Sufficient Conditions For Absolute Stability Research Articles

Absolute Stability and Synchronization in Second-Order Neural Fields with Conduction Delays

We consider a class of neural field models represented by a second-order nonlinear system of integro-differential equations with space-dependent delays. Such a system models interaction of populations of neurons, each with a continuum description. We justify the existence and uniqueness of solution for the system in a suitable function space. Global existence and boundedness of solutions for the system are confirmed. Two methodologies, the comparison argument and sequential contracting, are developed to establish sufficient conditions for absolute stability and synchronization among different populations of the system. Finally, we present some numerical examples to demonstrate the theoretical results.

Linear Matrix Inequalities in Stability Problems: Retrospective and Theoretical Aspects

Some aspects of the development of the theory of linear matrix inequalities are considered. A number of results obtained at the initial stage of the development of this theory, both in the development of numerical methods and in obtaining analytical conditions for their solvability, are highlighted. The main attention is focused on the system of linear matrix inequalities arising in solving the absolute stabi lity problem. E. S. Pyatnitskiy and his followers showed that the solvability of this system is a criterion for the existence of a quadratic Lyapunov function and a sufficient condition for absolute stability. The prerequisites leading to this result are considered here. The use of the considered system of inequalities for studying the stability of hybrid systems described by differential inclusions and switching systems is shown. An analysis is given of citing some works of Pyatnitskiy’s school on the theory of stability and the theory of systems of linear matrix inequalities, from which the relevance of the results of these works at the present time follows.In developing numerical methods, it was first shown in the work of Pyatnitskiy and Skorodinskiy that the solvability problem for a system of linear matrix inequalities reduces to a convex programming problem. An interesting gradient algorithm for finding solutions to such a system is also presented. In analyzing analytical conditions of solvability, an unsolvability criterion for the system of our interest obtained by Kamenetskiy and Pyatnitskiy is noted. In modern terms, this result can be considered as a description of an admissible set in the dual semidefinite programming problem. A similar result is given in the famous book by S. Boyd et al. The paper shows that the result of Boyd et al. is a simple corollary of the unsolvability criterion. Here the unsolvability criterion is generalized and refined.

On stability for sampled-data nonlinear ADRC-based control system with application to the ball-beam problem

This paper mainly concerns with the stability analysis of the sampled-data nonlinear active disturbance rejection control (ADRC)-based control system. Firstly, a class of single-input-single-output (SISO) continuous plant is discretized using zero-order-hold (ZOH), and several kinds of digital implementation methods for the nonlinear extended state observer (NLESO) are newly proposed. Then the sampled-data nonlinear ADRC (NLADRC) based closed-loop system is transformed into a discrete-time Lurie-like system, to which linear matrix inequality (LMI)-based sufficient conditions for absolute stability and robust absolute stability are obtained. The sufficient conditions provide convenient and effective methods for determining the stability and its relationship with the parameters of the controller, the plant and the sampling period. Using the ball-beam system as an example, the proposed results are verified in both simulations and experiments.

Aizerman's problem in absolute stability theory for regulated systems

A new method for investigating the absolute stability of regulated systems with limited resources is proposed. It is based on estimating improper integrals along the solution of the system. A nonsingular transformation is obtained which allows information about the nonlinearity properties to be taken into account. A class of regulated systems is distinguished for which Aizerman's problem is solvable. For this class a necessary and a sufficient condition for absolute stability are found. The proposed method for investigating absolute stability differs from the other available methods by the fact that conditions for absolute stability are derived without using the Lyapunov function and the frequency theorem. For systems with limited resources the phase variables are bounded, uniformly continuous functions. These properties were used in deriving a condition for stability and in estimating improper integrals. The estimate obtained allows the domain of absolute stability in the space of constructive parameters of the system to be greatly extended by comparison with the earlier known results, and in a number of cases a necessary and a sufficient condition for absolute stability can be obtained. Bibliography: 15 titles.

Absolute stability of time-varying delay Lurie indirect control systems with unbounded coefficients

This paper investigates the absolute stability problem of time-varying delay Lurie indirect control systems with variable coefficients. A positive-definite Lyapunov-Krasovskii functional is constructed. Some novel sufficient conditions for absolute stability of Lurie systems with single nonlinearity are obtained by estimating the negative upper bound on its total time derivative. Furthermore, the results are generalised to multiple nonlinearities. The derived criteria are especially suitable for time-varying delay Lurie indirect control systems with unbounded coefficients. The effectiveness of the proposed results is illustrated using simulation examples.

Adaptive dynamics of saturated polymorphisms.

We study the joint adaptive dynamics of n scalar-valued strategies in ecosystems where n is the maximum number of coexisting strategies permitted by the (generalized) competitive exclusion principle. The adaptive dynamics of such saturated systems exhibits special characteristics, which we first demonstrate in a simple example of a host-pathogen-predator model. The main part of the paper characterizes the adaptive dynamics of saturated polymorphisms in general. In order to investigate convergence stability, we give a new sufficient condition for absolute stability of an arbitrary (not necessarily saturated) polymorphic singularity and show that saturated evolutionarily stable polymorphisms satisfy it. For the case [Formula: see text], we also introduce a method to construct different pairwise invasibility plots of the monomorphic population without changing the selection gradients of the saturated dimorphism.

The absolute stability for general discrete Lurie systems with superposition of multiple nonlinearities

In this paper, with the application of the concept of absolute stability about the certain variable, the necessary and sufficient conditions of absolute stability are obtained, which are about general discrete Lurie systems with superposition of nonlinearities. Some sufficient conditions are also obtained via the discrete Gronwall inequality. A numerical example illustrates the effectiveness of the proposed results.

Stabilization of neutral-type indirect control systems to absolute stability state

This paper provides sufficient conditions for absolute stability of an indirect control Lur’e problem of neutral type. The conditions are derived using a Lyapunov-Krasovskii functional and are given in terms of a system of matrix algebraic inequalities. From these matrix inequalities a sufficient condition for linear state feedback stabilizability follows.

Absolute stability of a class of trilateral haptic systems.

Trilateral haptic systems can be modeled as three-port networks. We present a criterion for absolute stability of a general class of three-port networks. Traditionally, existing (i.e., Llewellyn's) criteria have facilitated the stability analysis of bilateral haptic systems modeled as two-port networks. If the same criteria were to be used for stability analysis of a three-port network, its third port termination would need to be assumed known for it to reduce to a two-port network. This is restrictive because, for absolute stability, all three terminations of the three-port network must be allowed to be arbitrary (while passive). Extending Llewellyn's criterion, we present closed-form necessary and sufficient conditions for absolute stability of a general class of three-port networks. We first find a symmetrization condition under which a general asymmetric impedance (or admittance) matrix Z3 × 3 has a symmetric equivalent Zeq from a network stability perspective. Then, via the equivalence of passivity and absolute stability for reciprocal networks, an absolute stability condition for the original nonreciprocal network is derived. To demonstrate the convenience and utility of using this criterion for both analysis and design, it is applied to the problem of designing stabilizing controllers for dual-user haptic teleoperation systems, with simulations and experiments validating the criterion.

Stabilization of a Class of Two-Dimensional Critical Control Systems

This paper uses the Lyapunov method to study the control system of a class of two-dimensional critical, and acquires the sufficient condition for absolute stability of the two-dimensional critical nonlinear control system.

Stabilization of Lur’e-type nonlinear control systems by Lyapunov-Krasovskii functionals

Abstract The paper deals with the stabilization problem of Lur’e-type nonlinear indirect control systems with time-delay argument. The sufficient conditions for absolute stability of the control system are established in the form of matrix algebraic inequalities and are obtained by the direct Lyapunov method. MSC:34H15, 34K20, 93C10, 93D05.

Stability analysis of the extended state observers by Popov criterion

The analysis and design of the extended state observer (ESO) involves a continuous non-smooth structure, thus the study of the ESO dynamic requires mathematical tools of the nonlinear systems analysis. This paper establishes the sufficient conditions for absolute stability of the ESO. Based on this study, a methodology to estimate several nonlinear functions in dynamics systems is proposed.

Systems with monotone and slope restricted nonlinearities

Abstract The paper starts from the suggestion of R. E. Kalman that additional information on nonlinearity slope may improve the sufficient conditions for absolute stability. This leads to the so called systems with augmented dynamics. Motivated also by the problem of the PIO II aircraft oscillations-self sustained oscillations induced by the saturation nonlinearities, which are both sector and slope restricted-the paper considers a generalization of the Yakubovich criterion to the case of the systems with critical and unstable linear part. The same generalization concerns a quite well known stability criterion where only slope restrictions are taken into account: the published version is improved by using all advantages of the Liapunov method and of the frequency domain stability inequalities. The results are illustrated by several applications.

Absolute stability of nonlinear systems with two additive time-varying delay components

In this paper, we present a new sufficient condition for absolute stability of Lure system with two additive time-varying delay components. This criterion is expressed as a set of linear matrix inequalities (LMIs), which can be readily tested by using standard numerical software. We use this new criterion to stabilize a class of nonlinear time-delay systems. Some numerical examples are given to illustrate the applicability of the results using standard numerical software.

Absolute stability of uncertain discrete Lur’e systems and maximum admissible perturbed bounds

The robust absolute stability problem for norm uncertain and structured uncertain discrete Lur’e systems is considered in this paper by using Lyapunov function method. A sufficient condition of absolute stability for discrete Lur’e systems is established in terms of linear matrix inequalities (LMIs) or the equivalent frequency-domain condition. We compare the result with the Popov-like criterion (Tsypkin criterion) and extended strictly positive real (ESPR) lemma. Furthermore, sufficient conditions on absolute stability for discrete Lur’e systems with norm and structured uncertainties are also presented based on linear matrix inequalities. Estimates of the maximum bounds of all admissible perturbations are given by generalized eigenvalue problems . Finally, several numerical examples are worked out to illustrate the efficiency of the main results.

Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay

In this Letter, the synchronization problem for a class of complex dynamical networks in which every identical node is a Lur'e system with time-varying delay is considered. A delay-dependent synchronization criterion is derived for the synchronization of complex dynamical network that represented by Lur'e system with sector restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the each nodes and the isolated node. Using a convex representation of the nonlinearity for error dynamics, the stability condition based on the discretized Lyapunov–Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.

Multi-path nonlinear dynamic compensation for rudder roll stabilization

The design of rudder roll stabilizers (RRS) for ships is complicated by yaw-axis coupling, non-minimum phase shift, and the angle and rate limits of the rudder. These saturations tend to limit the available feedback and restrict RRS implementation to vessels with very fast rudders. This paper describes the application of a Nyquist-stable RRS with a multiple feedback path nonlinear dynamic compensator to provide high performance with slower steering mechanisms used on larger vessels. The restrictive sufficient condition of absolute stability is satisfied for angle or rate saturation. Computer simulation results are included to show the efficacy of the approach.

Synchronization for chaotic Lur’e systems with sector-restricted nonlinearities via delayed feedback control

In this paper, the effects of time delay on chaotic master–slave synchronization scheme are considered. Using delayed feedback control scheme, a delay-dependent stability criterion is derived for the synchronization of chaotic systems that are represented by Lur’e system with sector-restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the master and the slave system. Using a convex representation of the nonlinearity, the stability condition based on the Lyapunov–Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.

ABSOLUTE STABILITY AND APPLICATION TO DESIGN OF OBSERVER‐BASED CONTROLLER FOR NONLINEAR TIME‐DELAY SYSTEMS

ABSTRACTIn this paper we present a new sufficient condition for absolute stability of delay Lure system. This condition improves the one given in [1]. We use this new criterion to construct an observer‐based control for a class of nonlinear time‐delay systems. Some examples are given to illustrate the results of this paper.

On maximum sector bounds for absolute stability of single-input single-output systems

We consider the classical problem of absolute stability, i.e. stability of a linear time-invariant system in feedback with a memoryless time-varying sector bound non-linearity. We derive necessary and sufficient conditions for the existence of periodic motion. For the special case of symmetric sector bounds, this condition yields the minimum sector bounds for which the system can sustain periodic motion. By guaranteeing the absence of periodic motion we derive non-conservative sufficient conditions for absolute stability for arbitrary sector bounds. We demonstrate by numerical examples the application of our results.