It is almost the same way as how you demonstrate the stability proof for the conventional sliding mode controller. What exactly is your problem?

No, it is not same because those controllers aren't lyapunov base as conventional smc .

If so, please provide the details of your high-order SMC law. Let's see if the Lyapunov stability theorem can be applied. I think you probably have used the non-Lyapunov control design method.

I mean quasi-continous high order smc

Please read the article as follow:

https://ieeexplore.ieee.org/document/1532412

Hi Behzad,

Lyapunov stability methods usually require some continuity conditions, which cannot be satisfied in sliding mode control.

However, based on some LaSalle's unknown works, new results eliminate the need to even mention continuity, and so, can be and have been applied to sliding mode control.

Look for:

Selina Pan and J. Karl Hedrick: Tracking Controller Design for MIMO Nonlinear Systems With Application to Automotive Cold Start Emission Reduction,

Journal of Dynamic Systems, Measurement, and Control, Oct. 2015,

and similar works by same authors.

A review paper on the new Lyapunov results is

I. Barkana: “Can stability analysis be really simplified? (From Lyapunov to the new theorem of stability – Revisiting Lyapunov, Barbalat, LaSalle and all that)” Mathematics in Engineering, Science and Aerospace (MESA),Vol. 8, No. 2, pp. 171-199, 2017

Regards,

Itzhak

I will read all of them

Thank you for answereing