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## Q. 12.5.3

Consider the system (12.5-1), where $u(kT) = 0, ∀k$, and

$x(k+1)=Ax(k)+Bu(k), x(0)=x_0$        (12.5-1a)

$y(k)=Cx(k)$        (12.5-1b)

$A=\left[\begin{matrix} 1& 0\\1&1\end{matrix}\right]$,                  $c=\left[\begin{matrix}0\\ 1\end{matrix} \right]$

The output sequence of the system is {y(0), y(1)} = {1, 1.2}. Find the initial state x(0) of the system.

## Verified Solution

The observability matrix R of the system is

$R=\left[\begin{matrix}c^{T}\\ c^{T}A\end{matrix}\right]=\left[\begin{matrix} 0& 1\\1&1\end{matrix}\right]$

which has a nonzero determinant. Hence, the system is observable and the initial conditions may be determined from Eq. (12.5-7) which, for the present example, is

$\left[\begin{matrix}C\\ CA\\\vdots\\CA^{q-1}\end{matrix} \right]x(0)=\left[\begin{matrix}y(0)\\y(1)\\ \vdots\\y(q-1)\end{matrix} \right]$      (12.5-7)

$\left[\begin{matrix}0&1\\1&1\end{matrix}\right] \left[\begin{matrix} x_1(0)\\x_2(0)\end{matrix}\right]=\left[\begin{matrix}1\\1.2\end{matrix}\right]$

From this equation, we obtain $x_2(0)=1$ and  $x_1(0)=0.2$.