Conditions For Absolute Stability Research Articles

Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions

A generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and/or uncertainty. This paper will further enhance absolute stability analysis by using the composite quadratic Lyapunov function whose level set is the convex hull of a family of ellipsoids. The absolute stability analysis will be approached by characterizing absolutely contractively invariant (ACI) level sets of the composite quadratic Lyapunov functions. This objective will be achieved through three steps. The first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems. The second step establishes stability conditions for linear difference inclusions and then for saturated linear systems. The third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability. Based on the conditions for absolute stability, optimization problems are formulated for the estimation of the stability region. Numerical examples demonstrate that stability analysis results based on composite quadratic Lyapunov functions improve significantly on what can be achieved with quadratic Lyapunov functions.

Absolute exponential stability of a class of recurrent neural networks with multiple and variable delays

In this paper, we derive some new conditions for absolute exponential stability (AEST) of a class of recurrent neural networks with multiple and variable delays. By using the Holder's inequality and the Young's inequality to estimate the derivatives of the Lyapunov functionals, we are able to establish more general results than some existing ones. The first type of conditions established involves the convex combinations of column-sum and row-sum dominance of the neural network weight matrices, while the second type involves the p-norm of the weight matrices with p∈[1,+∞].

Absolute Stability Criteria for a Generalized Lur'e Problem with Delay in the Feedback

A nonautonomous linear system controlled by a nonlinear sector-restricted feedback with a time-varying delay is considered. Delay-independent sufficient conditions for absolute stability and instability (expressed in the transfer function of the linear part and the sector bounds) are established. For a system with an exponentially stable linear part, an upper bound for the Lyapunov exponent is found. It is shown that if the transfer function is sign-constant, asymptotic stability of the system with the margin-linear feedback guarantees absolute stability of the considered system; thus, such systems satisfy the known Aizerman conjecture. They include, in particular, a closed-loop system consisting of any number of time-varying first order links and feedback with arbitrary delay. Under some additional condition (which is certainly true for a time-invariant linear block), the obtained stability criterion is precise. The approach employed in the proofs is based on a direct analysis of the corresponding Volterra equation which contains only the transfer function of the linear block and, therefore, embraces a wide range of control systems. As an example, a second order system is considered; it is shown that here the obtained stability bound is reached for a linear feedback with a periodic delay function.

Robust stability of time-varying nonlinear systems using popov criterion

This paper establishes sufficient conditions for absolute stability of time-varying control systems. The proposed conditions extend the classical Popov criterion to time-varying memoryless nonlinearities, using previous results of Kharitonov. They are expressed in terms of Linear Matrix Inequalitie (LMI). The results ensure in general local stability, however, the stability is global for linear time-varying systems. An example is presented to show the efficacy of these extensions within the MATLAB environment.

Absolute stability analysis in cellular neural networks with variable delays and unbounded delay

In this paper, the existence and uniqueness of the equilibrium point and absolute stability of a class of cellular neural networks with globally Lipschitz continue activation functions are investigated. The neural networks contain both variable and unbounded delays. The necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the neural networks is obtained. By constructing proper vector Liapunov functions, nonlinear integrodifferential inequalities involving both variable delays and unbounded delay, which are corresponding to the neural networks, are established. By using M-matrix theory and qualitative property of the integrodifferential inequalities, the sufficient conditions for absolute stability (global asymptotic stability) are obtained.

Absolute stability of a class of neural networks with unbounded delay

AbstractIn this paper, the existence and uniqueness of the equilibrium point and absolute stability of a class of neural networks with partially Lipschitz continue activation functions are investigated. The neural networks contain both variable and unbounded delays. Using the matrix property, the necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the neural networks is obtained. By constructing proper vector Liapunov functions and non‐linear integro‐differential inequalities involving both variable delays and unbounded delay, the sufficient conditions for absolute stability (global asymptotic stability) are obtained. Copyright © 2004 John Wiley & Sons, Ltd.

Absolute stability of lurie systems with impulsive effects

This paper studies absolute stability of Lurie systems with impulsive effects. Using the method of Lyapunov functions and the variation of parameters technique, we establish sufficient conditions for absolute stability.

Necessary and sufficient condition for absolute stability of normal neural networks

Globally convergent dynamics of a class of neural networks with normal connection matrices is studied by using the Lyapunov function method and spectral analysis of the connection matrices. It is shown that the networks are absolutely stable if and only if all the real parts of the eigenvalues of the connection matrices are nonpositive. This extends an existing result on symmetric neural networks to a larger class including certain asymmetric networks. Further extension of the present result to certain non-normal case leads naturally to a quasi-normal matrix condition, which may be interpreted as a generalization of the so-called principle of detailed balance for the connection weights or the quasi-symmetry condition that was previously proposed in the literature in association with symmetric neural networks. These results are of particular interest in neural optimization and classification problems.

Necessary and sufficient conditions for absolute stability: the case of second-order systems

We consider the problem of absolute stability of linear feedback systems in which the control is a sector-bounded time-varying nonlinearity. Absolute stability entails not only the characterization of the most destabilizing nonlinearity, but also determining the parametric value of the nonlinearity that yields instability of the feedback system. The problem was first formulated in the 1940s, however, finding easily verifiable necessary and sufficient conditions for absolute stability remained an open problem all along. Recently, the problem gained renewed interest in the context of stability of hybrid dynamical systems, since solving the absolute stability problem implies stability analysis of switched linear systems. In this paper, we introduce the concept of generalized first integrals and use it to characterize the most destabilizing nonlinearity and to explicitly construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for absolute stability of second-order systems.

A New Approach to the Lur'e Problem in the Theory of Absolute Stability

The classical Lur'e problem consists of finding conditions for absolute stability of a linear system with a nonlinear feedback contained within a prescribed sector. Most of the results obtained on this problem are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach providing a sufficient stability criterion for systems with time-variable coefficients, which is expressed in the transfer function of the linear block and the sector margins of the nonlinear block. The systems for which this criterion is precise are found. It is shown that stability of a system with a sign-constant transfer function is guaranteed by stability of the system with a limit linear feedback (so that, for such systems, the famous Aizerman conjecture is true). This, in particular, is the case for systems with a linear block consisting of an arbitrary number of first order time-dependent links. As an example, the stability criterion is applied to a second order system for which the obtained results are contrasted with ones delivered by the Popov criterion.

Algebraic criteria for absolute stability

Algebraic criteria for absolute stability of Luré systems, whose linear part has a diagonal system matrix, are derived. The conditions for absolute stability are expressed in the form of inequalities, which contain no variable other than system parameters, and therefore, can be checked readily.

Robust absolute stability of time-varying nonlinear discrete-time systems

This paper studies the problem of robust absolute stability of a class of nonlinear discrete-time systems with time-varying matrix uncertainties of polyhedral type and multiple time-varying sector nonlinearities. By using the variational method and the Lyapunov second method, criteria for robust absolute stability are obtained in different forms for the class of systems under consideration. Specifically, we determine the parametric classes of Lyapunov functions which define the necessary and sufficient conditions of robust absolute stability. We apply the piecewise-linear Lyapunov functions of the infinity vector norm type to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations. Some simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for several special cases. An example is presented as an application of these results to a specific class of systems with time-varying interval matrices in the linear part.

Aizerman's Problem for Discrete Systems

Absolute stability conditions for nonlinear difference equations are derived. In particular, it is proved that equations with positive Green's functions (impulse responses) satisfy the generalized Aizerman conjecture.

Criteria for robust absolute stability of time-varying nonlinear continuous-time systems

The present paper establishes results for the robust absolute stability of a class of nonlinear continuous-time systems with time-varying matrix uncertainties of polyhedral type and multiple time-varying sector nonlinearities. By using the variational method and the Lyapunov Second Method, criteria for robust absolute stability are obtained in different forms for the given class of systems. Specifically, the parametric classes of Lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability. The piecewise linear Lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations. Several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases. Two examples are presented as applications of the present results to a particular second-order system and to a specific class of systems with time-varying interval matrices in the linear part.

Absolute Stability of General Lurie Control Systems with Multi-Nonlinear Feedback Terms

In this paper, the problem of absolute stability for general Lurie control systems with multi-nonlinear feedback terms is investigated. From the Lyapunov function approach, new necessary and sufficient conditions and some practical sufficient conditions of absolute stability are derived, which improve the absolute stability for the set. A numerical example illustrates the effectiveness of the proposed results.

An additive diagonal-stability condition for absolute exponential stability of a general class of neural networks

This paper presents new results on the absolute exponential stability (AEST) of neural networks with a general class of partially Lipschitz continuous and monotone increasing activation functions under a mild condition that the interconnection matrix T of the network system is additively diagonally stable; i.e., for any positive diagonal matrix D/sub 1/, there exists a positive diagonal matrix D/sub 2/ such that D/sub 2/ (T-D/sub 1/)+(T-D/sub 1/)/sup T/D/sub 2/ is negative definite. This result means that the neural networks with additively diagonally stable interconnection matrices are guaranteed to be globally exponentially stable for any neuron activation functions in the above class, any constant input vectors and any other network parameters. The additively diagonally stable interconnection matrices include diagonally semistable ones and H-matrices with nonpositive diagonal elements as special cases. The obtained AEST result substantially extends the existing ones in the literature on absolute stability (ABST) of neural networks. The additive diagonal stability condition is shown to be necessary and sufficient for AEST of neural networks with two neurons. Summary and discussion of the known results about ABST and AEST of neural networks are also given.

Absolute stability of general lurie type indirect control systems

In this paper, by introducing a new concept of absolute stability for a certain argument, necessary and sufficient conditions for absolute stability of general Lurie indirect control systems are obtained, and some practical sufficient conditions are also given.

Algebraic Criteria of Absolute Stability

New algebraic criteria of absolute stability are obtained on the basis of a polynomial form of the frequency condition for absolute stability (Popov inequality). The analysis of the inequality is reduced to an analysis of the local properties of polynomials and a rational function on the negative real semiaxis. As a result, the stability criteria do not require the plotting of any sort of curves and can be expressed on the basis of algorithms of the Routh–Hurwitz type and calculations of the negative real roots of polynomials.

A solution of the absolute stabilizability problem on the basis of frequency inequalities

The absolute stabilizability problem is a generalization of the well-known absolute stability problem to controlled dynamical systems. The absolute stability problem was posed by Lur'e, who also suggested a method for solving this problem by constructing a global Lyapunov function of a special form (a quadratic form plus an integral of the nonlinearity). The mathematical techniques for the problem were comprehensively developed in [1, 2]; further they were extended to more general statements. Popov and later Yakubovich used sufficient conditions for absolute stability in the form of "frequency" inequalities, that is, inequalities containing the Fourier transforms of functions of time determined by given linear relations. This form of sufficient conditions proved to be convenient in automatic regulation and control problems. The present paper deals with the problem of absolute stabilizability via output variables in the general statement and gives a method for solving this problem on the basis of frequency inequalities and an auxiliary minimax problem (see Section 2). This method was suggested in [3, 4] for special

On Aizerman-Myshkis problem for systems with delay

Absolute stability conditions for nonlinear retarded systems are derived. It is proved that systems with positive Green's functions (impulse response) satisfy the generalized Aizerman conjecture.