Continuous finite-time stabilization of the translational and rotational double integrators

Abstract

A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator. Lyapunov theory is used to prove finite-time convergence. For the rotational double integrator, these controllers are modified to obtain finite-time-stabilizing feedbacks that avoid "unwinding."

On the finite-time stabilization of strings connected by point mass

“…Let T * the settling time of z. The same result is obtained for z(t), if we act by continuous feedback of the form [4] p := −sgn(ż)|ż| α − sgn(φ(z,ż))|φ(z,ż)| α 2−α ; α ∈ (0, 1), where φ(z,ż) = z + 1 2 − α sgn(ż)|ż| 2−α .…”

A note on finite-time and fixed-time stability

“…One is finite-time stability that means that the system converges within a finite time interval for any initial values; the other is fixed-time stability that means that the time intervals of convergence have a uniform upper-bounds for all initial values within the definitive domain. The previous works on this topic include Dorato 1961, Roxin 1966, Haimo 1986, Bhat and Bernstein 1998, 2000, Hong, et. 2002,.…”

A discrete tracking-differentiator design for active vibration control in maglev systems

“…where, 1 x is the tracking signal of the input signal () vk , 2 x is the differential approximation signal of () vk . r is the speed coefficient, h is the step, that is sample time.…”

“…Some scholars [2][3] have proposed many ways to obtain differential signals and linear differentiator is the simplest one. Linear differentiator with high-gain can provide n-1 order derivative, but this kind differentiator may contain disturbance directly in every level [4].…”