(1)
We rewrite the two equations to make our presentation more clear.
Equation (1): \dot x1 = x2;
Equation (2): \dot x2 = -k1 sign(x1) |x1|^alpha1 - k2 sign(x2) |x2|^alpha2
Proof:
Without loss of generality, we choose the Lyapunov function
V(x1, x2) = (1/2) x2^2 + (k1/(alpha1+1)) |x1|^(alpha1+1)
then we can obtain
Equation (3): \dot V(x1, x2) = x2 \dot x2 + k1 sign(x1) |x1|^alpha1 \dot x1
By substituting Equations (1) and (2) into Equation (3), we can obtain
\dot V(x1, x2) = x2 (-k1 sign(x1) |x1|^alpha1 - k2 sign(x2) |x2|^alpha2) + k1 sign(x1) |x1|^alpha1 x2
\dot V(x1, x2) = x2 (-k1 sign(x1) |x1|^alpha1 - k2 sign(x2) |x2|^alpha2 + k1 sign(x1) |x1|^alpha1)
\dot V(x1, x2) = x2 (- k2 sign(x2) |x2|^alpha2 )
\dot V(x1, x2) = - k2 |x2|^(alpha2+1) < 0
This correctly shows that the differential equation, \dot V(x1, x2), is asymptotically stable about the origin (refer to the following link).
http://en.wikipedia.org/wiki/Lyapunov_function
By LaSalle's theorem, the closed-loop system is asymptotically stable, and the origin is a finite-time stable equilibrium.
This guarantees finite time stability (using negative homogeneity) for double integrator system.
End of Proof.
(2) The paper [BB97] is a good reference for the proof.
[BB97] S.P. Bhat and D.S. Bernstein, "Finite-Time Stability of Homogeneous Systems," Proceedings of IEEE ACC, Albuquerque, New Mexico, June 1997, pp. 2513-2514.
(3) Recent discussions on control system design approaches can be found in the following RG link.
"What are trends in control theory and its applications in physical systems (from a research point of view)?"
https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2
Thanks Prof. Hong,
Further, What I read from literature that we cant calculate finite time by this method, Is there any scope to calculate the finite-time?
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