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Morning and Good Day to all!!

I am trying to Model control for a Micro-controller which (under Real Time Development) does not support Continuous time solver. So, instead i have to use Discrete Time Solver. How do i replace SECOND ORDER INTEGRATOR with DISCRETE TIME INTEGRATOR? The equation i am modelling is a Second Order Differential Equation.

Equation: m.y" = F_coil - Ks.x -mg

where

y" = d^2(x)/dt

F_coil = ((ni)^2)/(2.g^2)

{ where n=number of turns

i=current

g=air gap

x = distance travelled by the solenoid plunger }

m = mass of the plunger

g = gravitational constant

Intentionally this is easy using a second order integrator.But when i use the discrete time integrator the output is completely different. Continuous time Integrator Output: 0.067 Discrete Time Integrator Output: 3207

HOW DO I ACHIEVE THE SAME WITH DISCRETE TIME INTEGRATOR?

My Attempt: One of my ideas was to solve the Differential Equation with laplace transform, get the function which could replace the differential equation. Is this a right approach?
Also while solving the same i realised that F_coil is not a constant during the complete simulation period. F_coil is only constant for a given time step!!