Про проектування малочутливих прискорювально-фокусуючих систем методами практичної стійкості

Topicality. When solving a number of problems associated with the design of complex control systems [1-3], it often becomes necessary to adjust the motion in the vicinity of the calculated trajectories set at the initial moment of time and belong to some dynamic sets of phase space [2, 3]. In this case, the correction requirements may relate to the individual coordinate of the calculated trajectories and their linear combination. These tasks belong to the class of tasks of stabilization of motion and reduce to the construction of regulatory influences that provide the original system a certain type of stability [1, 2, 4]. This approach is used, for example, in modeling charged particles in order to control them in dynamics in the presence of appropriate sustainability requirements. To construct constructive algorithms, the stability analysis is carried out on a finite time interval, and the initial conditions are given in the structural form [3, 5-7]. The latter allows us to obtain numerical estimates for the stabilization of motion to practical stability in the presence of dynamic constraints on phase coordinates. Analysis of recent research and publications. With the analysis of practical stability and sensitivity of dynamic systems, problems of optimal control of a beam of trajectories are closely related [2, 3, 6]. Thus, numerical algorithms for calculating the optimal areas of practical stability are used to estimate the area of particle capture in the acceleration process. In this case, the field of initial conditions can be in the given structures (sphere, ellipsoid), and the maximum in volume (optimal by inclusion). Then the question is raised about optimization of estimates of such sets due to the proper choice of system parameters [7]. The tasks of constructing regulatory influences that provide the desired stability requirements are an important part of a set of tasks that arise in the design of technical systems, in particular accelerating-focusing [1, 2]. Unlike the classical statements [1], the analysis of the stability of parametric systems [2-4, 6, 7] allows us to study the system's performance in real modes, taking into account the requirements for sensitivity, and to solve such problems numerically from the standpoint of stability [3]. The purpose of the research ─ development of constructive algorithms for solving stabilization problems for practical stability for linear parametric systems with disturbances. Materials and methods of research. The paper uses methods of stability theory, differential equations and control theory. Research results and their discussion. For linear parametric systems of differential equations with perturbations, the sequence of stabilization problems to practical stability in the given domains of dynamic constraints is investigated. Considered cases of known and limited perturbation norm. For structurally-defined areas of initial conditions, numerical estimates of the stabilization of motion to practical stability have been obtained. Conclusions and perspectives of further research. Formation of stabilization problems for practical stability of linear parametric systems with perturbations is formulated. On the basis of practical stability algorithms, numerical criteria for stabilizing the system motion to a certain type of stability were obtained. Such an approach can be extended to the solution of problems of stabilization of motion to asymptotic stability in regions of a more general structure. Keywords: mathematical model, stabilization, regulator, parameters, practical stability, parametric system, dynamic constraints

In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h-stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h-stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h-stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

This paper investigates the problem of output feedback $h$-stabilization of nonlinear uncertain systems. We construct an output feedback controller that guarantees global uniform practical $h$-stability of the closed-loop system. Our original results generalize well-known fundamental results: practical stability, practical asymptotic stability and practical exponential stability for nonlinear time-varying systems. Finally, two numerical examples are presented to demonstrate the validity of the proposed method.

In this chapter we consider practical stability of discrete systems on the basis of maximum sets of initial conditions concept. We propose results concerning nonlinear discrete systems including topological properties of the maximum sets of initial conditions for internal practical stability, the necessary and sufficient conditions of internal practical stability using the optimal Lyapunov function. Further we offer the analytical forms (such as the Minkowski function, the inverse Minkowski function, and the support function) of the maximum sets of initial conditions representation in linear case, optimal ellipsoidal estimations of practical stability domains, and optimal estimations of phase constraints. In the last section we consider the problem of external practical stability of discrete systems.

This paper provides sufficient conditions for both practical stability and finite time stability of linear singular continuous time delay systems which can be mathematically described as Eẋ(t) = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> x(t) + A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> x(t - τ). Considering a finite time stability concept, new delay independent conditions have been derived using the approach based on the Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to have the properties of positivity in the whole state space and negative derivatives along the system trajectories. When the practical stability has been analyzed the above mentioned approach was combined and supported by the classical Lyapunov technique to guarantee the attractivity property of the system behavior.

Mean-square practical stability for stochastic large-scale dynamical systems is studied. The vector Lyapunov functions and the basic comparison principle of stochastic systems as well as the decomposition-aggregation method for large-scale systems were used to obtain results for various types of mean-square practical stability of nonlinear stochastic large-scale dynamical systems. The objective is to analyze stochastic large-scale systems in terms of their lower-order subsystems and in terms of their interconnecting structure. These results make it possible to determine the mean-square practical stability of stochastic large-scale dynamical systems by testing the corresponding practical stability of the auxiliary deterministic systems. >

Practical stability of discontinuous large-scale systems

In this paper, we introduce various definitions of practical stability and integral stability for nonlinear singular differential systems with maxima and give criteria of stability for such systems via the Lyapunov method and comparison principle.

Practical Stabilization of Nonlinear Control Systems

This paper is concerned with practical asymptotic and exponential stability analysis of time-varying nonlinear systems. With the help of the new notion of practical stable functions, some differential Lyapunov inequalities based necessary and sufficient conditions are derived for testing global uniform practical asymptotic stability and practical exponential stability of general time-varying nonlinear systems. Therefore, we generalised some works which are already made in the literature. Furthermore, some illustrative examples are presented.

This article examines the effect of changing the design length of the elements of frame-bar structural systems operating in a constrained bifurcation. For the criterion of the form of buckling of an element of a frame-bar structural system, the sign of the work of end moments and shear forces is taken. Using this criterion, an expression was obtained to assess the effect of changing the calculated length of the frame struts on the critical parameters of the entire system, which allows varying the geometry of the structure to influence its performance. A two-span frame is considered, in which the central pillar is loaded with a concentrated force Pcr, and the outer pillars are loaded with forces aRcr. The type of bifurcation of the rods (constrained or forced) and the critical parameters of the stability of the system before and after changing the calculated length of its elements are determined. Changes in the design length of elements that passively lose stability do not have a significant effect on the critical stability parameters of the frame-bar structural system. At the same time, a 30% decrease in the calculated length of the struts in an active bifurcation leads to a decrease in the critical force by 50%. The presented rather simple algebraic dependencies allow obtaining a qualitative and quantitative assessment of the effect of the calculated length coefficient on the stability of frame-rod structural systems made of wood. It has been established that the conditions for securing elements that passively lose stability do not have a significant effect on the critical parameters of the system.

Finite-time stabilization of linear systems by bounded event-triggered and self-triggered control

This note is devoted to study the stabilization problem of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. By introducing the notation of input-to-state practical stability and an event-triggered strategy, we establish the input-to-state practically exponential mean-square stability of the suggested system. Moreover, we investigate the stabilization result by designing the feedback gain matrix and the event-triggered feedback controller, which is expressed in terms of linear matrix inequalities. Also, the lower bounds of interexecution times by the proposed event-triggered control method are obtained. Finally, an example is given to show the effectiveness of the proposed method. Compared with a large number of results for discrete-time stochastic systems, only a few results have appeared on the event-triggered control for continuous-time stochastic systems. In particular, there have been no published papers on the event-triggered control for continuous-time stochastic delay systems. This note is a first try to fill the gap on the topic.

On the stabilization of the homogeneous bilinear systems

Practical stability, boundedness criteria and Lagrange stability of fuzzy differential systems