A SYMPLECTIC ALGORITHM FOR THE STABILITY ANALYSIS OF NONLINEAR PARAMETRIC EXCITED SYSTEMS

In this article, we provide a metaheuristic-based solution for stability analysis of nonlinear systems. We identify the optimal level set in the state space of these systems by combining two optimization phases. This set is in a definite negative region of the time derivative for a polynomial Lyapunov function (LF). Then, we consider a global optimization problem stated in two phases. The first phase is an external optimization to search for a definite positive LF, whose derivative is definite negative under linear matrix inequalities. The candidate LF coefficients are adjusted using a Jaya metaheuristic optimization algorithm. The second phase is an internal optimization to ensure an accurate estimate of the attraction region for each candidate LF that is optimized externally. The key idea of the algorithm is based mainly on a Jaya optimization, which provides an efficient way to characterize accurately the volume and shape of the maximal attraction domains. We conduct numerical experiments to validate the proposed approach. Two potential real-world applications are proposed.

This paper considers a fuzzy Lyapunov method for stability analysis of nonlinear systems subjected to external forces. The nonlinear systems under external forces can be represented by Tagagi-Sugeno (T-S) fuzzy model. In order to design a nonlinear fuzzy controller to stabilize this nonlinear system, the parallel distributed compensation (PDC). scheme is used to construct a global fuzzy logic controller. We then propose the robustness design to ensure the modeling error is bounded and some stability conditions are derived based on the controlled systems. Based on the stability criterion, the nonlinear systems with external forces are guaranteed to be stable. This control problem can be reformulated into linear matrix inequalities (LMI) problem.

This article concerns the problem of stability analysis of nonlinear systems under the sampled-data control (SDC) with actuator faults. The addressed nonlinear systems can be expressed by a number of linear arrangements, based on the Takagi–Sugeno (T–S) fuzzy technique. The time-dependent SDC with actuator faults of the addressed nonlinear systems delineates based on the input model of Markovian variability. The time-scheduled Lyapunov functional and the looped-functional are taken together in the production of a Lyapunov functional candidate. Based on a free matrix-based integral inequality and the Lyapunov functional, some sufficient linear matrix inequality conditions are delivered to guarantee the mean-square asymptotic stability under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance of the considered sampled-data T–S fuzzy systems with actuator faults. Finally, we demonstrate the feasibility and strength of our proposed method through some examples.

In this work we discuss some results on stability analysis for nonlinear systems with time-delays. A central theme of the work is to treat state variables with time-delays as disturbances and uncertainties. The advantage of such an approach is that by exploring the robust stability properties of a system with disturbances, one may obtain various properties for a system with time-delays.

This paper addresses the issue of constructing piecewise linear Lyapunov functions (PWLLF) for nonlinear systems satisfying the generalized sector condition, which is a generalization of the polytopic uncertainty. The main result of this paper is to propose a fast method to solving linear programming problems (LPs) which appear repeatedly in constructing PWLLF. The method uses a special structure of LPs which appear in constructing PWLLF.

This paper presents a constructive method for showing exponential stability of autonomous nonlinear systems consisting of state-dependent weighted linear systems. This kind of system representation is common in for instance fuzzy systems or is the result of an exact or approximative description of an arbitrary nonlinear vector field. Stability is shown by joining multiple local Lyapunov functions properly in the state-space. The overall Lyapunov function, consisting of the local ones, are allowed to be discontinuous at the states where the trajectory passes from one local region to another. By using local quadratic Lyapunov functions the stability conditions are formulated as linear matrix inequalities (LMIs), which can be solved efficiently by computerized methods.

In this brief, stability analysis of nonlinear systems on time scales is further investigated. The traditional stability theorems require the time-scale derivative of related Lyapunov function to be negative, which are conservative. Based on the induction principle on time scales, an improved stability theorem is proposed in this brief by introducing the time-scale type uniformly asymptotically stable function. It is shown that this improved stability theorem weakens the negativity restriction on the time-scale derivative of related Lyapunov function. Then, this improved stability theorem is applied to solve consensus problem of multi-agent systems on time scales. A numerical example is given to illustrate the effectiveness of the theoretic results.

Analysis of Nonlinear Systems with Stochastic Input and Stochastic Parameters—II

Lyapunov's direct method has been applied extensively to the stability analysis of nonlinear systems. The method enables one to directly judge stability without solving the nonlinear differential equations representing motor involved. In the paper, the effect of important parameters on trasient stability of a three-phase reluctance motors have been assessed from the data obtained for load and applied voltage changes by applying the Lyapunov's direct method.

— In this paper we introduce a new type of Lyapunov functions in a general framework particularly suitable for the analysis of systems with noise and uncertainty. These Lyapunov functions may depend on parameters possibly satisfying differential equations or inequalities. The main differences with respect to classical Lyapunov functions and classical tools for the design of composite Lyapunov functions are discussed through examples. A design tool for the design of composite filtered Lyapunov functions is given and examples show improvements over existing literature.

The author analyses results obtained by Megretski and Rantzer (ibid., vol.42, p.819-30, 1997) on the stability analysis of nonlinear systems. He establishes the connections and relations between their results and the abstract quadratic criterion developed by Yakubovich (Automation Remote Control, p.5-30, 1967).

Lyapunov stability analysis for nonlinear systems with state-dependent state delay

This paper presents a new energy function reflecting the damping effect in multimachine power systems. The Lyapunov direct method provides precise and rigorous theoretical backgrounds for stability analysis of nonlinear systems. Incorporating damping effects into accurate estimates of the domain of attraction, which is a minor but crucial point, has been attempted with long history to yield partial success for single machine systems. In this paper, a damping-reflected energy function has been generalized for application to multimachine systems. The generalized energy function is tested for the WSCC 9-bus system to show the semi-negativeness of its time derivative.

This paper mainly investigates the transmission projective synchronization among an array of identical coupled chaotic systems. Based on the stability analysis of nonlinear system with special antisymmetric structure, the control laws are designed to realize the transmission projective synchronization. Some simple stability criteria are then derived for reaching the transmission projective synchronization among multi-systems. Numerical examples are presented to verify the effectiveness and correctness of the synchronization within a desired scaling factor. For the multi-systems synchronization model, it seems to have more valuable than the usual one drive system and one response system synchronization model.

The persistency of excitation property is crucial for the stability analysis of parameter identification algorithms and adaptive control loops. We propose a relaxed definition of persistency of excitation and we use it to establish uniform global asymptotic stability (UGAS) for a large class of nonlinear, time-varying systems. Our relaxed definition is conceptually equivalent to the definition introduced in Loria et al. (1999), but is easier to verify. It is useful for stability analysis of nonlinear systems that arise, for example, in nonlinear adaptive control, stabilization of nonholonomic systems, and tracking control. Our proof of UGAS relies on some integral characterizations of UGAS established in Teel et al. (2000). These characterizations also streamline the proof of UGAS in the presence of (uniform) classical persistency of excitation.