(Single Input) Second Order Linear Time Invariant (LTI) Plant

The general form of a discrete time state space representation for a second order single input linear time invariant plant is as follow:

$$\begin{pmatrix}x_1 \\ x_2 \end{pmatrix}_{k+1} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix} \begin{pmatrix}x_1 \\ x_2 \end{pmatrix}_{k} + \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix} \begin{pmatrix} u \end{pmatrix}_{k}$$

where $\begin{pmatrix}x_1 \\ x_2 \end{pmatrix}_{k}$ is the state vector, and $\begin{pmatrix} u \end{pmatrix}_{k}$ is the input vector.