Robust optimal stabilization of the Brockett integrator via a hybrid feedback

Abstract

The problem of semi-global minimal time robust stabilization of the Brockett integrator (also called Heisenberg system) is addressed and solved by means of a hybrid state feedback law. It is shown that the solutions of the closedloop system converge to the origin, in quasi-minimal time (for a given bound on the controller), with a robustness property with respect to small measurement noises, external disturbances and actuator noises.

Hybrid Feedback Control and Robust Stabilization of Nonlinear Systems

“…This is due to the presence of a discrete state variable, part of the hysteresis mechanism, that makes jumps at certain locations depending on its own value and the value of the control system state. The benefits of hybrid control have been recognized in the nonlinear control literature for some time; see [23], [17], and also [16], [28], [31], and [30], where robustness of different types of hybrid feedbacks to unknown parameters or unmodeled dynamics was addressed, and [32] and [33], where quasi-optimal robust stabilization was achieved by means of hybrid feedbacks. Nevertheless, a general theory of hybrid systems-those arising from the implementation of hybrid feedbacks, those describing dynamical systems with different modes of operation (say a thermostat), and those where "continuous" variables may jump (say a bouncing ball)-is not yet complete.…”

Integral sliding mode control of an extended Heisenberg system

“…Therefore, an important issue for a practical control design is the robustness consideration. Most of robust stabilization results and investigations focus on the stabilization of chained form systems with matched uncertainties [13] (perturbations that enter the state equation at the same point as the control input), with sensor noise and external disturbances [12,14,15].…”

Nonlinear optimal control: approximations via moments and LMI-relaxations

“…More precisely, the gradients ∂T /∂x i , i = 1, 2, are discontinuous at every point (0, 0, x 3 ), x 3 = 0. For the interested reader, the level sets {(x 1 , x 2 , x 3 ) ∈ R 3 | T (x 1 , x 2 , x 3 ) = r}, with r > 0, are displayed in Prieur and Trélat [16]. 2) Numerical approximation: Recall that inf Q r ↑ as r increases, i.e., the more moments we consider, the closer to the exact value we get.…”