Method of Synthesis of Algorithms for Optimal Control of Nonlinear Objects

The paper considers the problem of analytical design of optimal controllers (ADOC) in the Letov-Kalman formulation for stable single-channel high-order objects whose motion is described by a system of differential equations with continuous nonlinearities from the phase coordinates of the object with a linear entry of the control signal. The studied class of control objects is relatively wide for applications, for example, it includes most electromechanical devices. The proposed method for synthesizing optimal controllers for objects of the specified class is based on the use of a known optimal control algorithm for a first-order nonlinear object. For this purpose, the initial description of a high-order object is transformed to a conditionally equivalent model of a first-order object using the so-called aggregated variable (macrovariable) of the object (the terminology of A. A. Kolesnikov is used), which is a certain function of the state vector of the original object. For the adequacy of the object models, this function must satisfy the corresponding linear partial differential equation, the solution of which can be found by known methods. The admissible set of such functions determines a whole set of simply calculated, analytical control algorithms for the original object. Methods for determining the macrovariable are proposed, ensuring the stability of the closed control system and its optimality according to the corresponding quality functional. For linear subobjects of the class under consideration, it is established that the solution of the partial differential equation describing the conditional adequacy of object models is equivalent to solving the well-known problem of determining the eigenvalues and eigenvectors of the transposed matrix of the object model with its state vector. The conditionally adequate first-order model obtained by these standard matrix calculations ensures the optimality of a high-order control system according to the corresponding quadratic quality functional.