Optimal Stabilization Control of an Inverted Pendulum with a Flywheel. Part 1
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International Applied Mechanics, Vol. 56, No. 4, July, 2020
OPTIMAL STABILIZATION CONTROL OF AN INVERTED PENDULUM WITH A FLYWHEEL. PART 1 V. S. Loveikin1, Yu. A. Romasevich1, and A. S. Khoroshun2
The problem of optimal control of stabilization of an inverted pendulum with a flywheel is stated. The criterion of the problem is represented in the form of a linear-quadratic integral functional. A control constraint is used as well. To solve the problem, the calculus of variations and numerical synthesis of the controller is used. For the first option, the problem is isoperimetric. Its solution is found, and an algorithm is developed for taking into account the change in the duration of the control of the system. For the second option, the initial problem is reduced to the minimization of a nonlinear function, which is solved by a modified particle swarm method. A brief comparative analysis of the results is carried out and the advantages and disadvantages of optimal controls are given. Keywords: inverted pendulum, optimal control, constraint, numerical optimization, controller Introduction. The problem of stabilizing a pendulum with a flywheel was first posed and solved in [13] and draws attention of many researchers. The problem allows us to evaluate the effectiveness of different methods of synthesis of motion controls for systems in which the number of degrees of freedom is greater than the number of controlled variables (controls). Such systems are quite common in robotics, load-lifting machines, aircraft, electric vehicles (segway, gyroboards), and so on. To solve the problem, various forms of controls have been proposed [3–6, 10, 14]. All of them are nonlinear functions of the phase coordinates of the system since these studies used a nonlinear model of an inverted pendulum with a flywheel. For the practical implementation of control, it is important to take into account the constraints on control and the phase coordinates of the system. The control constraints in the problem of stabilizing a pendulum with a flywheel were used in [12]. In [2], the Ricatti equation was obtained. The solution to the problem of optimal stabilization of the system follows from the Ricatti equation. The criterion is a linear quadratic integral functional that includes a vector of phase coordinates and control. The solution to the problem is obtained numerically. The problem of determining the coefficients of the optimal controller for stabilizing the system was solved in [9] using the particle swarm methods and genetic algorithm. From the point of view of the time required to solve the problem, the first method turned out to be more effective. In this study, we solve the problem of optimal control of the stabilization of an inverted pendulum with a flywheel. In the problem statement, an integral criterion and a control constraint are used, and the solution to the problem is obtained in the form of program control and feedback. 1. Flywheel Inverted Pendulum Model. Consider a system whose dynamic model is shown in Fig. 1. In this stud
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