Complex Lyapunov Functions Research Articles

Quasi-invariant and attracting sets of competitive neural networks with time-varying and infinite distributed delays

This paper focuses on the quasi-invariant set (QIS), global attracting set (GAS) and global exponential attracting set (GEAS) of competitive neural networks (CNNs) with time-varying and infinite distributed delays. For these purposes, based on the characteristics of nonnegative matrix and M-matrix, a new bidirectional delay integral inequality and a novel integro-differential inequality are first established. From the founded integral inequality, the existence conditions of the QIS and the GAS of the discussed system are obtained. Besides, the existence conditions of the GEAS are also given by the proposed integro-differential inequality, which gets rid of the construction of complex Lyapunov functions and functionals. The frameworks of the QIS, GAS and GEAS are also given. A numerical example is analyzed to confirm the validity of the obtained results in the end.

On the design of fixed-order output controllers for 2D systems via complex Lyapunov functions

This paper addresses the problem of designing fixed-order output stabilizing controllers for 2D systems with homogeneous or mixed dynamics. A strategy is proposed by searching for a complex Lyapunov function depending polynomially on a frequency and on the controller. The strategy consists of determining the controller candidate in a chosen region that maximizes the feasibility of the Lyapunov inequalities. This is realized by solving two semidefinite programs, the first one for constructing a feasibility index, and the second one for constructing the controller that achieves its maximum.

Dynamical behavior of a stochastic SIQR epidemic model with Ornstein–Uhlenbeck process and standard incidence rate after dimensionality reduction

Due to the continuous interference of environmental white noise, the dynamical behaviors of a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck process and standard incidence are considered. Several conclusions can be verified after dimensionality reduction. We study the existence and uniqueness of positive solution by construct a nonnegative Lyapunov function at first. Then, a sufficient condition for extinction of the diseases is derived by constructing a suitable Lyapunov function. In addition, we also obtain the stationary distribution of the model by constructing a complex Lyapunov function. Particularly, we propose the exact local expression of the density function of the random model near the unique endemic equilibrium. Finally, the numerical simulations illustrate our above theoretical results.

Uniform Stability of a Class of Fractional-Order Fuzzy Complex-Valued Neural Networks in Infinite Dimensions

In this paper, the problem of the uniform stability for a class of fractional-order fuzzy impulsive complex-valued neural networks with mixed delays in infinite dimensions is discussed for the first time. By utilizing fixed-point theory, theory of differential inclusion and set-valued mappings, the uniqueness of the solution of the above complex-valued neural networks is derived. Subsequently, the criteria for uniform stability of the above complex-valued neural networks are established. In comparison with related results, we do not need to construct a complex Lyapunov function, reducing the computational complexity. Finally, an example is given to show the validity of the main results.

Design of adaptive parameter observer and laser network controller based on sliding mode control technology

In this work, we extend the sliding mode control technology so that it can be applied to laser network synchronization. Firstly, the adaptive parameter observer is designed and the uncertain parameters in the synchronization target are effectively identified based on the extended sliding mode control technology. Furthermore, we design the laser network controller in order to achieve complete synchronization between the laser network and the synchronization target. In our strategy, the designed network controller is very simple. The complex Lyapunov function is not required, and the only work required is to determine the range of adjusting parameters for achieving synchronization. Therefore, this technology is simple and practical.

Impulsive Control of Complex-Valued Neural Networks with Mixed Time Delays and Uncertainties

This paper investigates the global exponential stability of uncertain delayed complex-valued neural networks (CVNNs) under an impulsive controller. Both discrete and distributed time-varying delays are considered, which makes our model more general than previous works. Unlike most existing research methods of decomposing CVNNs into real and imaginary parts, some stability criteria in terms of complex-valued linear matrix inequalities (LMIs) are obtained by employing the complex Lyapunov function method, which is valid regardless of whether the activation functions can be decomposed. Moreover, a new impulsive differential inequality is applied to resolve the difficulties caused by the mixed time delays and delayed impulse effects. Finally, an illustrative example is provided to back up our theoretical results.

Robust Structural Stability, Stability Margins, and Maximum Uncertainty Amplification for 2D Uncertain Systems via Structured Lyapunov Functions and Matrix Annihilators

This paper considers 2D uncertain systems with homogeneous or mixed dynamics, and addresses three problems: establishing robust structural stability, computing robust structural stability margins, and determining the maximum uncertainty amplification that preserves robust structural stability. Two sufficient linear matrix inequality (LMI) conditions are proposed for establishing robust structural stability obtained by introducing an equivalent closed-loop complex system and by searching for real or complex structured Lyapunov functions and matrix annihilators parameterized by the uncertainties and auxiliary quantities. Moreover, it is shown that lower bounds of the introduced robust structural stability margins and maximum uncertainty amplification can be obtained by solving quasi-convex optimization problems under some restrictions on the sought Lyapunov functions or on the set of uncertainties. Lastly, the nonconservatism of the proposed results is analyzed, showing that the proposed LMI condition based on the use of complex Lyapunov functions is not only sufficient but also necessary under some assumptions.

Nonquadratic Lyapunov function for continuous TS fuzzy models through their discretization

Since the beginning of the fuzzy control theory, results have been obtained independently for continuous and discrete models. It is still quite difficult to use nonquadratic Lyapunov functions for the continuous case, while this is much easier for the discrete case. This approach tries to put a bridge between the continuous and discrete cases for the class of continuous Takagi Sugeno fuzzy models that can be exactly discretized. Indeed, for this particular class, once the stability of the discrete model is ensured, the same control law applied to the continuous model will ensure the stability too. The interest of such an approach is that complex control laws and complex Lyapunov functions can be easily used. Simulation examples show that stabilizing feedbacks may be obtained with discrete control laws when purely continuous ones fail.

Necessary and Sufficient LMI Conditions for Stability and Performance Analysis of 2-D Mixed Continuous-Discrete-Time Systems

This paper proposes necessary and sufficient conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems that can be checked with convex optimization, in particular linear matrix inequalities (LMIs). Specifically, the first contribution of the paper is a condition for exponential stability based on the introduction of a complex Lyapunov function depending polynomially on a parameter and on the use of the Gram matrix method. It is shown that this condition is sufficient for any chosen degree of the complex Lyapunov function, and necessary for an a priori known degree. The second contribution is a non-Lyapunov condition for exponential stability based on eigenvalue products. This condition is necessary and sufficient, and has the advantage of requiring a significantly smaller computational burden for achieving necessity. Lastly, the third contribution is to show how upper bounds on the H∞ norm of 2-D mixed continuous-discrete-time systems can be obtained through a semidefinite program based on complex Lyapunov functions. A necessary and sufficient condition is provided for establishing the tightness of the found upper bounds.

On the ℋ∞ Norm of 2D Mixed Continuous-Discrete-Time Systems via Rationally-Dependent Complex Lyapunov Functions

This paper addresses the problem of determining the ℋ∞ norm of 2D mixed continuous-discrete-time systems. A novel approach is proposed based on the use of a class of complex Lyapunov functions with rational dependence on a parameter, which provides upper bounds on the sought norm via linear matrix inequalities (LMIs). It is also shown that the provided upper bounds are nonconservative by using rational functions in the chosen class with degree sufficiently large. Some numerical examples illustrate the proposed approach.

Synchronization Conditions for Multiagent Systems With Intrinsic Nonlinear Dynamics

This brief studies local and global synchronization in multiagent systems with nonlinear dynamics with respect to equilibrium points and periodic orbits. For local synchronization, a method is proposed based on the transformation of the original system into an uncertain polytopic system and on the use of homogeneous polynomial Lyapunov functions. For global synchronization, another method is proposed based on the search for a suitable polynomial Lyapunov function. The proposed methods exploit linear matrix inequalities and have several advantages. In particular, the proposed methods require the solution of convex optimization problems. Also, the proposed methods exploit more complex Lyapunov functions than the quadratic Lyapunov functions typically considered in the literature and included in this brief as a special case.

MODIFIED PROJECTIVE LAG SYNCHRONIZATION OF TWO NONIDENTICAL HYPERCHAOTIC COMPLEX NONLINEAR SYSTEMS

In this work, we introduce and investigate the modified projective lag synchronization (MPLS) of two nonidentical hyperchaotic complex nonlinear systems. The idea of an active control technique based on complex Lyapunov function with lag in time is used for an approach to investigate MPLS of hyperchaotic attractors of these systems. For illustration, this approach is applied to hyperchaotic complex Chen and Lü systems. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve MPLS.

Lag synchronization of hyperchaotic complex nonlinear systems

In this paper, we study the lag synchronization (LS) of n-dimensional hyperchaotic complex nonlinear systems. The idea of the nonlinear control technique based on the complex Lyapunov function with lag in time is used to propose a scheme to investigate LS of hyperchaotic attractors of these systems. Both complex Lyapunov and control functions are introduced. For illustration, the scheme is applied to two hyperchaotic complex Lorenz systems. The real and complex control functions are derived analytically to achieve LS and to show that the complex error dynamical systems are globally stable. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve LS of two identical hyperchaotic attractors.

Local Stability Analysis of Feedback Linearizing Schemes for a Class of Uncertain Nonlinear Systems

This work presents a (numerical) technique to the local stability analysis of feedback linearizing control schemes for a class of uncertain nonlinear systems. A nonlinear decomposition of vector functions is applied to transform the Lyapunov stability conditions into a set of state-and parameter-dependent LMI (linear matrix inequality) constraints, which are numerically solved in a finite set of points. The proposed method can deal with a large class of nonlinear systems as well as very complex Lyapunov function candidates. In addition, a maximized level set of the Lyapunov function is proposed as an estimate of the system stability region.

Complete synchronization of chaotic complex nonlinear systems with uncertain parameters

Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.