Design modern methods of nonlinear control systems of nonlinear objects in the majority assume transformation of initial object model to some special forms. In these cases, it is reasonable to use quasilinear models as they can be designed on condition only of differentiability of the nonlinearities of the initial objects models. These models allow to find control analytically, i.e. as a result of the solution of some equations system, if the object, naturally, meets the controllability condition. The quasilinear models are synthesized traditionally analytically, bytransformation of initial nonlinear models using operation of the taking of partial derivatives from the nonlinearities of the initial objects models and the subsequent integration of these derivatives on the auxiliary variable with application of the known formulas of differentiation and integration. However, in many cases, the objects nonlinearities have so complicated character, that the operations of the differentiation and, in particular, the integration are executed very difficult by shown way. This complexity can be overcome by application of the new numerical design method of the quasilinear models, which excludesneed of the analytical differentiation and integration, but demands considerable number of the arithmetic operations. But now it is not the big problem since the modern multiprocessor controllers can carry out all the necessary operations for a short time. The developed method allows to receive rather exact, approximate piecewise-constant quasilinear models for the objects with the complicated nonlinearities. It is convenient to apply such models at numerical control of the nonlinear objects. The efficiency of a numerical method is shown by comparison of phase portraits of piecewise constant quasilinear and nonlinear models of a simple object and also by comparison of the state variables values of these models. The offered method can be applied to nonlinear control systems design for the nonlinear, characterized by complicated characteristics objects ship, aviation, chemical, agricultural and other industries.

In this article, we consider the development of numerical methods of large deviations analysis for rare events in nonlinear stochastic systems. The large deviations of the controlled process from a certain stable state are the basis for predicting the occurrenceof a critical situation (a rare event). The rare event forecasting problem is reduced to the Lagrange-Pontryagin optimal control problem.The presented approach for solving the Lagrange-Pontryagin problem differs from the approach used earlier for linear systems in that it uses feedback control. In the nonlinear case, approximate methods based on the representation of the system model in the state-space form with state-dependent coefficients (SDC) matrixes are used: the state-dependent Riccati equation (SDRE) and the asymptotic sequence of Riccati equations (ASRE). The considered optimal control problem allow us to obtain a numerical-analytical solutionthat is convenient for real-time implementation. Based on the developed methods of large deviations analysis, algorithms for estimating the probability of occurrence of a rare event in a dynamical systemare presented. The numerical applicability of the developed methods is shown by the example of the FitzHugh-Nagumo model for the analysis of switching between excitable modes. The simulation results revealed an additional problem related to the so-called parameterization problem of the SDC matrices. Since the use of different representations for SDC matrices gives different results in terms of the system trajectory, the choice of matrices is proposed to be carried out at each algorithm iteration so as to provide conditions for the solvability of the Lagrange-Pontryagin problem.

The paper is devoted to the problem of fault diagnosis (isolation and identification) in linear dynamic systems under disturbances. The performances of fault diagnosis depend on the sensors which are in the system under diagnosis. To improve the performances, additional sensors can be applied. But sometimes it is impossible to use such sensors; besides they have low reliability. In this paper, we suggest to use so-called virtual sensors instead of additional ones. To obtain such sensors,Luenberger observers can be used. Such an observer is designed in two steps. On the first step, the model of minimal dimension invariant with respect to the disturbances and estimating a predetermined component of the system state vector and some other components of the system state vector is designed. The second components are necessary to provide stability of the observer by means of generating residual and using feedback. Such components are determined during t  he process of the problem solution which is based on the canonical form of matrices describing the model. On the second step, the feedback matrix is found based on the required quality of transient. To obtain this matrix, eigenvalues are selected and coefficients of the characteristic equation are calculated. The rule to find the predetermined component of the system state vector to be estimated by vir tual obser ver is suggested. Theoretical results are illustrated by practical example of well known three tank system.

The multi-robot formation control is an essential issue in robotics. This review focuses on important lines of research on current contr ol is s ues and s tr ateg ies on a g r oup of unmanned autonomous vehicles/r obots for mation. In this paper, we pr ovide a br ief description of each method characterizing its key benefits and drawbacks. A multilayered classification of both centralized and decentralized formation control methods is proposed. We consider the classification of robot communication topologies in terms of centralized control. Seminal works dedicated to the practical application of centralized approach are briefly discussed. The majority of centralized methodsare represented by a " leader-follower" approach, taking into account the robot’s dynamics models. Furthermore, the most common models of vehicle dynamics are mentioned. In the framework of decentralized approach, behaviour-based algorithms, as well as swarm algorithms, are discussed. Then, we present an outlook of both centralized and decentralized virtual structure methods used in robot formation control. The described modifications of these methods allow tracing the evolution of the virtual structure approach to hybrid algorithms used for cooperative movement of a group of robots. This paper deals with formation control approach considering communication delays and low carrying capacity in an inter-vehicular communication network as very few works discussed this issue despite its relevance. We pointed out the main development trends of formation control approaches. The most effective approach is the integration of various methods of the formation control so that their disadvantages are nullified. As the same time, the most common disadvantage of discussed formation control methods is their weak conceptual framework in terms of kinematic and dynamic constraints of robots.

This study is devoted to the consideration of a method for assessing the stability of systems with pulse-width modulation, based on the linearization of its equivalent system with pulse-width modulation. An approximate study of the dynamic modes of operation of systems with pulse-width modulation, taking into account the stability for the system of automatic control of the supply current of electromagnets under the conditions of external and internal interference, is carried out. Variants of execution of schemes of pulse-width regulators for the power supply of an electromagnet based on a unipolar and bipolar element with pulse-width modulation are presented. The possibility of linearization of systems with pulse-width modulation for the subsequent detailed assessment of the stability of such systems is shown. The prospects of using functional differential equations for stability analysis of automatic systems with pulse-width modulation are shown. The frequency characteristics of an equivalent pulse system are analyzed using the example of a current stabilization system of high-power electromagnets with a pulse-width regulator, taking into account the replacement of the latitude modulation by the amplitude one. Based on the analysis of the resulting transfer function, which is a stable linearized equivalent open system, the ways of evaluating the stability of the original system with pulse-width modulation using the Nyquist stability criterion are proposed. The conclusion is made about the advantage of a system with PWM, in relation to a system with AIM, in terms of stability, and recommendations are given for the use of the obtained data in the analysis oftransients in such systems.

An analytical solution to the optimal control problem of spacecraft reorientation from an arbitrary initial angular position into a required final angular position under the restrictions on control functions and phase variables is presented (the controlling moment and angular velocity are restricted). Time of slew maneuver is minimized. The specific case was considered when maximum admissible kinetic energy of rotation is significant restriction. Constructing the optimal control of reorientation is based on Pontryagin’s maximum principle and the quaternionic variables and models. It is shown that optimal mode is piecewise-continuous control when a direction of spacecraft’s angular momentum is constant relative to the inertial coordinate system during rotation of a spacecraft; for a per forming an optimal turn, the moment of forces is parallel to a straight line fixed in inertial space. Two types of optimal control are possible depending on the given initial and final positions and spacecraft’s moments of inertia — relay control with one switching point when the controlling moment is maximal over the entire time interval of control (segments of acceleration and braking), and relay control with two switching point consisting of intensive acceleration, motion by inertia with the absented moment and an exit onto restriction of rotation energy, and then final braking with the maximum controlling moment. The analytical equations and relations for a finding the optimal control program are written down. The calculation formulas for determining the time characteristics of maneuver and computing a duration of acceleration and braking are given. The proposed algorithm of control provides maximally fast implementation of spacecraft reorientation under the limited kinetic energy of rotation. For an axially symmetric solid body (spacecraft), the optimal control problem, in dynamical statement, was solved completely — we obtained the dependences as explicit functions of time for the control variables, and relations for calculating the key parameters of the law of control are derived. The numerical example and results of mathematical simulation of spacecraft motion under the optimal control are presented, demonstrating the practical feasibility of the developed method for control of spacecraft attitude.

For the linearized fourth-order model of the isolated lateral motion of a single-rotor helicopter as a MIMO system containing two inputs, the control is analytically synthesized, which ensures the invariance of the roll angle motion in the presence of disturbances in the control channels, as well as the required placement of the poles of the closed-loop system, given any specific values from the area of their stability. The approach to the synthesis of invariant control consists in finding a matrix of feedback coefficients of a linear system that satisfies the invariance conditions, which are a system of power matrix equations of a certain design. The synthesis is based on the application of theor ems based on the use of the regularization condition of the matrix equation and the invariance conditions under disturbances in the control channels, as well as theorems that make it possible to place the poles of the MIMO system using the original decomposition of the control object. Regularization of a matrix equation is understood as a solution to the problem of providing a given set of singular values for an inverted symmetric square matrix. The invariance of the MIMO system is considered with respect to unmeasured disturbances inthe control channels. The use of such an approach to the synthesis of invariant control made it possible to obtain an analytical solution that is versatile and can be applied in various flight modes of single-rotor helicopters with different dynamic properties. The results of the numerical synthesis of the lateral motion of a singlerotor helicopter using the obtained laws of invariant control, which confirm the reliability of the analytical expressions, areshown.