In many papers, we use homogeneous controller of negative degree, so that we proof the asymptotic stability to deduce the finite time stability.
Let us take, the pure double integrator system
\dot x = y; \dot y = u,
with u = -l*sign(y + |x|^0.5sign(x))
according to Levant and Emelyanov, for $l > 0.5$, our system is finite time stable. I agree.
Now, let us take $0
Negative homogeneity degree+asymptotic stability always imply finite-time stability.
In the considered case you probably get "twisting-like" convergence.
For a non-pure case, like
\dot x = y; \dot y = u +d,
with u = -l*sign(y + |x|^0.5sign(x))
what could be the way to finite time stability proof using homogeneity?
Just the same, if d is a constant. If d \in [-c,c], then one gets a homogeneous differential inclusion
\dot x = y;
\dot y \in [-c,c] -l*sign(y + |x|^0.5sign(x)),
\deg x = 2, \deg y = 1, hom. degree of the system = -1
Thanks Prof. Levant.
We can give proof for finite time convergence, but, could we know the finite time by any method?
Hello, Pyare Mohan. When c = 0, one may be able to compute the convergence time by integrating the trajectories. I have not tried this but it seems in the state-space, one could look at two partitions: one below the curve y+ |x|^{0.5}sign(x) and the other above the same. In a twisting-like convergence, there is no sliding mode other than that in the origin and hence integration is possible. Then, taking an arbitrary initial condition and integrating step-by-step may give you a geometric series giving a finite settling time. When c is not equal to zero, things become a little bit more tedious but this case can be tackled using majorisation approach. Other way to compute convergence time is by applying Lyapunov theory. There are a few Lyapunov functions proposed in the literature for cases similar to the one being discussed where \dot{V}\leq - K V^{\alpha}, \alpha \in (0,1) holds true. This way, one has the required convergence time when comparison principle is applied. The challenge in this route is, however, to find (or guess) such a Lyapunov function successfully. I will happily stand corrected by Professor Levant (and all others) on anything wrong in the above answer, though.
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