Nonlinear Nonstationary Systems Research Articles

Stability of solutions to impulsive differential equations in critical cases

We propose a new approach to constructing a piecewise differentiable Lyapunov function for some classes of nonlinear nonstationary systems of impulsive differential equations in the critical case. This approach allows us to obtain new sufficient conditions for the Lyapunov stability of solutions to this class of systems.

On partial stability theory of nonlinear dynamic systems

A stability problem with respect to a part of variables of the zero equilibrium position is considered for nonlinear non-stationary systems of ordinary differential equations with the continuous right-hand side. As compared to known assumptions, more general assumptions are made on the initial values of variables non-controlled in the course of studying stability. In addition, a stability problem is considered with respect to a part of variables of the "partial" equilibrium position, with similar assumptions made for initial values of variables that do not define the given equilibrium position. Conditions of stability and asymptotic stability of this type are obtained within the method of Lyapunov functions and generalize a number of existing results. The results are applied to the stability problem with respect to a part of variables of equilibrium positions of nonlinear holonomic mechanical systems. The problem of unification (to a certain extent) of the process of studying partial stability problems of stationary and non-stationary systems is discussed.

On partial detectability of the nonlinear dynamic systems

Conditions were obtained under which the uniform stability (uniform asymptotic stability) in one part of the variables of the zero equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position relative to another, larger part of variables. Conditions were also obtained under which the uniform stability (uniform asymptotic stability) in one part of variables of the "partial" (zero) equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position. These conditions complement a number of the well-known results of the theory of partial stability and partial detectability of the nonlinear dynamic systems. Application of the results obtained to the problems of partial stabilization of the nonlinear control systems was considered.

Synthesis of nonlinear nonstationary control systems by output under conditions of uncertainty

A problem of synthesizing the control laws for nonlinear nonstationary control systems by output being measured under uncertain external actions and nonlinearities that meet the sector constraints is considered.

Comparison equations in problems of motion stability

New results of the investigation of stability of solutions to differential equations describing the behavior of different nonlinear nonstationary systems are represented; the results are based on the joint use of the method of logarithmic matrix norms and the method of limit equations.

On the existence of periodic motion and the maximum sector bounds for absolute stability of nonlinear nonstationary systems

In this contribution we consider the absolute stability problem. We derive necessary and sufficient conditions for the existence of periodic motion using an operator approach. The results yield an efficient algorithm to approximate the maximum sector bounds for absolute stability numerically.

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.

Analysis of stability and synthesis of stable control systems for a class of nonlinear nonstationary systems

On the basis of sufficient condition of the stability of linear nonstationary discrete systems stated in this paper, we carry out an analysis of stability for a special class of discrete systems with nonstationary linear part, whose parameters satisfy constraints in the form of sets, and with a scalar nonlinear function satisfying linear or nonlinear restrictions. The problem of parametric synthesis of robustly stable control systems is solved for the same class of objects. The obtained results are generalized to the class of nonstationary systems with many nonlinearities of the same type.

RBF-ARX model-based nonlinear system modeling and predictive control with application to a NO x decomposition process

This paper considers the modeling and control problem for nonstationary nonlinear systems whose dynamic characteristics depend on time-varying working-points and may be locally linearized. It is proposed to describe the system behavior by the RBF-ARX model, which is an ARX model with Gaussian radial basis function (RBF) network-style coefficients depending on the working-points of a system. The RBF-ARX model is constructed as a global model, and is estimated off-line so as to avoid the possible failure of on-line parameter estimation during real-time control. A receding horizon predictive control (RBF-ARX-MPC) strategy based on the RBF-ARX model that does not require on-line parameter estimation for the nonlinear system is presented. The local linearization of the system at each working-point may be easily obtained from the global RBF-ARX model and so the use of nonlinear programming techniques to solve the on-line optimization problem with constraints in RBF-ARX-MPC is also avoided. A fast-converging estimation method is applied to optimize the RBF-ARX model parameters. A case study and example of an industrial experiment on the nitrogen oxide (NO x ) decomposition process in thermal power plants are given to demonstrate the modeling precision and control performance.

Sufficient and Necessary Conditions for the Stable Controllability of a Nonlinear Nonstationary System on the Plane in the Critical Case

Sufficient and Necessary Conditions for the Stable Controllability of a Nonlinear Nonstationary System on the Plane in the Critical Case

Modeling of nonlinear nonstationary dynamic systems with a novel class of artificial neural networks

This paper introduces a novel neural-network architecture that can be used to model time-varying Volterra systems from input-output data. The Volterra systems constitute a very broad class of stable nonlinear dynamic systems that can be extended to cover nonstationary (time-varying) cases. This novel architecture is composed of parallel subnets of three-layer perceptrons with polynomial activation functions, with the output of each subnet modulated by an appropriate time function that gives the summative output its time-varying characteristics. The paper shows the equivalence between this network architecture and the class of time-varying Volterra systems, and demonstrates the range of applicability of this approach with computer-simulated examples and real data. Although certain types of nonstationarities may not be amenable to this approach, it is hoped that this methodology will provide the practical tools for modeling some broad classes of nonlinear, nonstationary systems from input-output data, thus advancing the state of the art in a problem area that is widely viewed as a daunting challenge.

Parametric Synthesis of One Class of Dynamic Systems under Random External Disturbances

The problem of suboptimal parametric optimization of nonlinear nonstationary controllable systems under random external disturbances of white noise type is considered in the present paper. The method of solution incorporates V.F. Krotov's idea of global estimates.

Dynamic bifurcations in non-stationary systems: transitions with an unpredictable outcome

In nonlinear non-stationary systems, dynamic bifurcations result in a transition to a qualitatively new state. In this paper we examine how the dynamics of transition of such systems may be assessed using the concept of transient basins of attraction. We delineate the phenomenon of indeterminate dynamic bifurcations, where it is shown that the response, after the system passes through critical parameter values, may be extremely sensitive to the choice of initial conditions or parameter states. This new form of unpredictability in systems whose parameters vary with time, is clearly an important concept to be assimilated in the theory of non-stationary dynamics.

Higher-order pointwise optimality conditions

Necessary higher-order optimality conditions are given for nonlinear smooth nonstationary control systems, linear with respect to control and defined on a smooth finite-dimensional manifold; the admissible values of the control belong to a closed convex polyhedron, while the initial and final times and the initial and final points of the trajectory are not fixed. Unlike the previously known necessary optimality conditions, the paper does not require that the passage from one condition, of a given sequence of necessary conditions, to another one should be performed only when the preceding condition degenerates on a time interval of nonzero length.

The existence and uniqueness of the generalized solution of the inverse problem for the nonlinear nonstationary Navier-Stokes system in the case of integral overdetermination

The existence and uniqueness of the generalized solution of the inverse problem for the nonlinear nonstationary Navier-Stokes system in the case of integral overdetermination

GENERALIZED REDUCTION PROCEDURE AND NONLINEAR NONSTATIONARY DYNAMICAL SYSTEMS

In this paper a generalized reduction procedure is used to obtain nonlinear nonstationary dynamical systems out of free ones.

Solvability of a nonlinear nonstationary system of equations

Solvability of a nonlinear nonstationary system of equations

Controllability of nonlinear discrete systems without differentiability assumption

The paper gives another approach to studying controllability of nonlinear discrete systems, which is based on proving the support principle for some boundary problems involving locally Lipschitzian set-valued maps. The approach enables us to develop sufficient conditions for various types of local controllability of nonlinear non-stationary discrete systems without differentiability assumption. For linear discrete systems with constrained states and controls, necessary and sufficient conditions for local controllability are given.

Methods for Modeling a Class of Nonstationary Nonlinear Systems

Methods for Modeling a Class of Nonstationary Nonlinear Systems

Pulse response of non-linear non-stationary vibrational systems with N degrees of freedom

The paper deals with an approximate analysis of non-linear, non-stationary vibrational systems with multiple degrees of freedom subjected to a pulse excitation. The nonstationary system parameters, which may include masses, restoring forces or damping, are considered to be slowly varying functions of time. A general procedure for obtaining the first order approximate solution is presented through an application of the Bogoliubov-Mitropolsky technique. The special case of systems with two degrees of freedom is studied in detail. An illustrative example is included and the results are compared with fourth-order Runge-Kutta numerical solutions.