IMA Journal of Mathematical Control and Information - current issue

Adaptive stabilization of Kirchhoff's non-linear strings with output disturbances

Kobayashi, T. — Mon, 23 Nov 2009 04:05:14 PST

This paper is concerned with adaptive stabilization of an undamped non-linear string with disturbed outputs by boundary feedback control. The adaptive controller is constructed by the concept of high-gain adaptive feedback and the estimation mechanism for the unknown parameters of the measurement noise. The local existence and uniqueness of the solution of the closed-loop system are justified. Moreover, global existence and boundedness of the solution are shown for smooth and small initial data. The stability of the closed-loop system is proved such that the convergence of the system state to zero and the convergence of the estimated parameter to the unknown parameter are guaranteed for the smooth and small initial data.

An adaptive tracking controller design for non-linear singularly perturbed systems using fuzzy singularly perturbed model

Li, L., Sun, F. C. — Mon, 23 Nov 2009 04:05:14 PST

For a class of non-linear singularly perturbed systems with unknown dynamic and external disturbance, an adaptive tracking controller using fuzzy singularly perturbed model is developed. First, a series of dynamic Takagi-Sugeno fuzzy singularly perturbed subsystems are built to approximate a non-linear singularly perturbed system and a stable reference model with the same fuzzy premise is chosen to design both expected trajectory and dynamic performance. Then, a kind of controller including adaptive feedback term is developed to make the states of the closed-loop system follow those of the reference model. The linear feedback gain of the controller can be solved by the linear matrix inequality approach. The adaptive sliding term is used to compensate the uncertainty and alleviate the disturbance. Lyapunov constitute techniques can be used to prove the stability of the closed-loop systems. Finally, the simulations results illustrate the effectiveness of this approach.

Convex hull of two quadratic constraints is an LMI set

Yildiran, U. — Mon, 23 Nov 2009 04:05:14 PST

In this work, we are interested in the convex hull of the region determined by two quadratic polynomial constraints. The main result is that if this region is not empty, the convex hull is either Rⁿ or the feasible set of another pair of quadratic constraints which are, in fact, positive linear combinations of the original ones. Based on this result, a losslessness condition is also derived for the classical semidefinite programming relaxation. The characterization of the convex hull we found does not have to be composed of concave quadratic constraints. However, we propose an algorithm to convert them into linear matrix inequalities (LMIs) and explain how the LMI characterization can be employed to solve a certain class of non-convex optimization problems. It is shown that this approach may perform better than the available relaxation methods for the optimization problem considered. Lastly, we show how the results developed can be applied to a certain class of control problems.

Improved robust stability and feedback stabilization criteria for time-delay systems

Mahmoud, M. S. — Mon, 23 Nov 2009 04:05:14 PST

This paper develops novel robust stability and feedback stabilization criteria with guaranteed performance for a class of linear continuous time-delay systems with polytopic uncertainties. The time-varying delay function is unknown and differentiable within bounded interval and the input delay is constant. The criteria is derived based on the constructive use of a new Lyapunov–Krasovskii functional together with the integral inequality. The developed stability condition is expressed in terms of linear matrix inequality that manipulates fewer decision variables and requires reduced computational load. Through a comparison with other existing stability methods, it is established that the developed method retains some useful terms that are frequently dropped out and does not employ any free-weighting matrices to avoid redundancy. A state-feedback stabilizing controller is designed to ensure that the closed loop is robustly stable with guaranteed performance. Representative examples are simulated to illustrate the developed results.

Improved results for non-linear discrete-time systems with time-varying delays

Mahmoud, M. S., Xia, Y. — Mon, 23 Nov 2009 04:05:14 PST

In this paper, complete results for delay-dependent stability, feedback stabilization and linear filtering for a class of non-linear discrete-time systems are developed. The system under consideration has time-varying delays with Lipschitz-type non-linearities and subject to real convex bounded parametric uncertainties in all system matrices. A major thrust of the analysis is the constructive use of an appropriate Lyapunov functionals coupled with ‘Finsler's lemma’ and free-weighting parameter matrices. We establish a linear matrix inequality (LMI) characterization of delay-dependent conditions under which the non-linear discrete delay system is robustly asymptotically stable with an L₂ gain smaller than a prescribed constant level. Feedback stabilization schemes, based on state, static output or by using dynamic output feedback, are designed to guarantee that the corresponding closed-loop system enjoys the delay-dependent asymptotic stability with an L₂ gain smaller than a prescribed constant level. Finally, the developed approach is applied to linear filtering to design both H and L₂ – L filters. All the developed results are expressed in terms of convex optimization over LMIs and tested on several representative examples.

Stabilization and decay estimate of linear control systems in Hilbert space with non-linear feedback

Berrahmoune, L. — Mon, 23 Nov 2009 04:05:15 PST

We consider the problem of stabilization of linear control systems in Hilbert space with non-linear feedback. Extensions of known results in the linear feedback case to the non-linear one are given. Furthermore, under appropriate controllability assumptions, energy decay estimates are established. Applications to the saturating control case are treated.