Hi @Achraf
Without knowledge about the physical nature of the dynamics, your questions can only be adequately answered through extensive testing of the model's validity and an analysis of the simulated results. However, based on your description of the system's hysteretic response, it appears to exhibit behavior akin to a population growth model, where this "hysteretic-like" behavior is dependent upon the initial population value. It is important to note that the population growth model is autonomous, meaning it does not explicitly depend on an external control variable.
Your initial course of action should be to diligently research the nature of the dynamical system in existing literature. Subsequently, you should attempt to fit the proposed model with the available data. It might also be worthwhile to investigate whether the system can be accurately represented as a linear parameter-varying (LPV) system. If all other attempts prove unsuccessful, you can resort to estimating the transfer function model by employing techniques found in the System Identification Toolbox. Only then will you be able to reliably assess whether a tuned PI controller is adequate for handling the system or not.
y0 = [0.1 0.01 0.001 0.0001];
for j = 1:numel(y0)
F = ode;
F.ODEFcn = @(t, y) y*(1 - y);
F.InitialValue = y0(j);
tFinal = 20;
sol = solve(F, 0, tFinal);
plot(sol.Time, sol.Solution), hold on
end
hold off, grid on
xlabel('Time'), ylabel('Displacement')
