Practical Stabilization for Uncertain Pseudo-Linear and Pseudo-Quadratic MIMO Systems


In this paper the problem of practical stabilization for a significant class of MIMO uncertain pseudo-linear and pseudo-quadratic systems, with additional bounded nonlinearities and/or bounded disturbances, is considered. By using the concept of majorant system, via Lyapunov approach, new fundamental theorems, from which derive explicit formulas to design state feedback control laws, with a possible imperfect compensation of nonlinearities and disturbances, are stated. These results guarantee a specified convergence velocity of the linearized system of the majorant system and a desired steady-state output for generic uncertainties and/or generic bounded nonlinearities and/or bounded disturbances.

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L. Celentano, "Practical Stabilization for Uncertain Pseudo-Linear and Pseudo-Quadratic MIMO Systems," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 1, 2013, pp. 34-42. doi: 10.4236/ijmnta.2013.21004.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] G. Ambrosino, G. Celentano and F. Garofalo, “Tracking Control of High-Performance Robots via Stabilizing Controllers for Uncertain Systems,” Journal of Optimization Theory and Applications, Vol. 50, No. 2, 1986, pp. 239- 255. doi:10.1007/BF00939271
[2] H. Nijmeijer and A. J. Van der Schaft, “Nonlinear Dynamical Control Systems,” Springer-Verlag, Berlin, 1990. doi:10.1007/978-1-4757-2101-0
[3] J. J. E. Slotine and W. Li, “Applied Nonlinear Control,” Prentice-Hall, Upper Saddle River, 1991.
[4] B. Brogliato and A. T. Neto, “Practical Stabilization of a Class of Nonlinear Systems with Partially Known Uncertainties,” Automatica, Vol. 31, No. 1, 1995, 145-150. doi:10.1016/0005-1098(94)E0050-R
[5] R. A. Freeman and P. V. Kokotovic, “Robust Nonlinear Control Design,” Birkhauser, Boston, 1996. doi:10.1007/978-0-8176-4759-9
[6] A. Isidori, “Nonlinear Control Systems II,” Springer-Verlag, New York, 1999. doi:10.1007/978-1-4471-0549-7
[7] S. Sastry, “Nonlinear Systems, Analysis, Stability and Control,” Springer-Verlag, New York, 1999.
[8] X.-Y. Lu and S. K. Spurgeon, “Output Feedback Stabilization of MIMO Non-Linear Systems via Dynamic Sliding Mode,” International Journal of Robust and Nonlinear Control, Vol. 9, No. 5, 1999, pp. 275-305. doi:10.1002/(SICI)1099-1239(19990430)9:5<275::AID-RNC404>3.0.CO;2-F
[9] L. Moreau and D. Aeyels, “Practical Stability and Stabilization,” IEEE Transactions on Automatic Control, Vol. 45, No. 8, 2000, pp. 1554-1558. doi:10.1109/9.871771
[10] I. Karafyllis and J. Tsinias, “Global Stabilization and Asymptotic Tracking for a Class of Nonlinear Systems by Means of Time-Varying Feedback,” International Journal of Robust and Nonlinear Control, Vol. 13, No. 6, 2003, pp. 559-588. doi:10.1002/rnc.738
[11] L. Celentano, “A General and Efficient Robust Control Method for Uncertain Nonlinear Mechanical Systems,” Proceedings of the IEEE Conference on Decision and Control, Seville, 12-15 December 2005, pp. 659-665. doi:10.1109/CDC.2005.1582231
[12] L. K. Murray and S. Jayasuriya, “An Improved Non-Sequential Multi-Input Multi-Output Quantitative Feedback Theory Design Methodology,” International Journal of Robust and Nonlinear Control, Vol. 16, No. 8, 2006, pp. 379-395. doi:10.1002/rnc.1061
[13] E. Moulay and W. Perruquetti, “Finite Time Stability and Stabilization of a Class of Continuous Systems,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 2, 2006, pp. 1430-1443. doi:10.1016/j.jmaa.2005.11.046
[14] D. V. Efimov and A. L. Fradkov, “Input-to-Output Stabilization of Nonlinear Systems via Backstepping,” International Journal of Robust and Nonlinear Control, Vol. 19, No. 6, 2009, pp. 613-633. doi:10.1002/rnc.1336
[15] M. L. Corradini, A. Cristofaro and G. Orlando, “Robust Stabilization of Multi-Input Plants with Saturating Actuators,” IEEE Transactions on Automatic Control, Vol. 55, No. 2, 2010, pp. 419-425. doi:10.1109/TAC.2009.2036308
[16] S. F. Yang, “Efficient Algorithm for Computing QFT Bounds,” International Journal of Control, Vol. 83, No. 4, 2010, pp. 716-723. doi:10.1080/00207170903390161
[17] F. Amato, C. Cosentino and A. Merola, “Sufficient Conditions for Finite-Time Stability and Stabilization of Nonlinear Quadratic Systems,” IEEE Transactions on Automatic Control, Vol. 55, No. 2, 2010, pp. 430-434. doi:10.1109/TAC.2009.2036312
[18] S. Ding, C. Qian and S. Li, “Global Stabilization of a Class of Feedforward Systems with Lower-Order Nonlinearities,” IEEE Transactions on Automatic Control, Vol. 55, No. 3, 2010, pp. 691-696. doi:10.1109/TAC.2009.2037455
[19] L. Celentano, “New Robust Tracking and Stabilization Methods for Significant Classes of Uncertain Linear and Nonlinear Systems,” In: A. Mueller, Ed., Recent Advances in Robust Control—Novel Approaches and De- sign Methods, InTech, Winchester, 2011, pp. 247-270.
[20] L. Celentano, “Robust Tracking Controllers Design with Generic References for Continuous and Discrete Uncertain Linear SISO Systems,” LAP—Lambert Academic Publishing, Saarbrücken, 2012.
[21] L. Celentano, “Robust Tracking Method for Uncertain MIMO Systems of Realistic Trajectories,” Journal of the Franklin Institute, Vol. 350, No. 3, 2013, pp. 437-451. doi:10.1016/j.jfranklin.2012.12.002

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