It could be taken as u_sw=-(1/b)ksgn(s) where k is defined as in Applied nonlinear Control by slotine. Does the definition vary for advanced SMC like non-singular terminal SMC?
The first step is to define a "Sliding Surface" and make your system dynamic reach there and then stay there afterwards. We normally take
surface = S = sum of states = 0 and then define a control law that ensures it. Now one way to approach it is by defining a Lyapunov function as
V=1/2(S^2) so that V'=SS' and then take the control input U such that V' is negative definite (i.e. if V' = F(x) + G(x)U take U=1/G(x) [ -F(x) - Ksign(s) ] and your V' becomes
V' = -Ksign(s) which is negative definite and ensures that the system trajectory reaches the surface S=0 in a finite time and then stays there.
So we take our system dynamics to a predefined surface which ensures that origin of the closed loop system is asymptotically stable.
A very good reference on this topic is the Book "Non-Linear Systems" by Hassan K Khalil Chapter 14 (3rd Ed). I have it in pdf so ask me if you can't find it.
Hope i answered your question satisfactory.
Thank you very much for your response. I appreciate it if you could send me the book as I only found the 2nd edition.
Dear Muhammad and samira,
It's not necessry to design a sliding surface and this approch did not work in general if the surface are not lipschitz. The objective is to stabilize a perturbed integrator chain.
You can refer to my proposed approach wich you can stabilize a perturbed integrator chain, based on controller wich stalilize a pure integrator chain, and based on Lyapunov analysis.
https://www.researchgate.net/publication/261466137_Robust_and_adaptive_Higher_Order_Sliding_mode_controllers
Conference Paper Robust and adaptive Higher Order Sliding mode controllers
Dear Samira,
I think the response given By Muhammad Ahsan and Mohamed Harmouche, are OK, also the Book of Prof. Hassan K. Khalil is interesting.
However, for sliding mode I think you can refer to the following books :
Sliding mode, by UTKIN
SLIDING MODE CONTROL IN Engineering, by W. PERRUQUETTI & J.P. BARBOT
(http://books.google.fr/books/about/Sliding_Mode_Control_In_Engineering.html?id=_EeES0_bR0wC&redir_esc=y)
and if you need application, you also refer to my book :
Sliding mode based analysis and identification of vehicle dynamics
(https://www.researchgate.net/publication/233816738_Sliding_mode_based_analysis_and_identification_of_vehicle_dynamics?ev=prf_pub)
Book Sliding Mode Based Analysis and Identification of Vehicle Dynamics
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