Abstract:

The efficiency of the automatic control system of any technical device is largely determined by the ability to ensure the sustainable implementation of a given mode of operation. A rigorous solution of problems of stability can be obtained by the method of Lyapunov functions. The simplest case of stability is asymptotic stability in the first approximation. In the technical practice of controlled systems, one of the most effective methods to ensure the required stability is associated with solving the analytical design problem for optimal regulators (ACOR) formulated by A. Letov: to determine the linear control coefficients, which ensures the asymptotic stability in the first approximation of a given motion and minimizes this is some quality criterion. It was proposed to consider this task as a task of optimal control. The methods for solving optimal control problems are the Pontryagin maximum principle and the Bellman dynamic programming method. Krasovsky N.N. first noted the relationship of the dynamic programming method and the method of Lyapunov functions and proved the optimal stabilization theorem. He developed a practical method for determining the stabilizing control coefficients from solving a linear-quadratic problem for the linear system, which from the equations of perturbed motion is selected. The values of coefficients can be found from algebraic equations obtained from the Lyapunov-Bellman equation. Determining their numerical values is a rather complicated task, since in the general case the equations can only be solved numerically. The method of numerical determination of stabilizing control coefficients was justified by passing from algebraic equations to differential equations. This method was used to develop original software products that implement this approach in symbolic information processing software environments (Maple,2006, Python, 2011). Using this software, a study was made of the effect of wheel deformability on the dynamics of a mobile robot with incomplete information about the state. These software products were not brought to the user level and their use required quite high qualifications, and therefore their use in the educational process or in technical practice created certain difficulties.

From this disadvantage, there is a free method for unambiguously determining the stabilizing control coefficients for solving linear-quadratic problems available in the software package MatLab, the corresponding program module [x, l, g] = care (a, b, q, r), which calculates solution of the matrix algebraic Riccati equation. If the first approximation in the perturbed motion equations of a particular controlled object is highlighted, the solution to the problem of stabilizing a given motion using the care module can be obtained quite simply. Opportunities MatLab allow you to easily display graphs of transients in a closed system control found. Therefore, this method of determining stabilizing control and visualizing the results of its introduction into the resulting control action can be applied both by practical engineers and undergraduate students, which is especially important for the training and retraining of specialists in automation technological processes of food processing.

Keywords:

Automatic control system, sustainable implementation, ACOR, MatLab, numerical determination of stabilizing.