Yu.P. Kornushin1

1 Kaluga Branch of Bauman Moscow State Technical University (Kaluga, Russia)

Formulation of the problem. The problem of synthesis of quasi-optimal controllers for tracking systems with nonlinear objects is considered. Limits are imposed on control on the energy spent on its formation. The conditions for solving the problem in the presence of a constraint are determined - the condition of triviality and the condition of controllability. The quadratic criterion of optimality is used. The main mathematical apparatus is the Newton-Kantorovich linearization and the method of matrix operators.

Target. Design a regulator that forms in the system such a control action on a nonlinear object that ensures a change in its phase coordinates in accordance with an arbitrarily changing signal. In this case, the control must have a limitation on the expended resources (energy) throughout the entire tracking interval.

Results. A new method is proposed for the mathematical design of controllers of tracking systems, which include control objects, the mathematical models of which are described by nonlinear differential equations of the affine type. The result is regulators that allow control systems to monitor randomly varying signals arriving at its input. A feature of the regulators is that the control they generate is limited in energy. To achieve this goal, the following actions are sequentially performed (synthesis algorithm):

1. The conditions for solving the problem in the presence of a constraint are determined - the condition of controllability and the condition of triviality.

2. The linearization of the nonlinear mathematical model of the object is carried out according to the Newton - Kantorovich scheme.

3. Parametrization of the linearized mathematical model of the control object, quality functional and limitation is performed using the apparatus of matrix operators.

4. Assumptions on the constraint are accepted, which allow constructing the Lagrange function for the subsequent finding of the optimal program control.

5. An optimal program control is constructed according to the necessary condition for the minimum of the Lagrange function.

6. The critical values of the limitation are determined.

7. An algorithm for the functioning of the regulator of the tracking system is formed.

Practical significance. The proposed algorithm for the synthesis of regulators can be used in the design of servo drives for antenna installations and radio telescopes, computer vision systems, guidance and orientation systems for scientific instruments, weapons systems, electric power systems, systems for observing moving objects, in motion and navigation systems, etc.

Kornushin Yu.P. Synthesis of quasi-optimal controllers in the tracking problem for nonlinear plants with control constraints using the method of matrix operators. Nonlinear World. 2022. V. 20. № 1. P. 42-49. DOI: https://doi.org/10.18127/j20700970-202201-04 (In Russian)

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