Kibern. vyčisl. teh., 2018, Issue 2 (192), pp.
International Research and Training Center for Information Technologies
and Systems of the National Academy of Science of Ukraine
and Ministry of Education and Sciences of Ukraine, Kiev, Ukraine,
Acad. Glushkova av., 40, Kiev, 03187, Ukraine
ADAPTIVE STABILIZATION OF SOME MULTIVARIABLE SYSTEMS WITH NONSQUARE GAIN MATRICES OF FULL RANK
Introduction. The paper states and solves a new problem concerning the adaptive stabilization of a specific class of linear multivariable discrete-time memoryless systems with nonsquare gain matrices at their equilibrium states. This class includes the multivariable systems in which the number of outputs exceeds the number of control inputs. It is assumed that the unknown gain matrices have full rank.
The purpose of this paper is to answer the question of how the pseudoinverse model-based adaptive approach might be utilized to deal with the uncertain multivariable memoryless system if the number of control inputs is less than the number of outputs.
Results. It is shown that the parameter estimates generated by the standard adaptive projection recursive procedure converge always to some finite values for any initial values of system’s parameters. Based on these ultimate features, it is proved that the adaptive pseudoinverse model-based control law makes it possible to achieve the equilibrium state of the nonsquare system to be controlled. The asymptotical properties of the adaptive feedback control system derived theoretically are substantiated by a simulation experiment.
Conclusion. It is established that the ultimate behavior of the closed-loop control system utilizing the adaptive pseudoinverse model-based concept is satisfactory.
Keywords: adaptive control, multivariable system, discrete time, feedback, pseudoinversion, stability, uncertainty.
1 Dahleh M.A., Pearson J.B. l1 optimal-feedback controllers for MIMO discrete-time systems. IEEE Trans. Autom. Contr., 1987, vol. 32, no. 4, pp. 314–322. https://doi.org/10.1109/TAC.1987.1104603
2 McDonald J.S., Pearson J.B. l1 optimal control of multivariable systems with output norm constraints. Automatica, 1991, vol. 27, no. 2, pp. 317–329. https://doi.org/10.1016/0005-1098(91)90080-L
3 Maciejowski J. M. Multivariable Feedback Design. Wokinghan: Addison-Wesley, 1989.
4 Skogestad S., Postlethwaite I. Multivariable Feedback Control. UK, Chichester: Wiley, 1996.
5 Albertos P., Sala A. Multivariable Control Systems: an Engineering Approach. London: Springer, 2006.
6 Francis B., Wonham W. The internal model principle of control theory. Automatica, 1976, vol. 12, no. 5, pp. 457–465. https://doi.org/10.1016/0005-1098(76)90006-6
7 Brockett R. W. The invertibility of dynamic systems with application to control. Ph. D. Dissertation, Case Inst. of Technology, Cleveland, Ohio, 1963.
8 Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom. Contr., 1969, vol. AC-14, no. 3, pp. 270–276. https://doi.org/10.1109/TAC.1969.1099169
9 Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized inversion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5, pp. 733–739. https://doi.org/10.1080/00207177608932859
10 Seraji H. Minimal inversion, command tracking and disturbance decoupling in multivariable systems. Int. J. Control, 1989, vol. 49, no. 6, pp. 2093–2191. https://doi.org/10.1080/00207178908559765
11 Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of multivariable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no. 10, pp. 1695–1703. https://doi.org/10.1016/S0005-1098(02)00088-2
12 Liu C., Peng H. Inverse-dynamics based state and disturbance observers for linear time-invariant systems. ASME J. Dyn Syst., Meas. and Control, 2002, vol. 124, no. 5, pp. 376–381.
13 Lyubchyk L. M. Disturbance rejection in linear discrete multivariable systems: inverse model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–7926.
14 Pushkov S. G. Inversion of linear systems on the basis of state space realization. Journal of Computer and Systems Sciences International, 2018, vol. 57, vo. 1, pp. 7–17.
15 Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Operator Method. Kiev: Nauk. dumka, 1966 (in Russian).
16 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, no. 8, pp. 57–66, 2014 (in Russian).
17 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, Cape Town, South Africa, 2014, pp. 8140–8145.
18 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. https://doi.org/10.1134/S1064230717040189
19 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. https://doi.org/10.1615/JAutomatInfScien.v49.i5.30
20 Fomin V. N., Fradkov A. L., Yakubovich V. A. Adaptive Control of Dynamic Plants. Moscow: Nauka, 1981 (in Russian).
21 Goodwin G.C., Sin K.S. Adaptive Filtering, Prediction and Control. Engewood Cliffs, NJ.: Prentice-Hall, 1984.
22 Landau I. D., Lozano R., M’Saad M. Adaptive Control. London: Springer, 1997.
23 Kuntsevich V. M. Control under Uncertainty: Guaranteed Results in Control and Identification Problems. Kiev: Nauk. dumka, 2006 (in Russian).
24 Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Nonparametric Uncertainties. Kiev: Nauk. dumka, 2010 (in Russian).
25 Narendra K. S., Annaswamy A. M. Stable Adaptive Systems. NY: Dover Publications, 2012.
26 Ioannou P., Sun J. Robust Adaptive Control. NY: Dover Publications, 2013.
27 Astrom K. J., Wittenmark B. Adaptive Control: 2nd Edition. NY: Dover Publications, 2014.
28 Albert A. Regression and the Moore-Penrose Pseudoinverse. New York: Academic Press, 1972.
29 Kaczmarz S. Approximate solution of systems of linear equations. Internat. J. Control, 1993. vol. 57, no. 6. pp. 1269–1271. https://doi.org/10.1080/00207179308934446
30 Marcus M., Minc H. A Survey of Matrix Theory and Matrix Inequalities. Boston: Aliyn and Bacon, 1964.
31 Desoer C.A., Vidyasagar M. Feedback Systems: Input–Output Properties. New York: Elsevier, 1975.