Stabilization via homogeneous feedback controls

Abstract

In this paper, we provide an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists for an affine control system and satisfies an homogeneous condition. We use a modified version of the Sontag formula to achieve our main goal. Moreover, we prove that the existence of an homogeneous control Lyapunov function for an homogeneous affine system leads to an homogeneous closed-loop system by using the previous feedback control.

Oscillating system design applying universal formula for control

“…Among approaches for stabilizing control design it is worth to mention CLF method [2], [26], [6], [5] that gives a universal formula for the control laws. For the homogeneous system (3) this approach has been developed in [9], [17], [18], [19].…”

Homogeneous Feedback Control of Nonlinear Systems Based on Control Lyapunov Functions

“…Herein, it is not necessary to require that the CLF V ( x ) and system are homogeneous, that is, we can design a homogeneous controller stabilizing nonhomogeneous system (see, Example ). A similar method is also used in . Corollary Under Assumption and − τ < k 0 < 0, system is finite‐time stable under the feedback control , and the setting time where x 0 is the initial state of system , and σ and l are defined in . Proof When B ( x ) = ( b 1 ( x ), … , b m ( x )) = (0, … , 0), β ( x ) = || B ( x )|| 2 = 0. It follows that a ( x ) = L f V ( x ) < 0 by Definition .…”

“…Herein, it is not necessary to require that the CLF V(x) and system (1) are homogeneous, that is, we can design a homogeneous controller stabilizing nonhomogeneous system (1) (see, Example 2). A similar method is also used in [25].…”

Global finite‐time stabilization of planar nonlinear systems with disturbance

“…After they are filtered by a properly designed filter in the loop or the inertia of dynamic systems, they can be approximated by (7). After they are filtered by a properly designed filter in the loop or the inertia of dynamic systems, they can be approximated by (7).…”

“…Many engineering disturbances consist of both high frequency and low frequency parts. After they are filtered by a properly designed filter in the loop or the inertia of dynamic systems, they can be approximated by (7). Hence (7) represents a large variety of disturbances in engineering.…”