Homotopy Method in Applied Problems of the Anisotropic Control Theory

The work describes a numerical method of solving the specific systems of matrix equations emerging in the tasks of the modern theory of control. Since the standard tasks of the control theory demand making a number of assumptions about input effect, at the slightest non-compliance the synthesized laws of control become either extremely inefficient or too much power consumable. As opposed to these assumptions, while setting the problem of anisotropic theory of control, it is necessary to know only the average anisotropy level of the input sequence. Consequently, anisotropic regulators are always found to be no worse than standard ones. In synthesis of anisotropic regulator a rather complex algorithm of its construction is the only difficulty. When considering a problem of ensuring robust quality of the control object in case of the structured uncertainty there is a need to solve a system of four connected Riccati equations, equation of a special form, and Lyapunov equation. To solve it by standard methods of convex optimization is impossible. The work shows how the standard mean square Gaussian regulator allows us to obtain as anisotropic regulator to meet requirements of robust quality when there is an imperfect knowledge of mathematical model of object of control, a lack of exact stochastic characteristics of the input control, parametrical uncertainty, etc. The article offers an algorithm based on the homotopy method with the Newtonian iterations to solve a problem of anisotropic optimization. It presents a computing procedure to reach the objective. Using a task of searching the anisotropic regulator to minimize the maximum value of anisotropic norm of transfer function of the control object, the article describes required matrix derivatives of stabilizing solutions of Riccati equations, equation of a special form, and Lyapunov equation. Properties of Kronecker product and matrix differentiation with respect to matrix are given