Further results on the global continuous control for finite‐time and exponential stabilisation of constrained‐input mechanical systems: desired conservative‐force compensation and experiments

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

“…Similar observations arose through different tests with alternative values of A dj and dj , j = 1, 3, in (40). For instance, Figure 4 shows additional results obtained with A d1 = A d2 = ∕18, A d3 = ∕6, d1 = d2 = 0.6, and d3 = 1 (for which B dv = 0.534 and B da = 0.5273), where the experiments were run with the same values for a i and in K i , i = 1, 2-that is, a 1 = 11∕20 and a 2 = 22∕31 for the finite-time controller, a 1 = a 2 = 1 for the exponential control algorithm, and K 1 = diag [5,48,3] and K 2 = diag[0.5, 4.5, 0.0015] for both (finite-time and exponential) controllers-and, in accordance to (11), for every i = 1, 2: M i1 = 6, M i2 = 14, and M i3 = 1 were taken for both (finite-time and exponential) controllers. Smaller posttransient variations are again corroborated to arise in the finite-time controller case.…”

“…Furthermore, from expressions (17), one may corroborate, after basic developments, that < 0 ≤ min{ 1 , 2 } ⇒ min{p 11 , p 12 } > 0. From this and (20b), W 1 (q,q) is concluded to be positive definite and radially unbounded, while it is obvious that W 2 (q,q) has the same properties too.…”

“…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…”

“…ij . Let us note that by such defined functions ij we have B j = M 1j + M 2j , j = 1, 3 (see (11)). Furthermore, the experiments were run taking initial conditions at q(0) =q(0) = 0 3 , and the desired trajectory as…”

“…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.…”