Last edited by Kagor
Friday, July 24, 2020 | History

5 edition of Stability by Liapunov"s Matrix Function Method with Applications (Pure and Applied Mathematics) (Pure and Applied Mathematics) found in the catalog. # Stability by Liapunov"s Matrix Function Method with Applications (Pure and Applied Mathematics) (Pure and Applied Mathematics)

## by A.A. Martynyuk

Written in English

Subjects:
• Algebra,
• Applied mathematics,
• Number Systems,
• Analytic Mechanics (Mathematical Aspects),
• Mathematics,
• Science/Mathematics,
• System Theory,
• Science,
• Applied,
• Mathematics / Number Systems,
• Lyapunov stability

• The Physical Object
FormatHardcover
Number of Pages276
ID Numbers
Open LibraryOL8124484M
ISBN 100824701917
ISBN 109780824701918

I read the book A Linear Systems Primer by Antsaklis, Panos J., and Anthony N. Michel (Vol. 1. Boston: Birkhäuser, ) and I am confused by a part of the proof of a theorem about the Lyapunov Mat. Thus, the derivative is identically zero. Hence, the function $$V\left(\mathbf{X} \right)$$ is a Lyapunov function and the zero solution of the system is stable in the sense of Lyapunov. The condition of asymptotic stability is not satisfied (for this, the derivative $${\large\frac{{dV}}{{dt}}\normalsize}$$ must be .

In this lecture, we consider some applications of SDP: Stability and stabilizability of linear systems. { The idea of a Lyapunov function. Eigenvalue and matrix norm minimization problems. 1 Stability of a linear system Let’s start with a concrete problem. Given a matrix A2R n, consider the linear dynamical system x k+1 = Ax k; where xFile Size: KB. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations.

Stability By Liapunovs Direct Method by Joseph La Salle. Publication date Topics Liapunov's direct method, Stability, Control Collection folkscanomy; additional_collections Language English. Stability by Liapunov's Direct Method. by. Joseph La Salle, Solomon Lefschetz. Addeddate Identifier. if A is stable and Q > 0, then for each t, etATQetA > 0, so P = Z ∞ 0 etA T QetA dt > 0 meaning: if A is stable, • we can choose any positive deﬁnite quadratic form zTQz as the dissipation, i.e., −V˙ = zTQz • then solve a set of linear equations to ﬁnd the (unique) quadratic form.

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### Stability by Liapunov"s Matrix Function Method with Applications (Pure and Applied Mathematics) (Pure and Applied Mathematics) by A.A. Martynyuk Download PDF EPUB FB2

Get this from a library. Stability by Liapunov's matrix function method with applications. [A A Martyni︠u︡k] -- This book provides a systematic study of matrix Liapunov functions, incorporating new techniques for the qualitative analysis of nonlinear systems encountered in a wide variety of real-world.

Stability by Liapunov's Matrix Function Method with Applications (Chapman & Hall/CRC Pure and Applied Mathematics) 1st Edition by A.A. Martynyuk (Author) ISBN ISBN Why is ISBN important. ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book.

Cited by: Due to the concept of matrix-valued function developed in the book, the direct Liapunov method becomes yet more versatile in performing the analysis of nonlinear systems dynamics.

The possibilities of the generalized direct Liapunov method are opened up to stability analysis of solutions to ordinary differential equations, singularly perturbed. Stability by Liapunov's matrix function method with applications.

New York: Marcel Dekker, © (DLC) (OCoLC) Material Type: Document, Internet resource: Document Type: Internet Resource, Computer File: All Authors / Contributors: A A Martyni︠u︡k. As we will see later, the Lyapunov equations also arise in many other important control theoretic applications.

In many of these applications, the right-hand side matrix M is positive semi-definite, rather than positive definite. The typical examples are M = BB T or M = C T C, where B and C are, respectively, the input and output matrices.

The Lyapunov equations of the above types arise in. Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay Hardcover – June 1, by Krasovsky Nikolai N (Author) out of 5 stars 2 ratings. See all 2 formats and editions Hide other formats and editions.

Price Cited by: The method of Lyapunov matrix-valued functions is critically examined for its capability, applicability and overall functionality in the adequate construction and development of an appropriate. 1 Lyapunov theory of stability Introduction. Lyapunov’s second (or direct) method provides tools for studying (asymp-totic) stability properties of an equilibrium point of a dynamical system (or systems of dif-ferential equations).

The intuitive picture is that of a scalar output-function, often thoughtFile Size: KB. shall strive to prove global, exponential stability.

The direct method of Lyapunov. Lyapunov’s direct method (also called the second method of Lyapunov) allows us to determine the stability of a system without explicitly inte-grating the diﬀerential equation (). The method is a generalizationFile Size: KB.

Purchase Stability by Liapunov's Direct Method with Applications by Joseph L Salle and Solomon Lefschetz, Volume 4 - 1st Edition. Print Book & E-Book Book Edition: 1.

dynamical system, like frequency criteria and the method of comparing with other systems. The theory of Lyapunov function is nice and easy to learn, but nding a good Lyapunov function can often be a big scienti c problem.

Detecting new e ective families of Lyapunov functions can be seen as a serious advance. Example of stability problemFile Size: KB. The method of matrix-valued functions in the theory of stability of classical dynamic equations on a time scale is described in the book of Martynyuk.

The role of multicomponent Lyapunov-type functions in the stability theory of differential equations is well known , .Author: A.A.

Martynyuk, I.M. Stamova, Yu.A. Martynyuk-Chernienko. One the major stability technique for non linear dynamic system is called lyapunov. A detailed post on the Lyapunov Stability Criteria will be uploaded soon. A wikipedia page gives a general idea about the lyapunov stability.

Following posts gives a very basic example to hel user use the lyapunov function in. The Matrix Method for Stability Analysis The methods for stability analysis, described in Chapters 8 and 9, do not take into account the influence of the numerical representation of the boundary conditions on the overall stability of the scheme.

The Von Neumann method is based on the assumptions of the existence of a Fourier decomposition ofFile Size: 1MB. Asymptotic Stability of Linear Systems An LTI system is asymptotically stable, meaning, the equilibrium state at the origin is asymptotically stable, if and only if the eigenvalues of A have negative real parts For LTI systems asymptotic stability is equivalent with convergence (stability condition automatically satisfied)File Size: KB.

Definition of a Lyapunov function. A Lyapunov function for an autonomous dynamical system {: → ˙ = ()with an equilibrium point at = is a scalar function: → that is continuous, has continuous first derivatives, is locally positive-definite, and for which − ∇ ⋅ is also locally positive definite.

The condition that − ∇ ⋅ is locally positive definite is sometimes stated as. for a certain), then the solution is called asymptotically (respectively, exponentially) stable. If is a normed space, then this definition may be formulated as in 1 above, if as norm one takes any norm compatible with the topology on.

4) Let a differential equation (2) be given on a Riemannian manifold (for which a Euclidean or a Hilbert space can serve as a model) or, in a more general.

function V (x) that is 0 at the origin and p ositiv e elsewhere in some ball enclosing the origin, i.e. V (0) = 0 and (x) > for 6 _ this ball. Suc h a V (x) m y b e though t of as an \energy" function. Let denote the time deriv ativ e of V (x) along an y tra jectory the system, i.e.

its rate c hange as t v aries. Martynyuk, “Qualitative analysis of nonlinear systems by the method of matrix Lyapunov functions,”Rocky Mount. Math., 25, – (). zbMATH MathSciNet CrossRef Google Cited by: 4. CRITERION FOR STABILITY OF MATRICES 3. STABILITY CRITERIA In this section, we assume that A g MR.

n THEOREM For s A. - 0, it is sufficient and necessary that s Aw2x.- 0 and.y1 detn.A) 0. Proof. By the spectral property of Aw2x, the condition sA w2x.- 0im- plies that at most one eigenvalue of A can have a nonnegative real part. We thus may assume that all of the eigenvalues.

LINEAR SYSTEM STABILITY Lyapunov Stability of Linear Systems In this section we present the Lyapunov stability method specialized for the linear time invariant systems studied in this book.

The method has more theoretical importance than practical value and File Size: 66KB.In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with.function, and the solution trajectories are usually not known.

Or as Krasovski i wrote in [], pp. 11/ One could hope that a method for proving the existence of a Lyapunov function might carry with it a constructive method for obtaining this function.

This hope has not been realized.