V.F. Sokolov. Adaptive optimal stabilization of a discrete-time minimum-phase plant under output and input uncertainties ... P. 180-193

This paper addresses a problem of optimal robust, within the framework of the $\ell_1$-theory of robust control, stabilization of a discrete-time minimum-phase plant under large a priori uncertainty. A minimum-phase plant is a control system with stable zeros of the transfer function of the nominal model. The coefficients of the transfer function are assumed to be unknown and belonging to a known bounded polyhedron in the coefficient space. An upper bound of the external disturbance and the amplification factors of the uncertainties (disturbances) in the output and control are also assumed to be unknown. The performance index is the worst asymptotic absolute value of the output in the class of disturbances and uncertainties. The solution of the problem of adaptive optimal stabilization with a prescribed accuracy is based on the method of recurrent objective inequalities, the choice of the performance index of the control problem as the identification criterion, and the use of polyhedral estimates of all unknown parameters. The application of the method of recurrent objective inequalities provides the online verification of current estimates of the unknown parameters and a priori assumptions.

Keywords: adaptive control, optimal control, robust control, bounded disturbance, uncertainty

Received April 2, 2021

Revised May 19, 2021

Accepted May 24, 2021

Victor Fedorovich Sokolov, Dr. Phys.-Math. Sci., Institute of Physics and Mathematics of the Komi Science Center, Syktyvkar, 167982 Russia, e-mail: vfs-t@yandex.ru

REFERENCES

1.   Kosut R., Goodwin G., Polis M. Special issue on system identification for robust control design: Introduction. IEEE Trans. Automat. Control, 1992, vol. 37, no. 7, pp. 899–899.

2.   Ljung L., Vicino A. Guest editorial; special issue on system identification. IEEE Trans. Automat. Control, 2005, vol. 50, no. 10, pp. 1473–1473.

3.   Veres S.M. Bounding methods for state and parameter estimation. Int. J. Adaptive Control and Signal Processing, special iss., 2011, vol. 25, no. 3, pp. 189–190. doi: 10.1002/acs.1232 

4.   Casini M., Garulli A., Vicino A. A linear programming approach to online set membership parameter estimation for linear regression models. Int. J. Adaptive Control and Signal Processing, 2017, vol. 31, no. 3, pp. 360–378. doi: 10.1002/acs.2701 

5.   Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation under conditions of uncertainty]. Moscow: Nauka Publ., 1977, 392 p.

6.   Smith R.S., Dahleh M. (Eds.) The modeling of uncertainty in control systems, Ser. Lecture Notes in Control and Information Sciences, vol. 192, London, U.K.: Springer-Verlag, 1994, 391 p. ISBN: 0-387-19870-9 .

7.   Ljung L., Goodwin G., Aguero J.C. Model error modeling and stochastic embedding. IFAC-PapersOnLine, 2015, vol. 48, no. 28, pp. 75–79. doi: 10.1016/j.ifacol.2015.12.103 

8.   Delgado R.A., Goodwin G.C., Carvajal R., Aguero J.C. A novel approach to model error modelling using the expectation-maximization algorithm. In: IEEE 51st Conf. on Decision and Control (CDC), 2012, pp. 7327–7332. doi: 10.1109/CDC.2012.6426633 

9.   Lamnabhi-Lagarrigue F., Annaswamy A., Engell C., et al. Systems & Control for the future of humanity, research agenda: Current and future roles, impact and grand challenges. Annual Reviews in Control, 2017, vol. 43, pp. 1–64. doi: 10.1016/j.arcontrol.2017.04.001 

10.   Khammash M., Pearson J.B. Performance robustness of discrete-time systems with structured uncertainty. IEEE Trans. Automat. Control, 1991, vol. 36, no. 4, pp. 398–412. doi: 10.1109/9.75099 

11.   Khammash M., Pearson J.B. Analysis and design for robust performance with structured uncertainty. Systems & Control Letters, 1993, vol. 20, no. 3, pp. 179–187. doi: 10.1016/0167-6911(93)90059-F 

12.   Fomin V.N., Fradkov A.L., Yakubovich V.A. Adaptivnoe upravlenie dinamicheskimi sistemami [Adaptive control of dynamic systems]. Moscow: Nauka Publ., 1981, 448 p.

13.   Khammash M.H. Robust steady-state tracking. IEEE Trans. Automat. Control, 1995, vol. 40, no. 11, pp. 1872–1880. doi: 10.1109/9.471208 

14.   Khammash M.H. Robust performance: unknown disturbances and known fixed inputs. IEEE Trans. Automat. Control, 1997, vol. 42, no. 12, pp. 1730–1734. doi: 10.1109/9.650028 

15.   Sokolov V.F. Asymptotic robust performance of a discrete tracking system in the $\ell_1$-metric. Autom. Remote Control, 1999, vol. 60, no. 1, pp. 82–91.

16.   Sokolov V.F. Adaptive $\ell_1$ robust control for SISO system. Systems & Control Letters, 2001, vol. 42, no. 5, pp. 379–393. doi: 10.1016/S0167-6911(00)00110-9 

17.   Sokolov V.F. Closed-loop identification for the best asymptotic performance of adaptive robust control. Automatica, 1996, vol. 32, no. 8, pp. 1163–1176. doi: 10.1016/0005-1098(96)00044-1 

18.   Picasso B., Colaneri P. Non-minimal factorization approach to the $\ell_\infty$-gain of discrete-time linear systems. Automatica, 2013, vol. 49, no. 9, pp. 2867–2873. doi: 10.1016/j.automatica.2013.06.003 

19.   Sanchez-Pena R.S., Sznaier M. Robust systems theory and applications. John Wiley & Sons, Inc. 1998, 486 p. ISBN: 978-0-471-17627-5 .

Cite this article as: V.F. Sokolov. Adaptive optimal stabilization of a discrete-time minimum-phase plant under output and input uncertainties, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 180–193.