A Guide To The Generation of Lyapunov Functions [PDF]

Zitiervorschau

Okan Gurel and Leon Lapidus

ver the past eight years a vast outflow of research and publications has resulted from the use of Liapunov’s “second (or direct) method” of stability analysis. This 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. I n 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 h e difficulty, however, 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 an extent upon the experience, ingenuity, and, often, good fortune of the analyst. Unfortunately, the available material on the second method has evolved to the point where a n 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. This paper has thus been prepared to allow a novice in the field to approach this available literature in a rational manner. Consideration is given only to the deterministic problem described by ordinary differential equations. We include briefly the theorems necessary for discussing Liapunov’s second method and then approach the different methods for constructing Liapunov functions in a general manner. T h e main theme is to classify the constructive procedures into a few simple

0

30

INDUSTRIAL A N D ENGINEERING CHEMISTRY

A Guide to the Generation of Liapunov Functions This survey aids the engineering application of Liapunov’s second method of stability analysis by bringing together published work on the construction of Liapunov functions

2 . T h e system is asymptotically stable, if for any xo in Rt, lirn /lx(t)ll + 0. Stated in another fashion, this t+

OJ

says that the trajectory of x(t) eventually ends u p a t the origin. 3. T h e system is asymptotically stable in the large or completely stable if it is both asymptotically stable and the region RZ is the entire state space. Now the trajectory eventually ends u p a t the origin no matter where its starting point is. 4. T h e system is unstable, if for some xg in Rz with 0 small, lim (t)l/ > a. Now the system goes outside t+

categories and then to outline how the various methods fall within each category. Because of the almost overwhelming amount of material, only a few of the constructions will be explicitly detailed. Brief descriptions of all other methods are contained in five sections of a n appendix ; this appendix is, however, not published here but can be obtained directly from the authorsdetails 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. Further and 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 a n 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,and let R2 be a similar region for which llxll < @. Assume that a > @, then regions RIand Rz may be thought of as hyperspheres around the origin of radius CY and 8, respectively. If the state of the system a t time t = 0 is x = xo # 0, then we say that: 1. T h e system is stable, if for every xo in Rz there is a region RBdefined by 11x11 < 7,CY > 7 > 8, such that lirn \\x(t)ll < a. Stated in another fashion, this says t-+ m

that the system never goes outside the region Ra.

- XI/

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 and are continuous. 2. V(x) is positive definite. This means that V(x) > 0 for all x # 0, but that Y(x) = 0 for x = 0. 3. T h e 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 the same time, we note by the chain rule of differentiation that dV dxi - - = vv X ( t ) = vv . f(x) = V(x) = %=I dt grad V 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 semi0 for x # 0. definite. This means that V ( x ) 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., V ( x ) > 0. By combining certain of these six properties, we may now specify various features of the stability of the system given by Equation 1. Thus: Theorem 1. If a Liapunov function exists satisfying properties 1, 2, and 4, then the system is stable in the vicinity of the origin. Theorem 2. If a Liapunov function exists satisfying properties 1, 2, and 3, then the origin is asymptotically stable. e


O

which has the equilibrium state x = 0.

f(x) =

and note that

is symmetric. N o w we state Krasovskii’s theorem as Theorem 9 (Krasovskii). For Equation 11, with f(0) = 0 and f(x) differentiable, the equilibrium state x = 0 is asymptotically stable ifj(x) is negative definitive. A Liapuiiov function for the system is given by the quadratic in f

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 quadratic form

Y ( x ) = f’(x)Af(x) where A is a constant positive definite matrix. led to the requirement that

J’WA

+ AJ(x)

This

+ f’(x)f(x)

+ Jb)lf(x)

=

f’(x) IJ’(x)

r=

f’(x)j(x)f(x)

When J(x) is negative definite, p(x) is negative definite, and V(x) in Equation 13 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. Further, f(x) is used in the Liapunov function rather than the states, x(t), thernselves. The case where the x(t) are used in the form 36

1

x2

xZ3

=

[-“ -1 1

=

1

-

(-ax1

3x22

]

+

j(x) x2)2

+

=

[-2,

(x1

2

-

-2 x2

-

4

6x22 2

By examining the minors of j ( x ) , we can show that it is negative definite; further, f’(x)f(x)+ 00 as (1xl/+ co, and thus the equilibrium state (origin) is asymptotically stable in the large. Unfortunately, while Krasovskii’s method is relatively easy to use, experience seems to indicate that its greatest application holds for “slightly” nonlinear systems. As a result, there have been many attempts such as those by Ingwerson (11.73),Szego (11.25), K u and Puri (I1.29),and others to generalize the procedure. As a typical illustration of a generalization, we consider briefly the work of Ingwerson. In this approach, it is required that the gradient VV(x) satisfy the condition that the curl of a vector is equal to zero. I n particular, it is known that the necessary and sufficient condition for a vector function, g, to be the gradient of a scalar is that the curl matrix must be zero, where the (i,j) element of the (n x n ) curl matrix is defined by

For the curl matrix to be zero, the following ‘/z [ n ( n - 1)] conditions on the gl,gz,. . . ,g, must be satisfied

(12b)

f(x) = J(x)X = J(x)f(x) V(x) = f’(x)f(x)

+-

(13b)

be negative definite. W e note that

and thus

--ax1

x1 - x 2

Here

and

f’(x)f(x) =

x2

k 2 = x1 - x 2 - xZ3

J(d J(x)

+

-“XI

INDUSTRIAL A N D ENGINEERING CHEMISTRY

Further, these conditions are necessary and sufficient for the scalar whose gradient is g to be independent of the path of any line integration. Ingwerson’s method starts with the idea already outlined for a linear or linearized system. Thus, if we write our linearized equation as

York Scient$c Center of the I B M Corp. Leon Lapidus is Professor of Chemical Engineering at Princeton University. They coauthored “Stability via Liakunov’s Second Method,” which appeared on page 72 of I&?EC, J u n e 1968. AUTHORS O k a n Gurel is a Stuz Member ut the N e w

and select a Liapunov function as in Equation 4 V(x) = x’Ax

(16 )

then, for a truly linear system, Equation 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). With this complication in mind, we follow the basic idea of the linear problem of choosing a C and then calculating a n A. But now we note that if A is constant,’then

or that the elements of A are related to the second derivatives of V(x). Thus, once we have A, one integration 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 a t 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) =

J0

Adx

(18)

where the explicit integration is carried out for x t as though the other variables were constant-Le., VV(X) =

1

UildXl

+

IZu,, +

=

1

dxz

.

*

.

+

where the component of vV(x) in the x t direction is v V ~ . T o summarize, Ingwerson’s method calculates J(x) and 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 variables 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. This 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 the important matrices. Szego (11.25) and a series of investigators beginning with Ku and Puri (11.29)and including Puri (11.33), Puri and Weygandt (11.3#),and Haley (11.27) have all approached the problem of forming generalized quadratics in either the pure Krasovskii form [involving f(x)] or in the state form [involving x(t)]. As a n 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 considered a form of Krasovskii’s approach. Thus, Mangasarian 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 the function fi(x) as follows:

and second

V(X)

vV(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 . The simplest path for the integration is given by

VVz(x1,hz,O,. . .,O)dXz

J2

+ ... +

If the quadratic form is used as a measure, Krasovskii’s form is obtained. Zu bov-Type Methods

Zubov-type methods start with derivatives of V(x) and then proceed back to the function itself. Thus, in this section we consider methods based upon the use of ?(x) rather than V(x) directly. I n other words, a type of indirect 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 gradient 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) VOL. 6 1

NO. 3

MARCH

1969

37

Kate that v(x1,xz) vanishes only at X I = x2 = 0 and is positive everywhere else; as such, it fulfills the requirement of a Liapunov function. Also

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 = fl(Xl,X2) x 2

= fz(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 24 becomes

-@(xi,xz) [1 - V(XI,XZ)](26) Actually, Zubov also considered a more general righthand 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 : 1. If Equation 26 can be solved for v, and if 0 5 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. 2. 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. Example. As an illustration of a system which can be solved by Zubov’s method, we consider 21

= -x1

+

2X12X2

2 2

=

=

fz(X1,Xd

-22

= fl(x1,xz)

with the same properties. When ~ 1 x 2= 1, we see that v(x1,x~) = 1, and thus x ~ x z = 1 is the boundary of stability; in other words, when xlxz < 1, the system is asymptotically stable. Obviously, the main difficulty in using Zubov’s method is the problem of solving the partial differential equation in closed form and the need to choose +(x1,xZ) in an intelligent fashion to facilitate the solution. Authors such as Szego (11.26) have reinterpreted the approach in an effort to make this selection easier. But even here the problem remains a most difficult one which cannot be recommended for general applicability. The variable gradient method, by contrast, tends to develop a relatively straightforward procedure for tailoring a specific Liapunov function to each particular nonlinear system. It does not start with the assumption of a quadratic form for the Liapunov function but rather defines an arbitrary gradient function with coefficients to be determined. With this gradient, an integration of the form previously discussed as used in Ingwerson’s method is performed to yield V(x)--i.e., we have from Equation 2 ,

V ( x ) = VV(x)’X(t) and V(x)

=

1

vV(x)’dx

The coefficients in the gradient are determined so as to make $“(x) negative semidefinite. Note that in this procedure one gets away from the purely quadratic Liapunov function which may not exist for some systems. The first step is to assume a completely arbitrary column vector VV(x)--i.e.,

VV(x) =

(27)

Since +(x~,xz)must be positive definite, we make the obvious choice +(Xl,XZ) = x?

+

The coefficients ai,(x) are functions of x and, in particular, may have the explicit form of a constant plus a function of the state variables,

x22

such that using Equation 27, Equation 26 becomes

+ a,&>

a t ) = at]&.

-(XI’

+ xz2)(l - V)

(28)

The solution to this equation can be obtained by elementary means as

38

I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY

These coefficients are to be determined from constraints put on $”(x), by the curl conditions, by obvious inspection and even by the necd for V(x) to be positive definite. This feature will be seen shortly. Once vV(x) has been assumed, it follows directly that $“(x) = V V ( x ) ’ x can be calculated; v ( x ) is constrained to be at least negative semidefinite, that is,

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. Example. T o illustrate this method, we choose the system 21 =

x2

i 2= - x 2

- x13

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 the form VV(x) = [A(x)lx

We choose the gradient as

Peczkowskii and Liu start with

[A(x)lf(x)

=

and calculate, using azz(x) = 2 for simplicity, and dropping the x functionally notation for ease in writing,

V(x) = VV(x)’X = (UllXl

+ a12x2)21 + (azm

=

X1XZ(U11

+

UzzX2)22

- 2x12 - a21) + ~ 2 ~ ( ~-1 2) 2 - az1x14

Lur’e-Postn ikov-Type Method

(30)

T o make V(x) at least negative semidefinite we put a11

- 2x12 - a 2 1 0

=

0

< a12 < 2

I n particular, we choose a12 = 1. As a result, Equation 30 becomes

V(x) =

-x22

-

(31)

~21x4

and

with the only unknown coefficient left being we determine from the curl equations

a21.

This

bVVl - dVV2

axz

where VV,

=

axl

bV/bxl, or using Equation 32

When we recall that a21 is really a function of x in the two parts discussed previously, then

1

=

a218

aUzl2 + a211 + x1 ax1

This identity can be satisfied if we choose a212 = 1. Thus

~

Reiss and Geiss (11.35) have suggested essentially a n iterative technique for forming V(x) as linear combinations of the squares of the individual states. The usual test for definiteness is used to determine the iteration and to yield an approximate Liapunov function.

The methods to be detailed here originated in the work of Lur’e and Postnikov although Lur’e is frequently referenced singly. The 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’ePostnikov class. These are the so-called indirect control and the direct control cases. The difference is due to the manner in which the input (control) variable, u, is 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 convert it into canonical form. Then a Liapunov function may be defined which is an extension of the quadratic type we have already discussed. I n particular, consider a scalar nonlinear element whose input is given by u and whose output is a ( u ) , and which satisfies the requirements

1 La(+ a(u)du

2 0,u

a ( 0 ) = 0,

2

=1

0~ and

# 0

cr = a3,

0

u+

(41) 03

This nonlinearity is included in the system equation as X

=

Bx

+ ua(u)

(42)

u = v’x

and the line integration of VV(x) yields 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

(33)

x

=

VOL. 6 1

Ty NO. 3

(43) MARCH

1969

39

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 Equation 43 converts Equation 42 to

y = Ay Q

=

+ T-lua!(u)

v’Ty

(44)

with A the diagonal matrix having elements Xi. O n this basis, Lur’e and Postnikov suggested the Liapunov function

V ( x ) = y’Ay

+la(,)&

functions for systems of deterministic ordinary differential equations. I n addition, surveys which appeared between 1960 and 1967 have also been cited. As seen, a type of classification is possible within which almost all the different methods can be contained. I n a basic sense, very little work has been done since Krasovskii proposed his generalized quadratic construction. I t is hoped a new- approach to this problem might lead to fruitful results.

APPENDIX

(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 ) ~+

I

A five-part appendix containing details of the five methods for constructing Liapunov functions (Chetaevtype, 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.

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 a t least a negative definite V ( x ) . Further work in this area has been detailed by Letov (11.3),Yakubovich (11.g), Popov (ZI.I 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. Miscellaneous-Type Methods

Here, we have a number of different methods which do not seem to fit conveniently into our previous categories. I n general, these methods do not introduce basic changes in the development of Liapunov function generation. They can be viewed as either energy-type analogies which fall back into Chetaev-type methods or analytic-type constructions using various mathematical techniques to form a suitable function, which fall into the Krasovskii-type group. I n particular, there are the methods of Zubov (11.4,Barbashin (11.72)) Karendra-Ho-Goldwyn (11. 75), Harris (11.28),Antipenko (11.36), Puri (Z1.39), Boyanovich (11.47), Ponzo (ZI. 42), and Kinnen-Chen (11.47). Conclusion

This paper summarizes the historical development and classification of methods for generating Liapunov 40

INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY

Bibliography

This bibliography is made up of three parts. The first part contains those survey papers which are currently available. As seen, the first surveys were published in 1960 by three Russian scientists (1.7, 1.2). The most comprehensive survey is by Drake and associates (1.6) in 1965 as a NASA report. I n reading some of these surveys the reader should be aware that a bias seems to exist, in the sense that certain papers strike the authors’ favor. The second part of this bibliography lists chronologically all the papers and reports of interest. The first paper in connection with construction of a Liapunov function was written half a century after the original treatise of Liapunov (ZIZ. 7) by Lur’e and Postnikov (11.7)in 1944. The first work in the Western world appeared 16 years later in 1960 in a thesis by Ingwerson

(ZZ.73). The third part of this bibliography presents a list of books in the English language. Except for one, all of these originally appeared in Russian and have now been translated. The first book by Liapunov is a French translation. I.

SURVEY PAPERS

1960

(1.7) Barbashin, E. A., “ T h e Construction of Liapunov Functions for Non-linear Systems,” Vol. 2 , p p 943-7, Proc. First Intern. Congr. of Int. Fed. Auto. Cunt.,

Moscow, 1960, Butterworths, London. (1.2) Lur’e A. I. and Rorenvasser E. K. “ O n Methods of Constructing Liapunov Functions in tge Theory of Non-]:near dontrol Systems,” ibid., Vol. 2, pp 928-33. 1964

(1.3) G . R . Geiss, “ T h e Analysis and Design of Nonlinear Control Systems via Liapunov’s Direct Method,” Air Force Flight-Dynamics Laboratory Research and Technologv Division U. S. Air Force Wright-Patterson Air Force Base, Ohio, Tech. Do; R e p . No.’RTD-TDR-63-40?6, August 1964. (1.4)Parks, P. C., “Stability Analysis for Linear and Nonlinear Systems Using Liapunov’s Second Method,” “Progress in Conrrol Engineering,” (Ed, R . H, 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 Generating Liapunov Functions,” (N66-35556), pp 114-32, Aero-Astrodvnamics Research Review No. 2, July I-Dec. 30, 1964, N66-35546*, 156 pp, Naiional Aeronautics and Space Administration, Marshall Space Flight Center, Huntsville, Ala., 1 April 1965. (1.6)Drake, R. L., “Methods for Systematic Generation of Liapunov Functions,” Parts I and 11, NASA CR-67863 and NASA CR-6?864,1965. (1.7) Lefferts, E. J., “ A G u i i e of the Application of the Liapunov’s Direct Method to Flight Control Systems, NASA CR-209, April 1965.

(18) Letov A. M., “Liapunov Theory of Stability of Motion, Disciplines and Techniqdes of System Control,” (Ed. J. Peschon), p p 267-314, Blaisdell, New York, N. Y., 1965. (I 9) Schultz, D. G., “ T h e Generation of Liapunov Functions, Advances i n Control Systems,” (Ed. C. T. Leondes), p p 1-64, Vol. 2, Academic, New York, N. Y.,

(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 Liapunov Functions for Time-Invariant Control Systems.”)

1765. 1969 1966

(I 10) Salah, M . M., “Investigating Stability of Differential Equations b Liap-

unov’s Direct Method,” Middle East Technical University, Ankara, qurkey, M.S. Thesis, 1966. (1.77) SzegB, G . P.,“Liapunov Second Method,” Appl. Mech. Rev., 19 (lo), 833-8

(1766). 1967 (1.72)Gurel, O.,and Salah, M . M “ A Surve of Methods of Conatructing Liapunov Functions,” IBM New York’Scientific &enter, Rept. No. 39-022,February

1967. ORIGINAL PAPERS 1944 (11.7)Lur’e, A. I., and Postnikov, V. N., “ O n the Theory of Stability of Control Systems,” P M M , 8 (1944).

11.

1949 (112)Aizerman M A “On a Problem Concerning the Stability in the Large ofDynamical $ystkms;i” Us$. Mat. Nauk., 4 (4), 187-8 (1947). 1950 (1I.3) Letov, A. M.,“ Inherently Unstable Control Systems,” P M M , 14 (1950). 1953 (11.4)Zubov V I. “Some Sufficient Conditions of Stability of Nonlinear Systems of Differen;ial’Ec;uations,” ibid., 17 (1953). 1954 (I15) Krasovskii, N. N.,,YOn the Stability in the Large of a System of Nonlinear Differential Equations, rbid., 18, 735-7 (1954). (See also 11.7below.) 1955 (11.6)Zubov V. I. “Problems in the Theor of the Second Method of Liapunov, Constructich of h e General Solution in tLe Domain of Asymptotic Stabilit$,” ibid., 19, 179-210 (1955).

(11.27)Haley, 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 Routh Criterion to Phase-Space Stability ” Master’s TLesis,’ Moore School of Electrical Engineering, University of Penns;lvania, 1963. (11.29),Ku, Y. H., and Puri, N , N., “ O n Liapunov Functions of Higher O r d e r Nonlinear Systems,” J . Franklin Inst., 276, 349-64 (1763). (11.30)Lei hton W “On t h e . 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 Equations,” Contributions20 Diferentiul Equations, 2(1-41, 367-83 (19631.1 (11.39)Mangasarian 0. L “Stability Criteria for Nonlinear Ordinary Differential Eguations,” SIAM’J. C o n h , Ser. A , 1(3), 311-13 (1963). (11.32)Puri, N . N., “NASA Proposal for Study and Research in New Methods for Systematic Generation of Liapunov Functions for Control Systems,” Submitted .to NASA. October 1763. (See 1.6.) (11.33)Puri, N. N., “ N S F Proposal for Study and Research in the Generation of Liapunov Functions and the Design of Optimal Systems,” Submitted to NSF, 1963. [See 1.6.1 (See also 11.43below.) (11.34)Pari, N. N., and Weygandt C. N. “Second Method of Liapunov and Routh’s C h o n i c a l Form,” J . Frankiin Inst., 576,365-83 (1763). (1135)Reiss, R., and Geiss, G., “ T h e Construction of Liapunov Functions,” IEEE ?ratis. Auto. Cont., 8 , 382-3 (1963).

1964 (ZI.36) h t i p e n k o V I “New Method of Determining Liapunov Functions,” Avtomatiku, 9 (Zj,7i-5”(1764). (English translation NASA Report N66-11716.) (11.37)Brockett, R . W., “ O n the Stability of Nonlinear Feedback Systems,” IEEE Trans. on Appl. andIndurtry, 83, 443-7 (1764). (11.38)Infante, 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 ASME, Ser. E , 86,315-20 (1964). (11.39),pu:i, N . N . , “ O n the Global Stability of a Class ofNonlinear Time-Varying Systems, Presented a t Dubrovnik, IFCA Symposium on Sensitivity Analysis, September 1764. (See also 11.43below.) (11.40)Walker J. A. “ A n Integral Method of Liapunov Function Generation for Nonlinear A h o n o k o u s Systems,” Ph.D. Thesis, University of Texas, 1964. (See also 11.44 below.)

1957 (I17) Krasovskii N. N . “Stability in the Case of Large Initial Disturbances,” i’btd., 21, 309-{9 (17573. (11.8)Letov, A. M., “ D i e Stabilitat von Regelsystemen mit nach ebender Ruck-

1965 (11.41)Boyanovich, D., “ O n the Application of Hydrodynamics to the Study of

Munich, 1957. (11.9)Yakubovich, V. A. “ O n a Class of Nonlinear Differential Equations,” Dokl. Akad. Nauk SSSR,’ 117, 44-6 (1957). [Engl. Trans.: AMS Translations Series 2, pp 1-4, Vol. 25 (1963).1

tions,”-IEEE Trans. Auto. Cont., 10,470-2 (1965). (11.43)Ruri, N. N., the Liapunov Functions for a Class of Nonlinear Nonautonomous System, Proc. 7965 Joint Auto. Cant. Conf. Rensselaer Polytechnic

fuhrung,” Regelunstechnick, Moderne Theorien und ihre gerwendbarkeit,

1958 (11.70)Popov, V. M “Relaxing the Sufficiency Conditions for Absolute Stability,” Automat. i Telerneh.,”l9, 1-7 (1958). (See also 11.78 below.) (11.77)Pozharitskii, G. K., “ O n the Construction of the Liapunov Functions from the Integrals of the Equations for Perturbed Motion,” P M M , 22,145-54 (1958). 1960 (II.72) Barbashin E. A

“ O n Constructing Liapunov Functions for Nonlinear Systems,” Proc.’of IFA&, Moscow, 1960 (Butterworths 1961). (11.73)Ingwerson, D . R., “ A Modified Liapunov Method for Nonlinear Stabilit Problems,” Ph.D. Thesis, Stanford University, November 1960. (See also 11.

rb

below.)

(11.74)Lefschetz, S., “Controls: An Application of the Direct Method of Liapunov,” Bol. Sac. Maternat. Mex., p p 139-43, 1760. (See also 11.20 below.) (11.75)Narendra, K . S., and Ho, Y. C., “On the Construction of Liapunov Functions for Nonlinear Systems,” Cruft Lab. Tech. Rep. N. 328, Harvard University Cambridge, Mass., 1960. 1961 (11.76)Chang, S. ,?. L. “Kinetic Function for Stabilit Analysis of Nonlinear Control Systems, J . ojBaszc Engineering, ASME, 83, 91-i (1961). (11.17) Ingwerson, D. R., “ A Modified Liapunov Method for Nonlinear Stability Analysis,” IRE Trans., pp 199-210, 6 (2) (1961). (21.78) Poppv V

M “Absolute Stability of Nonlinear Systems of Automatic Control, A h m a t . i kelernch., 22, 961-77 (1761).

1962 (11.19)Infante, E. F., “ A New Approach of the Determination

:!the Domain of

Stability of Nonlinear Autonomous Second Order Systems,

Ph.D. Thesis,

University of Texas, 1962. (See also 11.38below.) (11.20) Lefschetz, S., “Some Mathematical Considerations on Nonlinear Automatic Controls,” Contributionsto Differential Equations, 1(1), 1-28 (1962). (11.27) Mufti, I. M . “On the Stability of Nonlinear Controlled Systems,” J . Math. Anal. Appls.,’4,257-75 (1962). (11.22)kosenbrock, H. H., “ A Liapunov Function with Applications to Some Nonlinear Physical Systems,” Autornatica, 1, 31-53 (1962). (11.23) Schultz,. D. G., and Gibson, J. E., “ T h e Variable Gradient Method for Generating Liapunov Functions,” AZEE Trans. Part 11,Appls. 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. [Trans. ASME, S e r . 6 , 85023, 137-42 (1963).] (11.25)SzegB, G. P., Contribution to Lia unov’s Second Method Nonlinear Autonomous Systems Traw ASME Ser J Basic Eng 84 573-8 (1962).

‘‘k

8 (Presented a t the Widter A k a 1 Meekng of ASME, NewYo;k,

G. Y.)

the Stability of Singular Points of Differential Equations: Autonomous Systems,” P ~ G c7965 . Joint Auto. Cont. Conf., Rensselaer Polytechnic Institute,

(11.42) Ponzo, P. J., “ O n the Stability of Certain Nonlinear Differential Equa-

“qn

Institute. ....... (11.44)Walker J. A and Clark, L. G “ A n Integral Method of Liapunov Function Generation tor N&linear Autonorr;bus Systems,” J . Appl. Mechanics, Tranr. of ASME,Ser. E , 32 (3),569-75 (1765). ~

1966 (11.45)George, J. H., “ O n the Construction and Interpretation of Liapunov Functions,” Ph.D. Thesis, University of Alabama, 1766. (11.46)Peczkowski, J. L., “ A Format Method of Generating Liapunov Functions,” Ph.D. Thesis, University o f N o t r e Dame, April 1766. (See also 11.49 below.) 1967 (11.47)Kinnen E

and Chen C. S “?,apunov Functions for a Class of n-th Order NonliAea;bifferential’Equazons, NASA CR-687, January 1967. (N.48)Kinnen, E., and Chen, C. S “Liapunov Functions from Auxiliary Exact Differential Equations,” NASA C R 7 7 7 , M a y 1767. (11.49)Peczkowski J. L., and Liu, R. W., “ A Format Method for Generating Liapunov Functions,” Trans. ASME, J . Basic Eng., Series D , 89,433-7 (1967).

COMPREHENSIVE BOOKS 1949.

111.

(211.7) Liapunov, A. M “Problbme General d e la Stabilite du Moiivemmt,” Princeton University P’iess, Princeton, N . J. 1949. (Russian Edition 1892.)

1957 (111.2)Lur’e A. I., “Some Nonlinear Problems in the Theory of Automatic Control,” Her Majesty’s Stationery Office, London, 1957. (Russian Edition 1952.) 1961 (111.3) Chetaey, N. G., “Stability of Motion,” Pergamon Press, London, 1961. (Russian Edition 1946,1950.) (111.4)Letov, A. M., “Stability in Nonlinear Control Systems,” Princeton University Press, Princeton, N. J., 1961. (Russian Edition 1955.) 1963 (111.5) Hahn, W., “Theory and Application of Liapunov’s Direct Method,” Prentice Hall, Inc., Englewood Cliffs, N. J., 1963. (German Edition 1759.) (111.6)Krasovskii N. N “Stability of Motion,” Stanford University Press, Stanford, Calif., 196j. (RAssian Edition 1959.) (121.7)Z u p y , V. I., “Mathematical Methods for the Study of Automatic Control Systems, Pergamon Press, New York, 1963. (Russian Edition 1757.)

1964 (111.8) Aizerman, M. A., and Gantmacher, F. R.,“ Absolute Stability of Regulator Systems,” Holden Day, Inc., San Francisco, Calif., 1964. (Russian Edition 1763.)

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