Hello estimated colleagues, I am studying Sliding mode controllers for a nonlinear plant, which I have linearized.
Now I have a second-order system described as follow: A=[0 1 ; -a2 -a1], B=[0 ; b0], C=[1 0].
I want the system to follow the dynamics described as: A=[-13.6 -64 ; 1 0]. B=[1 ; 0], C=[0 64].
How can I design an SMC for that case?
Thanks in advance,
Marcelo
Hi Marcelo Vinicius de Paula
The reference trajectory is given by:
xᵣ = 1 – e–6.8·t·cos(4.21426·t) – 1.61357·e–6.8·t·sin(4.21426·t)
The system
x'' = b₀·u – a₁·x' – a₂·x
The error dynamics
e = x – xᵣ
e' = x' – xᵣ'
e'' = x'' – xᵣ''
e'' = b₀·u – a₁·x' – a₂·x – xᵣ''
The sliding surface
s = e' + e
The sliding mode control law, u can be obtained by solving
d/dt {(d/dt + 1)·e} = – k·sign(s)
e'' + e' = – k·sign(s)
b₀·u – a₁·x' – a₂·x – xᵣ'' + e' = – k·sign(s)
u = (– k·sign(s) + a₁·x' + a₂·x + xᵣ'' – e')/b₀
You seemed to have co-written a paper related to sliding mode control design.Conference Paper Wind Distributed System Based on Switched Reluctance Generat...
That is true Yew-Chung Chak but that was not my part on the work. And now I am trying on my own. Anyway, thanks for your answer.
In addition, a Liapunov function V = (1/2) * s ^ 2 is chosen to demonstrate stability.
The sign function can also be replaced with a sigmoid function to reduce the chattering effect.
Thanks for the addition Marcel Nicola !! I Changed the sign function by the switching surface itself as in:
d/dt {(d/dt + 1)·e} = – k·s
d/dt {(d/dt + 1)·e} = -k((d/dt + 1)·e)
However, I think using the sigmoid function may be more appropriate. I'll try both approaches and compare them.
Thanks again
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