Dear all,
In traditional sliding mode control, the approach is so clear. we design the sliding surface firstly and then find a control law that satisfy the condition of sliding mode existance. But in high order sliding mode, I think it should follow some principle that is similar to 1-sliding mode control.
From my point of view, in high order sliding mode control, we design a sliding function which is the expression of s and its r-1 time derivatives equal to zero. while in 1-sliding mode control, just s=0 . So in 1-sliding mode control, just 1-order accuracy while in high order sliding mode control, cos s=dot(s)=dot(dot(s))....... r-1 time derivatives..=0, then it will have a higher order accuracy. What's more, high relative degree makes the real control law to the system a smooth function,Then we base on some principles to design control law U.
Here is my problem, what principles we should take in high order sliding mode control to design control law? In 1-sliding mode control, the principle is s*dot(s)＜0, or dot(Lyfun) is negative defined. In some papers of Prof.Levant and Prof.Slotine, They gave some principle for high order SMC. But I not so clear, I want to communicate with people and see how you think of it.
Thank you very much!
Hi Dawn,
Consider a simple affine nonlinear system
- x1' = x2
- x2' = f(x) + Bu
- x2'' = f'(x) + Bu'
Design a sliding surface, s
- s = x2 + λx1
- s' = x2' + λx2
- s'' = x2'' + λx2'
Design a dynamic sliding surface, σ
- σ = s' + Λs
- σ' = s'' + Λs'
- σ' = x2'' + λx2' + Λ(x2' + λx2)
- σ' = x2'' + (λ + Λ)x2' + Λλx2
- σ' = f'(x) + Bu' + (λ + Λ)[f(x) + Bu] + Λλx2
To satisfy V' = σσ' < 0, then σ' must be chosen based on the reaching law
- σ' = − k sgn(σ).
- f'(x) + Bu' + (λ + Λ)[f(x) + Bu] + Λλx2 = − k sgn(σ)
Rearrange it to obtain the dynamic controller, u'
- u' = B−1{− k sgn(σ) − f'(x) − (λ + Λ)[f(x) + Bu] − Λλx2}
- u = ∫〈B−1{− k sgn(σ) − f'(x) − (λ + Λ)[f(x) + Bu] − Λλx2}〉
Consequently, a continuous dynamic sliding mode controller u is realized and the chattering effect should reduce significantly.
Hello Yew-Chung Chak,
Thank you very much! Amazing! Now, I can understand it more clear. It seems there is another way contributes to the principle I mentioned above. I remember Prof.Slotine and Prof.Levant putted forward different approaches to high sliding mode control. I try to find it out and share it here. Thank you again!
Please refer to these articles:
https://www.researchgate.net/publication/313576646_From_PID_to_Nonlinear_State_Error_Feedback_Controller
https://www.researchgate.net/publication/312046377_Improved_Sliding_Mode_Nonlinear_Extended_State_Observer_based_Active_Disturbance_Rejection_Control_for_Uncertain_Systems_with_Unknown_Total_Disturbance
https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design
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