The Gradient-Based Algorithm for Parametric Optimization of a Variable Structure PI Controller with Dead Band

In the automatic system, the presence of an object with a delay that exceeds the value of the maximum time parameter of the controlled object reduces the operating quality of generic controllers (integral, proportionally-integral, proportionally-integral-differential). The occurrence of this kind of delay in the system requires addressing a particular class of regulators that compensate for the negative effects of the delay. This paper examines the PI controller known for its advantages with variable or switchable parameters, which belongs to the class of controllers with variable structure (henceforward — VSC) that do not use sliding mode. Due to the fact that the controller used contains switchable parameters and the object with delay is considered, it is extremely difficult to use analytical approaches to parametric optimization of the system. This lays one under a necessity to use algorithmic methods. This work employs a gradient-based algorithm in which the components of the gradient are calculated using sensitivity functions with their known advantages. The generated Automatic Parametric Optimization (APO) Algorithm calculated the optimal VSC parameters for a given object, based on the minimum of the integrated quadratic criterion. The reliability of the found vector of the controller setting, formed by the APO algorithm, is confirmed by the computational methodology. With accuracy sufficient for practice, the APO algorithm solved the problem of parametric optimization. The positive experience of optimizing the PI controllers with variable parameters allows one to apply it to other VSC, which do not use a sliding mode, and thus further expand the practice of using a gradient-based algorithm based on sensitivity functions for such a class of VSC under various laws of switching structures of the controller.

1. Guretsky H. Analysis and synthesis of control systems with delay. Transl. from Polish, Мoscow, Mashinostroenie Publ., 1974, 328 p. (in Russian).

2. Yanushevsky R. T. Controlling objects with delay. Series "Theoretical Foundations of Technical Cybernetics", Moscow, Nauka Publ., 1978, 416 p. (in Russian).

3. Denisenko V. V. Computer control of the technology process, experiment, equipment, Мoscow, Goryachaya liniya — Telecom Publ., 2009, 608 p. (in Russian).

4. Govorov A. A. Methods and construction tools for controllers with advanced functionality for continuous technology processes: Dr. Sc. (Engineering) diss: 05.13.06: defended on 15.11.02. Moscow, 2002, 499 p. (in Russian).

5. Shigin E. K. Automatic control of an object with pure delay by a controller with switchable parameters II, Automation and Telemechanics, 1966, no. 6, pp. 72—81.

6. Åström K. J., Hägglund T. The future of PID control, Control Engineering Practice, 2001, vol. 9, iss. 11, pp. 1163—1175.

7. Ramírez A., Mondié S., Garrido R. Proportional integral retarded control of second order linear systems, 2013 IEEE 52nd Annual Conference on Decision and Control (CDC), pp. 2239—2244.

8. Ramírez A., Garrido R., Mondié S. Integral Retarded Control Velocity Control of DC Servomotors, In IFAC TDS Workshop, Grenoble, France, 2013, pp. 558—563.

9. Suh I. H., Bien Z. Proportional Minus Delay Controller, IEEE Transactions on Automatic Control, 1979, vol. 24, pp. 370—372.

10. Villafuerte R., Mondié S., Garrido R. Tuning of Proportional Retarded Controllers: Theory and experiments, IEEE Trans-actions on Control Systems Technology, May, 2013.

11. Ramírez A., Mondié S., Garrido R. Integral retarded velocity control of dc servomotors, 11th IFAC Workshop on TimeDelay Systems, 46 (3), pp. 558—563.

12. Ramírez A., Mondié S., Garrido R. Velocity control of servo systems using an integral retarded algorithm, ISA Transactions 58, pp. 357—366.

13. Arousi F., Schmitz U., Bars R., Haber R. PI controller based on first-order dead time model, Proceedings of the 17th World Congress, Seoul, Korea, July 6—11, 2008, pp. 5808—5813.

14. Airikka P. Extended predictive proportional-integral controller for typical industrial processes, 18th IFAC World Cogress, Milano, Italy, August 28 — July 2, 2011, pp. 7571—7576.

15. Larsson P., Hagglund T. Comparison between robust PID and predictive PI controllers with constrained control signal noise sensitivity. 2nd IFAC Control Conference on Advances on PID Control, Brescia, Italy, pp. 175—180, March 28, 2012.

16. Kutsyi A. P., Kutsyi N. N., Malanova T. V. Determination of the Area of Robust Stability of the System on the Basis of V. L. Kharitonov’s Theorem, Mekhatronika, Avtomatizatsiya, Upravlenie, 2020, vol. 21, no. 4, pp. 208—212 (in Russian).

17. Kutsyy N. N., Malanova T. V., Kutsyy A. P. Synthesis of low-sensitivity systems, Transport infrastructure of the Siberian region, 2019, vol. 1, pp. 350—355 (in Russian).

18. Airikka P. Robust predictive PI controller tuning, 19th World Congress, IFAC, Cape Town, South Africa, August 24—29, 2014, pp. 9301—9306.

19. Kutsyi N. N. Automatic parametric optimization of discrete control systems: Dr. Sc. (Engineering) diss.: 05.13.06: defended on 11/26/97, Irkutsk, 1997, 382 p. (in Russian).

20. Kutsyi N. N., Malanova T. V. Optimization of automatic systems with pulse-width modulation with parametric mismatch, Mathematical modeling and information technology: the materials of the IX seminar school, Irkutsk, coll. of research papers. ISDCT SB RAS, 2007, pp. 97—101 (in Russian).

21. Gorodetsky V. I., Zakharin F. M., Rosenwasser E. N., Yusupov R. M. Methods of the theory of sensitivity in automatic control, Мoscow, Energoizdat Publ., 1971, 343 p. (in Russian).

22. Kostyuk V. I., Shirokov L. A. Automatic parametric optimization of control systems, Мoscow, Energoizdat Publ., 1981, 96 p. (in Russian).