Abstract
A difficult but important problem in optimal control theory is the design of an optimal feedback control, i.e., the design of an optimal control as function of the phase (state) coordinates [1,2]. This problem can be solved not often. We study here the autonomous nonlinear system of second order in general form. The constraints imposed on the control input can depend on the phase (state) coordinates of the system. The goal of the control is to maximize or minimize one phase coordinate of the considered system while other takes a prescribed in advance value. In the literature, optimal control problems for the systems of second order are most frequently associated with driving both phase coordinates to a prescribed in advance state. In this statement of the problem, the optimal control feedback can be designed only for special kind of systems. In our statement of the problem, an optimal control can be designed as function of the state coordinates for more general kind of the systems. The problem of maximization or minimization of the swing amplitude is considered explicitly as an example. Simulation results are presented.
Highlights
Let the motion of the studied object under control be governed by a system of two nonlinear autonomous differential equations of the form x f1 x, y,u, y f2 x, y,u
A difficult but important problem in optimal control theory is the design of an optimal feedback control, i.e., the design of an optimal control as function of the phase coordinates [1,2]
We study here the autonomous nonlinear system of second order in general form
Summary
Let the motion of the studied object under control be governed by a system of two nonlinear autonomous differential equations of the form x f1 x, y,u , y f2 x, y,u ,. A controllable mechanical system with one degree of freedom is described by similar differential equations. In this case, x is positional coordinate and f1 x, y,u y (2). Let for each piecewise continuous vector function u(t) , system (1) with initial conditions from some region of the phase plane x, y has a unique solution x(t) , y(t). We assume that the control parameter u belongs to a given set U (x, y) depending on the state coordinates x and y. If set U (x, y) depends on state coordinates x, y, condition (3) can be checked for a given piecewise continuous control function u(t) , in general, only by finding the solution of the system (1) with this control. We formulate new assumptions during the problem consideration as need arises
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