On motion stabilization of nonstationary controlled system

In this paper, a solution to the problem of optimal motion stabilization of nonstationary controlled system basing on the constant-positive Lyapunov function is represented.     

On the sufficient and necessary conditions for robustness of the nonlinear dynamic systems in terms of stability retention

Consideration was given to the conditions under which the nature of stability of the equilibrium of an arbitrary-order nonlinear dynamic system does not change under any sufficiently small linear and nonlinear perturbations in its right side (robustness in the sense of the retention of the nature of stability). Sufficient and necessary conditions for such robustness were presented.     

Essays on the absolute stability theory

A brief historical background of the development of the absolute stability theory was presented, and some of its methods and approaches, as well as the results obtained using them, were discussed. The kinds of systems under consideration are named, and applications of the methods and results of the absolute stability theory to other scientific and practical fields for solution of the engineering, mechanical, physical, and other problems were presented.     

An adaptation algorithm for stabilization of nonlinear system in the absence of sector constraints

The possibility for applying the control method designed in [1] for stabilization of nonlinear systems consisting of a linear minimal phase block and a nonlinear static feedback block is examined. Unlike in [1], the nonlinear characteristic of the static nonlinearity is not assumed to belong to a sector, but is the output function raised to some power.     

On maximum sector bounds for absolute stability of single-input single-output systems

We consider the classical problem of absolute stability, i.e. stability of a linear time-invariant system in feedback with a memoryless time-varying sector bound non-linearity. We derive necessary and sufficient conditions for the existence of periodic motion. For the special case of symmetric sector bounds, this condition yields the minimum sector bounds for which the system can sustain periodic motion. By guaranteeing the absence of periodic motion we derive non-conservative sufficient conditions for absolute stability for arbitrary sector bounds. We demonstrate by numerical examples the application of our results.     

On absolute stability of discrete systems

Modernization is considered of the criterion of absolute stability of discrete systems [1, 2], which uses the method of quadratic transformation of the state vector. The modernization aim is to obtain the criterion form that does not explicitly depend on the notions of this method. The new formulation permits performing all calculations in the space of the initial dimension, which reduces the complexity of the problem solution. This work represents the development of the approach suggested in [3] for continuous systems.     

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.     

Adaptive observer-based synchronization of the nonlinear nonpassifiable systems

The general approach to synchronization of the dynamic systems on the bassis on the adaptive observer and the passification method was extended to the nonpassifiable nonlinear systems—in particular, to those whose model has the relative order higher than one. Two schemes of synchronization relying on the extended-error adaptive observers and the high-order tuning algorithms were proposed. Solution of the problem of synchronization relies on a new canonical form of the adaptive observer. The conditions for convergence of the parameter estimates to the true values in the case of no system noise were established, and also robustness of the adaptive synchronization to the bounded measurement error was proved. The feasibility of information transmission by modulation of the chaotic signal with the use of the proposed method was demonstrated by the example of the controllable Lorentz system.     

Design of nonlinear systems based on the controllable jordan form

The controllable Jordan form of the control-affine system equations was shown to exist besides the controllable Frobenius form. Explicit expressions of the stabilizing control of continuous and discrete one-control plants whose equations are represented in the controllable Jordan form, as well as the conditions for reduction to the controllable Jordan form of the control-affine second-order equations of nonlinear plants were presented. Examples of design of nonlinear continuous control systems were given.     

Approximate optimal controls for a class of Volterra systems with nonlinear delay

Necessary and sufficient conditions of approximate optimal controls for integral linear in state Volterra systems with nonlinear delays are derived. They are used for studying approximate optimal controls obtained by the control parameterization method as an important application.     

Stabilization of program motion and kinematic trajectories in dynamic systems in case of systems of direct and indirect control

In this paper, analysis methods for self-vibrating processes in automatic control systems are represented. A qualitative analysis of the behavior of transient processes in automatic control systems is conducted. Accuracy of control system functioning is estimated. Problems of noise immunity of control systems are solved. Ways of control system implementation by applying dynamic self-vibrating regimes, by logical nets, and digital automatic devices are considered.     

An optimal control for random-structure nonlinear systems under incomplete state vector information

New sufficient conditions for the optimality of control for random-structure nonlinear stochastic systems under incomplete state vector information are formulated. Relations for determining the optimal control for the general minimized functional and for the mean optimal control are derived.     

External bounds and step controllability of the linear interval system

The differential equations describing double-sided bounds of the fundamental matrix of solutions for the linear control system with variable interval coefficients are obtained. Based on those bounds, the controllability problem, i.e., the possibility to transfer the system from a given parallelepiped to another in finite time in the class of step (piecewise constant) controls, is examined. Sufficient conditions of controllability in the form of solvability of an auxilliary linear programming problem are established. This problem results in step control transferring a bundle of system trajectories from an initial parallelepiped to a minimal neighborhood of a final parallelepiped.     

On retention of impulsive system stability under perturbations

Consideration was given to the nonlinear system of differential equations with impulsive action at the fixed time instants and perturbed right-hand sides. Criteria were obtained for retention of the uniform asymptotic stability of the invariant set of the impulsive system under vanishing perturbations.     

Adaptive control of linear nonstationary objects output

A problem of linear nonstationary systems output control with unknown rapidly varying parameters is studied. Assuming that an object of control is subject to the influence of an external bounded unknown disturbance, an approach that guarantees a solution to the problem of command signal output follow-up with the given accuracy is presented.     

Causality and stability conditions for the control of a linear distributed system

For a linear distributed system controlled by nonlinear feedback, a sufficient condition for the existence, causality, and boundedness of an operator that transforms the external action and initial state of the system to the behavior of the output of a closed-loop system is formulated. A procedure is designed for determining the class of feedbacks ensuring the causality and stability of the closed-loop control system. It is applicable to a linear one-dimensional heat conduction system controlled by nonlinear temperature feedback to the boundary action.     

Duality of the notions of controllability and observability for degenerate linear hybrid systems

Consideration was given to the linear system of equations in continuous and discrete time with constant matrices of coefficients which is not explicit in the derivative of the continuous component of the desired vector function. The notion of conjugate system was introduced, and an analog of the Kalman duality theorem relating the notions of total controllability and observability was proved.     

Generalized synthesis for singular nonlinear optimal control problems

We consider the problem of minimizing a quadratic noncoercive functional along the trajectories of a control-affine system. Due to lack of coercivity, existence of “classical” minimizers cannot, in general, be guaranteed. Under appropriate commutativity assumptions the problem can be extended into the space of generalized controls of class W _?1,∞ and reduced into a new problem which is generically coercive but nonconvex. We show how to extend further the problem in order to include generalized controls which are “generalized derivatives of one-parameter families of regular probability measures,” thus achieving convexification. Generalized trajectories for this type of controls exist only in a weak sense. We discuss a version of the maximum principle suitable to this class of problems and show how a generalized synthesis can be obtained.     

Principle of maximum in the problem of control under limited phase coordinates

Consideration was given to the autonomous problem of optimal control with phase constraints for which the necessary optimality conditions in the form of the Pontryagin principle of maximum were obtained. The result obtained features completeness of the conditions of the maximum principle and their nondegeneracy.     

Stability margins of the systems of optimal and modal control

Consideration was given to the systems of H _∞-suboptimal and modal control and the causes of their small phase and magnitude margins.