SMC.md

SMC

DEfine the next states: $$ \dot{x}_1 = x_2\ \dot{x}_2 = \frac{1}{I}(W\cos{x_1}-\beta x_2+F_hl_m) $$

Can be written as: $$ \dot{x}_1 = x_2\ \dot{x}_2 = \frac{1}{I}[W\cos{x_1}-\beta x_2+(K_f\cdot v+b_f)l_m] $$

• $h(x) = \frac{1}{I}(W\cos{x_1}-\beta x_2+b_fl_m)$
• $f(x) = \frac{1}{I}(K_fl_m)$

Now, we define an error:

$$ x_e=x_1-x_r\\ \dot{x}_e=\dot{x}_1 $$

and the next sliding manifold: $$ S = ax_e+x_2 $$

where its derivative is: $$ \dot{S} = a\dot{x}_e+\dot{x}_2\ \dot{S} = a\dot{x}_1+h(x)+f(x)v $$

and the signal control law is: $$ v=-\rho\ \text{sign}(S) $$

To the system stability, we need to define the following: $$ \rho \geq \left|\frac{ax_2+h(x)}{f(x)}\right|\ \rho \geq \left|\frac{ax_2+\frac{1}{I}(W\cos{x_1}-\beta x_2+b_fl_m)}{\frac{1}{I}(K_fl_m)}\right| $$

Taken as limits $x_2\leq\pi$ and, we get:

$$ \rho \geq \frac{a\max{x_2}+\frac{1}{\min{I}}(\max{W}+\max{\beta}\max{x_2}+\max{b_fl_m})}{\frac{1}{\max{I}}\min{K_fl_m}}\\ \rho \geq \frac{a\pi+\frac{1}{I_0}(\max{W}-\max{\beta}\pi+\max{b_fl_m})}{\frac{1}{I}\min{K_fl_m}} $$