For the system of Problem 9.6-5, use a transfer function approach to show that the transfer function $Q\left(z\right)/U\left(z\right)$ is first order (even though the system is second order) and is equal to $X\left(z\right)/U\left(z\right)$. Hence the mode of $Q\left(z\right)$, which is $(A-GC{)}^k$, is not excited by the input signal $u\left(k\right)$.

Problem 9.6-5

Consider the first-order plant described by

and a prediction observer described by (9-38).

x(k + 1) = Px(k) + Qu(k) y(k) = Rx(k)