The condition for the maximum of the generalized power function in the problem of forming a multi-mode control law with limitation

. A quasi-optimal control law is developed based on the condition for the maxi-mum of the generalized power function taking into account the stationarity of the Hamiltonian on the switching line for control objects that can be represented by the Lagrange equations of the second kind. The comparative analysis is carried out based on the mathematical simulation using the optimal nonlinear control laws with respect to several criteria. We found that the modes of the proposed control law provide high accuracy of approximation to the optimal performance laws and the Fuller laws, reducing energy costs for control by eliminating more frequent switching. The choice of the parameters of the developed control law makes it possible to implement a wide range of both nonlinear and linear operating modes, which allows to classify the obtained control law as multimode law.


Introduction
The synthesis of control laws that are used in the construction of multi-mode systems can be based on the use of the Lyapunov methods, the methods of optimal synthesis using the principles of L. Pontryagin and R. Bellman, the empirical solutions using and generalizing the control experience that require methods of intellectualization [1][2][3][4][5]. The latter are not effective enough in the case when it is possible to build an accurate mathematical model [6][7][8].
We can assert that the set of solutions that ensure the stability of the synthesized system is infinite, while there are significantly fewer optimal solutions to the synthesis problem satisfying a given criterion. Their set is limited, therefore, it is not always possible to use them to construct multi-mode systems. So it is necessary to search for quasi-optimal control laws that are close to optimal laws with respect to a given criterion [9].
Works [10][11][12][13][14] consider an approach that allows to obtain quasi-optimal solutions to the synthesis problem based on the condition for the maximum of the generalized power function for a wide class of dynamical systems that satisfy the Hamilton-Ostrogradskii principle and, accordingly, can be represented by the Lagrange equations of the second kind.
The aim of this work is to construct a multi-mode control based on the proposed approach and to compare the obtained solutions with classical optimal solutions.

Formulation of the control problem
Let us consider a class of controlled systems which motion in independent coordinates The specified class of controllable Lagrangian systems is given by the set U and by the constant parameters 0 K and 1 . K The specific system of this class is distinguished by the definition of the quadratic form T in accordance with (2). Any such system is considered to belong to this class.
We assume that the system can be affected by any forces included in the set of the limited controls , u t U , i.e. the system is fully controllable [15]: The optimal control problem is to transfer system (1), (2) from the initial state where F q is a positive definite convex function of generalized coordinates.
This paper formulates the problem to construct a multi-mode control for the system (1), (2) subject to condition (3), (6) on the set of the quasi-optimal control laws obtained from the condition for the maximum of the generalized power function [10][11][12][13][14][15].

Constructing the multi-mode control law
A quasi-optimal control law, constructed using the condition for the maximum function of the generalized power, can be represented in the form [15]: , , where , s q q P is the synthesizing function, one of the requirements for which is having definite sign, taking into account the limitation of the control signal (3) , , , Then, taking into account the expression (17) and the condition of definite sign of the synthesizing function [15], we have on the switching line Therefore, on the switching line the velocities are limited by the kinematic constraints. When implementing the multi-mode control, to ensure the holding kinematic constraints it is required, according to the release principle, to apply the reaction forces of the constraints [16]

Mathematical simulation
Let us consider the object control problem The optimal solution for the performance problem in comparison with mode which is optimal mode with respect to performance. The behavior of the system on the phase plane and the control signal are shown in Figure 1. The obtained control law makes it possible to provide quasi-optimal nonlinear control modes close to optimal modes in terms of various efficiency criteria, for example, the performance laws and the Fuller laws. In addition, the proposed law not only ensures proximity to the optimal law, but also delivers some gain in energy costs for control. Moreover, on the basis of the proposed structure, a wide range of linear controllers can be implemented by choosing the parameters of the law.
The structure of the control law, synthesized based on the condition for the maximum of the generalized power function, allows the implementation of linear and nonlinear operating modes of the controller by changing the shape of the switching line in accordance with the selected quality criterion. So we can get results close to optimal with respect to various efficiency criteria while reducing control costs.
Further research is related to the need to take into account random actions and to analyze combinations of various control modes based on the condition for the maximum of the function of generalized power in accordance with the principles of fuzzy control.