Abstract
Continuous homogeneous controllers are utilized in a full state feedback setting for the uniform finite time stabilization of a perturbed double integrator in the presence of uniformly decaying piecewise continuous disturbances. Semiglobal strong $\mathcal{C}^1$ Lyapunov functions are identified to establish uniform asymptotic stability of the closed-loop planar system. Uniform finite time stability is then proved by extending the homogeneity principle of discontinuous systems to the continuous case with uniformly decaying piecewise continuous nonhomogeneous disturbances. A finite upper bound on the settling time is also computed. The results extend the existing literature on homogeneity and finite time stability by both presenting uniform finite time stabilization and dealing with a broader class of nonhomogeneous disturbances for planar controllable systems while also proposing a new class of homogeneous continuous controllers.
Highlights
Continuous finite time stabilization of linear and nonlinear control systems is an active area of research
Earlier results on asymptotic stabilization [18, 31] of continuous homogeneous systems are based on the definition of a class of dilations where each state is dilated with a different weight [18]
A detailed literature review on the topic of geometric homogeneity is presented in [9], where it is established that geometric homogeneity leads to finite time stability if the homogeneity degree of the asymptotically stable continuous homogeneous system is negative
Summary
Continuous finite time stabilization of linear and nonlinear control systems is an active area of research. A detailed literature review on the topic of geometric homogeneity is presented in [9], where it is established that geometric homogeneity leads to finite time stability if the homogeneity degree of the asymptotically stable continuous homogeneous system is negative. Homogeneous approximations have been studied [3] that led to the development of tools to establish global asymptotic (and in some cases finite time) stability of nonlinear systems. This result used previous results on the so-called homogeneous domination approach (see [30], [3, section 5], and references therein for a detailed literature review).
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