Positivity and Linearization of a Class of Nonlinear Continuous–Time Systems by State Feedbacks

Tadeusz Kaczorek

doi:10.1515/amcs-2015-0059

.blurhash-client-img { display: none !important; }

Positivity and Linearization of a Class of Nonlinear Continuous–Time Systems by State Feedbacks

International Journal of Applied Mathematics and Computer Science

Volume 25 (2015): Issue 4 (December 2015)

By: Tadeusz Kaczorek

Open Access

|Dec 2015

Abstract

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.

References

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.
Search in Google Scholar
Brockett, R.W. (1976). Nonlinear systems and differential geometry, Proceedings of the IEEE64(1): 61–71.10.1109/PROC.1976.10067
Search in Google Scholar
Charlet, B., Levine, J. and Marino, R. (1991). Sufficient conditions for dynamic state feedback linearization, SIAM Journal on Control and Optimization29(1): 38–57.10.1137/0329002
Search in Google Scholar
Daizhan, C., Tzyh-Jong, T. and Isidori, A. (1985). Global external linearization of nonlinear systems via feedback, IEEE Transactions on Automatic Control30(8): 808–811.10.1109/TAC.1985.1104040
Search in Google Scholar
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.
Search in Google Scholar
Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, NewYork, NY.10.1002/9781118033029
Search in Google Scholar
Isidori, A. (1989). Nonlinear Control Systems, Springer-Verlag, Berlin.10.1007/978-3-662-02581-9
Search in Google Scholar
Jakubczyk, B. (2001). Introduction to geometric nonlinear control: Controllability and Lie bracket, Summer School on Mathematical Control Theory, Triest, Italy.
Search in Google Scholar
Jakubczyk, B. and Respondek, W. (1980). On linearization of control systems, Bulletin of the Polish Academy Sciences: Technical Sciences28: 517–521.
Search in Google Scholar
Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer Verlag, London.10.1007/978-1-4471-0221-2
Search in Google Scholar
Kaczorek, T. (2011). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuit and Systems58(6): 1203–1210.10.1109/TCSI.2010.2096111
Search in Google Scholar
Kaczorek, T. (2012). Selected Problems of Fractional System Theory, Springer Verlag, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar
Kaczorek, T. (2013). Minimum energy control of fractional positive discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences61(4): 803–807.10.2478/bpasts-2013-0087
Search in Google Scholar
Kaczorek, T. (2014a). Minimum energy control of descriptor positive discrete-time systems, COMPEL33(3): 1–14.10.1108/COMPEL-04-2013-0111
Search in Google Scholar
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 Sciences62(1): 85–89.10.2478/bpasts-2014-0010
Search in Google Scholar
Kaczorek, T. (2014c). Minimum energy control of fractional positive continuous-time linear systems with bounded inputs, International Journal of Applied Mathematics and Computer Science24(2): 335–340, DOI: 10.2478/amcs-2014-0025.10.2478/amcs-2014-0025
Search in Google Scholar
Malesza, W. (2008). Geometry and Equivalence of Linear and Nonlinear Control Systems Invariant on Corner Regions, Ph.D. thesis, Warsaw University of Technology, Warsaw.
Search in Google Scholar
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.
Search in Google Scholar
Marino, R. and Tomei, P. (1995). Nonlinear Control Design— Geometric, Adaptive, Robust, Prentice Hall, London.
Search in Google Scholar
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.
Search in Google Scholar
Taylor, J.H. and Antoniotti, A.J. (1993). Linearization algorithms for computer-aided control engineering, Control Systems Magazine13(2): 58–64.10.1109/37.206986
Search in Google Scholar
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.
Search in Google Scholar