8 research outputs found
Further Constructions of Control-Lyapunov Functions and Stabilizing Feedbacks for Systems Satisfying the Jurdjevic-Quinn Conditions
For a broad class of nonlinear systems, we construct smooth control-Lyapunov
functions whose derivatives along the trajectories of the systems can be made
negative definite by smooth control laws that are arbitrarily small in norm. We
assume our systems satisfy appropriate generalizations of the Jurdjevic-Quinn
conditions. We also design state feedbacks of arbitrarily small norm that
render our systems integral-input-to-state stable to actuator errors.Comment: 15 pages, 0 figures, accepted for publication in IEEE Transactions on
Automatic Control in October 200
Constructions of Strict Lyapunov Functions for Discrete Time and Hybrid Time-Varying Systems
We provide explicit closed form expressions for strict Lyapunov functions for
time-varying discrete time systems. Our Lyapunov functions are expressed in
terms of known nonstrict Lyapunov functions for the dynamics and finite sums of
persistency of excitation parameters. This provides a discrete time analog of
our previous continuous time Lyapunov function constructions. We also construct
explicit strict Lyapunov functions for systems satisfying nonstrict discrete
time analogs of the conditions from Matrosov's Theorem. We use our methods to
build strict Lyapunov functions for time-varying hybrid systems that contain
mixtures of continuous and discrete time evolutions.Comment: 14 pages. Accepted for publication in Nonlinear Analysis: Hybrid
Systems and Applications on September 6, 200
Further Results on Lyapunov Functions for Slowly Time-Varying Systems
We provide general methods for explicitly constructing strict Lyapunov
functions for fully nonlinear slowly time-varying systems. Our results apply to
cases where the given dynamics and corresponding frozen dynamics are not
necessarily exponentially stable. This complements our previous Lyapunov
function constructions for rapidly time-varying dynamics. We also explicitly
construct input-to-state stable Lyapunov functions for slowly time-varying
control systems. We illustrate our findings by constructing explicit Lyapunov
functions for a pendulum model, an example from identification theory, and a
perturbed friction model.Comment: Accepted for publication in Mathematics of Control, Signals, and
Systems (MCSS) on November 20, 200
Uniform global asymptotic stability of a class of adaptively controlled nonlinear systems
We give a new explicit, global, strict Lyapunov function construction for the error dynamics for adaptive tracking control problems, under an appropriate persistency of excitation condition. We then allow time-varying uncertainty in the unknown parameters. In this case, we construct input-to-state stable Lyapunov functions under suitable bounds on the uncertainty, provided the regressor also satisfies an affine growth condition. This lets us quantify the effects of uncertainties on both the tracking and the parameter estimation. We illustrate our results using Rössler systems. © 2009 IEEE