Consider the unconstrained, SISO version of MPC in Eq. 20-65. Suppose that the controller is designed so that the control horizon is M=1 and the weighting matrices are Q = I and R =1. The prediction horizon P can be chosen arbitrarily. Demonstrate that the resulting MPC controller has a simple analytical form.
K _{c} \triangleq\left( S ^{T} Q S + R \right)^{-1} S ^{T} Q (20-65)
The unconstrained MPC control law has the controller gain matrix:
K _{c}=\left( S ^{T} Q S + R \right)^{-1} S ^{T} Q
For this exercise, the parameter values are: m=r=1( SISO ), Q = I , R =1 and M=1
Thus (20-65) becomes
K _{c} \triangleq\left( S ^{T} Q S + R \right)^{-1} S ^{T} Q (20-65)
K _{c}=\left( S ^{T} Q S + R \right)^{-1} S ^{T} Q
Which reduces to a row vector: K _{c}=\frac{\left[S_{1} S_{2} S_{3} \ldots S_{P}\right]}{\sum\limits_{i=1}^{P} S_{i}^{2}+1}