a guide to the generation of lyapunov functions

8/16/2019 A Guide to the Generation of Lyapunov Functions

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Okan

Gurel

and Leon Lapidu

ver the past eight years a vast outflow of research

0

nd publications has resulted from the use of

Liapunov’s “second (or direct) method” of stability

analysis. Th is work stems from the appearance of the

original work of Liapunov in 1892, more than half a

century ago, but only recently has this concept been

appreciated to the point where workers in the area

of

stability of dynamic systems and automatic control are

aware of its potentialities.

In its simplest form this method treats the stability

of ordinary differential equations and tries to answer

the question of whether the solution remains arbitrarily

close to an equilibrium solution after being disturbed.

This is carried out via a function V x) , named the

Liapunov function, and its total derivative, p(x),which

are examined for certain properties.

When applied to specific systems, this method may be

used either for analysis or for synthesis. I t is the former

use, namely that of analysis, which is of interest here.

I n such a case, the application of the method lies in

constructing the function, V x), and its derivative such

that they possess certain properties. When these

properties of

V x)

and

p(x)

are shown, the stability

behavior of the system is known . T he difficulty, how-

ever, arises when the necessary conditions cannot be

exhibited, for then no conclusion can be drawn about

stability. Each problem is a new challenge, for the

functions must be shaped anew for each given system,

or class of systems. The proper choice of V x) depends

to a n extent upo n the experience, ingenuity, and, often,

good fortune of the analyst.

Unfortunately, the available material on the second

method has evolved to the point where an engineer who

wishes to use the method for the analysis of a specific

physical system finds himself confronted with an immense

job of merely searching the literature in many diverse

fields to ascertain the recent developments. Thi s pape r

has thus been prepared to allow a novice in the field

to approach this available literature in a rational

man ner. Consideration is given only to the determinis-

tic problem described by ordina ry differential equations.

We include briefly the theorems necessary for dis-

cussing Liapunov’s second method and then approach

the different methods for constructing Liapunov func-

tions in a general manner. Th e main theme is to

classify the constructive procedures into a few simple

A Guide to

the Generation

of

Liapunov

Functions

T hi s survey a ids the

engineering application

of Liapunov’s second method

of

s tabi li ty analy s is

by bringing together published

work on the construction

of

L i a p u n o v f u n c t i o n s

30 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y


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categories and then to outline how the various methods

fall within each category. Because of the almost over-

whelming amount of material, only a few of the con-

struc tions will be explicitly detailed. Brief descriptions

of all other methods are contained in five sections of an

appendix

;

this appendix is, however, not published

here but can be obtained directly from the authors-

details are given on page

40.

Liapunov Theorems

I n this section we shall outline, in a simple manner,

those definitions and theorems necessary for the rest of

the discussion. Furt her a nd more explicit details can

be found in various books, listed in Section I11 of the

References.

We consider the unforced, continuous-time, dynamic

system

X ( t )

=

f(x) (1)

which has an equilibrium point

at

the origin x(0)

=

0.

Now let R1 be a region in the state space of

x(t)

for

which the norm

Ijxl(

<

CY

nd let R2 be

a

similar region

for which llxll <

@.

Assume that a

> @

then regions

R I

nd Rz may be thought of as hyperspheres around the

origin of radius CY and 8, respectively. If the state of th e

system at time

t = 0

is x

=

xo 0, then we say that:

Th e system is stable, if for every

xo

in

Rz

there is

a region RBdefined by 11x11

< 7

Y >

7 >

8, such that

lirn

\\x(t)ll

< a. Stated in another fashion, this says

th at the system never goes outside the regionRa.

1.

t-+

m

2 . Th e system is asymptotically stable, if for an y

xo

in

Rt,

lirn /lx(t)ll

+ 0.

Stated in another fashion, this

says that the trajectory of x(t) eventually ends up at the

origin.

3.

T he system is asymptotically stable in the large or

completely

stable if it

is

both asymptotically stable and the

region

RZ

s the entire state space. Now the trajectory

eventually ends up at the origin no matter where its

startin g point is.

T h e system is unstable, if for some xg in Rz with

0

small, lim

X I

(t)l/ > a. Now the system goes outside

the region Rs.

With these definitions in hand, we now turn to a

variety of properties for a real-valued scalar function,

V(x),

to be called the Liapunov function. These

properties are :

1. V(x)

is continuously differentiable-Le., all the

first partial derivatives of V x) exist an d a re continuous.

2. V(x) is positive definite. This means th at V(x)

>

0 for all x

0,

but that

Y(x)

= 0 for x = 0.

3. The derivative of V(x), p(x), is negative definite.

This means that

p(x)

<

0

for all x

0

but that

v(x)

=

0

for x

= 0.

At th e same time, we note by the chain rule

of differentiation th at

t+ OJ

4.

t+

dV dxi

vv

(x)

=

X ( t )

= vv .

f(x)

=

% = I dt

grad

V

e f(x)

2)

where n is the number of states in the vector

x,

V V is the

gradient vector of

V,

and the dot indicates the dot

product

of

the two vectors.

4.

The derivative of

V(x), v(x),

is negative semi-

definite.

5.

As the norm of

x t )

goes to 00, llxll +

Q),

V(x)

also goes to 00.

6.

T h e derivative of

V(x), +(x),

is positive-Le.,

By combining certain

of

these six properties, we may

now specify various features of the stability of the sys-

tem given by Equation 1.

If a Liapunov function exists satisfying

properties

1, 2,

and

4,

then the system

is

stable in the

vicinity of th e origin.


properties 1, 2 , and 3, then the origin is asymptotically

stable.

This means tha t V x ) < 0 for x 0.

V x ) > 0.

Thus:

Theorem 1.

Theorem 2.

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Theorem 3.


the properties

1,

2 and

3

and

4,

hen the origin is asymp-

totically stable in the large.

If a Liapun ov function exists satisfying

the properties 1, 2,

4,

and 5 and also V x )

=

0 only at

the origin, then the origin is asymptotically stable in the

large.


the properties

1, 2,

and

6,

then the origin is unstable.

Consider a bounded region

Ro

n which

V x) satisfies properties 1 and

2 ,

then any solution of

Equation 1 (any trajectory) which starts in

Ro is

a. stable, and remains in

Ro

if V x) satisfies

property 4 n the region

b. asymptotical ly stable if

* x)

satisfies property

3

in this region

While these theorems may be extended in many

directions, this will not benefit subsequent discussions.

However, it does seem worthwhile to discuss further

some of the points above in a qu alita tive fashion.

First, it

is

noted that property

5

is concerned with

global properties since it considers the entire state

space. I n effect, property 5 implies that the surfaces

V x) = constant are closed and bounded in the whole

space and clustering about the origin. Th e properties

2

and the nonsign equivalence of

V x)

and

p x)

imply

that the trajectories of the system cross all

V x) =

con-

stant surfaces in the direction

of

the origin, and global

asymptotical stability results. Prope rty

3

or, better,

Equation

2,

measures the relative slope between

V x)

and the local tan gents to the system trajectories.

Second, it is noted that all of these stability condi-

tions are merely sufficient and they may not be neces-

sary. Th e word “necessary” must be interpreted as

“there exists at least one such pair

V x)

and V(x).”

Thu s, if one cannot find a pai r which satisfies the

theorems, this does not mean tha t the system is unstable.

Third, it is noticed that the Liapunov function may

not be unique and that a number of “pairs” may satisfy

a theorem. In such a case, each pair may yield different

quantitative information about the stability characteris-

tics of the system.

Finally, we wish to point out that most of the condi-

tions above hold, in an analogous form, when the system

is represented by a discrete model rather than a con-

tinuous model. Wh en the system equations are given

as difference equations, integro-differential equations, or

partial differential equations, certain equivalent results

may also be quoted. However, the reader is referred

to books given in the bibliography, such as

111.6)

nd

(IIZ.7) for these extensions.

Theorem 4.

Theorem 5.

Theorem

6.

Construction of Liapunov Functions

Since the first publication on the second method by

Liapun ov himself (which was based upo n the concept

of the energy of

a

system), there has been

a

stream of

papers related to extensions, applications, and discuss

of this idea. On e of the main themes in these pape

the means of generating the Liapunov function w

divorcing it from the restriction that it relates to

system energy. This follows because Liapunov’s orig

theorems do not reveal a general method for genera

such functions. I n the next section we shall prese

detailed bibliography of all these works and, while t

are not completely definitive, five major categories

constructive methods will be suggested. These c

gories are:

1. Chetaev-type methods (integral methods)

2. Krasovskii-type methods (quadratic €orms)

3.

Zubov-type methods (partial differential eq

4. Lur’e-Postnikov-type methods (canonical form

5. Miscellaneous-type methods

I n the Chetaev-type methods Liapunov functions

obtained, either directly or indirectly, from the f

integral of the system equations. I n the Krasovs

type methods, the construction is based upon quadr

forms derived directly from the system equatio

Zubov-type methods are quite powerful and are ba

upon solving certain partial differential equatio

Lur’e-Postnikov-type methods are quite specific

based upon certain canonical equations. Those ni

ods of construction which d o no t conveniently fit

the first four categories are placed in the final catego

We stress the point that this subdivision into catego

is not inclusive, but merely represents

a

conven

means of handling the problem of representing all

different methods. Obviously, there are alternate w

to define the constructive methods but the above se

the most general.

A

typical alternate might be ba

upon the first selection of a trial function; thus, so

methods start by selecting V x), others by selecting

gradient, vV(x), and others by selecting V(x).

Anot

possible approach might be to categorize the diffe

methods into those which form a Liapunov funct

directly from the system equations and those wh

allow more flexibility in the choice of the function.

will be observed that there are essentially two ba

approaches to constructing a Liapunov function :

a trial function which is candid ate for a Liapunov fu

tion is first formed ; if i t does not satisfy the neces

properties it is abandoned.

2)

A

trial function

formed, and to satisfy the desired properties ei

certain conditions are imposed on the system equati

or certain components of the candidate function

adjusted to make the function a Liapunov function.

T o aid the re ader in seeing the differences between

methods and categories, some simple examples will

detailed where it is felt to be specially instruct

Further, most of the discussion within a category

relate to one possible approach, wi th all other approac

and extensions within tha t category described briefly

tions)

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We star t our discussion with the special linear system

X ( t ) =

Bx(t)

( 3 )

where B is a constant nonsingular matrix. I n his

original treatise, Liapunov

(III. )

considered the

Liapunov function as

a

quadrat ic in the state variables

x

to obtain a positive definite function. Her e we select

the Liapunov function as a more general quadratic form

V(x)

=

x’Ax (4)

where

A

is a square positive definite matrix and the

prime denotes the transpose of the vector. I t follows

immediately tha t properties 1, 2, and 5 mentioned in the

preceding section are satisfied. Now forming the de-

rivative of V x) yields

‘(x) =

X’Ax + X’AX

=

-x’Cx

5)

6)

where we have used Equatio n

3

and

- C

=

B’A

+

AB

But we see tha t if C is positive definite, then property

4

s

satisfied a nd from Theorem 4 the origin is asymptotically

stable in the large.

Theorem 7. A necessary and sufficient condition

for the complete stability of the orig in of the system

given by Equatio n 3 is that there exists a positive definite

A which

is

the solution of the linear algebraic equations

of Equation G where

C

is any positive definite matrix .

As a result, we see that in the linear dynamic case we

can select any

C

(nonuniqueness) which is positive

definite (say

C

=

I),

solve Equation 6 for the elements

of

A,

and then test

A

for positive definiteness. Thus,

in this special linear case, we have a straightforward

method for constructing a Liapunov function.

It

should be mentioned here that in the linear dynamic

system just considered, local stability implies equivalent

asymptotic stability in the large. I n a nonlinear system,

however, an equilibrium point can be locally asymptoti-

cally stable without being stable in the large.

We can now proceed with the general discussions on

the various types of methods.

Chetaev Type Methods

In 1946, Chetaev (111.3) roposed a method for con-

structing a Liapunov function based on some combina-

tion of the known integrals of motion--i.e., linear

bundles of first integrals. I n essence, he made use of

Liapunov’s original idea tha t the total energy of a

conservative system could serve to define stability of a n

equilibrium point.

Stated in a different manner, if one can find a first

integral for which V x )

=

k

=

constant, then it follows

that

p(x)

= 0 and

V(x)

can be used to specify stability

conditions within the region k . Thus, for the nonlinear

conservative system

I n fact, it is possible to state :

Figure

7.

History of the construction of

Liapunov

unctions

At this point we can introduce a chart, Figure 1

summarizing the history of the construction of Liapunov

functions.

It

is clear that following the formulation of

the stability theorems by Liapunov in 1892 (motivated

by the physical phenomena of energy), th e first extension

was to use the sum of squares of the state variables

xi

as

the simple form of the Liapunov function. Exactly

half a century later while Lur’e and Postnikov were

formulating a n extension of this idea to automatic control

problems, Chetaev presented a more formal framework

for the construction of Liapunov functions. A decade

later Krasovskii extended this idea

so

that instead of

the state variables, the right-hand side of the system

equations or the velocity variables were suggested as pa rt

of the Liapunov function. This represented a consider-

able extension of th e previous proposal.

About the same time Zubov formulated those Liap-

unov theorems which allowed the solution of certain

partial differential equations to serve as a means for

finding a Liapunov function. This involves the concept

of first looking at the part ial derivatives of

V, VV,

rather

than the function itself-Le., this is now termed the

gradient approach. Since the 1950’s it is probab ly fair

to say that all the methods presented fall into the cate-

gories of either energy-type analogies (Chetaev) or some

analytic-type construction (generalized Liapunov func-

tions via Krasovskii-type construction).

Table I presents a summary in tabular form of the

different methods in the literature. This table shows

when each method was first discussed and where each

method has been included in surveys in the literature.

Th e symbols used in this table indicate (1) * he main

category in which the specific method is considered, 2)

other categories in which the method can also be

cross-referenced, and

(3)

12 urvey papers where the

method is included and discussed.

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TABLE I. METHODS

OF

GENERATING LIAPUNOV FUNCTIONS

t h e Method

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one can suggest tha t the total energy is

8)

X

Z2

2

= - -

x14

+ 2 x 1 ~

Further, it follows directly t hat k(x) =

0.

Thus, if we

select E x ) = V x ) as the Liapu nov function, it behaves

as indicated above. I n fact, it is a simple matt er to

show that E x )

=

V(x) from Equation 8 predicts stability

of the origin.

I n a more rigorous manner, we may state Chetaev’s

idea by first assuming that for a nonlinear dynamic

system of dimension

n

there a rep first integrals

@ < n

which vanish for x = 0.

If the given time-dependent

integrals are holomorphic functions of the variables,

then the constants U I ,

. .

., u p ,

c1,

. . . c p are selected

in such a way t ha t the expansion of the function

Q(U1, . . ., Up)= alU1

+

. . +

Ul(X,O, ’ ’

‘

UP X,t)

Th en we state the theorem:

Theorem 8 (Chetaev).

a,Up

+

c1U12

+

.

.

. +

cpUpz (9)

begins with a definite quadrati c form.

It

was Pozharitskii’s work in 1958 11 . 77 ) which

specified further conditions on the constants

ul, .

.

.,

u p

and

c1,

.

,

., c p such that Q(U1, .

.

.,U,) could be definite

with respect to x and thus could serve as a Liapunov

function.

We also point out the more recent work of Infante

and Clark in 1962 and 1964 11.79, 38), of Walker and

Clark in 1964 and 1965 11.40,4 4 , and of Kinnen and

Chen in 1967 (11.48))ll of which bear a relationship to

the work of Chetaev, in the sense that integrals of the

system are used to construct Liapunov Functions.

Th e work of Infante and Clark, termed the nearby

integral method for reasons which shall be obvious

shortly, dealt largely with two-dimensional systems

(and thus only one time-independent integral).

If

such

an integral g(x1,xz)

=

k exists, it must satisfy the relation

A sufficient condition t hat such an integral exists is tha t

Unfortunately, most systems do not satisfy this condi-

tion; Infante an d Clark showed how to modify the

system

so

that a Liapunov-like integral could actually

be constructed. Thus, for the system

21 = X Z = f l X 1 , X Z )

22 = i X1,XZ)

Equation 10 becomes bfz/bx2 = 0. In this case, an

integral exists, and the stability question can be ap-

proached. If no such integral exists, however, then

dfz/dxz

= 3 0 and the system is modified to

l l

21 = xz -

J ~ S X , X ~ ) ~ X

+ 4 ~ i , ~ z )

22 = fz(X1,Xz) +

s X 1 , X z )

0

where th e new functions f4 and 5 have been introduced

such that

df4

df6

- + - = o

dXl bxz

For this new system a Liapunov-like integral can be

found and, iff4 and f 5 can be selected properly, the main

properties of the original system can be retained . Ob -

viously, the proper selection of these two functions is the

key to the method. Such a selection can be obtained

rather easily in the two-dimensional case by some

geometric considerations which look at neighborhood

integrals U1 =

c1,

Us = cz,. . . in the x1,xz phase plane.

As such, the method is relatively easy to use.

Walker and Clark and Kinnen and Chen extended

this method to nth order systems by constructing

“nearby” systems containing a set of new functions wi.

In each case the main question is how to select these new

functions such that a Liapunov function for the original

system can be obtained. Both these approaches appea r

promising but largely untested to date.

Krasovskii-Type

Methods

Of all the methods for constructing a Liapunov func-

tion, the one attributed to Krasovskii and its extensions

have seen the most applica tion. Th e reason for this is

that the method is quite general, the motivation behind

the method being based on the following reasoning.

The right-hand sides of the system equations of first

order determine the equilibrium point as f = 0.

I t is

then intuitively obvious th at these right-hand sides play

an important role in determining the stability behavior

of the trajectories in the neighborhood of the equilibrium

point. Thus, it follows th at f should be used in the

Liapunov function itself. Moreover, to guarantee the

positive definiteness of V x ) , a quadratic form in f is

imperative. Just as in the Chetaev approach, the

simplest quadratic form in

f

(rather than in x ) can be

used in man y applications. However, for complicated

systems it is necessary to turn to more sophisticated

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forms of Krasovskii’s app roach. I n fact, most recent

efforts have been in the predicted direction of generaliz-

ing the quadratic form in f.

I n the present analysis, we include all methods which

use a suitable quadratic form (or weighted square) of

the system equations as the Liapunov function. T o

illustrate the features of this method, we start with the

usual nonlinear equation

X ( t )

=

f(x)

11)

where it is assumed that

f(0)

= 0. There may be

multiple equilibrium states which satisfy Equation

11,

bu t we assume that any such state c an be transferred to

the origin. We indicate the Jacobian matrix for this

system as

J(x)

=

and note that

is

symmetric.

Theorem

9

(Krasovskii). For Equation 11, with

f(0) = 0

and

f(x)

differentiable, the equilibrium state

x

= 0

is asymptotically stable i fj (x ) is negative definitive.

A Liapuiiov function for the system is given by the

quadratic in

f

Now we

state Krasovskii’s theorem as

V(x) = f’(x)f(x) 1

3 4

Further, if V(x)+- 03 as ] ; X I ’+ m , the equilibrium state

is asymptotically stable in the large.

Actually, Krasovskii used

a

generalized quadrat ic form

Y(x ) = f’(x)Af(x) (13b)

where A is

a

constant positive definite matrix.

led to the requirement that

This

J ’ W A

+

AJ(x) (12b)

be negative definite.

W e

note that

f(x) = J(x)X = J(x)f(x)

and thus V(x)

=

f’(x)f(x)

+

f’(x)f(x)

=

f’(x)IJ’(x)

+

Jb)f(x)

r= f’(x)j(x)f(x)

When J(x) is negative definite, p(x) is negative definite,

and

V(x)

in E quatio n 1 3 is

a

Liapunov function.

Krasovskii uses the Jacobian matrix of the usual

linearization procedure, but it does not limit trajectories

to the vicinity

of

the origin. Furthe r, f(x) is used in the

Liapunov function rather than the states, x(t), thern-

selves.

Th e case where the x(t) are used i n the form

V(x) = x(t)’x(t) (

is usually referred to

as

Liapunov’s original form ,

select the system

Example.

To illustrate the use of this theorem,

k1 = - “ X I + x2

k2

=

x 1

- x 2 -

x Z 3

a > O

which has the equilibrium state x

=

0.

Here

-ax1

+

x2

x 1 - x 2 - Z 3

f(x)

=

and

2 -2 -

6x

] j (x ) = [ -2 ,

J d = [-“

- 1

- 3x22

f’(x)f(x)

= - a x 1

+ x 2 ) 2

+

x1

-

x2

-

4

By examining the minors of j (x) , we can show that

negative definite; further, f’(x)f(x)+ 00 as (1xl/+

and thus the equilibrium state (origin) is asymptotic

stable in the large.

Unfortunately, while Krasovskii’s method is relativ

easy to use, experience seems to indicate that its grea

application holds for “slightly” nonlinear syste

As a result, there have been many attempts such

those by Ingwerson

(11.73),

Szego

11.25),

Ku and P

(I1.29),

and others to generalize the procedure.

As a typical illustration of a generalization, we c

sider briefly the work of Ingwerson. In this appro a

it is required t hat the gradient

VV x)

satisfy the condit

that the curl of a vector is equal

to

zero. I n particu

it is known th at the necessary a nd sufficient condition

a vector function, g, to be the gradient of a scalar is

the curl matrix must be zero, where the ( i , j ) elem

of the

( n

x

n ) curl matrix is defined by

For

the curl matrix to be zero, the following /z

[ n ( n

-

conditions on the

gl,gz,. .

.

,g,

must be satisfied

Further, these conditions are necessary and suffic

for the scalar whose gradient is

g

to be independent

the path

of

any line integration.

Ingwerson’s method star ts with the idea alre

outlined for a linear or linearized system. Thu s, if

write our linearized equation as

AUTHORS Okan Gure l is a S t u z Me m be r u t t he N e w Y

Scient c Center

of

the IBM Corp. Leon Lapid us is Profe

o f Chemical Engineering at Princeton University. T

coauthored “Stability via Liakunov’s Second Method,” w

appeared on page 72 o

I ?EC,

J u n e 1968.

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an d select

a

Liapunov function as in Equation

4

V(x)

=

x’Ax

16 )

then, for

a

truly linear system, Equat ion

6

J’A + AJ = -C

17 )

must be met to ensure stability. But this presupposes

that A is a constant independent of x; in the nonlinear

case this is not true and, in fact, A = A(x). Wit h this

complication in mind, we follow the basic idea of the

linear problem of choosing a

C

and then calculating an

A.

But now we note th at if

A

is constant,’ hen

or that the elements of

A

are related to the second

derivatives of V(x). Thus, once we have A, one integra-

tion yields the gradient

of V(x), vV(x),

and a second

integration yields the Liapunov function

V(x)

itself.

The real question is how to perform this integration,

and it is here that the curl conditions come to bear.

Note at the same time that this assumes a quadratic

form, Equation

16,

for the Liapunov function. As

such it then falls into the broad category of Krasovskii’s

method.

Once the matrix C is chosen, the matrix A follows,

and the integrations leading to VV(x) and then to V(x)

are carried out. Thus, first

n x

VV(x) = J Adx

(18)

where the explicit integration is carried out for x t as

though the oth er variables were constant-Le.,

VV X) =1 i l d X l +

IZu,,

xz +

.

. +

and second

V X) =1V(x)’dx (20)

I n this last integral, the unique scalar V(x) is obtained

by a line integration

of VV(x)

along any path. For

this integral to be independent of the path, the curl of

VV(x) must vanish or

This relation can be satisfied if the a t j are allowed to

contain only the variables

x i

and

x j .

Th e simplest path

for the integration is given by

J Vz x1,hz,O,. .

.,O)dXz

+ . .

+

where the com ponent of vV(x) in the x t direction is v V ~

T o summarize, Ingwerson’s metho d calculates J(x)

an d then chooses a symmetric, definite (or semidefinite)

C. Now A is calculated from Equation

17

but all

terms which violate

u i j

= a j i are crossed out and all vari-

ables in

a i j

are set to zero except x { and

x , .

This

A

matrix is integrated twice and the resulting

V x )

tested

for its appropriate properties.

If

V x) is definite, then

the solution to the problem is known.

Of

specific interest is that Ingwerson has tabulated

solutions of Equation

17

for J, a constant matrix, up to

the 4th order. Thi s helps in calculating A after having

chosen

C.

However, it must be mentioned that the

method is not completely general and since the A matrix

is not unique, considerable ingenuity may be required

in some cases to make the proper choice of th e impor tant

matrices.

Szego

11.25)

and a series

of

investigators beginning

with Ku and Puri (11.29) nd including Puri (11.33),

Puri and Weygandt

(11.3 ),

nd Haley 11.27) have all

approached the problem of forming generalized quad-

ratics in either the pure Krasovskii form [involving

f(x)] or in the state form [involving x(t)].

As

an

example, the Liapunov function is taken as

V(x) = x’A(x)x

and a set of conditions is set up in terms of a specific

form for

A(x)

such that the definite or semidefinite

conditions on V(x) and ?(x) are established. Sufficient

details on these methods are given in the Appendix .

Finally, we mention the work of Mangasarian (11.31)

and Rosenbrock (11.22). Each of these may be con-

sidered a form of Krasovskii’s approa ch. Thus , Man -

gasarian proposed certain conditions on

x’f(x)

and was

able to relax the differentiability requirement on f(x)

at x = 0. Rosenbrock suggested a Liapunov function

given as the sum of some measures on th e function fi(x)

as follows:

If the quadratic form is used as a measure, Krasovskii’s

form is obtained.

Zu

bov-Type Methods

Zubov-type methods sta rt with derivatives of

V(x)

and

then proceed back to the funct ion itself. Thus, in this sec-

tion we consider methods based upon th e use of ?(x)

rather than V(x) directly. I n other words, a type of in-

direct approach is used. I n Zubov’s method itself

(11.6),

a partial differential equation is solved (hopefully in

closed form) to yield stability information; in the

variable gradie nt method of Schultz and Gibson

(11.23)’

the analysis proceeds through the gradient

V V P )

which

is related directly to

?(x).

I n Zubov’s method the solution of the equation

V(x) = VV’X = VV’f(x)

V O L . 6 1

NO.

3

M A R C H

1 9 6 9

37


9/12

for V will yield the Liapunov function where +(x) is a

definite or at least semidefinite function of x. For the

two-dimensional case, as an illustration,

21 = l X l , X 2 )

x 2 =

z x1,xz)

Equation 23 becomes

Zubov solved this problem by a change of variable

V(x1,xd = -In

[I

- v(xl,xz>l

25)

so that from

Equation 2 4 becomes

-@(xi,xz) [1 - V(XI,XZ)]26)

Actually, Zubov also considered a more general right-

hand side of Equation 26, but we shall not

go

into this

here. Of particular importance was that Zubov was

able to show, under mild restrictions on the differential

equations, the following almost sweeping results :

If Equation 26 can be solved for v , and

if

0

v

< 1,

v

is a Liapunov function, and this is a necessary

and sufficient condition for complete stability of the

origin, X I =

x2

= 0.

If v = 1, assuming it exists, it is an integral curve

of

the system equations and is the boundary of the

region of asymptotic stability.

As an illustration of a system which can

be solved by Zubov’s method , we consider

1.

2.

Example.

2 = -x1 +

2 X 1 2 X 2 = fl(x1,xz)

2 2

= - 2 2 =

z X 1 , X d 2 7 )

Since +(x~,xz)must be positive definite, we make the

obvious choice

+(Xl,XZ)

=

x?

+

x22

such that using Equation 27, Equation

26

becomes

-(XI’ + xz2) l V) 28)

The solution to this equation can be obtained by ele-

mentary means as

Kate

that v(x1,xz) vanishes only at X I =

x2

= 0 and

positive everywhere else; as such, it fulfills the requir

ment of a Liapunov function.

Also

with the same properties. When ~ 1 x 2= 1, we see th

v(x1,x~)=

1,

and thus x ~ x z = 1 is the boundary

stability; in other words, when xlxz < 1, the system

asymptotically stable.

Obviously, the main difficulty in using Zubov

method is the problem of solving the partial differenti

equation in closed form and the need to choose

+ x1,x

in an intelligent fashion to facilitate the solution. Autho

such as Szego (11.26) have reinterpreted the approac

in an effort to make this selection easier. But eve

here the problem remains a most difficult one whic

cannot be recommended for general applicability.

The variable gradient method, by contrast, tends

develop a relatively straightforward procedure for tailo

ing a specific Liapunov function to each particular no

linear system. I t does not start with the assumptio

of a qu adratic form for the Liapunov function but rathe

defines an arbitrary gradient function with coefficien

to be determined. With this gradient, an integratio

of the form previously discussed as used in Ingwerson

method is performed to yield V(x)--i.e., we have fro

Equation

2 ,

V x )

= VV(x)’X(t)

and

V(x)

=

vV(x)’dx

Th e coefficients in the gradient are determined so as

make “(x) negative semidefinite. Note that in th

procedure one gets away from the purely quadrat

Liapunov function which may not exist for some system

The first step is to assume a completely arbitra

column vector VV(x)--i.e.,

1

VV(x) =

The coefficients ai,(x) are functions

of

x and, in pa

ticular, may have the explicit form of a constant plus

function of the s tate variables,

a t ) =

at]&.

+

a, >

These coefficients are to be determined from constrain

pu t on ”(x), by the curl conditions, by obvious inspe

tion and even by the necd for V(x) to be positive defini

This feature will be seen shortly.

Once vV(x) has been assumed, it follows direct

that “(x) = V V x ) ’ x can be calculated; v x ) is co

strained to be at least negative semidefinite, that

38

INDUSTRIAL A ND ENG INEERI NG CHEMISTRY


10/12

possess the same negative sign throughout state space

except a t isolated points [this determines some of the

aij(x)

above]. Now the curl equations are invoked to

determine the remaining unknown

aij(x)

and allow

the calculation of

V(x) = svV(x)’dx.

T o illustrate this method, we choose t he

system

Example.

21

= x2

i

- x 2

-

x13

We choose the gradient as

and calculate, using

azz(x)

= 2 for simplicity, and drop-

ping the x functionally notation for ease in writing,

V(x) = VV(x)’X = (UllXl + a12x2)21+

(azm +

U z z X 2 ) 2 2

= X 1 X Z U 1 1 2x12 a21) +

~ 2 ~ ( ~ 1 22) -

az1x14

(30)

To make V(x)

at

least negative semidefinite we pu t

a11 - 2x12 - 2 1 = 0

0 <

a12

< 2

I n particular, we choose

a12

= 1.

30 becomes

As

a result, Equation

V(x) =

- x 2 2

- ~ 2 1 x 4

(31)

and

with the only unknown coefficient left being

a21.

we determine from the curl equations

This

bVVl

dVV2

axz axl

where

VV, = bV/bxl,

or using Equation 32

When we recall that

a21

is really a function of x in the

two parts discussed previously, then

aUzl2

1 = a 2 1 8 +

a211

+

x1

x1

This identity can be satisfied if we choose ~ 2 1 ~0 and

a212 = 1. Thus

and the line integration of

VV(x)

yields

V(x),

33)

It is not difficult to show that Equation 33 is a valid

Liapunov function for the nonlinear system.

Note that a

nonquadratic in

x1,x2

has resulted.

Extensions and generalizations of the gradient method

have been proposed by Puri 11.32) and by Szego 11.24).

Details are presented in the Appendix.

Of interest,

however, in the present context is the recent work of

Peczkowskii and Liu 11.46). Whereas the variable

gradient method starts with th e form

VV(x) = [A(x)lx

Peczkowskii and Liu start with

= [A(x)lf(x)

Reiss and Geiss 11.35) have suggested essentially

an iterative technique for forming V(x) as linear com-

binations of the squares of the individual states. The

usual test for definiteness is used to determine the itera-

tion and to yield a n approximate Liapunov function.

Lur’e-Postn kov-Type Method

The methods to be detailed here originated in the

work of Lur’e and Postnikov although Lur’e is fre-

quently referenced singly. Th e methods apply to a

special class of systems suitable for feedback control

applications with a single, special type of nonlinearity.

Because of the lengthy details of the development, we

shall merely present the necessary highlights.

First, however, we wish to point out that there are

two forms of the equations which belong to the Lur’e-

Postnikov class. These are the so-called indirect control

and the direct control cases. Th e difference is due to

the manner in which the input (control) variable,

u,

s

defined. However, since one case can be shown to be

transformable into the other, we shall not bother with the

distinction.

The basic idea of the present approach is to take the

system equation with its special nonlinearity and con-

vert it into canonical form. Th en a Liapunov function

may be defined which is an extension

of

the quadratic

type we have already discussed. I n particular, con-

sider a scalar nonlinear element whose input is given by u

and whose output is a u ) , and which satisfies the re-

quirements

1(u)du 2 0 0

a (0 ) = 0 r = 0 (41)

La +

3

u +

3

This nonlinearity is included in the system equation as

X = Bx + u a u )

42)

u = v’x

where B is a constant n

X n

matrix and

u

and

v

are

constant vectors.

T o analyze this system, we first make a transformation

to diagonal form by

x = Ty 43)

V O L . 6 1

NO. 3 M A R C H

1 9 6 9 39


11/12

where

T

is the Vandermonde matrix containing

the

eigenvalues

(A,)

of

A.

These A are assumed to be real,

distinct, and nonzero. The transformation of Equat ion

43

converts Equation 42 to

y

= Ay + T-lua (u)

Q

=

v’Ty

(44)

with

A

the diagonal matrix having elements

Xi.

Liapunov function

O n this basis, Lur’e an d Postnikov suggested the

V x )

=

y’Ay

+ l a ( , )

(45)

which is seen to be a quadratic term in the states

y

plus

an integral term involving the system nonlinearity.

After some manipulation this leads to

V X) = -y‘Cy

+ CY(~U’T’-’A+

v ’ T A ) ~ O.*V’U

(46)

where

C

has the form we have previously encountered,

viz:

- C

= (A‘A + AA)

Because of the special character of the eigenvalues,

if C

is positive definite then

A

is positive definite and

vice

versa. Thus, Lur’e and Postnikov further suggested that

C be chosen by

C

= bb‘

which, when substituted into Equation 46, leads to a

set of n algebraic equations for the components of b.

Assuming these equations can be solved, we see the

result is a positive definite V x) and at least a negative

definite

V x ) .

Further work in this area has been detailed by Letov

(11.3),Yakubovich 11.g), Popov (ZI. O , Lefschetz

Z 1 . 7 4 , and Mufti

11.27),

in particular. However, we

do not wish to detail these in the present writeup, but

details are given in the Appendix .

Mi sce l l a n e o u s - T yp e Me t h o d s

Here, we have a number of different methods which

do

not seem to fit conveniently into our previous cate-

gories. In general, these methods do not introduce

basic changes in the development of Liapunov function

generation. Th ey can be viewed as either energy-type

analogies which fall back into Chetaev-type methods

or analytic-type constructions using various mathe-

matical techniques to form a suitable function, which

fall into the Krasovskii-type group. In particular, there

are the methods of Zubov (11.4, arbashin 11.72))

Karendra-Ho-Goldwyn

(11.5 ,

Harris

11.28),

nti-

penko 11.36), Puri Z1.39), Boyanovich 11.47), Ponzo

ZI.

42), and Kinnen-Chen 11.47).

C o n c l u s i o n

This paper summarizes the historical development and

classification of methods for generating Liapunov

functions for systems of deterministic ordinary differe

tial equations. In addition, surveys which appeare

between 1960 and 1967 have also been cited. As seen,

type of classification is possible within which almost a

the different methods can be contained. In a bas

sense, very little work has been done since Krasovsk

proposed his generalized quadratic construction. I t

hoped a new- approach to this problem might lead

fruitful results.

APPENDIX

A

five-part appendix containing details of the five

methods for constructing Liapunov functions (Chetaev-

type, Krasovskii-type, Zubov-type, Lur’e-Postnikov-type,

and miscellaneous-type) can be obtained by citing this

article and writing

Dr.

Okan Gurel,

IBM

Carp., New York

Scientific Center, 410

E.

62nd St., New York, N. Y.

10021.

B i b l i o g r a p h y

Th

first part contains those survey papers which are cu

rently available. As seen, the first surveys were pu

lished in 1960 by three Russian scientists

1 . 7 , 1.2).

Th

most comprehensive survey is by Drake and associat

1.6) in 1965 as a NASA report. In reading some

these surveys the reader should be aware that a bi

seems to exist, in the sense that certain papers strike th

authors’ favor.

The second part of this bibliography lists chronolog

cally all the papers and reports of interest. Th e fir

paper in connection with construction of a Liapuno

function was written half a century after the origin

treatise of Liapunov

ZIZ.

7) by Lur’e and Postniko

(11.7)

n 1944. The first work in the Western wor

appeared 16 years later in 1960 in a thesis by Ingwerso

The third part of this bibliography presents a li

of books in the English language. Except for one, a

of these originally appeared in Russian and have no

been translated. Th e first book by Liapunov is

French translation.

This bibliography is made up of three parts.

ZZ 3).

I. SURVEY P A P E R S

1960

1.7) Barbashin, E. A., “The Construction

of

Liapunov Functions for Non-lin

Systems,” Vol. 2 , pp 943-7, Proc . First Intern. Congr. o In t. Fed. Auto. Cu

Moscow, 1960, Butterworths, London.

1.2)Lur’e A. I. and Rorenvasser E . K. On Methods of Constructing Liapu

Functions in tge Theory

of

Non-]:near do ntr ol Systems,” ib id . , Vol. 2 , pp 928-3

1964

(1.3)

G .

R. Geiss, “T he Analysis and Design of Nonlinear Control Systems

Liapunov’s Direct Method,” Air Force Flight-Dynamics Laboratory Resea

and Technologv Division U. . Air Force

Wright-Patterson

A i r

Force Ba

Ohio, Tech . Do; R e p . No.’RTD-TDR-63-40?6, August 1964 .

(1.4)

arks, P. C., “Stability Analysis for Linear and Nonlinear Systems Us

Liapunov’s Second Method,” “Progress in Conrrol Engineering,”

(Ed,

R .

Macmillan), Vol.

2,

pp 29-64, Academic, New York, N. Y. , 1964.

1965

(1.5)Derman, C. C., and LeMay, A. R., ”A Survey of Methods for Generat

Liapunov Functions,” (N66-35556), pp 114-32, Aero-Astrodvnamics Resea

Review No. 2, July I-Dec. 30,

1964,

N66-35546*, 156 pp, Naiional A eronau

and Space Administration, Marshall Space Flight Center, Huntsville, A

1 April 1965.

(1.6)Drake,

R.

L., “Methods for Systematic Generation of Liapunov Function

Parts I and

11,

NASA CR-67863 and NASA CR-6?864,1965.

1.7) Lefferts, E.

J.

“ A Gui i e of the Application

of

the Liapunov’s Direct Meth

to Flight Control Systems, NASA CR-209, April 1965.

4 0 I N D U S T R I A L A N D E N G I N E E R I N G C H E MI S T RY


12/12

18)

etov A. M., “Liapuno v Theory of Stability of Motion, Disciplines and

Techn iqdes of System Control,” (Ed. J. Peschon), p p

267-314,

Blaisdell, New

York, N. Y., 1965.

I9) Schultz, D. G., “Th e G eneration of Liapunov Functions, Advances in Control

Systems,” (Ed. C. T. Leondes), pp 1-64, Vol. 2,Academic, New York, N. Y.,

1765.

1966

(I10) Salah, M . M., “Investigating Stability of Differential Equations b Liap-

unov’s Direct Method,” Middle East Technical University, Ankara, qur key,

M.S. Thesis, 1966.

(1.77)SzegB, G. P. ,“Li apun ov Second Method,”

Appl. M ech. Rev., 19

lo),

833-8

1766).

1967

(1.72)Gurel,

O.

nd Salah, M. M

“ A Surve of Methods of Conatructing Liap-

unov Functions,” IBM New York’Scientific &enter, Rept. No. 39-022, ebruary

1967.

11. ORIGINAL PAPERS

1944

11.7) ur’e, A.

I.,

and Postnikov, V. N., “O n the Theory of Stability of Control

Systems,” P M M ,

8 1944).

1949

(112)Aizerman M A

“On

a Problem Concerning the Stability in the Large

ofDynamical ystkms;i” Us . Mat . N au k., 4 4), 187-8 1947).

1950

(1I.3)Letov, A. M.,“ Inherently Unstable Control Systems,” P M M , 14 1950).

1953

(11.4) ubov V I. “Some Sufficient Conditions of Stability of Nonlinear Systems

of

Differen;ial’Ec;uations,”

ibid., 17 1953).

1954

(I1 ) Krasovskii, N. N.,,YOn the Stability in the Large of a System of Nonlinear

1955

Differential Equations, rbid., 18, 735-7 1954). (See also 11.7 elow.)

(11.6) ubov V. . “Problems in the Theo r of the Second Metho d of Liapunov,

Constructich of h e General Solution in tLe Domain of Asymptotic Stabilit ,”

ibid., 19, 179-210 1955).

1957

(I1

) Krasovskii N. N. “Stability in the Case of Large Initial Disturbances,”

i’btd., 21, 309-{9 17573.

(11.8)

etov, A. M., “Die Stabilitat von Regelsystemen mit nach ebender Ruck-

fuhrung,” Regelunstechnick, Moder ne Theorien und ihre gerwendb arkeit,

Munich, 1957.

(11.9)Yakubovich, V. A. “On a Class of Nonli near Differen tial Equations,”

Dokl. Akad. Nauk SSSR,’ 117, 44-6 1957). [Engl. Trans.: AMS Translations

Series 2, p 1-4, Vol. 25 1963).1

1958

11.70) opov, V. M “Rela xing the Sufficiency Conditions for Absolute Stability,”

Automat. i Telerneh.,”l9,1-7 1958).

(See also 11.78below.)

(11.77)

ozharitskii, G. K.,

“ O n

the Construction of the Liapu nov Functions from

the Integrals of the Equations for Perturbed M otion,” P M M , 22,145-54 1958).

1960

II .72) Barbashin E. A

“ O n

Constructing Liapunov Functions for Nonlinear

(11.73) ngwerson, D. R., “ A Modified Liapunov Method for Nonlinear Stabilit

(See also 11.

Systems,” Proc.’of IFA &, Moscow,

1960

(Butterworths

1961).

Problems,” Ph.D . Thesis, Stanford University, Nove mber 1960.

below.)

(11.74)Lefschetz, S., “Controls: An Application of the Direct Method of Liap-

unov,” Bol.

Sac.

Maternat. Mex., p p 139-43, 1760.

(11.75)Narendra, K. S., and Ho , Y. C., “On the Construction of Liapunov Func-

tions for Nonlinear Systems,” Cr uft Lab. Tech. R ep. N. 328, Harvard University

Cambridge, Mass., 1960.

(See also 11.20 below.)

1961

(11.76) hang, S .

,?.

L. “Kinetic F unction for Stabilit Analysis of Nonlinear

11.17)

Ingwerson, D. R., “ A Modified Liapunov Metho d for Nonlinear Stability

21.78)

Poppv V M “Absolute Stability of Nonlinear Systems of Automatic

Contro l Systems, J . ojBaszc Engineering, AS M E, 83, 91-i 1961).

Analysis,” I R E

Trans.,

pp 199-210, 6 2) 1961).

Control, A h m a t . i kelernch., 22, 961-77 1761).

1962

(11.19) nfante, E.

F.,

“A New Approach of the Determination : he Domain

of

Stability of Nonlinear Autonomous Second Orde r Systems, Ph.D. Thesis,

University of Texas,

1962.

(See also 11.38 elow.)

(11.20)Lefschetz, S., “Some Mathematical Considerations onNonlinear Automatic

Controls,” Contributions o Differential Equations,

1 1), 1-28 1962).

(11.27)

Mufti, I. M .

“ O n

the Stability

of

Nonlinear Controlled Systems,” J .

Math. Anal.

Appls.,’4,

57-75 1962).

(11.22)kosenbrock, H. H., “A Liapunov Function with Applications

to

Some

Nonl inear Physical Systems,” Autornatica, 1, 31-53 1962).

(11.23)Schultz,. D. G., and Gibson, J. E., “The Variable Gradient Method for

Generating Liapun ov Functions,” AZEE

T ra n s .

Part

11,

ppls. and Ind., 81, 203-10

1

962).

(11.24)SzegB, G. P., “On the Application of the Zubov M,ethod for Construction

of Lia unov’s Functions for Nonlinear Control S stems Proc. 7962 Joint

Auto.

Cont.

C’onf.,

New York, N. Y.

(11.25) zegB, G. P.,

‘ ‘k

Contribution to Lia unov’s Second Method Nonlinear

Autonomous Systems

84 573-8 1962).

(Presen te d a t the Wi dte r A k a 1 Meekn g of ASME , NewYo;k,

G .

Y.)

[Tr an s . AS M E,

S e r . 6 ,

85023, 137-42 1963).]

Traw ASME Ser

8

J

Basic

Eng

2I .26 )

SzegB, G. P., “ O n New Partial Differential Equations for the Stability

Analysis of Time-Invariant Control Systems,”

J. SIAM Control, Ser . A,

l 1)

63-7 1962). (Same work has appeared in Proc. of the

2nd

Congress of the Int:

Fed. of Auto. Cont., Basle, Switzerland, 1763, under the title: “New Methods

for Constructing Liapuno v Functions for Time-Invariant Control Systems.”)

1969

(11.27)

aley, R. L. “Generation of Liapunov Functions for Certain Classes of

Nonlinear S stems’” Ph.D. Thesis, Moore School of Electrical Engineering,

University o8Penniylvania, 1963.

21.28) Harris

S.

“Application of Rout h Criterion to Phase-S pace Stability ”

Master’s TLesis,’ Moor e Scho ol of Electrical Engineering, University

of

Penns;l-

vania, 1963.

(11.29)

Ku,

Y. H., and Puri , N, N., “ O n Liapunov Functions of Higher Order

Nonlinear Systems,”

J . Franklin Inst., 276, 349-64 1763).

(11.30) ei hton W “On the . Construction of Certain Liapunov Functions,”

Proc.

Nat?. Acab. Scl 50 1763). ,[See also W. Leighton “ O n the Construction

of

Liapunov Functjbns for Certain Autonomous NonliAear Differential Equa-

tions,”

Contributions

0

Diferentiul Equations,

2 1-41, 367-83 19631.1

11.39)Mangasarian 0 L “Stability Criteria for Nonlinear Ordinary Differential

Eguations,”

S I A M ’ J . C o n h ,

Ser.

A , 1 3), 311-13 1963).

(11.32) uri, N. N., “NASA Proposal for Study and Research in New Metho ds for

Systematic Gener ation of Liapun ov Functions for Control Systems,” Sub mitted

.to NASA. Octob er 1763. (See 1.6.)

(11.33)

uri, N. N. , “N SF Proposal for Study and Research in the Generation of

Liapunov Functions and the Design of Opti mal Systems,” Submitted t o NSF,

1963. [See1.6.1 (See also 11.43 elow.)

(11.34) ari, N. N., and Weygandt C. N.

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(1135) eiss, R., and Geiss, G., “T he Construction of Liapunov Functions,” I E E E

Routh’s Cho nical Form,” J . Frankiin Inst., 576,365-83 1763).

?ratis. Auto. Cont., 8 , 382-3 1963).

1964

ZI.36)

h t i p e n k o V

I

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Avtomatiku, 9 Zj, i-5” 1764). (English translation NASA Report N66-11716.)

(11.37) rockett, R. W., “O n the Stability of Nonlinear Feedback Systems,”

IEEE

Trans. on Appl. andIndurtry, 83, 443-7 1764).

(11.38) nfante, E. F., and Clark, L. G. , “A Method

for

theDetermination

of

the

Domain of Stability of Second-Order Nonlinear Autonomous Systems,” J .

Appl. Mech., Trans. of A S M E , S er . E , 86, 15-20 1964).

(11.39)

pu:i, N. N., “On the Global Stability of a Class ofNonlin ear Time-Varying

Systems, Presented at Dubrovnik, IFCA Symposium on Sensitivity Analysis,

September 1764.

11.40)

alker J. A. “An Integral Method of Liapunov Function Generation

for

Nonlinear Ah on ok ou s Systems,” Ph.D. Thesis, University of Texas,

1964.

(See also 11.44below.)

(See also 11.43 elow.)

1965

(11.41) oyanovich, D.,

“ O n

the Application of Hydrodynamics to the Study of

the Stability of Singular Points of Differential Equations: Auton omou s Systems,”

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10,470-2 1965).

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autonomous System, Proc. 7965 Joint Auto. Cant. Conf. Rensselaer Polytechnic

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A S M E , S e r . E , 32 3), 569-75 1765).

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the Construction and Interpretation of Liapunov

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Functions,” Ph .D . Thesis, University of Alab ama, 1766.

Ph.D. Thesis, University ofNo tre Dame, April

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(See also 11.49 below.)

1967

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E

and Chen C. S “?,apunov Funct ions for a Class of n-th

(N.48) innen, E., and Chen, C. S “Liapunov Functions from Auxiliary Exact

(11.49)Peczkowski

J.

L., and Liu, R. W., “A Format Method for Generating

Order NonliAea;bifferential’Equazons,NASA CR-687, January 1967.

Differential Equations,” NASA

CR777,

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211.7)

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Princeton University P’iess, Princeton, N. J.

1949.

(Russian Edition 1892.)

1957

(111.2)Lur’e A.

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rol,” He r Majesty’s Stationery Office, Lond on,

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1961

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of

Motion,” Pergamon Press, London, 1961.

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(Russian Edition 1946, 950.)

sity Press, Princeton, N. J.,

1961.


1963

(111.5)Hah n, W., “Theory and Application o f Liapunov’s Direct Method,”

111.6)Krasovskii N. N “Stability of Motion,” S tanford University Press, Stan-

(121.7)Z u p y , V .

I.,

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Prentice Hall, Inc., Englewood Cliffs, N.

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ford, Calif., 196j.

Systems, Pergam on Press, New York ,

1963.

(Russian Edition

1757.)

(German Edition 1759.)

(RAssian Edition 1959.)

1964

(111.8)

Aizerman, M. A., and Gantmacher, F. R.,“Absolute Stability of Regulator

Systems,” Holden Day, Inc., S an Francisco, Calif., 1964. (Russian Edition

1763.)

VOL. 6 1 NO. 3 M A R C H 1 9 6 9 41

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