PLC ST programming

Question

I have a mathematical model of two dynamic systems which I need to implement in PLC, but I don't know how to implement differential equations in structured text language. Any suggestions on that will be helpful.

I have been thinking of converting differential equations to difference equation, but I don't have any idea how to do this.

$$\ddot{\Theta}=\dfrac{1-K_d\cdot\dot{\Theta}}{J} \tag {1} $$

$$\dot{i} = \dfrac{1}{L}(V-K_g k_f \dot{\Theta}-R i) \tag {2}$$

theta_dot_dot = 1/J*(1 - Kd*(theta_dot)) -------> Equation 1
i_dot = 1/L*(V - (Kg)(kf)(theta_dot) - R*i) --------> Equation 2

Answer 1

$$\ddot{\Theta}=\dfrac{1-K_d\cdot\dot{\Theta}}{J} $$

$$J\ddot{\Theta}+K_d\cdot\dot{\Theta}-1=0 $$ $$\ddot{\Theta}+\dfrac{K_d}{J}\cdot\dot{\Theta}-\dfrac{1}{J}=0 $$ Discriminant : \$ \dfrac{K_d^2}{J^2} -\dfrac{4}{J}\$ is it positive?

$$\dot{i} = \dfrac{1}{L}(V-K_g k_f \dot{\Theta}-R i)$$

$$i = \dfrac{1}{L}\int(V-K_g k_f \dot{\Theta}-R i)dt$$ You solve the second equation with trapz rule.

$$Sample_{act}=\dfrac{1}{L}(V-K_g k_f \dot{\Theta}-R i)$$ $$i = i+\dfrac{1}{2}\cdot T_{sample}\cdot (Sample_{act}+Sample_{old})$$ $$Sample_{old}=Sample_{act}$$