A new method has been developed for control systems design of nonaffine in control plants with differentiable nonlinearities and a measurable state vector. It is assumed that the equations of a nonaffine nonlinear plant are given in the Cauchy form. To solve the design problem, these equations, by including an integrator into the system, are converted to equations of an extended virtual affine in control plant. For this purpose, a quasilinear model of a nonaffine in control object is first created. This model describes the nonlinear plant with the same accuracy as the Cauchy equations. Then the quasilinear model of the extended virtual plant, which is affine in control, is formed on basis of the quasilinear model of the initial nonaffine plant, taking into account the added integrator. It is shown that if the quasilinear model of the initial object is controlled on the state, then the extended plant quasilinear model has the same property. This makes it possible to apply an algebraic polynomialmatrix method of nonlinear control systems design for the resulting extended model. The resulting closed nonlinear system is Hurwitz systems, its equilibrium is asymptotically stable, and it can be provided with the required duration of the transition under certain conditions. The controllability criterion output of the nonaffine nonlinear plant is established. This criterion differs from the similar criterion of the affine nonlinear plants only in that it takes into account the nonaffinity. This criterion is determined solely by the properties of the quasilinear model of the initial nonaffine plant. If the condition of this criterion is met, it is possible to ensure a zero value of the static error of a closed system according to the setting impact. The design procedure of nonaffine nonlinear control systems by the proposed method is analytical and consists in determining of quasilinear models, several polynomials and solving SLAE with functional coefficients. The procedure for applying the proposed method and its effectiveness are shown by the example design control system of the autonomous underwater vehicle motion. This method can be used to create control systems for nonaffine nonlinear plants of various purposes. 

One of the actual problems of modern control theory is the study of the stability of systems with switching modes of operation. Such systems are widely used for modeling technological processes, automatic control systems, mechatronic systems, etc. In recent years, they have been effectively used in problems of controlling formations of mobile agents. The main approach to solving this problem is the Lyapunov direct method. For families of subsystems corresponding to the studied hybrid systems, either common or multiple Lyapunov functions are constructed. However, the methods and algorithms for finding such functions are well developed only for linear systems. The problem of analyzing the stability of nonlinear systems with switching has not been sufficiently investigated. It should be noted that its solution is significantly complicated in the case when the systems under consideration contain a delay. In this paper, we study a class of complex systems with switching and distributed delay, modeling the interaction of linear and nonlinear homogeneous subsystems. These systems correspond to the Lyapunov critical case of several zero eigenvalues of the matrix of the linearized system. Note also that under certain additional restrictions on nonlinearities, they are Lurie indirect control systems. The presence of distributed delay can be due to the use of PID controllers. For the considered hybrid systems, new approaches to constructing Lyapunov—Razumikhin functions and Lyapunov—Krasovsky functionals are proposed, which guarantee stability under any switching law. In the case when such functions and functionals cannot be selected, using multiple functionals, classes of switching laws are determined under which stability is preserved. The efficiency of the developed approaches is demonstrated on the example of a controlled mechanical system with a PID controller. 

The paper considers the problem of analytical design of optimal regulators (ADOR) in the formulation of A. A. Krasovsky for the studied stable multidimensional objects described by a matrix differential equation with polynomial nonlinearities in phase coordinates. This class of control objects, called polynomial, is quite wide for applications: these models are used to describe the movement of systems of a very different nature, for example, electromechanical devices, chemical reactors, industrial facilities with recycling, biological and environmental systems, etc. To solve this problem of ADOR in the early work of the author, a method is proposed for synthesizing quasi-optimal controllers, which largely weakens the disadvantages of the power series method (a large volume of operations with polynomials that are not suitable for programming) through the use of a procedure for multidimensional linearization of the description of polynomial objects, carried out by expanding the space state of the object with new coordinates, which are products of the original phase coordinates, and the application of the apparatus of matrix theory with the Kronecker (direct) product. In this paper, a modification of this method of system synthesis is carried out by using a reduced rather than a full Kronecker degree of the object state vector and taking into account the block-diagonal structure of the parameter matrix of the applied linearized model of the object, which ensures a further multiple reduction in the volume of calculations in the synthesis of control systems. The proposed modified synthesis method makes it possible to find, in the form of a polynomial function, an approximate solution to the ADOR problem with high accuracy, and its implementation is extremely simple due to the use of mainly well-known software for solving linear-quadratic optimal control problems (procedures for solving Lyapunov and Sylvester matrix equations).

The features of the method are illustrated using a specific example of the synthesis of a quasi-optimal control system. 

Continuum robots are flexible robots capable of maneuvering in spaces with complex geometry, such as the insides of complex devices. High maneuverability of continuum robots are ensured due to the elastic bending deformation of the robot’s own body and the linear displacement of the base. Bending deformation can be described by two assumptions: absence of torsion and piecewise constant curvature. The absence of torsion eliminates torsional deformation in the robot. Piecewise constant curvature assumption allows us to describe the shape of the robot’s neutral line. To do this, the bending section is divided into subsections, the neutral line of which can be represented by an arc of a circle. However, this approach complicates the inverse kinematics. The presence of a movable base is also an obstacle to solving the inverse kinematics. This paper presents a solution to the inverse kinematics for a continuous robot with a movable base and variable curvature, which uses the tangent line to the robot’s workspace to determine the amount of base displacement. To determine the tangent, the robot’s work area is divided into several sites. Each site has its own center. A tangent is defined as a perpendicular to a line drawn through the center of the site and to a point in the work area for which the tangent needs to be determined. A comparison of the proposed method with an analogue in numerical experiments shows that the proposed method more accurately determines tangents and is capable of solving a larger portion of inverse kinematics problems than the analogue. 

The task of motion simulation of flying manipulation robots used in space modules in near-earth orbit is considered. As part of this task, a nonlinear mathematical model of flying robot dynamics was obtained, in which manipulator mass and dimensions are significantly less than those of the base link. The resulting mathematical model is used to motion control synthesis of the flying robot based on sliding modes. The proposed solution consists in providing such control that ensures the absence of oscillations in the vicinity of sliding surface. For this purpose, the control law of the flying robot is implemented using a continuous hyperbolic tangent function, which is an approximation of the discontinuous switching function. This approach makes it possible to effectively implement the robot’s movement along a given trajectory, its reorientation and stabilization. The approbation of methods and approaches proposed in the paper was carried out in virtual environment complex VirSim created by the authors. In this software package, the manipulation robot dynamics is performed using a universal sequential impulses method, which allows to handle various types constraints that arise due to the joints of robot links, body collisions, friction, etc. The flying robot control is implemented by means of a created functional diagram scheme that computes the control actions applied to the robot’s actuators based on virtual sensors readings and flight program’s input commands. Regarding this, the robot manipulator control in this complex is performed using PD controllers and inverse kinematics solving to ensure the required position of robot’s end effector (gripper). Simulation results showed the adequacy of the solutions proposed in the paper, which can be further used to flying robot control when solving more complex tasks related to the grip and transfer load inside the orbital station. 

This article is devoted to the problem of "targeted" fixing of the measurement results of spatial position of the blades’ tips of compressor or turbine of gas turbine engine (GTE) to a specific struct ural element of the power plant (blade) in monitoring the state of engine’s gas-air channel under limitations on use of diagnostic equipment and, in particular, the impossibility of installing the shaft’s angular position sensor, that ensures the synchronization of primary converters polling with the rotation period of the turbomachine rotor. The diagnosis of disruptive and surging phenomena in GTE compressor, detection of foreign objects entering the gas-air path of the engine, determination of fatigue failure of the blades are the typical examples of the tasks that require binding of the measurement results to each individual blade on a rotor wheel. The proposed solution is based on the analysis of a unique set of radial clearance values for the complete ensemble of rotor blades with the comparison of the current and previously obtained "reference" images of the turbo compressor’s bladed wheel. Two algorithms for blades’ number determining are considered. The algorithms do not require a separate channel for synchronization and use the minimum Euclidean distance and the maximum value of the cross-correlation function between the elements of the "reference" and current arrays of codes as criteria for blades’ identifying. The results of the performance test of both algorithms on a laboratory bench with a real compressor wheel are presented. It is shown that the criterion based on cross-correlation function calculation has higher noise immunity, and therefore it is the most preferable for practical implementation in actual practice measurement systems. Low computational complexity of the criterion allows to use it at the microprocessor level and to reduce the measuring system hardware.