Consider a discrete-time single input linear regulator problem where the control is designed via a linear quadratic cost function. Say that the input is required to satisfy a constraint of the form |u(k)| ≤ ∆. Use a receding horizon MPC
set-up for this problem. Hence show that if,in the solution to the optimization problem,on ly the first control is saturated then
u(k) = −sat(Kx(k)) (23.10.1) is the optimal constrained control where u(k) = −Kx(k) is the optimal unconstrained
control (Hint: use Dynamic Programming and examine the last step of the argument).