Analysis of Singular Perturbations for a Class of Interconnected Homogeneous Systems: Input-to-State Stability Approach

An Analysis of Convergence Properties of Finite-Time Homogeneous Controllers Through Its Implementation in a Flexible-Joint Robot

“…To the best of our knowledge, there is not available literature regarding stability analysis of the consider class of systems and controllers. Thus, inspired by the results of [16], we show that the design of a controller making the whole system homogeneous avoids the undesired chattering and recovers the ideal finite-time convergence properties for the case under study. Regrettably, in most cases, information on the states of the fast dynamics is not available, then the presented strategy is not applicable at all.…”

“…Now, let us recall the results of [16] about the stability of a particular class of singularly perturbed homogeneous systems with different HD. For this, consider the interconnected system…”

“…Under the aforementioned hypothesis, the main result of [16] establishes a trade-off between the kind of stability of the interconnected system (2)-( 3) and the relation of the homogeneity degrees of each subsystems.…”

“…Inspired in the results of [16] (despite they cannot be applied for the consider case), the objective of this work is to verify if a "homogenization" of the closed-loop system ( 9)-( 14) can avoid the oscillatory steady-state response of its trajectories, i.e., if the first point of Theorem 2 can be realize for this particular case.…”

“…Recently, [16] has presented an analysis of the stability properties of singularly perturbed systems in the framework of homogeneity notions. Three types of stability were discovered by depending on the relation between the homogeneity degrees (HD) of the fast dynamics (FD) and slow dynamics (SD): global asymptotic stability (GAS) when both dynamics have the same HD and the singular perturbation parameter (SPP) is sufficiently small, practical GAS when the SD has a smaller HD, and local asymptotic stability when the SD has a greater HD.…”