Probably, it would needed such a knowledge about bounds  or some estimation of  them. For instance, giving a certain structure to the upper-bounding contribution  of the unmodelled dynamics  with a couple of parameters acting on signals :

parameter1 * signal + parameter 2

Then estimate both parameters in an extended estimation scheme. This strategy worked in other control approaches to estimate upper-bounds of contribution of  unmodeled dynamics + noise.

There are many SMC strategies using adaptive gain tuning. They do not require the bound on the uncertainty.

Please have a look at this chapter penned by Utkin and Poznyak. (http://link.springer.com/chapter/10.1007%2F978-3-642-36986-5_2)

The  non-requirement of the uncertainties bound in designing a SMC can be accomplished via using the Adaptive  SMC. In this approach  the goal being to ensure a dynamic adaptation of the control gain to be as small as possible whereas sufficient to counteract the uncertainties and perturbations. For more details one can refer to the reference “Advances in Sliding Mode Control” Springer-Verlag Berlin Heidelberg 2013,  Chapter two which written by Vadim I. Utkin and Alexander S. Poznyak.

Designing SMC or any other control approaches without using the information of the uncertainties bound seems to be impossible if you need to guarantee the closed loop stability. This is clear because in general case when the uncertainty in the model is increased the controller has to put more effort into practice. 

Hello,

I solved partially the problem of adaptive HOSMC based on Lyapunov Analysis in CDC 12, and completely in IEEE CST2015.

https://www.researchgate.net/publication/261466137_Robust_and_adaptive_Higher_Order_Sliding_mode_controllers?ev=prf_pub

https://www.researchgate.net/publication/269710083_Control_of_PEMFC_Air-Feed_System_Using_Lyapunov-Based_Robust_and_Adaptive_Higher_Order_Sliding_Mode_Control

These controllers ensure for instance a convergence to neighborhood of zero.

feel free to consult my page on researchgate.

I'm working on impruving these results.

Conference Paper Robust and adaptive Higher Order Sliding mode controllers

Article Control of PEMFC Air-Feed System using Lyapunov-based Robust...

Thank you very much for all the answers!

I will peruse the book chapter and the papers recommended, and if there are any more questions, I will ask again.

Thank you.

Hi

I have read the 'Adaptive Sliding Mode Control' written by Utkin and Poznyak.

However, I found that they still use the 'sign' function which may result in chattering even though the gain K is minimized.

Is there any paper or article that does not use the 'sign' function, but use a continuous function? I am also wondering if there is an article that handles a MIMO (Multi-Input, Multi-Output) system.

Thank you very much.