Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

Control of wheeled platforms straight motions taking into account jerk restrictions under speeding-up from the state of rest

I. Sh. Nevliudov — 2022-11-18

The generalized mathematical model of wheeled platforms straight motions on the ideal horizontal plane under speeding-up from the state of rest mode is proposed, and the controls satisfying the restrictions of motion jerks are find. The pure mechanical and electromechanical wheeled platforms are considered, as well as the computer simulations of the researched processes are made. The jerks restrictions are reduced to limiting the value of the wheeled platform acceleration time derivative. The proposed approaches are based on the holonomic systems mechanics and on the electromechanical analogies allowing to consider the different kinds of the wheeled platforms taking into account the electric on-board systems like the drive electric motors and the control systems by using the Lagrange equations of second kind. The examples of the proposed approaches using to define the controls satisfying the jerks restrictions under speeding-up from the state of rest are considered for the pure mechanical and electromechanical wheeled platforms. It is obtained the inequality allowing to chose the instantly supplied driving mechanical couple which will provide the admissible jerks of the motion of the wheeled platform under speeding-up from the state of rest. It is shown that the rolling friction and the viscous damping are the principal causes of the wheeled platforms jerks under speeding-up from the state of rest. It is obtained the inequality defining the voltage instantly supplied on the drive electric motors which will provide the admissible jerks of the motion of the electromechanical wheeled platform during speeding-up from the state of rest, and it is shown that the proposed general approaches are suitable for considering the different kinds of wheeled platforms. The computer simulations of the processes of speeding-up from the state of rest for the electromechanical wheeled platform are considered to show results correctness and to illustrate satisfying the restrictions of the motion jerks. The obtained results of the computer simulations are in the full agreement with the well-known fundamental property inherent for the wheeled platforms. The results for the jerks show that the maximum value of the jerk is really at the initial time as was suggested before, and it is shown that the jerks values at the initial time obtained by using the computer simulations are in full agreement with the theoretically defined correspondent exact values. The big jerks of the considered electromechanical wheeled platform are due to the voltage instantly supplying on the drive electric motors at the initial time, and it is understandable that limiting of such instantly supplied voltage value cannot provide any wished small jerks. The smooth time depending for the voltages supplying on the drive electric motors are required to provide any wished small jerks of the electromechanical wheeled platforms.

Homogeneous approximation for minimal realizations of series of iterated integrals

D. M. Andreieva — 2022-12-24

In the paper, realizable series of iterated integrals with scalar coefficients are considered and an algebraic approach to the homogeneous approximation problem for nonlinear control systems with output is developed. In the first section we recall the concept of the homogeneous approximation of a nonlinear control system which is linear w.r.t.\ the control and the concept of the series of iterated integrals. In the second section the statement of the realizability problem is given, a criterion for realizability and a method for constructing a minimal realization of the series are recalled. Also we recall some ideas of the algebraic approach to the description of the homogeneous approximation: the free graded associative algebra, which is isomorphic to the algebra of iterated integrals, the free Lie algebra, the Poincar\'{e}-Birkhoff-Witt basis, the dual basis and its construction by use of the shuffle product, the definition of the core Lie subalgebra, which defines the homogeneous approximation of a control system. In the third section we show how to find the core Lie subalgebra of the systems that is a realization of the one-dimensional series of iterated integrals without finding the system itself. The result obtained is illustrated by the example, in which we demonstrate two methods for finding the core Lie subalgebra of the realizing system. In the last section it is shown that for any graded Lie subalgebra of finite codimension there exists a one-dimensional homogeneous series such that this Lie subalgebra is the core Lie subalgebra for its minimal realization. The proof is constructive: we give a method of finding such a series; we use the dual basis to the Poincar\'{e}-Birkhoff-Witt basis of the free associative algebra, which is built by the core Lie subalgebra, and the shuffle product in this algebra. As a consequence, we get a classification of all possible homogeneous approximations of systems that are realizations of one-dimensional series of iterated integrals.

The explicit form of the switching surface in admissible synthesis problem

V. I. Korobov — 2022-12-24

In this article we consider the problem related to positional synthesis and controllability function method and more precisely to admissible maximum principle. Unlike the more common approach the admissible maximum principle

method gives discontinuous solutions to the positional synthesis problem. Let us consider the canonical system of linear equations $\dot{x}_i=x_{i+1}, i=\overline{1,n-1}, \dot{x}_n=u$ with constraints $|u| \le d$. The problem for an arbitrary linear system $\dot{x} = A x + b u$ can be simplified to this problem for the canonical system. A controllability function $\Theta(x)$ is given as a unique positive solution of some equation $\Phi(x,\Theta) = 0$. The control is chosen to minimize derivative of the function $\Theta(x)$ and can be written as $u(x) = -d \text{ sign}(s(x,\Theta(x)))$. The set of points $s(x,\Theta(x)) = 0$ is called the switching surface, and it determines the points where control changes its sign. Normally it \mbox{contains} the variable $\Theta$ which is given implicitly as the solution of equation $\Phi(x, \Theta) = 0$. Our aim in this paper is to find a representation of the switching surface that does not depend on the function $\Theta(x)$. We call this representation the explicit form. In our case the expressions $\Phi(x, \Theta)$ and $s(x, \Theta)$ are both polynomials with respect to $\Theta$, so this problem is related to the problem of finding conditions when two polynomials have a common positive root. Earlier the solution for the 2-dimensional case was known. But during the exploration it was found out that for systems of higher dimensions there exist certain difficulties. In this article the switching surface for the three dimensional case is presented and researched. It is shown that this switching surface is a sliding surface (according to Filippov's definition). Also the other ways of constructing the switching surface using the interpolation and approximation are proposed and used for finding the trajectories of concrete points.

Vyacheslav O. Rezunenko (1941 - 2022) Obituary

V. I. Korobov — 2022-12-07

On August 28, 2022, a docent at the Department of High mathematics and informatics of V. N. Karazin Kharkiv National University Vyacheslav Oleksijovich Rezunenko passed away. The bright memory of Vyacheslav Oleksijovich Rezunenko, a real scientist and a wonderful man, forever remains in the hearts of his colleagues, students and friends.