Positivity and linearization of a class of nonlinear continuous-time systems by state feedbacks

Tadeusz Kaczorek

International Journal of Applied Mathematics and Computer Science, Tome 25 (2015), p. 827-831 / Harvested from The Polish Digital Mathematics Library

Résumé

The positivity and linearization of a class of nonlinear continuous-time system by nonlinear state feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of nonlinear systems by nonlinear state feedbacks is presented. It is shown that by a suitable choice of the state feedback it is possible to obtain an asymptotically stable and controllable linear system, and if the closed-loop system is positive then it is unstable.

Détail

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:275905

Bibliographie

[000] Aguilar, J.L.M., Garcia, R.A. and D'Attellis, C.E. (1995). Exact linearization of nonlinear systems: Trajectory tracking with bounded control and state constrains, 38th Midwest Symposium on Circuits and Systems, Rio de Janeiro, Brazil, pp. 620-622.

[001] Brockett, R.W. (1976). Nonlinear systems and differential geometry, Proceedings of the IEEE 64(1): 61-71.

[002] Charlet, B., Levine, J. and Marino, R. (1991). Sufficient conditions for dynamic state feedback linearization, SIAM Journal on Control and Optimization 29(1): 38-57. | Zbl 0739.93021

[003] Daizhan, C., Tzyh-Jong, T. and Isidori, A. (1985). Global external linearization of nonlinear systems via feedback, IEEE Transactions on Automatic Control 30(8): 808-811. | Zbl 0666.93054

[004] Fang, B. and Kelkar, A.G. (2003). Exact linearization of nonlinear systems by time scale transformation, IEEE American Control Conference, Denver, CO, USA, pp. 3555-3560.

[005] Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, New York, NY. | Zbl 0988.93002

[006] Isidori, A. (1989). Nonlinear Control Systems, Springer-Verlag, Berlin. | Zbl 0693.93046

[007] Jakubczyk, B. (2001). Introduction to geometric nonlinear control: Controllability and Lie bracket, Summer School on Mathematical Control Theory, Triest, Italy. | Zbl 1017.93034

[008] Jakubczyk, B. and Respondek, W. (1980). On linearization of control systems, Bulletin of the Polish Academy Sciences: Technical Sciences 28: 517-521. | Zbl 0489.93023

[009] Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer Verlag, London. | Zbl 1005.68175

[010] Kaczorek, T. (2011). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuit and Systems 58(6): 1203-1210.

[011] Kaczorek, T. (2012). Selected Problems of Fractional System Theory, Springer Verlag, Berlin.

[012] Kaczorek, T. (2013). Minimum energy control of fractional positive discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 61(4): 803-807.

[013] Kaczorek, T. (2014a). Minimum energy control of descriptor positive discrete-time systems, COMPEL 33(3): 1-14.

[014] Kaczorek, T. (2014b). Necessary and sufficient conditions for minimum energy control of positive discrete-time linear systems with bounded inputs, Bulletin of the Polish Academy of Sciences: Technical Sciences 62(1): 85-89.

[015] Kaczorek, T. (2014c). Minimum energy control of fractional positive continuous-time linear systems with bounded inputs, International Journal of Applied Mathematics and Computer Science 24(2): 335-340, DOI: 10.2478/amcs-2014-0025. | Zbl 1293.49042

[016] Malesza, W. (2008). Geometry and Equivalence of Linear and Nonlinear Control Systems Invariant on Corner Regions, Ph.D. thesis, Warsaw University of Technology, Warsaw.

[017] Malesza, W. and Respondek, W. (2007). State-linearization of positive nonlinear systems: Applications to Lotka-Volterra controlled dynamics, in F. Lamnabhi-Lagarrigu et al. (Eds.), Taming Heterogeneity and Complexity of Embedded Control, John Wiley, Hoboken, NJ, pp. 451-473.

[018] Marino, R. and Tomei, P. (1995). Nonlinear Control Design - Geometric, Adaptive, Robust, Prentice Hall, London. | Zbl 0833.93003

[019] Melhem, K., Saad, M. and Abou, S.C. (2006). Linearization by redundancy and stabilization of nonlinear dynamical systems: A state transformation approach, IEEE International Symposium on Industrial Electronics, Montreal, Canada, pp. 61-68.

[020] Taylor, J.H. and Antoniotti, A.J. (1993). Linearization algorithms for computer-aided control engineering, Control Systems Magazine 13(2): 58-64.

[021] Wei-Bing, G. and Dang-Nan, W. (1992). On the method of global linearization and motion control of nonlinear mechanical systems, International Conference on Industrial Electronics, Control, Instrumentation and Automation, San Diego, CA, USA, pp. 1476-1481.