Alexey P. Zhabko

Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems

In this paper, a procedure for construction of quadratic Lyapunov-Krasovskii functionals for linear time-delay systems is proposed. It is shown that these functionals admit a quadratic low bound. The functionals are used to derive robust stability conditions.

Linear quadratic Gaussian controller design for plasma current, position and shape control system in ITER

This paper is focused on the linear quadratic Gaussian (LQG) controller synthesis methodology for the ITER plasma current, position and shape control system as well as power derivative management system. It has been shown that some poloidal field (PF) coils have less influence on reference plasma-wall gaps control during plasma disturbances and hence they have been used to reduce total control power derivative by means of the additional non-linear feedback. The design has been done on the basis of linear models. Simulation was provided for non-linear model and results are presented and discussed.

Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems

In this paper, a necessary and sufficient condition for the exponential stability of linear systems with several time-delays is presented. Such a condition is based on the construction of quadratic lower bounds for the Lyapunov-Krasovskii functionals on the special Razumikhin-type set of functions. The result reveals a constructive procedure for the stability analysis whose application is illustrated with examples.

On the asymptotic stability of solutions of nonlinear systems with delay

Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin’s approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.

Mathematical methods of plasma vertical stabilization in modern tokamaks

The paper presents the application of modern computational methods for tokamak plasma control system analysis. Several different approaches for feedback controller synthesis are described. General positions of the modern robust analysis theory are briefly formulated. The technique of robust features comparative analysis for feedback controllers is presented. The application of these computational methods is illustrated by the example of the MAST tokamak plasma vertical feedback control system.

Delay-Independent Stability Conditions for Some Classes of Nonlinear Systems

Some classes of nonlinear time-delay systems are studied. It is assumed that the zero solution of a system is asymptotically stable when delay is equal to zero. By the Lyapunov direct method, and the Razumikhin approach, it is shown that in the case when the system is essentially nonlinear, i.e., the right-hand side of the system does not contain linear terms, the asymptotic stability of the trivial solution is preserved for an arbitrary positive value of the delay. Based on homogeneous approximation of a time-delay system some stability conditions are found. We treat large-scale systems with nonlinear subsystems. New stability conditions in certain cases, critical in the Lyapunov sense, are obtained. Three examples are given to demonstrate effectiveness of the presented results.

On stability of the solutions of a class of nonlinear delay systems

Consideration was given to a class of systems of nonlinear differential equations with retarded argument. It was assumed that in the absence of delay the zero solutions of the systems under study are asymptotically stable. Using the method of Lyapunov functions in the form of B.S. Razumikhin, it was proved that if the right-hand sides of these equations are free of the linear terms relative to the phase variables, then the asymptotic stability is retained for any delay.

Lyapunov-Krasovskii Approach to Robust Stability of Time Delay Systems

Abstract In this papar a procedure is proposed for construction a more general class of Lyapunov Krasovskii quadratic functionals, whose time derivative includes terms which depend on some past state of the systen and it is shown how one can use these functionals for the robust stability analysis of uncertain time delay systems.

Asymptotic Stability Conditions and Estimates of Solutions for Nonlinear Multiconnected Time-Delay Systems

This paper examines certain classes of multiconnected (complex) systems with time-varying delay. Delay-independent stability conditions and estimates of the convergence rate of solutions to the origin for those systems are derived. It is shown that the exponents in the obtained estimates depend on the parameters of Lyapunov functions constructed for the corresponding isolated subsystems. The problem of computing parameter values that provide the most precise estimates is investigated. Some examples are presented to demonstrate the effectiveness of the proposed approaches.

Constructive Method of Linear Systems with Delay Stability Analysis

Abstract In this paper, the finite constructive method for linear differential-difference systems stability and instability analysis is proposed, and its convergence is proved. This method is based on obtaining of the quadratic lower bound for the quadratic Lyapunov – Krasovskii functionals on some special set of functions. The method proposed is applied to the stability domain construction in a parameter space, and to the critical delay values numerical determination.