Local-homogeneity-based global continuous control for mechanical systems with constrained inputs: finite-time and exponential stabilisation

Abstract

A global continuous control scheme for the finite-time or (local) exponential stabilization of mechanical systems with constrained inputs is proposed. The approach is formally developed within the theoretical framework of local homogeneity. This has permitted to solve the formulated problem not only guaranteeing input saturation avoidance but also giving a wide range of design flexibility. The proposed scheme is characterized by a Saturating-Proportional-Derivative type term with generalized saturating and locally-homogeneous structure that permits multiple design choices on both aspects. The work includes a simulation implementation section where the veracity of the so-cited argument claiming that finite-time stabilizers are faster than asymptotical ones is studied. In particular, a way to carry out the design so as to indeed guarantee faster stabilization through finite-time controllers (beyond their finite-time convergence) is shown.

A robustness study of a finite‐time/exponential tracking continuous control scheme for constrained‐input mechanical systems: Analysis and experiments

“…for constants a i ∈ (0, 1] and M ij > 0. An example is illustrated in Figure 2 for fractional and unitary values of a i ; further examples are illustrated for instance in the work by Zamora-Gómez et al; 11 a discussion about the different forms adopted by the functions ij in the finite-time and exponential convergence cases, and their effect on the control actions they are involved in, is found in the work by Zavala-Río et al 8 For each i = 1, 2, the functions in Equation (39) prove to be bounded strongly passive functions for…”

“…Lemma 4. For every j = 1, n, let j be a bounded strongly passive function for ( , a, b, ,ā, b), k j be a positive constant, k m = min j {k j }, k M = max j {k j } and S 0 as defined through (8). Then, in addition to item 1 of Lemma 3,…”

“…Actually, further efforts in the just-mentioned supporting direction have been made in the context of the control of mechanical systems; finite-time continuous stabilization of mechanical systems has been addressed under divers design contexts and analytical formulations. [6][7][8][9][10][11][12] In particular, the work by Zavala-Río and Zamora-Gómez, 8 where the problem of global finite-time continuous stabilization under input constraints was solved, carried out-within the developed control design context-a computer-simulation-based study on the veracity of the so-cited argument claiming that finite-time controllers achieve faster stabilization than asymptotic ones. This was actually shown to depend on the specific functions involved to achieve the required finite-time stability and the precision used to practically evaluate the stabilization time; furthermore, based on such observations, the study shows that the referred functions can be defined in a way through which finite-time controllers indeed prove to give rise to faster stabilization than their exponentially stabilizing analog versions.…”

Finite‐time consensus of Euler‐Lagrange agents without velocity measurements via energy shaping

“…There are, however, other equivalent versions that can be consulted in the available literature. 53,55,58,59 Lemma 2. Suppose that system (6) admits an h-approximation of the form .…”

“…The next lemma is a direct consequence of Lemma . There are, however, other equivalent versions that can be consulted in the available literature …”