Research Papers

Convergence Rates of a Class of Uncertain Dynamic Systems

[+] Author and Article Information
Juan Ignacio Mulero-Martínez

Departamento de Ingeniería de
Sistemas y Automática,
Universidad Politécnica de Cartagena,
Campus Muralla del Mar,
Cartagena 30203, Spain

The notion of hyperring should not be confused with that of domain of attraction.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received September 15, 2010; final manuscript received March 14, 2013; published online May 14, 2013. Assoc. Editor: Eugenio Schuster.

J. Dyn. Sys., Meas., Control 135(5), 051001 (May 21, 2013) (8 pages) Paper No: DS-10-1268; doi: 10.1115/1.4024078 History: Received September 15, 2010; Revised March 14, 2013

The problem of stabilization of uncertain systems plays a broad and fundamental role in robust control theory. The paper examines a boundedness theorem for a class of uncertain systems characterized as having a decreasing Lyapunov function in a ringlike region. It is a systematic study on stability that embraces both the transient and steady analysis, covering such aspects as the maximum overshoot of the system state, the stability region and the exponential convergence rate. The emphasis throughout is on deriving dominant time constants and explicit time expressions for a state to reach an invariant set. The central theorem provides a complete treatment of the time evolution of trajectories depending on the specific compact set of initial conditions. Toward this end, the comparison lemma along with a particular Riccati differential equation are essential and conclusive. The scope of questions addressed in the paper, the uniformity of their treatment, the novelty of the proposed theorem, and the obtained results make it very useful with respect to other works on the problem of robust nonlinear control.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Khalil, H. K., 1992, Nonlinear Systems, Macmillan, New York.
Dawson, D. M., Qu, Z., Lewis, F. L., and Dorsey, J. F., 1990, “Robust Control for the Tracking of Robot Motion,” Int. J. Contr., 52(3), pp. 581–595. [CrossRef]
Qu, Z., 1998, Robust Control of Nonlinear Uncertain Systems (Wiley Series in Nonlinear Science), John Wiley and Sons, New York.
Corless, M., and Leitmann, G., 1981, “Continuous State Feedback Guaranteeing Uniform Ultimate Boundness for Uncertain Dynamics Systems,” IEEE Trans. Auto. Contr., 26, pp. 1139–1143. [CrossRef]
Hinrichsen, D., Plischke, E., and Pritchard, A. J., 2001, “Liapunov and Riccati Equations for Practical Stability,” Proc. 6th European Control Conference, Porto, Portugal, pp. 2883–2888.
Hinrichsen, D., Plischke, E., and Wirth, F., 2002, “State Feedback Stabilization With Guaranteed Transient Bounds,” Proc. Mathematical Theory of Networks and Systems, Notre Dame, IN.
Qu, Z., Dorsey, J. F., Zhang, X., and Dawson, D. M., 1991, “Robust Control of Robots by Computed Torque Law,” Syst. Contr. Lett., 16, pp. 25–32. [CrossRef]
Qu, Z., and Dorsey, J. F., 1991, “Robust PID Control of Robots,” Int. J. Robotic. Auto., 6(4), pp. 228–235.
Qu, Z., and Dawson, D. M., 1996, Robust Tracking Control of Robot Manipulators, IEEE Press, New York.
Rouche, N., Habets, P., and Laloy, M., 1977, Stability Theory by Liapunov's Direct Method, Springer-Verlag, New York.
Matrosov, V. M., 1962, “On the Stability of Motion,” J. App. Math. Mech., 26, pp. 1337–1353. [CrossRef]
Rouche, N., and Mawhin, J., 1980, Ordinary Differential Equations II: Stability and Periodical Solutions, Pitman Publishing Ltd., London.
Slotine, J.-J., and Li, W., 1990, Applied Nonlinear Control, 1st ed., Prentice Hall, Englewood Cliffs, NJ.
Kelly, R., Santibanez, V., and Loria, A., 2005, Control of Robot Manipulators in Joint Space, Springer-Verlag, London.
Ghorbel, F., Srivinasan, B., and Spong, M. W., 1998, “On the Uniform Boundedness of the Inertia Matrix of Serial Robot,” J. Robot. Syst., 15(1), pp. 17–28. [CrossRef]
Mulero-Martínez, J. I., 2007, “Functions Bandlimited in Frequency are Free of the Curse of Dimensionality,” Neurocomputing, 70(7–9), pp. 1439–1452. [CrossRef]
Silvester, J. R., 2000, “Determinants of Block Matrices,” Math. Gaz., 84(501), pp. 460–467. [CrossRef]
Horn, R. A., and Johnson, C. R., 1999, Matrix Analysis, Cambridge University Press, Cambridge, UK.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In