Homogeneous State Feedback Stabilization of Homogenous Systems

Abstract

We show that for any asymptotically control-

Global stability results for systems under sampled‐data control

“…Consider the planar control system i) results concerning globally Lipschitz systems (see [13]) cannot be applied since system (4.2) is not globally Lipschitz, ii) results concerning homogeneous systems (see [8]) cannot be applied since system (4.2) is not homogeneous, iii)…”

“…Recent research takes into account performance and robustness issues as well (see [10,21,23,37]). * exploiting special characteristics of the system such as homogeneity (see [8]), global Lipschitz conditions (see [13]) or linear structure with uncertainties (see [2] as well as the textbook [42]). * making use of Linear Matrix Inequalities in the context of hybrid systems (see [14,15,27,29,48]).…”

Input‐to‐state stability, numerical dynamics and sampled‐data control

“…When proceeding this way we face the problem that the discrete time system Φ u h induced by (7.3) is hardly ever known exactly, because it is the solution of a nonlinear ordinary differential equation. Therefore, typically one has to design the feedback controller for a family of approximations Φ u k , k ∈ N, to Φ u h , see, e.g., [4,5,9,10,11,12,18,21,25,27,26] for examples of such design methods. Since a feedback law constructed on basis of an approximation Φ u k will depend on k, we need to consider sequences of feedback laws u k in the following definition of the approximating systems:…”

“…Of course, the regularity of the maps u k strongly depends on the sampled data feedback design method employed. While for series based (re-)design methods as presented, e.g., in [11,21,25] or for the Euler backstepping method [27,18] the Lipschitz property can be expected or even rigorously proved, optimal control based design methods as in [5,9,10,12,26] typically yield discontinuous feedback laws u k , for which convergence (7.6) cannot be expected in general.…”

Stability analysis of discrete–time general homogeneous systems

“…Therein, D-stability was extended for nonlinear systems and some results are established for different classes of positive nonlinear systems, like homogeneous cooperative systems. Non-autonomous continuous-time homogeneous systems were considered in (Grüne, 2000) and , and it was shown therein that any asymptotically controllable homogeneous control system admits a homogeneous control Lyapunov function and a stabilizing, possibly discontinuous, homogeneous state feedback law.…”