Functions’ algebra in nonlinear control: computational aspects and software; pp. 89–107Full article in PDF format
The paper describes the Mathematica-based software for studying nonlinear control systems. The software relies on an algebraic method, called functions’ algebra. The advantage of this approach, over well-known linear algebraic and differential geometric methods is that it is applicable to certain non-smooth systems. The drawback is that the computations are more complicated since the approach manipulates directly with the functions related to the system equations and not with the differential one-forms/vector fields that simplify (linearize) the computations. We have implemented the basic operations of functions’ algebra, i.e., partial order, equivalence, summation, and multiplication, but also finding the simplest epresentative of an equivalence class. The next group of functions is related to the control system and involves binary relation, operators m, M, and computation of certain sequences of invariant vector functions on the basis of system equations. Finally, we have developed Mathematica functions, allowing us to solve the following control problems in case of discrete-time systems: checking accessibility, static state feedback linearization, and disturbance decoupling.
1. Aranda-Bricaire, E., Kotta, Ü., and Moog, C. H. Linearization of discrete-time systems. SIAM J. Contr. Optim., 1996, 34(6), 1999–2023.
2. Belikov, J., Kaparin V., Kotta, Ü., and Tõnso, M. NLControl website. Online, www.nlcontrol.ioc.ee, 2016 (accessed 20 June 2016).
3. Conte, G., Moog, C. H., and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Springer, London, 2007.
4. Hartmanis, J. and Stearns, R. E. The Algebraic Structure Theory of Sequential Machines. Prentice-Hall, New York, 1966.
5. Isidori, A. Nonlinear Control Systems. Springer, London, 1995.
6. Kaldmäe, A., Kotta, Ü., Shumsky, A. Ye., and Zhirabok, A. N. Measurement feedback disturbance decoupling in discrete-time nonlinear systems. Automatica, 2013, 49(9), 2887–2891.
7. Kaldmäe, A., Kotta, Ü., Shumsky, A. Ye., and Zhirabok, A. N. Disturbance decoupling in nonlinear hybrid systems. In 12th IEEE International Conference on Control & Automation (ICCA), Kathmandu, Nepal. 2016, 86–91.
8. Kaparin, V., Kotta, Ü., Shumsky, A. Ye., Tõnso, M., and Zhirabok, A. N. Implementation of the tools of functions’ algebra: First steps. In Proceedings of MATHMOD 2012 – 7th Vienna International Conference on Mathematical Modelling, Vienna, Austria. 2012, 1231–1236.
9. Kotta, Ü. and Tõnso, M. Linear algebraic tools for discrete-time nonlinear control systems with Mathematica. In Nonlinear and Adaptive Control, NCN4 2001. Lecture Notes in Control and Information Sciences, 2003, 281, 195–205.
10. Kotta, Ü., Tõnso, M., Belikov, J., Kaldmäe, A., Kaparin, V., Shumsky, A. Ye., and Zhirabok, A. N. A symbolic software package for nonlinear control systems, In Proceedings of the 2013 International Conference on Process Control (PC), Štrbské Pleso, Slovakia. 2013, 101–106.
11. Kotta, Ü., Tõnso, M., Shumsky, A. Ye., and Zhirabok, A. N. Feedback linearization and lattice theory. Syst. & Contr. Lett., 2013, 62(3), 248–255.
12. Nijmeijer, H. and van der Schaft, A. J. Nonlinear Dynamical Control Systems. Springer, New York, 1990.
13. Zhirabok, A. N. and Shumsky, A. Ye. The Algebraic Methods for Analysis of Nonlinear Dynamic Systems. Dalnauka, Vladivostok, 2008 (in Russian).
Back to Issue