I am new to the MPC idea and I am trying to understand the key concepts but there are two things which I found confusing and I didn't find answers regarding to them.
The first one is about the optimized control signal sequence which can be computed from the cost function. If we want to predict, say five steps further, then we will have 5 control signals. After the calculation is done, we will apply only the first signal from the sequence to our system, and them remaining four will be "wasted" (I read that it can be used as an initial guess for the next optimization but thats not my point here). My first question is why don't we just predict one step further, and instead of optimizing five signals, we restrict our optimization to just one, which makes the computation faster?
The second question is about constraints. Lets say that I have some restriction on my input signal, say $0 < u < 5$. With some math, we can include these constraints to the optimization task but it takes more time to solve. Why don't we just do an unconstrainted optimization and after our input signals are ready we apply our contraints on them? Obviously this can not be done for state contraints, but i am interested about input constraints.
Thanks for your answers in advance.