Stability Analysis of a Class of Nonlinear Switched Systems with Distributed Delay

One of the actual problems of modern control theory is the study of the stability of systems with switching modes of operation. Such systems are widely used for modeling technological processes, automatic control systems, mechatronic systems, etc. In recent years, they have been effectively used in problems of controlling formations of mobile agents. The main approach to solving this problem is the Lyapunov direct method. For families of subsystems corresponding to the studied hybrid systems, either common or multiple Lyapunov functions are constructed. However, the methods and algorithms for finding such functions are well developed only for linear systems. The problem of analyzing the stability of nonlinear systems with switching has not been sufficiently investigated. It should be noted that its solution is significantly complicated in the case when the systems under consideration contain a delay. In this paper, we study a class of complex systems with switching and distributed delay, modeling the interaction of linear and nonlinear homogeneous subsystems. These systems correspond to the Lyapunov critical case of several zero eigenvalues of the matrix of the linearized system. Note also that under certain additional restrictions on nonlinearities, they are Lurie indirect control systems. The presence of distributed delay can be due to the use of PID controllers. For the considered hybrid systems, new approaches to constructing Lyapunov—Razumikhin functions and Lyapunov—Krasovsky functionals are proposed, which guarantee stability under any switching law. In the case when such functions and functionals cannot be selected, using multiple functionals, classes of switching laws are determined under which stability is preserved. The efficiency of the developed approaches is demonstrated on the example of a controlled mechanical system with a PID controller. 

1. Vassilyev S. N., Kosov A. A. Analysis of hybrid systems’ dynamics using the common Lyapunov functions and multiple homomorphisms, Avtomatica i Telemechanica, 2011, no. 6, pp. 27—46 (in Russian).

2. Liberzon D. Switching in Systems and Control, Boston, MA, Birkhauser, 2003.

3. Moradvandi A., Lindeboom R. E. F., Abraham T., De Schutter B. Models and methods for hybrid system identification: a systematic survey, IFAC-PapersOnLine, 2023, vol. 56, no. 2, pp. 95—107.

4. Eremin E. L., Smirnova C. A., Shelenok E. A. Structural and parametric synthesis of a hybrid repetitive control system for uncertain multi-mode plant, Mekhatronika, Avtomatizatsiya, Upravlenie, 2024, vol. 25, no. 9, pp. 447—457 (in Russian).

5. Shorten R., Wirth F., Mason O., Wulf K., King C. Stability criteria for switched and hybrid systems, SIAM Rev. 2007, vol. 49, no. 4, pp. 545—592.

6. Aleksandrov A. Yu., Kosov A. A., Platonov A. V. On the asymptotic stability of switched homogeneous systems, Systems Control Letters, 2012, vol. 61, no. 1, pp. 127—133.

7. Aleksandrov A. Yu., Kosov A. A., Chen Y. Stability and stabilization of mechanical systems with switching, Avtomatica i Telemechanica, 2011, no. 6, pp. 5—17 (in Russian).

8. Aleksandrov A., Andriyanova N., Efimov D. Stability analysis of Persidskii time-delay systems with synchronous and asynchronous switching, Int. J. Robust Nonlinear Control, 2022, vol. 32, no. 6, pp. 3266—3280.

9. Fridman E. Introduction to Time-delay Systems: Analysis and Control, Basel, Birkhauser, 2014.

10. Zubov V. I. Stability of Motion, Moscow, Vysshaya Shkola, 1973 (in Russian). 11. Lefschetz S. Stability of Nonlinear Control Systems, New York, Academic Press, 1965.

11. Kosov A. A., Kozlov M. V. On the asymptotic stability of homogeneous singular systems with switchings, Avtomatica i Telemechanica, 2019, no. 3, pp. 45—54 (in Russian).

12. Aleksandrov A. Y. On the asymptotic stability and ultimate boundedness of solutions of a class of nonlinear systems with delay, Differentsial’nye Uravneniya, 2023, vol. 59, no. 4, pp. 435—445 (in Russian).

13. Formal’sky A. M. On a modification of the PID controller, Dynamics and Control, 1997, vol. 7, no. 3, pp. 269—277.

14. Anan’evskii I. M., Kolmanovskii V. B. On stabilization of some control systems with an aftereffect, Avtomatica i Telemechanica, 1989, no. 9, pp. 34—43 (in Russian).

15. Zhao C., Guo L. Towards a theoretical foundation of PID control for uncertain nonlinear systems, Automatica, 2022, vol. 142, pp. 110360.

16. Kulikov V. V., Kutsyi N. N., Osipova E. A. Parametric optimization of the PID controller with restriction based on the method of conjugate Polak—Polyak—Ribier gradients, Mekhatronika, Avtomatizatsiya, Upravlenie, 2023, vol. 24, no. 5, pp. 240—248 (in Russian).

17. Aleksandrov A., Efimov D., Fridman E. Stability of homogeneous systems with distributed delay and time-varying perturbations, Automatica, 2023, vol. 153, pp. 111058.

18. Rosier L. Homogeneous Lyapunov function for homogeneous continuous vector field, Systems Control Letters, 1992, vol. 19, pp. 467—473.