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## Q. 20.7

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)

## Verified Solution

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: $K _{c}=\frac{\left[S_{1} S_{2} S_{3} \ldots S_{P}\right]}{\sum\limits_{i=1}^{P} S_{i}^{2}+1}$