Cybernetics and Computer Engineering, 2019, 3(197), pp.
Zhiteckii L.S.1, PhD (Engineering),
Acting Head of the Intelligent Automatic Systems Department
Azarskov V.N.2, DSc. (Engineering), Professor,
Chief of the Aerospace Control Systems Department,
Assistant of the Department of Computer Information Technologies and Systems
1International Research and Training Center for Information Technologies
and Systems of the National Academy of Sciences of Ukraine
and Ministry of Education and Science of Ukraine,
40, Acad. Glushkov av., Kyiv, 03187, Ukraine
2National Aviation University, Kyiv, Ukraine.
1, Kosm. Komarova av., Kyiv, 03680, Ukraine
3Poltava National Technical Yuri Kondratyuk University, Poltava, Ukraine.
24, Pershotravneva av., Poltava, 36011, Ukraine
SOLVING A PROBLEM OF ADAPTIVE STABILIZATION FOR SOME STATIC MIMO SYSTEMS
Introduction. The adaptive stabilization of some classes of uncertain multivariable static plants with arbitrary unmeasurable bounded disturbances is addressed in this article. The cases where the number of the control inputs does not exceed the number of the outputs are studied. It is assumed that the plant parameters defining the elements of its gain matrix are unknown. Again, the rank of this matrix may be arbitrary. Meanwhile, bounds on external disturbances are supposed to be known. The problem stated and solved in this work is to design adaptive controllers to be able to ensure the boundedness of the all input and output system’s signals in the presence of parameter uncertainties.
The purpose of this article is to show that it is possible to stabilize any uncertain multivariable static plant which gain matrix may be either square or nonsquare and may have an arbitrary rank remaining unknown for the designer.
Methods. The methods based on recursive point estimation of unknown plant parameters are utilized to design the adaptive inverse model-based controller.
Results. The asymptotic properties of the adaptive controllers have been established. Simulation results have been presented to support the theoretic studies.
Conclusion. The adaptive control laws proposed in this article can guarantee the boundedness of all the signals generated by the feedback control systems. However, this important feature will achieve via an “overparameterization” of these systems. Nevertheless, the simulation experiments demonstrate their efficiency.
Keywords: adaptive control, boundedness, discrete time, estimation algorithm, feedback, multivariable system, uncertainty.
- Maciejowski J. M. Multivariable Feedback Design. Wokinghan: Addison-Wesley, 1989.
- Skogestad S., Postlethwaite I. Multivariable Feedback Control. UK, Chichester: Wiley, 1996.
- Albertos P., Sala A. Multivariable Control Systems: An Engineering Approach. London: Springer, 2006.
- Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Operator Method. Kiev: Nauk. dumka, 1966. (in Russian).
- Lyubchyk L. M. Disturbance rejection in linear discrete multivariable systems: inverse model approach. Prep. 18th IFAC World Congress (2011, Milano, Italy). Milano, 2011, pp. 7921–7926.
- Skurikhin V. I., Gritsenko V. I., Zhiteckii L. S., Solovchuk K. Yu. Generalized inverse operator method in the problem of optimal controlling linear interconnected static plants. Dopovidi NAN Ukrainy. 2014. no. 8, pp. 57–66. (in Russian).
- Polyak, B.T., Shcherbakov, P. S. Robust Stability and Control. Moscow: Nauka, 2002. (in Russian).
- Kuntsevich V. M. Control under Uncertainty: Guaranteed Results in Control and Identification Problems. Kyiv: Nauk. dumka, 2006. (in Russian).
- Sokolov V.F. Robust Control with Bounded Disturbances. Syktyvkar: Komi Scientific Center, Ural Branch of the RAS, 2011. (in Russian).
- Zhiteckii L. S., Solovchuk K. Yu. Pseudoinversion in the problems of robust stabilizing multivariable discrete-time control systems of linear and nonlinear static objects under bounded disturbances. Journal of Automation and Information Sciences. 2017. vol. 49. no. 5, pp. 35–48.
- Zhitetskii L. S., Skurikhin V. I., Solovchuk K. Yu. Stabilization of a nonlinear multivariable discrete-time time-invariant plant with uncertainty on a linear pseudoinverse model. Journal of Computer and Systems Sciences International. 2017. vol. 56, no. 5, pp. 759–773.
- Zhiteckii L. S., Azarskov V. N., Solovchuk K. Yu., Sushchenko O. A. Discrete-time robust steady-state control of nonlinear multivariable systems: a unified approach. Proc. 19th IFAC World Congress. (2014, Cape Town, South Africa). Cape Town, 2014, pp. 8140–8145.
- Bunich A.L. On some nonstandard problems of the synthesis of discrete systems. Autom. Remote Control. 2000. no. 6, pp. 994–1002.
- Fomin V. N., Fradkov A. L., Yakubovich V. A. Adaptive Control of Dynamic Plants. Moscow: Nauka, 1981. (in Russian).
- Goodwin G.C., Sin K.S. Adaptive Filtering, Prediction and Control. Engewood Cliffs, NJ.: Prentice-Hall, 1984.
- Landau I. D., Lozano R., M’Saad M. Adaptive Control. London: Springer, 1997.
- Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Nonparametric Uncertainties. Kyiv: Nauk. dumka, 2010. (in Russian).
- Narendra K. S., Annaswamy A. M. Stable Adaptive Systems. NY: Dover Publications, 2012.
- Ioannou P., Sun J. Robust Adaptive Control. NY: Dover Publications, 2013.
- Aström K. J., Wittenmark B. Adaptive Control: 2nd Edition. NY: Dover Publications, 2014.
- Bakan G.M., Volosov V.V., Salnikov N.N. Adaptive control of a linear static plant by a model with unknown parameters. Kibernetika. 1984, no. 2, pp. 63–68.
- Lublinskii B.S., Fradkov A.L. Adaptive control of nonlinear statistical processes with an implicit characteristic. Autom. Remote Control. 1983, no. 4, pp. 510–518.
- Bakan G.M. Adaptive control of a multi-dimensional static process under nonstatistical uncertainty. Autom. Remote Control. 1987, no. 1, pp. 76–88.
- Zhiteckii L. S., Solovchuk K. Yu. Adaptive stabilization of some multivariable systems with nonsquare gain matrices of full rank. Cybernetics and Computer Engineering. 2018, no. 2, pp. 44–61.
- Zhiteckii L. S., Solovchuk K. Yu. Robust adaptive pseudoinverse model-based control of an uncertain SIMO memoryless system with bounded disturbances. Proc. IEEE 2nd Ukraine Conference on Electrical and Computer Engineering (UKRCON-2019), Lviv, Ukraine, 2019, pp. 628–633.
- Azarskov V.N., Zhiteckii L.S., Solovchuk K.Yu. Parametric identification of the interconnected static closed-loop system: a special case. Proc. 12th All-Russian Control Problems Council (VSPU-2014), Moscow, 2014, pp. 2764-2776.
- Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu. Adaptive robust control of inter-connected static plants with nonsquare gain matrixes. Proc. 13th All-Russian Control Problems Council (VSPU-2019), Moscow, 2019.
- Anderson B.D.O., Bitmead R.R., Johnson C.R., Kokotovic P.V., Kosut R.L., Mareels I.M.Y., Praly L., and Riedle B.D. Stability of Adaptive Systems: Passivity and Averaging Analysis. USA, Mas.: MIT Press. 1986.
- Marcus M., Minc H. A Survey of Matrix Theory and Matrix Inequalities. Boston: Aliyn and Bacon, 1964.