for
determining the stability (in the sense of Liapunov) of a
trajectory (unperturbed trajectory) with respect to its
neighboring trajectories.
The method proposed in this work
22
LIMIT CYCLE
REGION R
Figure 2o5
Geometric Interpretation of the
Poincare-Bendixson Theorem
Is to let the unperturbed trajectory be the limit cycle so
that the Second Method of Liapunov can be used to examine
Its stability properties.
This formulation presents two
conditions not encountered In the application of the Second
Method to stationary equilibrium states.
The first Is the
restrictiveness of stability in the sense of Liapunov
(Liapunov stability) compared to orbital stability, and the
second condition is the incomplete knowledge of the equi­
librium state. I.e., of the limit cycle.
These conditions are
discussed here only to the extent of showing how they arise
in the theory.
A complete investigation is deferred to the
later chapters where the necessary groundwork is developed.
The section concludes with a presentation of the fundamental
concepts of the Second Method of Liapunov.
Consider first the meaning of stability in the sense
of Liapunov and its relation to other types of stability,
namely, orbital stability.
Since stability is meaningful
physically only if the unperturbed trajectory is an equi­
librium state, let x^(t) represent an equilibrium state of
the system
x = f (jc).
(2.14)
Then the stability of xfi(t) in the sense of Liapunov is de­
fined in the following manner:
24
Definition 2.4
Stability In the Sense of Liapunov
The equilibrium state
(t ) Is stable in the sense
of Liapunov if for every e > 0, there exists a
6(e) > 0 such that any solution x(t) of the system
(2.14) for which
x(0) - xe (0)||
< 6,
(2.15)
< e
(2 .16 )
satisfies
||a(t) - xe (t)||
for all t ^ 0.
Note that Liapunov stability implies a time dependent com­
parison between the perturbed solution x(t) and the equi­
librium solution x (t ). This comparison is not evident when
e
defining the stability of a singular point, since the equi­
librium solution is a constant.
For oscillatory equilibrium
states, such a time dependent comparison classifies, for
example, nonlinear conservative systems as unstable.
This
is true because for nonlinear conservative systems adjacent
orbits have different periods so that the distance between
two orbits may be quite large when measured at specific
instants of time, whereas, in reality the orbits are geo­
metrically close together.
To include such cases in the
definition of stability, orbital stability is defined as
follows:
25
Definition 2. 5
Orbital Stability (as t — » oq)
Let Ce : x = xe (t) be an equilibrium state of the
system (2.14).
Then
Is orbltally stable as t
»
If for every € > 0, there exists a 6(e) > 0 such
that If xQ Is within a distance 6 of Ce , then all
solutions C+ : x = x (t) of (2.14) remain within a
o
o
distance e of C for t ^ 0.
e
Thus, orbital stability removes the time dependent comparison
In Definition 2.4 and compares the nearness of the two tra­
jectories geometrically.
In summary, the most general form of stability for
oscillatory equilibrium states is orbital stability.
But
for systems which have a single limit cycle enclosing one
singularity at the origin, a condition commonly satisfied
in physical systems, the limit cycles are Liapunov stable.
This statement is verified for second order systems in
Chapter 3 and for third order systems In Chapter 4.
The
Second Method, where applicable, offers an important ad­
vantage over the classical procedures of estimating the
limit cycle and determining its stability properties, be­
cause it is more systematic, and therefore, easier to apply.
After discussing the second condition associated with limit
cycle problems mentioned in the introduction, the remainder
of the section is devoted to the presentation of the funda­
mental concepts of the Second Method.
26
As stated previously, the method of investigating
limit cycle behavior in this work is to let the limit cycle
be the unperturbed trajectory and apply the Second Method of
Liapunov.
Consider first a method of reducing a general
unperturbed trajectory to a normalized form (Krasovskil
1963).
Let
Z = &(z)
(2.17)
represent the system of interest, and &(t) the unperturbed
trajectory.
Then the transformation
x = 2
-X
(2.18)
takes system (2.17) Into
x = f(x),
(2.19)
where
f(x) = &(x +X)- £(X) ,
(2.20)
x =0
(2.21)
so that
is the new form for the unperturbed trajectory.
Hence, in
principle, all problems can be transformed to a form which
requires only the investigation of a singular point at the
origin.
However, if the unperturbed trajectory is a limit
cycle, such a transformation is not meaningful practically
27
because It requires that the limit cycle be known before­
hand.
In most systems not only Is the limit cycle unknown,
but It cannot even be expressed In a closed form.
Thus, the Liapunov Theory, I.e., the Second Method,
formulated for the general unperturbed trajectory Is valid
in a practical sense only If the unperturbed trajectory Is
known a priori. as for example, a singular point of some
given system.
For the case where the unperturbed trajectory
Is a limit cycle, the theory is only formal and does not
indicate any methods of Investigating the limit cycle be­
havior.
The fundamental concepts of the Second Method are
presented next where the system Is assumed to have the form
(2.19) with Equation (2021) as the unperturbed trajectory.
General references for this material are Hahn 1963,
Krasovskll 1963 , LaSalle and Lefschetz 1961, and Malkin 1952.
The Second Method of Liapunov is based on finding a
function, V(x), which is a measure of the distance from
any point in the state space to the equilibrium state.
By
evaluating V(x) along the system trajectories, x(t),
stability can be established if
V(t) = V(x(t))
decreases monotonically with time.
(2.22)
A principal advantage of
the Second Method is that V(x) is not unique.
Functions which are definite with respect to sign
are suitable as distance functions.
28
Definition 2.6
Positive (Negative) Definite
A scalar function, V(x), Is positive (negative)
definite In a region N If for all x In N, x ^ 0,
V(x) > 0
(< 0)
(2.23)
and V(0) = 0.
V(x) is called positive (negative) semldefInlte if the
strict Inequality (2 .23 ) Is replaced by the weaker condition
V(x) ^ 0
U
0).
Furthermore, definite functions from which
information about the system stability can be determined are
called Liapunov functions.
Definition 2.7
Liapunov Function
A scalar function, V(x), defined on a neighborhood,
N , of the equilibrium state x = 0 Is called a
Liapunov function for the system
x = f(x),
f (0) = 0,
(2.24)
If the following conditions are satisfied;
1)
V(%) is
of class C |
2)
V(jc) is
positive definite;
3)
V(jc) is
negative semldef inlte along each
trajectory in N.
The name Liapunov function originally referred to the con­
ditions imposed on a definite scalar function, V(x), in the
Stability Theorem of Liapunov (Theorem 2.2) to show stability
in the sense of Liapunov.
But a Liapunov function as now
29
used In the literature means a definite function associated
with the hypotheses of various theorems which present re­
finements of the Stability Theorem of Liapunov.
Hence,
conditions (2) and (3) In the above definition are often
altered to conform with the hypothesis of the particular
theorem under consideration.
Normally, these changes appear
as qualifying conditions In the statement of the Liapunov
function. I.e., ”a Liapunov function satisfying . . .".
Geometrically, the Liapunov function V = V(x) Is
represented as a well behaved n + 1 dimensional surface, L ,
with an Isolated minimum at the origin of the (x, V) space.
V(%), when evaluated along a trajectory. Is represented as a
curve on this surface.
Lefschetz (1963) states that In a
sufficiently small neighborhood of the origin, £ can be pro­
jected in a continuous manner onto the state space as a
family of nested level surfaces,
V(i) = C,
enclosing the origin.
(2.25)
Thus the stability problem has a com­
plete geometrical representation In the state space where
evaluating V(%) along a solution takes on the significance
of the trajectory crossing successively smaller surfaces of
constant V.
The next theorem represents a sufficient condition
for stability.
Theorem 2.2
Stability Theorem of Liapunov
If there exists a Liapunov function, then the
singular point x = 0 of system (2.24) Is stable
(In the sense of Liapunov).
It Is Instructive to consider a geometrical proof of this
theorem; a two dimensional Interpretation Is shown In
Figure 2.6.
As preliminary notation, let S denote a
h
sphere consisting of points jc satisfying J|x||
< h.
Assume that there exists a Liapunov function V(x) defined
on a neighborhood N.
Then there exists a sufficiently
small constant, C > 0, such that
V(%) = C
represents
(2.26)
a closed surface, S, which lies wholly In N.
It
also follows from the above assumption that
V = VV'f < 0.
For any
(2.2?)
on S, V V represents the outward normal of S, and
f the direction of the trajectory on S.
Thus, geometrically,
condition (2.2?) signifies that the direction of every tra­
jectory onS is either toward the interior
to S.
face S.
Now
of S or tangent
choose £so that the sphere Se contains the sur­
Then clearly any 6, where
will satisfy the stability definition.
Is contained in S,
31
S: V(x) = C
Figure 2.6
Geometric Interpretation of the
Liapunov Stability Theorem
32
Liapunov stability is physically meaningful only for
the case where the trajectories eventually approach the
equilibrium state.
This type of behavior of the system tra­
jectories is called asymptotic stability.
Definition 2.8
Asymptotic Stability
The origin of system (2.24) is called asymptotically
stable if in addition to being Liapunov stable„ the
trajectories also tend to the origin as t —> «>.
The next theorem gives a sufficient condition for asymptotic
stability of a singular point.
Theorem 2.3
If there exists a Liapunov function V(x) such that V
is negative definite along the system trajectories„
then the singular point x = 0 of system (2.24) is
asymptotically stable.
This hypothesis can be weakened to having V a negative semidefinite function which is never Identically zero along the
system trajectories.
The discussion thus far has been concerned with local
stability.
Theorem 2.3 insures only that the origin is
asymptotically stable for sufficiently small perturbations In
initial conditions but gives no indication of their allowable
magnitude.
Clearly, it is of considerable importance to be
able to estimate the domain of asymptotic stability for a
given system.
This concept is used in Chapter 2 to find an
inner bound for limit cycles in second order systems.
33
Definition 2.9
Region of Asymptotic Stability
The set of all Initial points,
xq ,
In the state
space for which the singular point, x = 0, Is asymp­
totically stable Is called the region of asymptotic
stability of x = 0.
Other names used for this definition are domain of asymp­
totic stability and domain of attraction.
LaSalle and
Lefschetz (1961) give the following theorem for estimating
the region of asymptotic stability.
Theorem 2.4
Let V(x) be a scalar function of class C*.
designate the region where V(x) < I .
Let A fi
Assume that
Aj% is bounded and that within A ^ :
1)
V(z) >
0 ,for x ^ 0, V(0) = 0;
2)
V(x) <
0, for all x ^ 0.
Then the origin is asymptotically stable, and every
solution In
A^ tends to the origin as t
In conclusion,
the theorems given inthis
<®.
section are
concerned only with sufficient conditions for stability.
Thus, the failure to find a Liapunov function to prove sta­
bility in no way implies that the system is unstable.
Theo­
rems for proving instability are also sufficiency theorems
(see for example, Hahn 1963 or Krasovskli 1963 ).
Also,
different Liapunov functions for a given system imply dif­
ferent regions of stability; hence, one function may give a
34
better answer than another«
The next section gives prac­
tical methods of finding Liapunov functions„
2o5
Methods of Constructing Liapunov Functions
In the previous section it was shown that the Second
Method is based on the concept of examining stability in
terms of a Liapunov function.
Thus the application of the
Second Method is dependent upon the construction of an
appropriate Liapunov function.
Unfortunately, the theory
presented in Section 2,4 specifies only the existence of
such a function but not a method for its construction.
It
is the purpose of this section to present two methods of
constructing Liapunov functions so that a region of asymp­
totic stability of a singular point can be found.
Consider first the Zubov method (Kerr 1964, Margolis
and Vogt 1963» Zubov 1961),
This method is the result
of an existence theorem (Theorem 2,5) which gives both
necessary and sufficient conditions for a region A to be a
region of asymptotic stability.
In this ease the required
Liapunov function is a solution of a first order partial
differential equation (the Zubov equation),
The discussion
given here follows Margolis and Vogt (1963),
Theorem 2,5
Let A be an open domain containing the origin, and
let 6A be the boundary of A,
Then A is the region
of asymptotic stability of the system
If, and only If, there exists two functions v(x)
and p(x) satisfying:
1)
v(%) Is positive definite and continuous with
continuous first partial derivatives.
2)
p(x) Is positive definite and continuous.
3)
For 2 In
A, x ^ 0, 0 <
4)
v(jc) — ►!
as x-*»6A.
5)
v(^) satisfies
v(x)
v(x) < 1.
= -p(x)(l - v(x))•
(2.29)
The condition that p(x) be positive definite can be relaxed
to p(x) just being positive semidefinlte (Kerr 1964).
Insight into the
established in light
meaning of
Equation (2.29)iseasily
of the previous stabilitytheorems.
In
a neighborhood of the origin, Equation (2.29) is approximated
by
v(x) « -9(x).
(2.30)
Thus, v(x) is negative definite and v(x) is positive
definite.
That is, v(x) is a Liapunov function, and by
Theorem 2.3 the origin is asymptotically stable.
In the
region where v(j[) > 1, v(x) is positive definite; hence, the
system is
unstable. For v(][) = 1, v(x) = 0implies that the
system trajectory remains on the surface v(x)
t ^ 0.
=1 for all
These conditions are shown in Figure 2.7.
36
REGION OF
ASYMPTOTIC
STABILITY, A
Figure 2.7
Geometric Interpretation
of the Zubov Equation
37
The transformation
V = -ln(l - v ) ,
0 ^ v <
1,
(2.31)
reduces Equation (2.29) to
V(x) = -3(x),
(2.32)
which requires that V(x) be negative definite in the entire
space.
The boundary of the region ofasymptoticstability
is given by
V( x) — ► 00.
(2.33)
Form (2.32) is sometimes more convenient to use than (2.29).
The following example from Hahn (1963 ) illustrates
the Zubov procedure,
Let the system be
*1 = -X1 + 2i12i2
x2 = -x2 ,
(2.34)
and choose
p(a) = 2(x12 + *22 ).
(2 .35 )
The Zubov equation then becomes
+ 2*i 2x 2) + %
(-I2 ) = -2(X12 + x22)(1 • V)> (2*36)
or in terms of Equation (2.32),
The solution for the latter equation is
x 2
V(x) = x 22 + -- -3=---- .
X1X2
(2.38)
Hence, the stability boundary, obtained by letting V
approach infinity. Is
x1x2 = 1 .
(2.39)
This example is Illustrated in Figure 2.8, after the same
problem is solved using the variable gradient method.
For systems having stability boundaries not expres­
sible in closed form, and where f(x) in Equation (2.28) can
be expanded in a convergent power series about the origin,
Zubov (1961 ) proposed the following iteration method of
solution.
The procedure is given in terms of second order
systems only for convenience»
Theoretically, it is valid
for systems of any order.
Let the system equations be written as
^
= f = ^
+ f 2 + f 3 + ...
x 2 = g = e1 + 62 + g 3 +
where f^, g^represent the
terms, etc..
(2.40)
linear terms, f , g^ the quadratic
The solution v(x^,x^) of Equation (2.29) is
approximated by the series
39
v(x1 ,x2 ) = v 2 ^x! »x2 ^ + v 3*xi ,x2^ + •e e,
(2.41)
where v (x ,x ) Is a homogeneous form of the mth degree,
m 1 2
Also, let the sum of the first m - 1 terms of the series
(2.41) be denoted as
m
(x1 ,xp ) = ^
1=2
v (x ,x0),
1
■L
(2.42)
^
and assume that 3(x ^ ,x^) In Equation (2.29) Is a quadratic
form.
The quadratic approximation Is found by the following
procedure.
Let
the quadratic form
V2*X1 ,X2^ = aXl 2 ^ bXlX2 + CX22,
where a, b, and c are unknown constants.
(2.43)
Then substitute
Equation (2.43) into Equation (2.29) and keep only the
quadratic terms, i.e.,
= -9(x).
o ^
Equation (2.44)
(2.44)
2
is solved for the unknownconstants,
and c,to give the
desired form of v^. For
a, b,
the next approx­
imation, substitute
v (3) = v 2 + v3
into Equation (2.29) to get
(2.45)
40
+ f x ~ ^ gl + g2^ = -9(2) (1 - v^3 h . (2.46)
The quadratic terms will cancel, since they satisfy Equation
(2.44).
Now, discard all terms in Equation (2.46) which are
not homogeneous of the third degree to obtain
which is solved for the unknown constants in v .
The mth order approximation is
+ | ^ si - W
3^ *
( 2 -48)
where R (x.,x_) is the homogeneous form of the mth degree
m 1 2
given by
m-1
m-1
V h 1* , ’ - » v 2
l2 -49>
i=2
i=2
The approximate region of asymptotic stability for any iter­
ation is obtained by finding the largest closed curve,
v^m ^(x^,x^) = constant
about the origin which is contained by the curve,
(2 .50 )
Margolls and Vogt (1963 ) present the required theorems which
establish this procedure.
In summary, the Zubov method Is an exact formulation
for finding region of asymptotic stability.
Its major dis­
advantage is that no criteria exist for choosing 0 in
Equation (2.29).
This problem is still not solved as evi­
denced by the recent literature (Fall side et, al. 1965 )•
The approximation scheme is systematic and is easily imple­
mented on a digital computer, with the major disadvantage
being the slow convergence of the series solution (Margolls
and Vogt 1963 , Rodden 1964)0
In contrast to the Zubov method the variable gra­
dient method is an approximate scheme for finding Liapunov
functions which are tailored to the nonllnearlty of the sys­
tem so that the largest region of stability can be deter­
mined with a minimum of effort.
It is based on choosing n
undetermined elements in the gradient function,VV, such
that
v = v v ’f
(2.52)
is forced to be at least semidefinlte in some region about
the origin.
A general outline of the method is given next
(Schultz and Gibson 1962 ).
42
Let V V have the following form:
aH (z)xi + ai 2 (z)l2 +
+ ain(z)Zn
a21(z)ll + a 22(z)X2 +
+ a2n(z)ln
an i ^ ) xi + "
+ an n (z ) x
VV =
where the functions a^j(x) are unknown.
(2.53)
Then V(x) is formed
as
n
V(x) = V V » f = V " (VV) f ,
i=l
where the (VV)^ are the rows of VV.
(2.54)
The next and most cru­
cial step of the procedure is to choose the unknown a
so that V(x) is at least negative semidefinite.
The
Liapunov function, V(x), is found by integrating V V ,
2
i.e0 ,
i^ •(x2~ • ••=xn=:0)
V(x) = ^ V V « d x = ^
o
(x)
(VV)1dx1
+
o
x1 ,x2 ,(x^=...=xn= 0 )
< W ) 2d x 2 + . . .
n
+
\
( W ) n dx n .
(2.55)
43
subject to the condition
j(v v ),
d(vv)
n, 1 / j.
(2.56)
This result insures a unique V and also insures that the path
of integration is arbitrary.
Once V and V have been found,
the theorems of the previous section can be applied to de­
termine the stability properties of a region about the
singular point.
Consider the solution of the system given by Equations
(2.34) using the variable gradient method.
In this case the
unspecified V(jc) takes the form
x22(-a22 + 2a12xl2*"
If
and
(2.57)
are made zero. Equation (2.57) becomes
V(jc) = - X j ^ a ^ d - 2x 1 x 2 ) - x22 (a22).
(2.58)
Thus, V(2 ) is negative definite for
(2.59)
For convenience, let
a
11
* a
22
2
.
(2.60)
44
Then the integration In Equation (2.55) gives the Liapunov
function
V(x) =
(2.61 )
+ x/.
The approximate region of asymptotic stability for this
Liapunov function is the largest circle
x^2 + x^2 = constant
(2.62)
which is inscribed in the curve
xlx2 ’ 2*
This result,
bility
(2.63)
along with the exact region ofasymptotic sta­
asdetermined by the integration of the
Zubov equa­
tion, is shown in Figure 2.8.
Rodden (1964) solved the above example using the
Zubov iteration scheme.
His results, carried up to the
tenth degree in v, clearly indicate the slow convergence of
the iteration scheme.
In fact, the v ^ ^ approximation is
(2 )
not significantly better than the v
approximation. Also,
the v
(2 )
approximation is only slightly better than the re­
gion obtained using the variable gradient method.
The two methods of finding the region of asymptotic
stability of a singular point presented in this section are
applied to the limit cycle problem in Chapter 3*
The re­
sults carry over directly since a two dimensional unstable
limit cycle is actually the boundary of the region of
i+5
EXACT
BOUNDRY
APPROXIMATE
BOUNDRY
Figure 2.8
Approximate Region of Asymptotic Stability
Determined by Variable Gradient Method
asymptotic stability of the origin.
A Liapunov function is
developed in the next chapter which is referred directly to
the limit cycle instead of the singular point at the origin.
CHAPTER 3
CONSTRUCTION OP LIAPUNOV FUNCTIONS
FOR SECOND ORDER LIMIT CYCLE PROBLEMS
3.1
Introduction
In the previous chapter two general procedures are
indicated which, when combined with the Second Method of
Liapunov, provide a convenient means for solving second
order limit cycle problems.
These procedures are s
(1) the
estimation of the region of asymptotic stability of a sin­
gular point and
(2) the construction of the annular
region, R, used in the Poincare-Bendlxson Theorem.
Section 3.2 is concerned with estimating the region
of asymptotic stability via the Zubov and variable gradient
methods, while in Section 3.3 Liapunov methods are used to
find contact curves for the construction of the annular
region, R,
Unfortunately, the Liapunov methods considered in
Sections 3.2 and 3.3 are dependent on the structure of the
two dimensional space and are inadequate for the study of
limit cycles in third order systems.
In Section 3*4 a new
formulation is proposed where the Liapunov function is de­
fined directly in terms of the limit cycle.
This formulation
represents a restricted form of the Poincare-Bendlxson
Theorem and is valid for third order systems.
It is shown.
48
for the second order limit cycle problem, that this formu­
lation reverts to the solution of the Zubov equation.
The
application to third order problems is considered in
Chapter 4 e
3®2
Estimation of the Region of Asymptotic Stability
In this section the methods discussed in Section 2,5
are applied to the second order limit cycle problem.
More
specifically, the variable gradient and the Zubov methods
are used to find an interior bound to the limit cycle.
The
Zubov method, being an iterative scheme, specifies a bound
which tends to the limit cycle from the interior as more
iterations are taken,
A fundamental property of second order systems,
stated in the previous chapter, is as follows:
the limit
cycle divides the state plane into two disjoint regions, an
inner region R
and an outer region R , For the systems of
1
o
interest, i,e., systems which have only one limit cycle and
one singular point Interior to the limit cycle, the sta­
bility properties Of R^ are the same as those of the singu­
larity,
In this case the stability properties of R^ can be
reversed simply by replacing t with -t in the system equa­
tions,
Hence, the methods of Section 2,5 are equally appli­
cable when the limit cycle is either stable or unstable.
49
Consider first the Zubov method.
The Zubov equation,
V(x) = -ts(x) (1 - V(x) ),
(3.1)
is valid for finding the region of asymptotic stability of a
stable singular point.
If the system under consideration has
a stable limit cycle, then the enclosed singular point is
unstable.
In this case either the system equations can be
reversed in time or else t can be replaced with -t in
Equation (3.1) to obtain
v(jc) = P(x)(l - V(x))
(3.2)
In any case the limit cycle represents the boundary of the
region of asymptotic stability and is given by
V(x) = 1
(3.3)
This discussion is summed up in the following theorem
Theorem 3.1
Let
2. = I (*)
(3.4)
represent a second order system having one stable
limit cycle which encloses one singularity.
Then
the limit cycle is given by
V(x) = 1
which Is a solution of
(3.5)
50
v(x) = e (x )(l - v (x ))
(3.6)
For the systems which satisfy Theorem 3.1, the limit cycle
can always be found either by the direct integration of the
Zubov equation, if this is possible, or by applying the
Zubov Iteration procedure discussed in Section 2.5.
Both Margolis and Vogt (1963 ) and Rodden (1964)
have implemented the Zubov iteration procedure on a digital
computer and have given examples which show the solution of
the Zubov equation approaching the limit cycle as more
iterations are taken.
Unfortunately, the convergence of
the iterative solution is slow with the rate of convergence
depending on the form chosen for p(x).
The variable gradient method, on the other hand, is
not a systematic Iterative procedure»
The principal advan­
tage of the variable gradient method is that a reasonably
good bound, in the sense that it approximates the form of
the limit cycle orbit, can usually be found with a minimum
of effort.
To Illustrate the method, consider the van der Pol
equation.
(3.7)
where e is taken as unity for convenience.
Since the damp­
ing term in Equation (3.7) is negative for small x, and
positive for large x, then it appears that a stable limit
cycle exists and the origin is unstable.
It should be noted
51
that the previous statement is Just a guess and not a proof
for the existence of the limit cycle.
To apply the variable gradient scheme, Equation (3.7)
must first be expressed in state variable form.
Again for
convenience choose the system coordinates to be the phase
variables, i.e.,
X1 = X2
•
2
x2 = -x^ + *2 ~ X1 X2"
(3.8)
The first step of the variable gradientprocedure is to ob­
tain
thegeneral form for V which in this case is
V = ^
(-a21 ) +
" a 22 + a 21 “ *21*1 ^ +
X22(ai2 + a 22 " a22Xl2)-
(3*9)
It is easily shown that the origin is unstable by examining
the sign of the eigenvalues of the linearized part of system
(3.8).
Thus V is forced to be positive semidefinite.
This
is accomplished by letting the coefficients of the x^x^ and
2
x2 terms be zero, and then determining the region where the
2
coefficient of the x^ term is positive, i.e., by letting
*12 = -*22 + *22X1 2
(3*10)
a22 = constant
(3.11)
The relation between
and
is determined from Equation
(2 .56 ), which in this case reduces to
al2 = a ^ (a21Il )*
Hence,
(3-13)
becomes
a—
g
a21 = -*22 + “ f2*! *
Eliminating
in Equation (3.12) gives
(3-l4)
, in terms of
*22' 88
-fh-
*11 = 2*22 - 3a 22Xl2 +
Thus,
constant
, CL2\ t ai2 are
(3*15)
specified in terms of the
Substituting Equations (3 .10), (3.14), and
(3.15) into Equation (3.9) gives
2
V = CL22X1 d
x 2
3
(3.16)
Clearly, V is positive semldefinite for
xJ < v 5.
For convenience, let
“ I*
(3.17)
Then the gradient of V is
Integrating y v as shown in Equation (2.55) results in the
Liapunov function
+ I 1?2,
(3.19)
The appropriate inner bound to the limit cycle is obtained
by finding the largest value of V * constant for which the
closed curve (3.19) is contained in the region specified by
the inequality (3.17).
This bound is shown in Figure 3.1.
Another common form for expressing the van der Pol
equation is in terms of Lienard coordinates.
In this case
the van der Pol equation becomes
(3 .20 )
The van der Pol equation (3.20) is actually a special case
of the more general Lienard equation, for which the coor­
dinates were originally developed.
and McRuer (1961).
See, for example, Graham
The advantage of using Lienard coor­
dinates for the van der Pol equation is that the limit cycle
becomes a smoother curve and approximates a circle fairly
closely.
This property is due to the fact that x^ is
54
LIMIT
C YC LE
INNER
BOUND
Figure 3.1
Inner Bound for van der Pol Equation
via Variable Gradient Method
55
determined by an integration process in Equation (3e20)„
whereas in the phase coordinates of Equation (3«8) it is the
result of a differentiation.
Carrying this idea a step far­
ther » it would appear that the variable gradient method
would give a simpler curve for the inner bound if applied to
the Lienard coordinates.
This is exactly what happens , with
the inner bound taking the form of a circle.
The variable gradient method is considered again in
the next section where both an inner bound and an outer
bound are obtained for the limit cycle,
3,3
Application of the Poincare-Bendixson Theorem
This section is concerned with a method of bounding
the limit cycle between two .concentric closed curves.
The
procedure employs the concept of contact curves, originally
proposed by Poincare„ for finding a suitable annular region,
R, required in the application of the Polncare-Bendixson
Theorem (Hayashi 1964, Stoker 1950),
By combining the vari­
able gradient method with the concept of contact curves, a
method is developed for bounding a limit cycle between two
closed curves representing Liapunov functions.
The idea of
applying Liapunov functions in this manner is due to Szego
(1962)9 but the development of the scheme around the vari­
able gradient method is original.
First, the general prop­
erties of contact curves are given, and then they are
applied to the limit cycle problem.
56
For presenting the general properties of contact
curves, let
xi = fi (-)
= f2(Z>
(3.21)
represent the second order system under consideration, and
let a(a) be a scalar function.
The slope of the system
trajectories is given by
dx9
f9
-T-2 = 7^.
dxl
fl
(3.22)
and the slope of <x(x) = C, C a constant. Is
<ix
is.
ax
^.23)
ax2
With this background, a contact curve can be defined in the
following manner:
Definition 1*1
Contact Curve
The locus of points at which the direction field
of system (3.21) is tangent to a scalar function,
a(i), i.e., where
(3.24)
is called a contact curve.
57
Note that If
(3.25)
then a(x) = C represents a solution of (3.21).
Given next is the geometrical interpretation for
various forms of (%(%).
If a(x) is the linear function.
a(x) = ax
+ big
(3 .26 )
then the contact curve is an isocline, since at every point
of the contact curve, the direction field is tangent to
Equation (3.26) which has a constant slope (see Figure 3.2).
Each set of constants a,b in Equation (3.26) defines an
isocline.
For the case where 0 (5 ) is a quadratic form, let
<1(5) = x ^ + x22
(3.2?)
and assume that the contact curve is a closed curve contained
in a bounded region of the state plane (note that a(x) = C
is also a closed curve).
In this case the contact curve
represents a curve along which the direction field varies as
If the curve
(3.29)
58
TRAJECTORIES
CONTACT
CURVE
<*= CONSTANT
CURVES
Figure 3.2
Contact Curve when a(x) Is a Linear Form
59
inscribes the contact curve, and the curve
a(x) = CBai
(3.30)
circumscribes it, then every periodic orbit of system
(3.21) must necessarily lie in the annular region
Cmln < *(3)
<
(3-31)
Otherwise there would exist points of the contact curve
outside a(x) = C
(or inside a(x) = C , ), contradicting
max
min
the definition of these curves (see Figure 3.3). This prop­
erty of contact curves also applies to closed curves,
a(%)
* C,of
closed
higher degree.If the contact
andbounded,then only an inner
can be found.
curve
is
not
bound, <x(x)
=C . ,
mm
Examples illustrating this classical method
are found in Hayashi (1964) , Stoker (1950), and MacFarlane
(1964).
If a(]c) is a Liapunov function, then the concepts of
the Second Method of Liapunov can be applied to construct
contact curves.
Although the approach is straight forward,
it was not until 1962 that Szego (1962 ) used Liapunov func­
tions to find a contact curve for the van der Pol equation.
Szego1s ideas are extended here to include the variable
gradient method as a means of generating contact curves.
To obtain a contact curve using the Second Method, a
Liapunov function has to be found such that its time deriva­
tive, evaluated along the trajectories of the system, can be
60
CONTACT
z CURVE
m ax
LIMIT
' CYCLE
Figure 3.3
Bound for Second Order Limit Cycle
Using Contact Curves
61
expressed in the form,
v(z) = ;(z)(i - U(2 )).
(3 .32)
U(z) = l
(3.33)
where
is a closed curve and 3(z) is a semldefinite function.
Equation (3.33) represents the desired contact curve since
along this curve,
V
- %
fi +
(3-34)
is zero* which implies that the direction field is tangent
to the closed curve given by the scalar function
(3.35)
V(x) = C.
Note that if the constant, C, in Equation (3.35) is made
sufficiently large so that this curve completely encloses
the
contactcurve,
trajectories
then V is negativesemldefinite and all
crossV(^) = C In an inwarddirection.
this curve must also enclose the limit cycle.
Hence,
The smallest
outer bound for the limit cycle, denoted by the curve
V U > = Cmax.
(3.36)
is found by choosing C = C
so that the curve (3.35) clrmax
cumscribes the contact curve. A similar argument is used to
62
find the Inscribed curve,
v(— > = cm l n .
(3.37)
i.e., the largest inner bound for the limit cycle.
It should be noted that the above inner and outer
bounds depend on the contact curve.
Since contact curves
are not unique, then the bounds obtained using one curve may
give a better answer than for another.
It should also be
noted that for some systems the contact curves may be un­
bounded so that only an inner bound, as determined in the
previous section, can be found (Hayashi, 1964).
The variable gradient scheme is considered next as a
means for finding a suitable form for V.
Since the method
to be presented is basically a trial and error scheme, it is
best demonstrated by an example.
Consider the van der Pol
equation examined in Section 3.2, i.e..
(3.37)
Following the usual variable gradient procedure, V in terms
of the unknown a's is
(3.38)
63
Since the simplest form a closed curve can assume Is a
quadratic form without the cross term, it seems logical to
look first for a contact curve of this form.
V =
x 12 (-Ax 1 2
- B x 22 + 1).
Thus assume
(3.39)
where
U = Ax 2 + Bx 2 .
(3.40)
Equation (3*39) requires that the a's in Equation (3.38)
satisfy
“ll + *21 " *22 " *21Xl2 " 0
(3.41)
*12 + *22 ' *22=1^ = -Bll2
-&2i e
Since a,, appears
11
(3‘42)
(3.^3)
only in Equation (3.41),let
*11 = *22 " *21(1 " Xl2)>
(3>4)
= a 22 + 1 • x^2 (l + A) +
(3.45)
which becomes
after Equation (3.43) is substituted into Equation (3.44).
Note that <x^ is only a function of x^.
If
is assumed
to be a constant. Equation (2.56) of Section 2.5 can be
used to find
64
*12 = 5 ^ (a2ixi) = 3Axi2 ' 1*
Then Equation (3.42), In terms of
(3.^6)
• is
3Ax1 2 - 1 + a 22 - a 22x1 2 =-Bx1 2.
To satisfy Equation (3.47) choose
(3.47)
to be unity.
B = 1 - 3A.
Then
(3.48)
Hence, Equation (3.40) becomes
U = Ax
2 + (1 - 3A)x2 = 1
(3.49)
which represents a contact curve for all
0 < A < 1.
(3.50)
For the sake of illustration, let
A = J.
Then the contactcurve
(3. 51)
is the circle
l xi 2 + i x22 = ^
(3-52)
and the Liapunov function, determined by integrating V, as
indicated in Equation (2.55) Is
V = i?Xl6 - A X/
+ 4X!3X2 + X2 " X1X2 + K
2*
( 3
'
5 3 )
65
The inscribing and circumscribing Liapunov functions» as
well as the contact curve in Equation (3<>52), are shown in
Figure 3°^°
Note that in the extreme case, 1,6*, for
A = i
(3.54)
3
the contact curve (3.49) degenerates into two parallel lines
and the problem reverts to the one already solved in Section
3.2,
The Liapunov functions considered in both this sec­
tion and Section 3.2 treat the limit cycle problem in terms
of the singularity at the origin.
In the next section a
Liapunov function Is proposed which is defined in terms of
the limit cycle,
3.4
Liapunov Functions Definite with Respect
to t M Limit .Cycle
The methods presented in the previous sections of
this chapter are basically methods for examining the sta­
bility of the singular point at the origin.
The Liapunov
functions for this case are positive definite with respect
to the origin and give meaningful results for the second
order limit cycle problem only because of the simple struc­
ture of the two dimensional plane,
Furthermore, these
methods cannot be carried over to third order limit cycle
problems.
This aspect is discussed in Chapter 4,
66
V(x) =
CONTACT
CURVE
m ax
m in
LIMIT CYCLE
Figure 3.4
Bound for van der Pol Example
of Section 3*3
6?
In this section a Liapunov function definite with
respect to the limit cycle is defined and its properties
developed.
It is proposed that this Liapunov function
represents a general approach to the limit cycle problem,
but in this section only second order limit cycles are
studied.
Discussion of the proposed Liapunov function for
third order systems is deferred until Chapter 4.
The geometrical properties of a Liapunov function for
studying second order limit cycles are established by examin­
ing the Poincare-Bendixson Theorem.
From this theorem it
is seen that the desired Liapunov function should represent
two closed curves bounding the limit cycle, with all trajec­
tories crossing these curves in a direction toward the limit
cycle.
Furthermore, the Liapunov function must have an iso­
lated minimum along the limit cycle since this is the stable
state to which the system trajectories tend as t
Unfortunately, this last condition cannot be satis­
fied since the limit cycle is not known & priori.
To cir­
cumvent this difficulty the Liapunov function is expressed
as a known function
V = V(w - 1)
(3.55)
of an unspecified closed curve
W(z) = 1.
(3.56)
such that for each value of V, Equation (3.55) represents
68
two closed curves which bound the curve (3*56)•
Furthermore,
It is assumed that
V(0) = 0
(3.57)
The curve (3*56) is forced to be the limit cycle by imposing
the condition on the Liapunov function that all trajectories
tend to this curve as t —►«», i.e., V is negative definite
with respect to this curve.
In terms of equations the
Liapunov function, when differentiated with respect to time,
takes the form
(3.58)
If W(a) is chosen to have the form
W(z) = h(j),
(3.59)
such that
(3.60)
is negative definite with respect to the curve (3.56), then
Equation (3.59) specifies the limit cycle.
A Liapunov func­
tion of the form described above is shown in Figure 3.5.
The Justification of the procedure outlined above is
considered next.
After formally defining a Liapunov function
definite with respect to the limit cycle, it is shown that
such a function can indeed be used to examine the stability
69
LIAPUNOV
SURFACE
W(x) =
Figure 3.5
Liapunov Function Positive Definite
with Respect to the Closed Curve W (x)
70
of limit cycles.
The proposed method is then verified.
With this background it is shown that the class of second
order problems outlined in Chapter 2 has limit cycles which
are Liapunov stable so that the Second Method approach is
valid.
Finally, second order limit cycles are examined to
illustrate the proposed method.
Assume that the nth order system,
Z = 1 (z)
(3.61)
has one limit cycle which encloses only one singular point
located atthe origin.
a Liapunov
In a
manner similar to Definition 2.7#
function definite with respect to the limit cycle
is defined as follows:
Definition 1.2
Liapunov Function Definite
with Respect to the Limit Cycle
Let L represent the limit cycle of system (3.6l).
The scalar function V(,$) is a Liapunov function
definite with respect to the limit cycle in a region
N, if it satisfies:
1)
V(^) is positive definite with respect to the
limit cycle, i.e.,
V(z)
=0, for x
eL
V(z)
>0, for x
f L;
(3 .62 )
2)
V(%) is of class C1 ;
3)
V(%) is negative definite with respect to the
limit cycle.
71
The following theorem. If properly Interpreted, shows that
a Liapunov function definite with respect to the limit cycle
Is valid for examining stability of limit cycles,
theorem JU2
If there exists a Liapunov function definite with
respect to the limit cycle, then the limit cycle
is stable.
To prove this theorem, assume that there exists a Liapunov
function V(%) definite with respect to the limit cycle
L :^ =
(t ), and choose a number €. > 0 such that V(x) is
defined in the region
IU ‘ ijl
< E.
(3.63)
From Definition 3.2, V(a) > 0 on the surface
II* - * e ll
{3-6k)
=e*
Hence, let
V(js) ^ X > 0 for
Since V(%)
||x - xe || = €
•
(3.65)
is continuous, there exists a6 > 0, with 6 < e
such that
V(x) <: | for
||z - x^||
^ 6.
(3.66)
Choose ^(t) and a suitable %^(0) such that
||z(0) - z e(0)||
$ 6.
(3.6?)
72
Since V(x) Is a nondecreasing function along x(t), then
V(x(t)) ^ V(x(0)) < ^
(3 .68 )
as long as
x(t) - xe (t) || < e.
(3.69)
But £(t ) cannot penetrate the surface (3.64) since on this
surface
v(%) a x.
(3.70)
Thus,
Z(t)
- JEe (t)
^ 6 for all t ^ 0
and Definition 2.4 Is satisfied.
(3.7D
To verify that the tra­
jectories tend to the limit cycle as t — ► *>, note from
Definition 3.2 that
V < 0.
(3,72)
v(t) « V(z(t))
(3.73)
Then
decreases monotonlcally along z(t) when condition (3*67 ) Is
satisfied.
Thus V(t) tends to a constant a ^ 0 as t — ►«>.
Clearly,
V(t) ^ a for t ^ 0.
Suppose that a > 0.
(3-74)
Then x(t) Is bounded away from ^ ( t ) .
73
and hence, V is bounded away from zero.
Denote a lower
bound on V for all t ^ 0 as
V < - 3 < 0.
(3.75)
Then
t
V(t)
= \ V(s)ds <
-3t.
But this result contradicts statement (3.7*0.
(3.76)
Hence, a = 0
and (assuming a proper choice of 2 ^(0 ))
Z(t) — ► x e (t) as t —♦ ®°.
(3.77)
To show that the proposed procedure defines the limit
cycle, the following theorem is given.
Theorem 3.1
Let
V = V(W - 1)
(3.78)
be a Liapunov function of system (3.61) definite with
respect to the closed curve
W(a) = 1,
(3.79)
and form
(3*80)
If
W(z) = h(x)
(3.81)
74
Is chosen such that
1■
» - 8z>
Is negative definite In the neighborhood of the
curve (3.79), then Equation (3.81) defines the
limit cycle of the system.
The proof of this theorem follows directly from the reali­
zation that If In the neighborhood of the curve (3.79) the
Liapunov function satisfies the conditions
V > 0, for W ^ 1
V = 0, for W = 0,
(3.83)
then the following must be true:
aV
> 0, for W > 1
M w - i)
Av
M W - 1)
= 0, for W = 1
a(wV- i) <
° ’f°r w < 1-
(3-84)
Consequently, any function h(x) of Equation (3.81) which
has the form
h(jt) <
0,for W > 1
h(jc) =
0,for W = 1
h(z) >
0,for W < 1
(3.85)
guarantees that V Is negative definite with respect to
75
curve (3*79).
In particular choose
W = e(x)(W - 1),
(3.86)
where 9 (x) Is positive definite In the neighborhood of
curve (3.79).
But Equation (3.86) Is the Zubov equation.
Thus, by Theorem 3.1* Equation (3.79) Is the limit cycle of
the system.
Note that Equation (3.86) Is not unique, but
It does specify a unique limit cycle.
The properties considered above are general and are
not restricted to any particular system order.
However, the
remainder of this section is devoted to only second order
systems.
The next theorem Justifies that the limit cycles
of the class of systems of interest are indeed Liapunov
stable.
Theorem 2*4
If system (3.61) is a second order system, then its
limit cycle is Liapunov stable.
This theorem is verified by using the properties developed
above.
Assume that the limit cycle given by
W(j) = 1
is stable
the
(orbitally
or otherwise).
(3.8?)
Then by Theorem 3.1
Zubov equation,
W = -9(W - 1),
must be satisfied.
(3.88)
Recall that the Zubov equation implies
76
only that
the singularity at the origin exhibitsLiapunov
stability
and is not in any way related to thetype
bility of
the limit cycle,
U =
Now
of sta­
let
(3.89)
w
be a function definite with respect to the limit cycle.
u - (.w +..i ) (w -
(3 .90 )
i)w.
w2
Then
By substituting Equation (3 .88 ) into Equation (3-90), U
becomes
u = - i i « —iJ L l.(w
-
I) 2
(3 .9 1 )
tr
which is negative definite with respect to the limit cycle.
Hence, U is a Liapunov function.
Summarizing the above
proof, the limit cycle of system (3*61) is completely de­
fined by the Zubov equation whether the limit cycle is
orbltally stable or Liapunov stable.
The Zubov equation
implies the existence of a Liapunov function definite with
respect to the limit cycle, and the Liapunov function in
turn implies the limit cycle is Liapunov stable.
The solution of the general second order problem via
the proposed Liapunov function is outlined next.
Because
of the simple structure of the two dimensional space, it is
possible to specify a form for the Liapunov function which
is suitable for examining local and global stability for
77
any second order limit cycle enclosing only one singularity.
Such a form is
2
v = (" - 1) .
(3.92)
Note that another possible choice for the Liapunov function
is
V = (w - l)2 ,
(3.93)
but this equation is restricted to
V < 1,
(3.94)
and hence, suitable for examining only local stability.
In
Chapter 4 where it is difficult to obtain a third order
global form analogous to Equation (3*92), Liapunov func­
tions valid only in the local sense are used.
The form
given in Equation (3.92) is used In the following discussion.
For every constant V > 0, the level curves of Equation
(3 *92 ) given by
W(z) - 2_±_z ± J(V-±-Z)Z . 1
(3.95)
represent two closed curves which constrain the limit cycle
to the annular region
,-------j
j—
-
U-W)
14-2 -
- 1 «
1
(see Figure 3.6)*
To show the existence of a limit cycle.
78
LIMIT CYCLE
V(x) = CONSTANT
Figure 3.6
Annular Region Determined by
Proposed Liapunov Function
79
It must be shown that there exists a function W(x) which
makes V negative definite with respect to the limit cycle.
Differentiating Equation (3.92) with respect to time gives
v = - (1 +
W)w .
(3. 97)
w2
This equation assumes the negative definite form
V = - P U t W )(1 w
W)2
(3.98)
if W is chosen to be
W * 3(1 - W).
(3.99)
Thus, Equation (3.99) represents the auxiliary equation
which in this case is simply the Zubov equation, and the
limit cycle is
W(x) = 1.
(3.100)
Since the formulation is reduced to the solution of the
Zubov equation which was already considered previously, no
specific examples are considered in this section.
It is seen from the above discussion that a dis­
tinctive property of the proposed formulation is that the
given Liapunov function is reduced to a set of auxiliary
equations which define the limit cycle.
For the second
order case the auxiliary equation is the Zubov equation
which represents the stability of a singular point at the
80
origin.
Two auxiliary equations are obtained In the third
order case which can be Interpreted as representing the
stability of two surfaces whose Intersection defines the
limit cycle.
Third order systems are considered In Chapter
4.
To conclude this section, an Interesting property of
the proposed Liapunov function Is discussed which Is not
apparent In Liapunov functions defined with respect to the
origin.
In the general Liapunov formulation Liapunov func­
tions defined with respect to the origin need not be posi­
tive definite but only satisfy the condition that V and V
have opposite signs, I.e., that the Inequality
VV < 0
Is satisfied (Krasovskll, 1963 )♦
(3.101)
Since for this case the
Liapunov function Is always definite, Inequality (3.101)
Implies only that one sign Is associated with the Liapunov
function when It Is evaluated along a system trajectory.
In the case of Liapunov functions definite with respect to
the limit cycle. Inequality (3.101) takes on a broader
meaning.
For example, a Liapunov function which is not
positive definite usually takes one sign for all closed
curves interior to the limit cycle and the opposite sign for
curves exterior to the limit cycle (see Figure 3.7).
However, the previously developed geometrical meaning is
lost in this process.
81
LIMIT CYCLE :
< V = - 1/2
' (W = 2)
V =1
(W =1/ 2)
Figure 3.7
V = Constant Curves for the
Liapunov Function given In
Equation (3.102)
82
To illustrate this property, the method developed in
this section is applied to the general Liapunov function
v -
I ■~ w.
(3-102)
Differentiating Equation (3.102) with respect to time gives
V =
(3.103)
wz
which satisfies the inequality (3.101) if W is chosen as
W = 9(1 - W).
(3.104)
It is interesting to note that the auxiliary Equation
(3.104) is again the Zubov equation; hence, the limit cycle
is
W(a) = 1.
(3.105)
If 9 (5 ) in Equation (3.104) is made equal to W(x), then
Equation (3.103) becomes
v = - U "- w),
(3.106)
and
(3.107)
V * -V
is satisfied.
The latter equation is considered in greater
detail in Chapter 4.
that if V =V(%)
is
Furthermore, it can easily be shown
a Liapunov function, which
nite in sign but satisfies Equation (3.107).
is not defi­
then
83
u = v2
(3.108)
Note that Equation (3.109) is similar to Equation (3=107).
In summary, a Liapunov function is proposed which is
referred directly to the limit cycle instead of the singu­
larity at the origin.
This formulation is basically a re­
stricted form of the Po1neare-Bendlxson Theorem where the
bounding curves of the annular region, R, take on the exact
shape of the limit cycle.
The Liapunov function, V(x)„ is
represented by the bounding curves, and for successively
smaller values of V = constant the geometrical represen­
tation of these curves is a sequence of nested annular re­
gions tending to the limit cycle.
For second order limit
cycles the proposed formulation does not lead to any new
methods of solution but does indicate how the limit cycle
problem Is reduced to an auxiliary one, i.e., the Zubov
equation.
This property is very important in the third
order ease where no established methods similar to the
Zubov equation exist.
The remainder of this work is con­
cerned with the third order limit cycle problem.
CHAPTER 4
LIAPUNOV FUNCTIONS FOR THIRD ORDER LIMIT CYCLE PROBLEMS
4.1
Introduction
In this chapter Liapunov methods for estimating
limit cycles in third order systems are considered.
In
\
Section 4.2 a survey of the literature related to this sub­
ject is presented.
This survey is followed by a discussion
showing that Liapunov functions positive definite with re­
spect to the singular point at the origin are very restric­
tive when applied to three dimensional limit cycle problems.
For the purposes of this work the third order limit
cycle problem is divided into two classes.
In Section 4.3
third order systems are considered which can be reduced to
second order systems where the existing Liapunov techniques
apply.
The reduction is effected by locating a stable sur­
face (If one exists) which contains the limit cycle (Szego
and Geiss 1963)0
Unfortunately, no criterion exists for
judging whether a stable surface exists for a particular
third order system.
In Section 4.4 the general limit cycle problem is
discussed in terms of Liapunov functions definite with re­
spect to the limit cycle.
It is shown for the third order
problem that the Liapunov function represents a toroid en­
closing the limit cycle and exhibits the properties of the
84
three dimensional version of the Poincare-Bendixson Theorem
discussed in Chapter 2, where the trajectories enter the
toroid and circulate around it as they tend to a fixed
closed curve.
This Liapunov function can also be inter­
preted as the sum of two Liapunov functions, each definite
with respect to a stable surface such that the intersection
of these surfaces defines the limit 6ycle„
Using the latter
interpretation, it is shown that the proposed formulation is
a generalization of the problem considered by Szego and
Gelss (1963) in Section 4.3.
Examples are given to illus­
trate the various aspects of the theory.
4.2
Previous Work
The literature on the investigation of limit cycles
in third order systems via Liapunov functions consists of
two works.
Both of the works are considered in this section
and their limitations examined.
The section is concluded
with a discussion of why the Liapunov schemes considered in
Sections 3.2 and 3.3 cannot be extended to the study of
limit cycles in third order systems.
In his book Zubov (1963) determined the region of
asymptotic stability for the limit cycle of the third order
system,
86
X1 = X1 + X2 ' X1 (X12 + X22)
X2 = -X1 + X2 - X2 (X1 2 + X22)
(4.1)
X3 = •X3 >
by solving Equation (2.29).
by Zubov is as follows:
The method of solution given
Using the transformation (2.31),
the Zubov equation is reduced to
(4.2)
V = -e(x).
Then 0 (x) is chosen as
_ 2 li - (X1
0(x) =
X2
}] [ l + xl 2
x22]
+ 21*
(4.3)
Xl2 + X 22
to obtain the partial differential equation
%
X1 + X2 - X1 (X1
-2 [l -
d x 3x3
+ X2 >
(XjZ +
-X1 + X2 - X2 (X12 + X22)
x 2 2 )]
[l +
2
"
X1
(4.4)
+ x 2 2]
2x 2 .
2
+ X2
The solution to this equation is
[l - ( : / + x ^ )
V
+ x.
xl2 + x22
(4.5)
8?
which represents a toroid in
x
in the x ^ , x^-plane.
2
1
about the limit cycle
+ x 2
2
1
(4.6)
The region of asymptotic stability,
determined by letting
V ( x ) «
(4.7)
is seen from Equation (4.5) to be the entire space exclusive
Although this result is significant, no indication
is given on how to formulate the equation
(4.8)
V = -e(x)
for a particular problem.
Recall that this situation is
similar to the difficulty apparent in the previous chapter
where no criterion exists for choosing P(%) in the two
dimensional Zubov equation.
For second order problems
other Liapunov methods such as the variable gradient scheme
can be used to develop insight in choosing p(x); but un­
fortunately, these methods cannot be used in conjunction
with the third order limit cycle problem.
This difficulty
is partially overcome in the Liapunov formulation given in
Section 4.4.
The approach taken by Szego and Geiss (19^3) is to
find an asymptotically stable surface which contains the
limit cycle.
Then by constraining the system to this
88
surface, the problem Is reduced to one of two dimensions.
Szego considered the third order system
x
2
x
3
x3 = '*1 " x2 • x3 + e(1 • xl2
2x1x2 )x2
"
+
(4.9)
where £ > 0 .
He then showed that all the system trajec­
tories approach the surface
x, = -Xj
+ €(1
-
x
1 2 )x 2
(4.10)
(the appropriate theorem to verify that the trajectories
Indeed approach this surface Is given In Section 4.3). The
system when constrained to the surface (4.10) becomes
(4.11)
Xg = -x1 + €(1 - x1 )x2
which Is the familiar van der Pol equation having a stable
limit cycle.
A few remarks are In order concerning the properties
and limitations of this formulation.
vided into two distinct parts:
ically stable surface, and
behavior on this surface.
The solution Is di­
(1 ) finding the asymptot­
(2 ) examining the limit cycle
Since the surface is
89
asymptotically stable, the system trajectories cannot Inter­
sect It but tend to It as t — ►<».
Hence, the method excludes
all systems In which the trajectories spiral around the
limit cycle as they tend to the limit cycle.
If a system
has a stable surface, then any motion constrained to this
surface must have the effect of reducing the degree of free­
dom of the system by one.
In some cases this property ap­
pears as a pole-zero cancellation In the system equations.
This Is exactly what happens In the example given by Szego
and Gelss (1963) since system (4.9), In terms of a single
third order equation. Is
x* + (1 - e )x + (1 - e )x + x = -e(x2x + 2xx2 + x 2x ) ,
(4.12)
which becomes
(s2 - €s + 1)(s + l)x = -€(s + 1 )x2x
(4.13)
when expressed In terms of the differential operator s.
Note that after the cancellation of the common terms.
Equation (4.13) results In the van der Pol equation (see
Figure 4.1).
Thus, system (4.9) represents a very artifi­
cial third order limit cycle problem.
This method Is con­
sidered further In the next section.
It Is clear from the above discussion that the appli­
cation of Liapunov functions to the study of third order
limit cycle problems Is Just beginning to be explored.
To
conclude this section, the difficulty of applying existing
R= 0
x
2y=x^x
s+1
(s2 -es+lXs+1)
Figure 4.1
Block Diagram of System
considered by Szego and
Geiss (1963)
91
Liapunov methods to third order systems with limit cycles
is discussed.
The properties of second order systems which
do not carry over to higher dimensions and the dependence
of Liapunov functions on these properties are considered
first.
Then a counter example is given to show that Liapunov
functions definite with respect to the origin are too re­
strictive in third order systems.
Assume that the second order system of interest has
only one limit cycle enclosing the origin and one singu­
larity at the origin.
A limit cycle in a two dimensional
space divides the plane into two disjoint regions, such that
no trajectory can pass from one region into the other.
Thus
it must enclose a singularity (a node or a focus), and all
trajectories which originate Inside the limit cycle are
either stable with respect to the origin or approach the
limit cycle as t — ► *>.
Similarly, all trajectories origi­
nating outside the limit cycle are unstable with respect to
the origin or approach the limit cycle.
Since there are
only two possible choices in each region, the stability of
the limit cycle is completely determined by examining the
stability of the trajectories with respect to the origin,
i.e., the singular point.
Consequently, Liapunov functions
which are definite with respect to the singular point are
appropriate in this case.
Now consider a Liapunov function definite with re­
spect to the origin in a three dimensional space.
By
0
92
analogy, the Liapunov function represents a closed surface
about the origin.
Thus„ the Liapunov problem becomes one of
bounding closed curves by closed surfaces.
Although such a
procedure can be used for certain systems, the fact remains
that only a limited amount of information is obtained on the
position of the limit cycle.
It should also be noted that
the singular point at the origin of a second order system
with a limit cycle exhibits some type of stability behavior.
However, in many third order systems with limit cycles, the
singularity at the origin is a form of saddle point for
which stability in the usual sense has no meaning.
For this
case, a Liapunov function definite with respect to the ori­
gin cannot be used for finding an inner bound to the limit
cycle.
An example illustrating this point is system (4,1),
i.e., the system considered by Zubov (1963)0
The eigen­
values of the linearized part of system (4,1) indicate that
the origin is a saddle-focus with the stable eigenvector
near the origin being the x^-axis.
By combining this result
with the information that the x_-axis is excluded from the
3
region of asymptotic stability of the limit cycle (see Equa­
tion
4*7)» it Is deduced that the origin is the stable
state for any trajectory originating on the x-axls.
Thus,
3
there does not exist a closed surface about the origin on
which the direction field of the system points In an outward
direction.
Typical trajectories for this system are shown
in Figure 4,2*
Consequently, Liapunov functions representing
93
LIMIT C Y C L E
Figure 4.2
Typical Trajectories for System (4.1)
94
closed surfaces about the origin are In general inadequate
for examining the behavior of third order limit cycles.Following the reasoning of Section 2.5 that the
Liapunov function be referred to the limit cycle» it be­
comes evident that the geometrical form required for the
third order problem is a toroid enclosing the limit cycle.
Liapunov functions representing toroids are considered in
Section 4.4.
In the next section the method proposed by
Szego and Geiss (1963) is examined in detail.
4.3
Reduction of Third Order Problem
The present section is concerned with third order
systems which have a limit cycle contained in a stable sur­
face.
By finding the stable surface„ the system is reduced
to a second order system for which the Liapunov techniques
discussed in Chapter 3 apply.
This procedure is a special
ease of a theorem (see Theorem 4.1) given by Szego and Geiss
(1963 ) which implies that if a stable n - 1 dimensional sur­
face M in E*1 can be found„ then the problem is reduced, for
stability purposes» to an n - 1 dimensional problem on the
surface M.
First the theorem of Szego and Geiss (1963) 'is
given as the foundation for the procedure developed later in
the section.
Then a Liapunov function definite with respect
to a surface is defined.
After formulating the third order
problem, the computational aspects of finding a stable sur­
face are considered.
The’section is concluded with a third
95
order example illustrating the techniques discussed.
Except
for Theorem 4,1 the techniques and properties developed in
this section are original.
The following theorem presents a Liapunov approach to
the problem of finding a stable surface of a given systems
Theorem 4,1
(Szego and Geiss 19^3)
Consider the nth order dynamical system
x = f(x),
(4.14)
f (0 ) = 0
Let
1)
V(x) be a continuous scalar function with
continuous first partial derivatives in the
n
whole space E
2)
6 (v ) be a continuous scalar function,
3)
M be the manifold on which V(x)
= 0,
Assume that:
4)
0 (V(jc)) = 0 in all points of M,
6 (V(x)) ^ 0 for x ^ M.
5)
The equation
VV'f(jc) = -e(v)
(4.15)
is satisfied in the whole space E*1.
6)
V(jc)9(V(x)) ^ 0
n
in the whole space E .
7)
The trivial solution V * 0 of equation
(4.16)
96
v = -e(v)
(4.1?)
is globally asymptotically stable.
8)
a)
a( ^(x.M)) <
||V(x)||
b)
a(f(x,M)) <
||V(x)|| < b(^>(x,M))
(4.18)
where ^ (%,M) is the Euclidean distance of
the point x from the set M, a(r) and b(r)
are positive definite scalar functions, and
a(r) is such that 11m a(r) = ».
r —*oo
Then if 8a) is satisfied,
11m ^(x(t), M) = 0
t -*■00
(4.19)
for all initial conditions, and if 8b) is satisfied,
M is globally asymptotically stable.
Since Liapunov methods are used to find M, it is con­
venient to define a Liapunov function related to this surface.
Definition 4.1
Liapunov Function Definite
with Respect to a Surface
The scalar function V(jc) is a Liapunov function
definite with respect to the surface M if
1)
V(^) ^ 0 , for x £
M
V(%) = 0, for x € M;
(4.20)
2)
V is of class C^;
3)
V is identically zero only for points of M,
and satisfies
VV ^ o.
(4.21)
97
Using this concept, the general formulation for find­
ing a stable surface in a third order system can be given
as follows:
Suppose that a third order system given by
*1= V
x2 = W
w
V
W
has a limit cycle contained in the surface
M :
= 0,
(4.23)
Suppose further that Equation (4.23) represents a stable
surface, i.e., it satisfies
V = -0(V ) ,
(4.24)
where V is a suitable Liapunov function definite with respect
to the surface.
Once Equation (4.23) is found, the follow­
ing method is used to constrain the system equations to this
surface.
say
Solve Equation (4.23) for one of its variables,
, to obtain
]y =
g (x 1 ,x 2 ).
(4.25)
It should be noted that G may represent more than one sur­
face.
Since Equation (4.25) represents the constraining
surface, then
must be Identical to
= f^(x1 ,x2 ,G).
(4.2?)
Therefore, the constrained second order system is
X1 =: fl^xl ,x2 #G^
x2 = f2 (x ^ •x2 »G).
(4.28)
System (4.28) can now be examined for limit cycles using the
methods of Chapter 3*
Certain computational aspects of the above formu­
lation are discussed next.
After presenting the difficulties
involved in obtaining a solution to the general problem, a
special class of problems is examined in detail and a method
of solution outlined.
example.
The method is illustrated with an
Szego and Geiss (1963) do not consider any methods
of solution.
Their contribution consists only of Theorem 4.1
and two examples illustrating its meaning.
Consider first the difficulties in formulating a
general solution to the problem of finding a stable surface
in a third order system.
Initially, the existence of a sta­
ble surface must be established.
Unfortunately, Just as in
the case of limit cycles, a stable surface represents a
property of the system trajectories which does not appear
explicitly in the system equation, and hence, its existence
99
Is very difficult to establish.
The existence of the stable
surface can be incorporated into the existence of a Liapunov
function so that if a Liapunov function definite with re­
spect to the limit cycle can be found, then the system has
a stable surface.
difficulty.
Note that this does not eliminate the
The next step in the general solution is to
find the required Liapunov function by solving
V = -0(V),
(4.29)
where the function 9(V) satisfies
V0(V) % 0.
(4.30)
Unfortunately, 6(V) is unspecified and no criterion exists
for choosing it.
This condition is reminiscent of the dif­
ficulty associated with the Zubov equation.
The approach
taken in this work is to treat a special case of Equation
(4.29) and to examine its properties so that a solution may
be obtained for a limited class of problems.
In the remainder of this section 9(V) is taken as
9(V) = XV,
where X
is a
considered
realnumber.
This form restricts the problems
tothose systems whosetrajectories
degree of symmetry.
(4.31)
exhibit some
Both of the third order systems con­
sidered in the literature fall into this class of problems
100
(see Section 4.2) ; thus, Equation (4.31) Is worthy of
further study.
The following discussion Is centered on developing
a method for finding a Liapunov function V(jc) satisfying
V = XV
(4.32)
for third order systems containing a limit cycle.
It is
shown that if V(2 ) contains linear as well as nonlinear
terms, then X is an eigenvalue of the linear part of the
system equations.
Furthermore, in the case of systems
representable in terms of the block diagram configuration
shown in Figure 4.1, this form of V(x) Implies that polezero cancellation occurs in the system.
The example of
Szego and Gelss (1963) considered previously is a special
case of this result.
Then an iterative scheme is developed
for solving Equation (4.32).
It should be noted from
Definition 4.1 that V(x) is not required to be positive
definite with respect to a surface since the product W
is
negative definite.
To show that pole-zero cancellation occurs in the
system if the stable surface contains both linear and higher
order terms, consider the system
x + (a + 0)x + apx = f(x,x),
(4.33)
where f is a nonlinear function expressible in a power
series starting with terms of the second degree, i.e., all
101
the linear terms of f have already been incorporated into
the left hand part of the equation.
The discussion is given
in terms of a second order system only for convenience.
It
is valid for nth order systems since none of the arguments
given is restricted only to second order systems.
Equation
(4.33), expressed in phase variables, is
%2 = -(& + P)%2 ” a ^xi + f(x1 ,x2 ).
(4.34)
Again, phase variables are chosen for convenience; the dis­
cussion is independent of the coordinates used.
Assume fur­
ther that the Liapunov function is given as
V(jc) * ax1 + bx2 + h(x1 ,x2 ),
(4.35)
where a, b are constants and h(x^,x2 ) an unknown function.
In the following discussion the relation which exists be­
tween f and h is determined subject to the condition that
Equation (4.35) represents the desired Liapunov function.
Substituting Equation (4.35) into Equation (4.32) gives
ax2 - b(a + 0)*2 “ bagx^ + bf + h =
Xax^
+ Xbx^ + Xh.
By equating like powers of x in (5*36), the equations
(4.36)
102
-baft = Xa
a - b(a + y) = Xb
(4.37)
are obtained, with the remaining unknown terms satisfying
bf + h = Xh.
(4.38)
Note that for Equations (4.37), a nontrivial solution exists
for a and b , only If X Is an eigenvalue of the linear part
of System (4.33).
-P.
In this case the eigenvalues are -a and
For X = -a. Equation (4.34) becomes
V(x) = x1 - 3x 2 + h(x^,x2 )
(4.39)
and Equation (4.38) is
-Pf + h = -ah.
(4.40)
Solving Equation (4.40) for f yields
r = h + flh,
(4.4i)
P
where h Is a general function.
If Equation (4.41) is
expressed in terms of the differential operator, s, and
substituted into Equation (4.33) also expressed in terms of
s , It Is seen that the result
(s + a ) (s + P)x = 1
exhibits pole-zero cancellation.
(s + a)h
(4.42)
The problem considered by
Szego and Gelss (1963 ) in Section 4.2 is a special case of
this result.
Summarizing, it is shown that if V(x) contains
both linear and nonlinear terms, then X is an eigenvalue of
the linear part of the system and f must be of such a form
that pole-zero cancellation occurs.
The method proposed for solving Equation (4.32 ) is
essentially the Iteration scheme discussed in Section 2.5*
To outline the method here, recall that a homogeneous form
of degree m is defined as
m
Vm
(4.43)
k=0
where the constants a^ are unspecified.
For convenience,
the discussion is given in terms of two variables and can be
easilyextended to
ing for now that
the general caseof n
X in Equation (4.32)
iteration consists of finding
.
variables.
is known,
Assum­
the first
This is done by:
letting
m = 1 in Equation (4.43), substituting it into Equation
(4.32), and determining the unspecified coefficients by
equating like terms.
For the next iteration,
y(2) = V1 + V2
is formed, where
ratic form.
is known and
(4.44)
is an unspecified quad­
Following the same procedure as in the previous
step, the unknown coefficients in
are found by equating
like terms of the second degree and discarding all higher
order terms.
This iterative procedure is continued until
104
the desired Liapunov function,
(4.45)
v = Z vi-
is found such that V satisfies Equation (4.32) with no excess
terms discarded.
Before this scheme can be applied, X must be determined.
It is shown next that X is specified by the
homogeneous form of lowest degree appearing in V.
Although
this statement is verified using a second order system, it
is valid for systems of any order.
X = X
Li
+ X
Thus let
N
(4.46)
represent the system, where the subscripts L and N designate
the linear and nonlinear terms, respectively.
Furthermore,
assume that the stable surface is specified completely by a
known function homogeneous of degree m and n with m < n
(such a simplified formulation is used only for convenience
and does not limit the generality of the discussion).
Thus,
a suitable Liapunov function is
(4.4?)
where V
m
and V
n
are defined as
(4.48)
105
and the
are unspecified constants.
This Liapunov func­
tion Is definite with respect to the surface If It satisfies
Equation (4.32), I.e.,
W
M v m + Vn ).
ikM)
where
\ -& XL+Ss +TTyL +FTyr
(U-50)
Equating terms of the same degree of homogeneity In Equation
(4.49) gives
S
XL +
= XVm
as the homogeneous form of smallest degree.
(4-51)
Equation (4.51)
can be expressed as
a (m - j )xm~^"’1y^x
T3T J
L
a x^'^jy^'^y =
J=0" J
L
x>
a xm"^y^.
J=0 J
(4.52)
By equating like powers of x and y In Equation (4.51), a
set of m + 1 equations Is obtained In terms of X and the
m + 1 unknown coefficients a^.
This set of equations has
the form
Aa * Xa,
(4053)
where A Is an (m + l)x(m + 1) constant matrix and a Is the
106
vector
a = col(aQ
(4.54)
Equation (4.53) represents an eigenvalue problem associated
with the homogeneous form (4.48) and specifies X.
Consequently, the above result replaces the problem
of specifying X by one of determining the smallest power of
x In V.
The determination of the smallest power of x Is
strictly a trial and error process.
Fortunately, the number
of trials can be minimized by evaluating beforehand all the
possible eigenvalues associated with a particular homo­
geneous form.
By doing this. It Is then possible to reverse
the process and consider all the possible homogeneous forms
which are associated with a particular eigenvalue.
In this
manner, various eigenvalues can be ruled out simply by
examining the associated homogeneous forms to see If they
are admissible as a solution to Equation (4.32).
The veri­
fication that the eigenvalues can be computed separately for
each homogeneous form Is given next.
Consider again the second order system (4.46) and
assume that the Liapunov function of the form
V = V
m
+ V,
n
m < n
Is adequate to specify the stable surface.
(4.55)
Substituting
Equation (4.55) Into Equation (4.32) gives
(4.56)
10?
or
J x 5*1! +
+ dym(yL + yN^ + d x (xL + XN^ +
S f irL + V
“ XVm + XVn*
(4-57)
By equating all terms of the mth degree of homogeneity, the
equation
+ It X
* xvm
( 4 -58)
Is obtained which defines X , and hence, specifies V •
m
Equating homogeneous terms of degree n gives
Un +
where
n
+
= XVn ’
n
(4'59)
represents those terms of
51 %
+ r r yN
which are homogeneous of degree n.
5 1 %
+ l f yN - °n + ^
(4-6o)
Note that
XN + 5 T yN ’ °-
(4-6l)
Equation (4,59) Is a nonhomogeneous linear algebraic equa­
tion whose solution can be given In terms of a homogeneous
solution and a particular solution (Hohn 1958).
homogeneous solution Implies that
But the
is satisfied.
Equation (4.62) is interpreted as the eigen­
value problem considered previously and defines a X which
must agree with one X specified in Equation (4.58).
Summarizing, if
V = V + V
m
n
(4.63)
satisfies
Vm + Vn = Xl (Vm + V -
(4-64>
then it must also be true that
Vm “ XlVm
V
n
«
in
.
(4.65)
where only the linear terms of the system equations are used
in Equation (4.65).
A method of obtaining the desired X is considered in
the following discussion.
The discussion is given in terms
of the second order system (4.46) but can easily be general­
ized to higher order systems.
As the first step in the pro­
cedure, the eigenvalues associated with the homogeneous form
in Equation (4.43) are evaluated for m = 1,2,...,p, where p
is sufficiently large.
Then starting with an eigenvalue of
V^, say X^, all the homogeneous forms found in the previous
step are considered which have the same eigenvalue.
Note
109
that n Is taken sufficiently large to Insure that all the
homogeneous forms associated with a particular X have been
found.
By using the reasoning developed in the example of
system (4.72), it can be determined qualitatively whether
or not this combination of homogeneous forms is admissible
as a solution of
(4.66)
V = x1v
This procedure is continued first for all eigenvalues of
then for the eigenvalues of V^, V^,..., etc., until an admissible combination is found.
For the purposes of lllus-
tration assume that the eigenvalue, X^, associated with the
homogeneous form V , is a suitable candidate, and form
(4.6?)
V = V2 + h
where h contains all the other admissible homogeneous forms
of the admissible combination.
Using the complete system
equations, substitute Equation (4.6?) into Equation (4.32)
to obtain
v2 + h - x2(v2 + h)
• Recall that V
and X
are known.
(4.68)
Since
(4.69)
then Equation (4.69) becomes
110
(4.70)
or
h = X
(4.71)
The remainder of the procedure reverts to the application of
the iteration scheme of Section 2.5# i.e., each of the homo­
geneous forms in Equation (4.67) is substituted, in increas­
ing order of homogeneity, into Equation (4.71), and like
powers of x and y are equated to determine the unspecified
constants.
The above procedure is illustrated by finding the sta­
ble curve for the system
*1 - 4 = 1 + =2
(4.72)
System (4.72) can be represented by the block diagram con­
figuration of Figure 4.1 and, since no pole-zero cancel­
lation is evident, then V(jt) contains no linear terms.
The
eigenvalues associated with the homogeneous form
(4.73)
are
*■ = -1. -
-2.
(4.74)
Ill
Since the poles of the linearized part of system (4.72)
(which incldently are the same as the eigenvalues associated
with the homogeneous form V ) are at s = -1, -1/2, then the
only admissible choices of the X associated with
are
X = ”3/2 and -2.
For Illustrative purposes consider X = -3/2 first.
The eigenvector associated with this value Is
V2 = 2V 2
+ X22
<4 -75)
and Equation (4.71) becomes
h = - ^h - 2 x ^ -
(4.76)
By letting h = h^ and substituting It Into Equation (4.76),
the problem can be examined In general terms.
h 3 = d ^ (" 2X1 + x2 ) + ^
Thus
("X2 + xl > =
- -|h^ - 2x14 - 2x 1^x 2.
(4.77)
3
^(x/)
X2 1
(4.78)
Since the term
)h
is of
thefifth degree and does not cancel
Equation(4.77),
then dh^/dXg Is forced
condition Is true for any h .
m
any other term In
to bezero.
This
But the remaining term
is of the third degree and will not cancel the fourth degree
terms on the right hand side of Equation (4.77)•
Therefore,
Equation (4.76) is never satisfied, and, consequently,
X = -3/2 is an incorrect choice.
Next try X = -2.
In this
case
V 2 = cx22 .
(4.80)
where c is a constant, and h is
h = -2h - 2x 2x ^
Since Equation (4.81)
c
.
(4.81)
contains afourth degree term, h is
taken as
h * h^.
(4.82)
Then Equation (4.81) becomes
+ x2 ) + ^ ( - % 2
+ : / ) „ -2h - ZXgX^c.
(4.83)
For the same reason as before, the condition
3h,
r— T" “ 0
(4.84)
2
is imposed.
Since all the terms in Equation (4.83) are of
the fourth degree, a solution is possible.
shown that
It can be easily
satisfies Equation (4.83)#
Thus a solution of
(4.86)
V = -2V
is
(4.8?)
To conclude this section a stable surface containing
a limit cycle for the third order system.
(4.88)
is found using the procedure outlined above.
It is a
straight forward process to show that the origin is stable
(X = -1, -1, -3), thus, a stable surface containing the
origin may exist.
Following the procedure given earlier,
consider the homogeneous form
+ d x ^
1 3
+ ex^Xj + fx
2
3 *
(4.89)
Using only the linear terms of system (4.88), the admissible
X •s are determined by solving the equations
114
-2a
= Xa
- 2b
= Xb
- 2c
l/3b
= Xc
- 4d
2 /3c
= Xd
- 4e
= Xe
/3e - 6f = Xf
(4.90)
simultaneously which give
X = -2, -2, -2, -4,
V(x) cannot have both V
and
-4,-6.
(4.91)
terms since none of their
associated eigenvalues are the same.
Next, the eigenvalues associated with
are explored
in a general sense to determine if any of them lead to an
admissible form for V.
Consider X = -6.
Assuming that
represents the homogeneous form of lowest degree in V, the
eigenvector associated with this X is
V2 = ox32 ,
(4.92)
V = cx 2 + h.
(4.93)
which results in
Substituting Equation (4.93) into
V » -6V
gives
(4.94)
115
h = h - 2cx x
3 2
(4.95)
Since the additional term In Equation (4.95) Is of the
sixth degree, let
h = h^
(4.96)
and substitute It Into Equation (4.95) to get
-^(-^
+
+
+
2Xlx2 ) + g ^ ( - x 2 + x3 )+
x2^) = -6h^ - 2cx^x.^*
(4.97)
Note that Equation (4.97) Is satisfied if
(4.98)
By inspection It is seen that
h6 = -x26
satisfiesEquation
(4.99)
(4.97)• and that the unknown constant In
Equation (4.93) Is
c = 3.
Thus asolution
(4.100)
of Equation (4.94) is
V - 3x 2 - x2 .
(4.101)
This equation implies that a stable surface is
*3 = ± -4 x 23.
(4.102)
Consider next the system motion on the surface
(4.102)e
duces
It is easily verified that Equation (4.102) re­
thelast
equation in (4.88) to anidentity,so that
the secondorder system
constrained tothis surface
is
x^ = -x^ + x ^ + 2x1x22
(4.103)
x2 « -x2 + x23 .
(4.104)
To determine if the surface (4.102) contains a limit cycle,
the stability properties of the constrained system above
are examined using the variable gradient method.
The
general form for V is
V = vv
2,
2 . _
2
’f “ *1 (-*11 + *11=1 + 2*11=2 > +
=1=2(-*12 + *12=/ -
*21
* *21=/ + 2*12=22) +
=2 ("*22 * *22=2 ^
(4.105)
which becomes
V = -a11x12(l - I^2 - Zg2) - a22I12(1 - x12 * x22)
when
(4.106)
117
By examining Equation (4.106), it is seen that if
= a 22 = constant,
(4.108)
and if the plus part of the surface (4.102) is used, then
V becomes
v=
-al l (x;L2
+
x22 )(1
- x 12 -
x22 ).
(4.109)
The corresponding Liapunov function is
r
x2
V = Wvidjc =t
x2
+ * ^ 2 '
(4.110)
^0
For convenience
is taken as
a
= 2.
(4.111)
Thus, Equations (4.109) and (4.110) take the forms
V = -2(x1 2 + x 2 2 )(1 - x12 + x22 )
and
V = X 2 + x22 ,
(4.113)
respectively, from which it is seen that the system has an
unstable limit cycle given by
x^
+ Xg^ = 1.
(4.114)
Consequently, a very unusual limit cycle is obtained in this
case (see Figure 4.3)•
The surface
118
x.
SURFACE
CONTAINING
LIMIT CYCLE
V
z
Vs.
Figure 4.3
Limit Cycle for System (4.88)
119
3
(4.115)
contains no singularities except the origin and appears in
the formulation only because it has the same slope at the
origin as does the positive part of the surface.
Note that the results of this section are valid only
for systems having a surface which divides the space into
two disjoint regions so that trajectories originating in one
region cannot cross into the other.
The reason for this is
that equations of the form
(4.116)
imply that V = V(t) tends to the surface
V ;« 0
exponentially as t
(4.117)
Also, since in the constrained
system the Liapunov equation used is of the Zubov form
(Equation (4.112) for example), then the trajectories tend
to the limit cycle in an exponential manner.
Therefore,
systems, whose trajectories spiral around the limit cycle as
they tend to it, are excluded by this method.
In the next
section a formulation is developed where such a system can
be treated.
4.4
LlapuQpy Formulation o£ the General Problem
For the purposes of this work the three dimensional
limit cycle problem is divided according to its formulation
120
into two classes„
The first class consists of systems which
can he solved in terms of the stable surface containing the
limit cycleo
Once the stable surface Is found» the problem
reduces to a two dimensional one and the Liapunov techniques
discussed in Chapter 3 are applicable.
The disadvantage of
this formulation is that it applies only to the restrictive
class of problems where the trajectories approach but do
not cross the stable surface.
This problem was treated in
detail in Section 4,3 where a method of solution was devel­
oped for systems whose trajectories exhibit a certain symme­
try as they tend to the stable surface.
Both of the third
order systems cited in the literature (see Section 4,2) are
examples of the class considered here and are solvable by
the method developed in Section 4,3,
In this section a method of enclosing the limit
cycle in a toroid is developed.
The formulation is basi­
cally an extension of the concepts of Section 2,5 to three
dimensions.
It is shown that this formulation is a gener­
alization of the one discussed in Section 4,3, and, hence,!
applies to systems whose trajectories spiral about the limit
cycle as they tend to it.
The existing Liapunov methods, i,e,, Liapunov func­
tions represented by closed surfaces about the origin, were
examined in Section 4,2 and shown to be restrictive with re­
gard to the third order limit cycle problem.
It was also
proposed that the desired representation for the Liapunov
121
function is a toroid enclosing the limit cycle*
This con­
clusion is not new since it is the "basic approach taken in
the mathematical literature in the study of limit cycles.
Unfortunately, the toroids considered in this literature are
not defined explicitly in terms of an equation as is needed
here but are studied ,in terms of topological properties.
The Liapunov functions considered in this section,
just as those developed in Section 2,5» are formulated with
reference to the Poincare-Bendixson Theorem,
As indicated
in the discussion of the Poincare-Bendixson Theorem, it is
3
no longer sufficient in E to simply show that the trajec­
tories cross the toroid in an Inward direction.
The ad­
ditional hypothesis needed is that the trajectories also
spiral around the toroid and tend to a fixed closed curve
inside it as t — o<x>e
The goal of this section is to find a
Liapunov function which represents a toroid and satisfies,
the Poincare-Bendixson Theorem,
Following the ideas given in Section 2*5, it is de­
sired to develop the properties required of a Liapunov func­
tion definite with respect to the limit cycle in three dimen­
sions,
Since the limit cycle is not known a priori. a
Liapunov function of known form must be constructed about
an unspecified closed curve in E
3 which
is forced to be the
limit cycle when additional restrictions are imposed on the
Liapunov function.
It has already been stated that the
desired form for the Liapunov function is one which
122
represents a toroid.
Thus, let
V = V(W - 1, S)
represent a Liapunov function which is a toroid in E
(4.118)
3
en­
closing the closed curve, C, given by
W(x) = 1
C t
S(z) = 0 .
(4.119)
The closed curve C Is defined by the Intersection of a
cylindrical type surface
W(i) = 1
(4.120)
and an unbounded open surface
S(x) = 0
passing through the origin.
(4.121)
For the Liapunov formulation it
is desired that
V(0,0) = 0,
(4.122)
i.e., the Liapunov function is zero on the limit cycle.
The Poincare-Bendixson hypotheses that the tra­
jectories cross the toroid in an inward direction and that
the trajectories spiral about the toroid and tend to a fixed
closed curve are examined next.
In terms of the Liapunov
formulation the latter condition is automatically satisfied
once it is shown that the trajectories cross the toroid in
123
an inward direction.
This is true for the following reason.
Because of the continuity and definiteness properties of the
Liapunov function, it can be represented as a sequence of
nested toroids closing down onto the closed curve.
If the
trajectories cross each surface in an Inward direction, then
the limiting toroidal surface must approach a limit cycle.
In terms of equations this means that V must be made nega­
tive definite with respect to the closed curve.
Thus, dif­
ferentiating Equation (4.118) with respect to time results in
(4.123)
Then
W = g(a)
S = h(2)
(4.124)
are chosen such that
* * a(wv- i y g(3) + -ih(z)
(4.125)
is negative definite with respect to the closed curve.
Theorem 3*3 Equations (4.124) define the limit cycle.
By
It
should be noted that since there are two terms in Equation
(4.125)• then Equations (4.124) may contain terms which can­
cel when the substitution is made into Equation (4.125).
It
is precisely this condition which reveals itself in terms of
the trajectories spiraling about the limit cycle.
If no such
124
coupling occurs, then the problem reverts to the type con­
sidered in the previous chapter.
The problem considered by
Zubov (1963 ), which is discussed in Section 4.2, is of the
noncoupled type.
Unfortunately, no proof is given here to show the
extent to which the Liapunov function approach is applicable
in third order systems.
However, based on the study of com­
puter solutions of several representative systems and on the
intuition developed in the second order case, it appears
that a significant number of systems are Liapunov stable.
The next topics of interest are the specification of
a form for the Liapunov function representing a toroid and
the examination of its properties.
The method used in this
work for generating the toroid is a generalization of the
concepts presented in Section 2 .5 .
It is shown in Section 2.5
that a suitable Liapunov function for studying second order
systems is
V = (W - I)2 .
(4.126)
Equation (4.126), for any number 0 < V < 1, represents two
concentric closed curves bounding the limit cycle.
The
Liapunov function used in three dimensions is
V = (W - l)2 + S .
(4.12?)
It is easily verified that this Liapunov function represents
a toroid when Equation (4.12?) is expressed in the form
125
w = 1 ± l/v - s .
(4.128)
It Is seen that as the surface
S = K,
(4.129)
where K is a constant, is translated above and below the
surface
8
=
0
(4.130)
the equation
W = 1 ±
(4.131)
for
(4.132)
represents two concentric closed curves on the surface
(4.129)•
Thus, for any given 0 ^ V < 1, the locus of all
curves, obtained by letting K take on all values (positive
and negative) satisfying Equation (4.132), is a toroid.
It
should be noted that the Liapunov function is valid for
examining only local properties of the limit cycle.
The time derivative for the Liapunov function takes
the form
V - 2(W - 1)W + 2SS,
(4.133)
and the auxiliary equations required for making V negative
definite with respect to the limit cycle are
W - -/$(U - 1) + p(2)
(4.134)
126
S = -f(S) + q(x) ,
(4.135)
where
(W - l)p(x)
P(S)S ^
are satisfied.
+ Sq(x) = 0
(4.136)
0
(4.137)
Then V becomes
V = -20(W - I)2 - 2Sf(S)
(4.138)
For the case when no coupling exists, I.e., when
p(x) = q(x) = 0,
(4.139)
Equations (4.134) and (4.135) become
W = -P(W
- 1)
(4.140)
S = -P(3)
,
(4.141)
and the problem Is of the type considered In Section 4.3
where
f(S) = 9(8).
(4.142)
Unfortunately, when coupling Is present, the problem becomes
very complex.
In the first place both W(jc) and p(x) can be
functions of all three variables.
The degree of added com­
plexity Is realized by recalling that the problem of choos­
ing p(%) has not yet been resolved for the two dimensional
case.
The other principal difficulty Is associated with
12?
the equation
S = -fts) + q(x)
(4.143)
In the uncoupled case It Is shown In Section 4.3 that for
0(8) = -XS,
(4.144)
the constant X Is an eigenvalue associated with the homo­
geneous forms present In S.
Once X Is known an Iteration
procedure can be used to find S.
In the coupled case
S = XS + q(x)
(4.145)
Is an appropriate form for the solution of third order sys­
tems having a saddle-focus as the singularity at the origin;
but unfortunately, X no longer represents the eigenvalue as­
sociated with the homogeneous forms in S (this point is il­
lustrated In the examples).
The reason for this Is that
both the XS and q(x) terms of Equation (4.145) have homo­
geneous forms of the same degree, and It is not possible to
determine a priori how the homogeneous forms are distributed
tributed between these terms.
Because these difficulties have not been resolved,
no method of solution Is presented here.
The basic contri­
bution In this section Is the formulation of the general
third order limit cycle problem.
It has been shown that the
examples treated in the literature are special cases of this
formulation.
Hence, by means of the theory developed here,
128
It Is possible to justify the arbitrary choices made by
Zubov (1963 ) In the solution of his problem discussed In
Section 4.2.
Furthermore, the formulation also revealed
the existence of a type of behavior which has not been con­
sidered previously In the literature on Liapunov methods.
Finally, it is felt that the formulation given is a natural
one, since It appeals directly to the Polncare-Bendixson
Theory.
Several examples are given next which Illustrate the
oscillatory type behavior of the trajectories about the
limit cycle and which consider W(jc) as a function of three
variables.
Consider first the system represented by
^
^
+ *2 + x3 -
+ x22 )
*2 " "X1 + X2 + X3 ' X2 (X1 2 + X22)
= x1 + x2 - x^ -
+ x2 )(x1 2 + x22 ),
(4.146)
and let the Liapunov function be
V = (W - l)2 + S2.
(4.147)
Differentiating Equation (4.147) with respect to time gives
V = 2(W - 1)W
By examiningthe eigenvalues of
determined that
+ 2SS.
(4.148)
the linearizedsystem it is
the origin is a saddle-focus;hence,
it
129
appears that an appropriate set of auxiliary equations are
W = - P U ) ( W - 1) + p(x)
(4.149)
S = + XS + q(z)
(4.150)
where
(w - l)p(z) + Sq(i) = 0.
(4.151)
By combining the condition (4.151) Into the auxiliary
equations, they become
W «
iz)(W - 1) - r(x)S
S = XS + r(%)(W - 1).
(4.152)
It can be shown that a solution to this set of equations Is
W(x) = x ^
+ x22
S(a) =
(4.153)
(4.154)
where
r(s) =
X = -1
B U ) - x12 + x22.
Thus, the limit cycle Is given by
(4.155)
(4.156)
(4.15?)
130
X.
1
2 + x 2
= 1
2
(4.158)
(4.159)
Several features Illustrated by this example should be noted.
Since coupling exists, the trajectories spiral around the
limit cycle as they approach it (see Figure 4.4).
Also,
-1 in Equation (4.156) is not an eigenvalue of the linear­
ized system equations, i.e., of the linear homogeneous form
S(a) •
The next example exhibits the properties of W(x)
being a function of all three coordinates.
Consider the
system
(4.160)
Using the Liapunov function
V = (W - i)2 + s2
(4.161)
V is
#
V - +2(W - 1)W + 2SS.
(4.162)
In this case the appropriate auxiliary equations are
W = 0(W - 1)
(4.163)
S = XS.
(4.164)
131
TRAJECTORY
LIMIT CYCLE
Figure 4.4
Typical Behavior for System (4.146)
132
A solution to this set of equations Is
2
i
2
(4 .165 )
W(z) =
s(a) =
(4.166)
- x2 .
where
(4 .167 )
6(a) =
X « -1.
(4.168)
Thus, the Liapunov function Is
V =
+ U 3 - x2 )
1 " Xl 2 - 2 (X2 + X 3 )
.
(4 .169 )
To show that Equation (4.169) represents a toroid write
this equation In the form,
2
1
+ l (x2 + X3 )
'
“ 1 * r
' (x3 " V
'
and assume that V Is some given positive number V
(4.170)
^ 1.
Also, let
x3 ' X2 “ K *
(4.171)
where K Is any number satisfying
(4.172)
Then for every constant K Equation (4.170) represents two
concentric closed curves,
133
xl 2 + 2 (x2 + x3 )2 “ 1
in the surface (4.171).
±\lVo ' k2,
(4.173)
The two curves in Equation (4.173)
coincide on the surfaces
x3 " X2 = * ^
which indicate the top and bottom of the toroid.
(4-174)
CHAPTER 5
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH
5*1
Summary
In this work methods are Investigated for construct­
ing Liapunov functions which can be used to determine limit
cycles and study their stability behavior.
In Chapter 3 two approaches are taken in the study
of the second order limit cycle problem.
First, the es­
tablished methods of constructing Liapunov ',functions are
applied to -limit', cycle problems.
The notable contribution
here is the application of the variable gradient method to
the generation of contact curves needed in the construction
of a suitable annular region, R, in the Poincare-Bendixson
Theorem.
This scheme provides a method for bounding the
limit cycle to a certain region of the state plane.
Second, a Liapunov function is constructed which
represents the hypothesis of the Poincare-Bendixson Theorem
directly.
In this formulation a Liapunov function is de­
fined as a known function of a generic closed curve.
The
form of the Liapunov function is specified by the requlre,
°
ments of the Poincare-Bendlxson Theorem. By forcing V to be
negative definite with respect to the limit cycle, i.e.,
satisfying the condition that all trajectories cross into
the interior of the.annular region, R, an auxiliary equation
.
' '13^.
\
135
Is obtained which specifies the limit cycle,
It is shown
that for second order systems the auxiliary equation is
the Zubov equation.
Chapter 4 is concerned with limit cycles in third
order systems.
The first topic considered is the possi­
bility of reducing the third order problem to one of two
dimensions by finding a stable surface which contains the
limit cycle.
The idea was expressed in a short paper by
Szego and Geiss (19&3) where they presented a theorem
(see Theorem 4.1) regarding such a reduction.
This problem
is discussed more thoroughly in this work where some im­
portant characteristics of the formulation are presented.
First, a Liapunov function definite with respect to the
stable surface is defined in which the aggregate of tra­
jectories making up the surface is considered as the unper­
turbed trajectory in the Liapunov stability definition.
Then by Szego*s theorem, the surface is stable if a Liapunov
function V can be found satisfying
V = -9(V)
(5.1)
V0(V) £ 0.
(5.2)
It is interesting to note that V need not be a positive
definite function when Equation (5.2) is satisfied.
Also,
it is shown that the theorem is valid only if the surface
3
separates E into two disjoint regions such that trajectories
'136
cannot cross from one region Into the other.
T h u s , the
method Is limited to the case where the trajectories ap­
proach the limit cycle in the direction of the surface.
Unfortunately, this geometrical structure cannot be recog­
nized from the system equations.
On the computational side of the problem, a method
of solution is given when the system satisfies
(5.3)
V = XV,
where X is some real number.
Equation (5*>3) imposes a sym­
metry restriction on the type of systems considered.
It is
shown that X is completely specified by the terms of lowest
degree in V.
Thus, X can be thought of as an eigenvalue
associated with a homogeneous form with respect to the
linearized system.
For the special case where the lowest
degree terms in V are the linear terms, then X is an eigen­
value of the linear form, and pole-zero cancellation occurs
in the given system if it can be represented in terms of the
usual feedback diagram.
By using the above method for find­
ing X, an iterative scheme is developed and used for solv­
ing Equation (5.3)
Also developed in Chapter 4 is a general Liapunov
formulation for the third order limit cycle problem.
This
development is an extension of the results of Section 5.4 to
the three dimensional case where the limit cycle is enclosed
by a toroid.
Because of the extra degree of freedom, the
;i37
three dimensional formulation is more complex than the one
given previously.
In the first place„ the Liapunov func­
tion is a function of both the surfaces W(x) = 0 and
S(x) = 0;;whose- intersection specifies the required closed
o
curve.
Furthermore„ in forcing V to be negative definite
with respect to the limit cycle, two auxiliary equations
o
o
in W (x) and S(x) are obtained instead of one as in Section
3.4.
Since coupling can exist between the auxiliary equa­
tions, the system behavior specified by this formulation is
more general than the, formulation proposed by Szego and
Gelss (1963).
5.2
Suggestions for Future Research
Because of the generality of the proposed formulation,
there are many possibilities for extending this work.
For
second order systems the principal area of research is in
the development of criteria for choosing an appropriate
p(x) in the Zubov equation.
Although the method proposed by Szego and Gelss
(1963) can treat only a special class of problems, it still
Is important because it appears in the general formulation.
Consequently, it is desirable to find methods of choosing
9(x) in Equation (4.15).
The development of a method of solution for the
general formulation represents a fruitful area of investi­
gation.
For the ease where
138
S = xs + p(x),
(5.4)
It Is desirable to find a method of determining X.
Once X
Is known, then It Is possible to develop an Iterative scheme
for solving the auxiliary equations,
This, of course, as­
sumes that a suitable choice is made for p (x ) .
It should be noted
an exact one in the sense
that the proposed formulation is
that the auxiliary equations
specify the limit cycle exactly.
Using the ideas of the
two dimensional formulation, it seems that the limit cycle
can be approximated by taking only a few terms of a series
solution of the auxiliary equations.
Although this is
possible, no criteria exists for estimating how close the
approximation is to the true solution.
Another approach to this problem is to formulate an
approximate scheme for estimating the limit cycle instead
of the exact scheme proposed here.
A means of accomplishing
this is to formulate the problem so that an equation is ob­
tained which does not specify the limit cycle exactly but
gives information regarding whether or not a particular
toroid encloses the limit cycle.
It should be noted that
this type of information is not used in the exact formu­
lation, because once the auxiliary equations are solved for
the limit cycle, the fact
that the Liapunov function repre­
sents a toroid is of secondary importance.
APPENDIX A
SECOND ORDER SYSTEMS WHICH EXHIBIT LIMIT CYCLES
EXPRESSIBLE IN CLOSED FORM
Api
Introduction
At present» there exists in the literature few exam­
ples of second order systems which have limit cycle solu­
tions expressible in closed form.
The examples which are
given exhibit a high degree of symmetry and can be deter­
mined by inspection from the system equations.
The Zubov equation (2.29) represents an ideal pro­
cedure for synthesizing systems with known limit cycles.
The procedure consists of selecting an equation to rep­
resent the desired limit cycle and then forcing it to be a
solution of the Zubov equation.
The latter step is a trial
and error procedure where the system equations are chosen so
that the Zubov equation is satisfied.
Unfortunately8 in
many cases a trial and error method applied directly to the
Zubov equation for some desired limit cycle equation cannot
be realized because of the number of simultaneous conditions
which must be satisfied.
Also the synthesized system which
exhibits the desired limit cycle solution may be very com­
plex even though the limit cycle is given by a simple
expression.
. \X39*
140
In this appendix a method is presented where the
limit cycle of a system is approximated or bounded by a
curve expressible in closed form; then a new system is
generated which has this closed curve as its limit cycle.
Using this procedure, the disadvantages given previously
are minimized.
An outline of the method is given, followed
by an illustrative example.
In third order systems there is no way of obtaining
a closed form approximation to the limit cycle for a given
system.
Hence, any procedure for finding a third order
system which has as its limit cycle a desired closed curve
is entirely a trial and error process with no guarantee that
such a system can be found.
For this reason third order
limit cycles are not considered here.
A.2
A Synthesis Method
Let
(A.l)
represent a system which exhibits a stable limit cycle, L,
not expressible in closed form.
By applying the variable
gradient method to system (A.l) it is possible to find an
inner bound
b(x) = 1
to the limit cycle which approximates its shape
(A.2)
141
(see Chapter 3)•
It is desired next to generate a system,
*1 = gi (xr
x2 )
X2 * 62^Xl e X2^•
■ (*.3)
which exhibits (A.2) as a limit cycle and resembles the
original system (A.l).
Thus, let
V = b(x).
(A.4)
and substitute Equation (A.4) into the Zubov equation to
obtain
o
1
+
2
= -P(t> - 1).
(A.5)
The positive semidefinite function p(x) is determined by the
variable gradient approximation.
Equation (A,5) are g^ and
The only unknowns in
and are determined by trial
and error using system (A.l) as a guide.
An example illustrating the procedure described above
is developed in detail.
Starting with the block diagram in
Figure A.l, a nonlinear element, f(x), is chosen along with
a transfer function, G(s), such that when f(x) is linearized
using the describing function method, the root locus of the
linearized system exhibits a ^-crossing (Graham and McRuer
1961, Truxal 1955)•
By choosing f(x) and the poles of G(s)
so that the system is unstable for low gain and the root
locus crosses into the left half plane for high gain, the
142
R=0
y = f(x)
Figure A.l
Block Diagram of
Nonlinear Feedback System
-X
143
system will then have a stable limit cycle.
The choices
3
f (x ) = -2^—
(A .6)
G (s ) = — jj
S
8^ - S + 1
(A.?)
represent such a system, which when expressed In phase
variablesv is
X1 = X2
;2 = "xi + x2 - xi % -
(A-8)
Using the variable gradient method the inner bound,
V = lxi6 - 2x^4 + 2il 2 +
2
x
^
*
2
-
2 x 1
x
2
+ x22 = 3.
is determined for the limit cycle (see Figure A.2).
(A.9)
In this
case
V * - ^x1 2 (x12 - 3)
(Ado)
so that
e = i12 .
(a .ii)
The required condition for (A.9) to represent a limit cycle
is the Zubov equation
! ^ Sl
where
- -*12(V - 3).
(A.12)
144
LIMIT
C YC LE
INNER
BOUND
Figure A.2
Inner Bound for
Limit Cycle of System (A.8)
1^5
B;
" 3X1 5 " § X1 3 + 4xi + 2xi2x2 • 2X2
U -13)
= 3X1 3 - 2X1 + 2x2
(A-14)
and
-x1 2 (V - 3 ) = - Ix ^8 + - x ^ - 2x^^ -
•|x1 5x2 + 2x13x2 - x12x22 + 3x1 2 .
(A.15)
Q
To obtain the x
term on the left hand side of Equation
(A.12) let
gl = aXl3
6 2 = bX
(A.16)
where a and b are unknown constants•
These constants are
evaluated by equating all termsof the eighth power. I.e.,
■^a + ^b «
-J
(A.17)
a = - ^ -
b.
(A.18)
to get
Thus, Equations (A.16) become
146
g1 = <- £ - b)x1 3
82
»
(A.19)
It must be remembered that Equations (A.19) have added
other terms Into Equation (A.12), but these terms are of
lower order and are considered in other iterations.
The
next highest order term on the righthand side
ofEquation
(A.12)
most general
is "! x ^ .
To eliminate thisterm, the
form that the system equations can assume are
gi = cxi
g
2
« dx 3.
1
(A.20)
As before, c and d are evaluated by equating the terms of
the sixth degree, i.e.,
■^c + -^d — — (— ^ — b ) - 2b » —
3
3
3
6
3
(A.21)
to give
c « ^ - b - d.
(A.22)
Thus, after two iterations the system equations are
g^ = (- ^ - b)x^3 +
- b - d)x^
8 2 * bx^5 + dx^3.
Each successive
term,
(A.23)
iteration eliminatesthenext highest order
including those introduced byg^ andg^.
At
times it
14?
is better to let g be zero for a particular iteration be­
cause it may introduce other terms which are difficult or
impossible to remove•
Most of the unknown constants ac­
cumulated in the system equations during the iteration
process are evaluated when it is necessary to eliminate the
remaining terms in Equation (A.12) after no more iterations
are possible.
For this example the system equations become
*1 ” ""(& +
+ (^ - b - d)x1 - (^ + i)x2
(A.24)
x2 = bx]L5 + dxl 3 + (- ^ + 2b - d)xi + lx1 2x2 ’ ^2 + 1^X2 t
subject to the conditions
6b - 21 = 3
24b + 6d = -1.
(A.25)
If the one remaining free parameter, b, is set to zero,
system (A.24) becomes,
*1 “
X2 =
W
-W - 2X12x2 "K
"
+ 2x i + x 2
+ X2
(A'26)
which has the limit cycle
lx & _ 2% 4 + 2% 2 +
3%
91
31
1
3 1 2
as a solution.
- 2x x
12
x 2 * 3
2
(A.27)
It is interesting to note system (A.26) and
its prototype» system (A,8), contain terms of only the first
and third degree.
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