Stability analysis of 2D Roesser systems via vector Lyapunov functions

Throughout this article, a novel control strategy for fractional-order gene regulation networks (FOGRN) of all categories is designed by using the vector Lyapunov function in combination with the M-matrix measure. Firstly, a series of puzzles surrounding the asymptotic stability of two-dimensional FOGRN are studied, and a new asymptotic stability control strategy is formulated based on the vector Lyapunov function in combination with the M-matrix measure, ensuring that the controlled FOGRN has a strong robust stability. In addition, the corresponding asymptotic stability criterion is deduced. On this basis, the problem of asymptotic stability of a three-dimensional FOGRN is studied. Based on the new method, a stabilization control strategy is also formulated with the corresponding asymptotic stability criterion deduced, ensuring that the controlled FOGRN has a strong robust stability as well. Finally, this novel method’s effectiveness and generality are authenticated via simulation experiments.

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the novel notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we present a universal decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system. These results are then used to develop decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling uncertainty

Uniform Parametric Convergence in the Adaptive Control of Mechanical Systems

For general finite-dimensional time-varying nonlinear systems we derive sufficient conditions for closed-loop asymptotic controllability and asymptotic stabilization by means of a smooth ordinary time-varying feedback integrator. The main hypotheses are based on the existence of an almost smooth time-varying control Lyapunov function. The corresponding results generalize earlier works in the literature concerning feedback stabilization for nonlinear autonomous systems that in general cannot be stabilized by continuous time-varying state feedback.

Some sufficient conditions for exponential stability of linear neutral functional differential equations

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

One of the actual problems for the theory and practice of control of dynamic objects is the development of methods for research and synthesis of control systems of multidimensional objects. The paper proposes a universal approach to construct Lyapunov vector functions directly from the equation of state of control system and a new gradient-speed method of Lyapunov vector functions to study aperiodic robust stability of linear control system with m inputs and n outputs. The study of aperiodic robust stability of automatic control systems is based on the construction of Lyapunov vector functions and gradient-speed dynamic control systems. The basic statements of Lyapunov's theorem about asymptotic stability and notions of stability of dynamic systems are used. The representation of control systems as gradient systems and Lyapunov functions as potential functions of gradient systems from the catastrophe theory allow to construct the full-time derivative of Lyapunov vector functions always as a sign-negative function equal to the scalar product of the velocity vector on the gradient vector. The conditions of aperiodic robust stability are obtained as a system of inequalities on the uncertain parameters of the automatic control system, which are a condition for the existence of the Lyapunov vector-function. A numerical example of synthesis of aperiodic robustness of a multidimensional control object is given. The example shows the main stages of the developed synthesis method, the study of the system stability at different values of the coefficients k, confirming the consistency of the proposed method. Transients in the system satisfy all requirements

While the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated, the theory for nonlinear positive systems is considerably less well-developed. This paper presents a set of conditions for establishing delay-independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, when the time delays have a known upper bound, we derive necessary and sufficient conditions for exponential stability of (a) continuous-time positive systems whose vector fields are homogeneous and cooperative, and (b) discrete-time positive systems whose vector fields are homogeneous and order preserving. We then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization. Finally, we extend the results to general linear systems with time-varying delays.

On exact and discretized stability of a linear fractional delay differential equation

Based on Lyapunov's theory of stability, a equation of sufficient condition for multivariable linear delayed systems with delayed independent stabilization is derived and another forms are also given. Based on this, several simple criterions for judging independent stabilization from linear delayed system are presented. We also discuss the exponential stability for delayed system and also provide the sufficient condition of exponential stability with any appointed convergent rate and its corresponding deductions. Using these conditions, we can choose a set of suitable parameters to reduce the conservation. By calculating and being compared with methods of the literature, the results show that our methods have less conservation.

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

International audience

Modification and extension of procedures of strict analysis of stability and estimates of domains of attraction based on the reduction method with sublinear vector Lyapunov function is given for sufficiently wide class of nonlinear stabilization systems of continuous objects using piecewise constant control formed using discrete state measurements. The new type of realization of heterogeneous comparison systems in the form of interconnected differential and discrete impulse subsystems with varying right-hand side allowing to avoid preliminary discretization of the original system and thus increasing accuracy of investigations is proposed. In the case of nonlinearities (including those with respect to control and measurements), limited by semi-homogeneous functions. Constructive conditions of exponential stability with necessary quantitative estimates are formulated. Application to investigation of stability of economic growth in the Phillips-Bergstrom model with discrete monetary regulation is presented. The paper consists of two parts. In this part, the studied models are described, the studied stability property is defined, procedures for construction of vector Lyapunov function and comparison system are given, and some of their specific features are established.

This technical note is concerned with exponential stability analysis of linear time-varying systems represented by the second-order vector differential equations with the singular leading coefficient matrix. Using bounding techniques on the trajectories of the linear time-varying systems, the stability problem of the time-varying systems is transformed to that of the time-invariant systems and a new sufficient condition for exponential stability is derived. The obtained condition can be applied to the nonsingular case and it is proven that the condition for the nonsingular case is superior to a test presented in the recent literature. Analogously, the proposed method is also extended to the stability analysis for a class of linear time-varying systems with delay. By illustrative examples, better results are obtained by the proposed criteria as compared with some results in the literature.