Fundamental limitations of the decay of generalized energy in controlled (discrete-time) nonlinear systems subject to state and input constraints

Selek, István ; Ikonen, Enso

International Journal of Applied Mathematics and Computer Science, Tome 29 (2019) no. 4, pp. 629-639.

Résumé

This paper is devoted to the analysis of fundamental limitations regarding closed-loop control performance of discrete-time nonlinear systems subject to hard constraints (which are nonlinear in state and manipulated input variables). The control performance for the problem of interest is quantified by the decline (decay) of the generalized energy of the controlled system. The paper develops (upper and lower) barriers bounding the decay of the system’s generalized energy, which can be achieved over a set of asymptotically stabilizing feedback laws. The corresponding problem is treated without the loss of generality, resulting in a theoretical framework that provides a solid basis for practical implementations. To enhance understanding, the main results are illustrated in a simple example.

Détail

Keywords: decay rate maximization, Lyapunov function, nonlinear control system
Mots-clés : rozkład wykładniczy, funkcja Lapunowa, układ sterowania nieliniowy

Bibliographie

[1] Al’brekht, E.G. (1961). On the optimal stabilization of nonlinear systems, Journal of Applied Mathematics and Mechanics 25(5): 1254–1266.

[2] Aranda-Escolástico, E., Salt, J., Guinaldo, M., Chacón, J. and Dormido, S. (2018). Optimal control for aperiodic dual-rate systems with time-varying delays, Sensors 18(5): 1–19.

[3] Bemporad, A., Torrisit, F.D. and Morarit, M. (2000). Performance analysis of piecewise linear systems and model predictive control systems, IEEE Conference on Decision and Control, Sydney, NSW, Australia, pp. 4957–4962.

[4] Boyd, S., El-Ghaoui, L., Feron, E. and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA.

[5] Buhl, M. and Lohmann, B. (2009). Control with exponentially decaying Lyapunov functions and its use for systems with input saturation, European Control Conference, Budapest, Hungary, pp. 3148–3153.

[6] Darup, M.S. and Mönnigmann, M. (2013). Null-controllable set computation for a class of constrained bilinear systems, European Control Conference, Zürich, Switzerland, pp. 2758–2763.

[7] Duda, J. (2012). A Lyapunov functional for a system with a time-varying delay, International Journal of Applied Mathematics and Computer Science 22(2): 327–337, DOI: 10.2478/v10006-012-0024-7.

[8] Feyzmahdavian, H. R., Charalambous, T. and Johansson, M. (2013). On the rate of convergence of continuous-time linear positive systems with heterogeneous time-varying delays, European Control Conference, Zürich, Switzerland, pp. 3372–3377.

[9] Fu, J. (1993). Families of Lyapunov functions for nonlinear systems in critical cases, IEEE Transactions on Automatic Control 38(1): 3–16.

[10] Grushkovskaya, V. and Zuyev, A. (2014). Optimal stabilization problem with minimax cost in a critical case, IEEE Transactions on Automatic Control 59(9): 2512–2517.

[11] Hu, T., Lin, Z. and Shamash, Y. (2003). On maximizing the convergence rate for linear systems with input saturation, IEEE Transactions on Automatic Control 48(7): 1249–1253.

[12] Kaczorek, T. (2007). The choice of the forms of Lyapunov functions for a positive 2D Roesser model, International Journal of Applied Mathematics and Computer Science 17(4): 471–475, DOI: 10.2478/v10006-007-0039-7.

[13] Lenka, B.K. (2019). Time-varying Lyapunov functions and Lyapunov stability of nonautonomous fractional order systems, International Journal of Applied Mathematics 32(1): 111–130.

[14] Li, W., Huang, C. and Zhai, G. (2018). Quadratic performance analysis of switched affine time-varying systems, International Journal of Applied Mathematics and Computer Science 28(3): 429–440, DOI: 10.2478/amcs-2018-0032.

[15] Polyak, B. and Shcherbakov, P. (2009). Ellipsoidal approximations to attraction domains of linear systems with bounded control, Proceedings of the American Control Conference, St. Louis, MO, USA, pp. 5363–5367.

[16] Prieur, C., Tarbouriech, S. and Zaccarian, L. (2011). Improving the performance of linear systems by adding a hybrid loop, 18th IFAC World Congress, Milan, Italy, pp. 6301–6306.

[17] Scokaert, P. and Rawlings, J.B. (1998). Constrained linear quadratic regulation, IEEE Transactions on Automatic Control 43(8): 1163–1169.

[18] Selek, I. and Ikonen, E. (2018). On the bounds of the fastest admissible decay of generalized energy in controlled LTI systems subject to state and input constraints, 15th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE2018), Mexico City, Mexico, p. ID: 19.