I'm studying Model Predictive Control, and basically, the most solid theory is developed for Discrete-Time Systems. For Continuous Non-Linear System is advised to linearize the system at a point of equilibrium and then discretize it or directly discretize the system and use discrete model predictive control theory, depending on the applications.
I would like to know more about how can be safe that the outputs we get with the discretized system coherent with the result we get with the continuous nonlinear system.
Particularly, I have the following questions.
I know that for linear systems the can have an exact discretization with Euler Discretization, so $x_d(0) = x_c(t=0)$, $x_d(1) = x_c(T)$ and so on, where $x_d$ is the discrete state, $x_c$ the continuous state and T the sample time. Still, what guarantees we have on points between the sample time, so between (0 and T, T and 2T, …. n*T and (n+1)*T)? How can be sure that these points are somewhat bounded?
Instead for a non-linear system, what can we conclude if we discretize the system? And what is the most common and safe procedure to have a successful discretization on a nonlinear ODE?
Best Answer
All digital control systems are discrete.
Sample time is chosen short enough so the system does not scape in between.
Majority of real systems tolerate noise and disturbances to an extend. So, small deviation is very normal.
Minor errors are corrected with the feedback and observation methods in the way that the error does not accumulate.
Linearization and discretization are not the same thing nor against each other. They are independent concepts.
If the system is nonlinear but linearization around the nominal point works, use it. But if the system is too mucn nonlinear and simple linearization is not sufficient think about using adaptive MPC or gain-scheduled MPC.