Systems Of General Form Research Articles

Peculiar behavior of the instability in systems with weak electron beam-plasma coupling

A study is developed, which allows investigating weakly coupled electron beam-plasma systems in general form regardless on geometry and specific parameters. In this case the instability acquires completely new aspects. In contrast to the well-known case of strong coupling, the role of the beam natural oscillations becomes decisive. The instability is due to the growth of the beam wave of negative energy. The study shows that the space-time evolution of an initial perturbation in weakly coupled beam-plasma systems is described by partial differential equation of second order. The equation is solved. Obtained expression for the fields’ space-time distribution gives most complete information on the instability: growth rates, velocities of unstable perturbations, influence of the dissipation on the instability, its character, (absolute/convective), etc. Most of the information is unavailable by conventional methods. In particular, the results clearly show that with an increase in the level of dissipation in the plasma, the instability transforms into a new, previously unknown type of dissipative streaming instability (DSI). Its growth rate depends on dissipation more critically than the growth rate of the single known DSI.

On a Class of Optimal Control Problems for Functional Differential Systems

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case where the control is chosen from a finite-dimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

Ultrafilters and maximal linked systems: basic relations

Ultrafilters and maximal linked systems with elements in the form of sets from the fixed $\pi$-system with “zero” and “unit” are investigated. Ultrafilters are maximal linked systems, but, among them, maximal linked systems that are not ultrafilters can be situated. In this paper, special attention is given to the description of the set of maximal linked systems that are not ultrafilters (they are called characteristic). In their properties these maximal linked systems differ from ultrafilters essentially. Necessary and sufficient conditions for existence of the above-mentioned systems (we mean conditions with respect to the initial $\pi$-system) and some topological properties for the set of all maximal linked systems of this type are obtained. In addition, for the construction of the corresponding equipment (just as in the ultrafilter case), the schemes ascending to procedures used under Wallman extension and Stone compactums are used; but the above-mentioned schemes are realized in the case when the anticipating measurable (by sense) structure is given by a $\pi$-system of general form. In particular, this allows one to envelop by a unique structure procedures for construction of spaces of ultrafilters and maximal linked systems for measurable and topological spaces. In the framework of this construction, bitopological spaces corresponding to Wallman and Stone variants of equipment arise naturally; in the first variant, for the case of maximal linked systems, the supercompact $T_1$-space is realized. Examples are given for which all maximal linked systems are ultrafilters; this corresponds to realization of supercompact ultrafilter space when the topology of Wallman type is used.

Robust Sliding Mode-Based Learning Control for MIMO Nonlinear Nonminimum Phase System in General Form.

The tracking control of a multi-input multioutput nonlinear nonminimum phase system in general form is discussed. This system is assumed to be suffering from parameter uncertainties and unmodeled dynamics, and the priori information of them is unknown. By considering both the exact model and uncertain model, the sliding mode-based learning controller is proposed. By designing an appropriate sliding surface and a learning controller, the stability of the closed-loop system is guaranteed for both the exact model and uncertain model. To overcome the disadvantage caused by parameter uncertainties and unmodeled dynamics, a fuzzy logical system is adopted here. A numerical simulation result carried on vertical takeoff and landing aircraft is taken as an example to validate the effectiveness of the presented controller.

The synthesis of adaptive multi-mode regulators based on combined control of the combined maximum principle

The problem of synthesis of the optimal controls using nonlinear dynamic systems in general form is not yet analytically solved. Fundamental results of L.S. Pontryagin leads to programs of optimal control, and the solution to the synthesis problem can be obtained using complex constructions only in special cases. The implementation of such solutions implies the need to take into account the sliding modes, the regimes of frequent switching, the negative influence of interference, and so on. This was the reason for the development of methods for synthesizing systems of variable structure, which allow to increase the accuracy of control, eliminate undesirable modes, and counteract the influence of external and parametric disturbances. The results of the research showed that using the combined maximum principle leads to obtaining a general structure of quasi-optimal control for different classes of constraints and allows to provide parametric and structural adaptation to of the state and the parameters due to the possibility of complex implementation of control principles with respect to deviation and disturbance.

The method of plane waves for a hyperbolic system with several spatial variables

In the 80s of the XXth century the Riemann method for hyperbolic equations of the second order was extended to hyperbolic systems in general form with one spatial variable. In this paper, this result is extended to hyperbolic systems in general form with several spatial variables with time dependent coefficients.

Linear System in general form for Hyperbolic type

In this paper, we consider the linear system of partial differential equations hyperbolic type Which is solved by T.V. Chekmarev [1] and solve the general form for this Hyperbolic type.

Time optimal control problems for some non-smooth systems

Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.

Geometrical and dynamical properties of general Euler top system

In this paper we present various geometrical and dynamical properties of the Euler top system in general form. Also the stability problem for the equilibrium states of them are discussed.

Robust ℋ∞ filter design for singular systems with time-varying uncertainties

This article deals with the design of robust ℋ∞ filter for singular systems in general form, subject to bounded time-varying uncertainties. The aim is to design linear admissible descriptor filters that ensure a guaranteed bound on the ℒ2 gain of the operator from the noise signals to the estimation error signal, irrespective of the time-varying uncertain parameters. A strict linear matrix inequality based method is proposed to the robust ℋ∞ filter design employing affine Lyapunov functions. A numerical example is presented to show the efficiency of the proposed method.

Minimum-order observers for discrete-time systems

We consider the synthesis of a minimum-order state or functional observer for a linear dynamical system. The synthesis problem is solved for completely certain systems of general form and for some classes of uncertain systems. Various approaches are described, which ultimately lead to the same task: finding a minimum-dimension Hurwitz solution for a system of linear equations with a Hankel matrix. For scalar and vector linear systems, prior upper and lower bounds on the observer dimension are derived, which makes it possible to switch to an iterative procedure of finding an optimal solution. The discussion is set out for discrete-time dynamical systems.

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.

High Tc metal oxide as Mott–Hubbard unstable paraelectric: General properties of density–mobility dependence

Coupling with large quasi momenta and presence of carriers of different signs of the effective mass, which are the main features of the high- T c superconductivity mechanism developed recently by Kopaev et al., have found the experimental evidence in our previous works. It has been shown that dc resistivity and Hall effect temperature behaviour can be explained by the model of the paraelectric close to the point of the Mott–Hubbard instability. This model possesses the above-mentioned characteristic features. Here we consider in more details the properties of the density–mobility dependence in such Mott–Hubbard unstable systems in general form and show that it reveals some common general features important for considerations of transport properties.

THEORETICAL BASIS FOR THE DETERMINATION OF RATIONAL PARAMETERS OF COMPLEX MECHANICAL SYSTEMS

The mathematical description of choosing the parameters of mechanical systems in general form and with eking account of the features of rail vehicles is presented. The examples of solution of vector optimization problems according to several indices are considered.

Diagonal dominance via eigenstructure assignment

This paper presents a new methodology for diagonal dominance of large-scale systems via eigenstructure assignment. For a given large-scale system in general form, an equivalent descriptor system in the input–output decentralized form is defined. Sufficient conditions for diagonal dominance of the closed-loop system are introduced. These conditions are in terms of the isolated subsystems. Based on them, interactions between subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used for disturbance attenuation via dynamical output compensators based on complete parametric eigenstructure assignment. By attenuating disturbances, closed-loop poles of the overall system are placed in a desirable region, by assigning the eigenstructure of each isolated subsystem appropriately. The presented algorithm alleviates the necessity of choosing a suitable frequency in designing a pre-compensator, as required by previous methods. The designed controller is in the decentralized form and plays the role of pre-compensator as well. An illustrative example is given to show the effectiveness of the proposed method.

On the Stability Degree

Let A be a nondegenerate operator: |A| 6 = 0 (This is equivalent to the condition |B| 6 = 0.) Then the eigenvalues of A can be of three types: real pairs ±a, pure imaginary pairs ±ib, and quadruples ±a± ib. It was shown in [1] that the linear system (2) with nondegenerate operator A that admits an integral in the form of a nondegenerate quadratic form (1) can always be reduced to a system of Hamiltonian equations. Therefore, the theory developed below applies to this (formally, more general) case. The instability degree u of system (2) is defined as the number of roots (counted according to their multiplicities) of the characteristic equation of A in the right half-plane, and the stability degree s is defined as the number of pure imaginary roots (counted according to their multiplicities) of the characteristic equation of A. By i and i− we denote the positive and negative indices of inertia of the quadratic form (1). By the Thomson theorem [2], u ≡ i−mod 2. (3) In particular, if i− (or i) is odd, then so is u. Consequently, the characteristic polynomial of A necessarily has a positive real root in this case; therefore, x = 0 is an unstable equilibrium. More precisely, the classical Thomson theorem holds for linear mechanical systems subjected to additional gyroscopic and dissipative forces. The congruence (3) was proved in [3] for systems of general form (with Ḣ ≤ 0). If Ḣ = 0, then the Thomson theorem is equivalent to the following assertion on the stability degree: s is even if and only if

Feedback passivity of nonlinear discrete-time systems with direct input–output link

This paper is devoted to the study of the feedback passivity property in nonlinear discrete-time systems. The relative degree and zero dynamics of the non-passive system are related to the feedback passivity of the system. Two main results are presented. First, some relative degree-related properties of passive systems in general form are stated. Second, sufficient conditions in order to render a multiple-input multiple-output (MIMO) system passive by means of a static state feedback control law are obtained.

Decentralized stabilization of large-scale systems via state-feedback and using descriptor systems

In this paper a new method for decentralized stabilization of a large-scale system in general form via state-feedback is presented. An appropriate descriptor system is defined for a large-scale system, such that the new system is in input-decentralized form. The interactions between the subsystems are considered as uncertainty. Sufficient conditions for stability of the closed-loop uncertain system are introduced. By appropriately assigning the eigenstructure of each isolated subsystem, these conditions are satisfied. This is accomplished by using the method suggested by Patton and Liu, such that the effects of the interconnections between the subsystems are compensated via the combination of genetic algorithms and gradient-based optimization.

Robust decentralized control of large-scale systems via H ∞ theory and using descriptor system representation

In this paper, a method for design of linear decentralized robust controllers for a class of uncertain large-scale systems in general form is presented. For a given large-scale system, an equivalent descriptor system in input–output decentralized form is defined. Using this representation, closed-loop diagonal dominance sufficient conditions are derived. It is shown that by appropriately minimizing the weighted sensitivity function of each isolated subsystem, these conditions are achieved. Solving the appropriately defined H∞ local problem for each isolated uncertain subsystem, the interactions between the subsystems are reduced, and the overall stability and robust performance are achieved in spite of uncertainties. The designs are illustrated by a practical example.

A class of nonlinear boundary value problems for the second-orderE 2 class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E2 class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods.