Method for Synthesis of Quasi-Optimal Regulators Based on Multidimensional Linearization

The paper considers the problem of analytical design of optimal regulators (ADOR) in the formulation of A. A. Krasovsky for the studied stable multidimensional objects described by a matrix differential equation with polynomial nonlinearities in phase coordinates. This class of control objects, called polynomial, is quite wide for applications: these models are used to describe the movement of systems of a very different nature, for example, electromechanical devices, chemical reactors, industrial facilities with recycling, biological and environmental systems, etc. To solve this problem of ADOR in the early work of the author, a method is proposed for synthesizing quasi-optimal controllers, which largely weakens the disadvantages of the power series method (a large volume of operations with polynomials that are not suitable for programming) through the use of a procedure for multidimensional linearization of the description of polynomial objects, carried out by expanding the space state of the object with new coordinates, which are products of the original phase coordinates, and the application of the apparatus of matrix theory with the Kronecker (direct) product. In this paper, a modification of this method of system synthesis is carried out by using a reduced rather than a full Kronecker degree of the object state vector and taking into account the block-diagonal structure of the parameter matrix of the applied linearized model of the object, which ensures a further multiple reduction in the volume of calculations in the synthesis of control systems. The proposed modified synthesis method makes it possible to find, in the form of a polynomial function, an approximate solution to the ADOR problem with high accuracy, and its implementation is extremely simple due to the use of mainly well-known software for solving linear-quadratic optimal control problems (procedures for solving Lyapunov and Sylvester matrix equations).

The features of the method are illustrated using a specific example of the synthesis of a quasi-optimal control system. 

1. Krasovskii A. A. et al. Handbook on the theory of automatic control, Moscow, Nauka, 1987, 712 p (in Russian).

2. Kolesnikov A. A. ed. Modern Applied Control Theory, Taganrog, Publishing house, TRTU, 2000 (in Russian).

3. Porter W. A. The Review of the Non-linear Systems Theory, TIIR, 1976, vol. 64, no. 1, pp. 23—30 (in Russian).

4. Sage A. P., White C. C. Optimum Systems Control, Moscow, Radio i svjaz’, 1982, 392 p (in Russian).

5. Afanasyev V. N., Kolmanovsky V. B., Nosov V. R. Mathematical Theory of Control Systems Design, Moscow, Vysshaja shkola, 1998, 576 p. (in Russian).

6. Krasovskii A. A., Bukov V. N., Shendrik V. S. Universal algorithms for optimal control of continuous objects, Moscow, Nauka, 1977, 272 p. (in Russian).

7. Bukov V. N. Adaptive predictive flight control systems, Moscow, Nauka, 1987, 232 p. (in Russian).

8. Kawasaki N., Kobayashi H., Shimemura E. Relation between pole assignment and LQ-regulator, Int. J. Contr., 1998, vol. 47, no. 4, pp. 947—951.

9. Filimonov N. B. The problem of the quality of management processes: a change in the optimization paradigm, Mekhatronika, Avtomatizatsiia, Upravlenie, 2010, no.12, pp. 2—10 (in Russian).

10. Lovchakov V. I., Lovchakov E. V., Suhinin B. V. Linearization Method of the Polynomial Control Systems, Izvestija TulGU, Seriya Tehnicheskie nauki, 2009, no. 1, Part 2, pp. 18—26 (in Russian).

11. Lovchakov V. I. Application of Multidimensional Linearization in Quasi-optimal Controllers Synthesis in the Functional of Generalized Work, Mekhatronika, avtomatizatsiia, upravlenie, 2019, vol. 20, no.3, pp. 131—142 (in Russian).

12. Lankaster P. Theory of matrices, Moscow, Nauka, 1982, 269 p. (in Russian).

13. Jelali M., Kroll A. Hydraulic servo-systems: Modelling, identification and control, Springer, 2002, 356 p.

14. Carravetta F., Germani A., Raimondi M. Polynomial Filtering for Linear Discrete Time Non-Gaussian Systems, SIAM Journal on Control and Optimization, 1996, vol. 34, no. 5, pp. 1666—1690.

15. Voevodin V. V., Kuznetsov Yu. A. Matrices and calculations, Moscow, Nauka, 1984, 320 p. (in Russian).