Represent cruise control system with PID controller as an ode. Issue with Derivative term

I have been tasked with representing a simple cruise control system with a PID controller as an ordinary differential equation and estimating KP, KI, and KD parameters (I use torchdiffeq library for differentiation and simulation). I know, I can add error to the state to calculate the integral, however, I struggle with the implementation of the derivative. I believe there is a better way to represent the system without the use of ‘global’ variables.

Here is my current code:

    def forward(self, t, x):              
        y, E = x
        dydt = torch.zeros_like(x)
        
        workaround_int = y + y.trunc().detach() - y.detach()
        error = self.Setpoint - workaround_int
        
        inte = E 
        div = (error - self.Error_previous)/self.Dt
        
        pid = self.Params[0]*error + self.Params[1]*inte + self.Params[2]*div
        F_ego = pid
        
        D_ego = self.CdA*(y ** 2)
        
        if y > 0:
            F_net = F_ego - D_ego
        else:
            F_net = F_ego + D_ego

        dydt[0] = F_net/self.M
        dydt[1] = error
        
        self.Error_previous = error
        return dydt

If you need any additional information, I am happy to provide!

I have tried to change the system to be of a more correct ‘ODE’ form by setting the setpoint to 0. This however resulted in an inaccurate system, because of the trunctuated velocity when calculating error. Moreover, it did not solve the issue of having to approximate the derivative term with a finite difference.

I looked at similar problems but they all were related to the higher-order differential equations. I think that technically velocity is of second order but with applied control, it is regarded as first(?).

Unfortunately, I am not very familiar with the control theory and Laplace transform thus any help would be welcomed.