We are given the time discrete system
$$
\begin{aligned}
x(k+1) &=\left(\begin{array}{cc}
0 & 1 \\
-2 & -3
\end{array}\right) x(k)+\left(\begin{array}{l}
0 \\
1
\end{array}\right) u(k) \\
y(k) &=\left(\begin{array}{cc}
2 & 1
\end{array}\right) x(k)
\end{aligned}
$$
Determine the transition matrix, impulse response function and the transfer
function of this system. Suppose the following periodic input signal is
applied to the system:
$$
u(k)= \begin{cases}0 & k<0 \\ (-1)^{k}, & k \geq 0\end{cases}
$$
What is the output response (take \(x(0)\) as the zero state)? Why is the output
signal not periodic?
The time-discrete, time-invariant, linear system characterized by matrices
\((A, B, C, D)\) is called controllable if for each \(x_{0}, x_{1} \in
\mathcal{R}^{n}\) a time \(k>0\) and a sequence \(u(0), u(1), \ldots\) exist such
that \(x\left(k, x_{0}, u\right)=x_{1} .\) The meaning of \(x\left(k, x_{0},
u\right)\) will be clear; the state at time instant \(k\), starting with initial
condition \(x(0)=x_{0}\) and having applied an input sequence \(u\). The system is
observable if a \(k>0\) exists such that for any sequence of controls \(u\) we
have:
$$
y\left(j, x_{0}, u\right)=y\left(j, x_{1}, u\right), \quad j=0,1, \ldots, k,
\text { implies } x_{0}=x_{1}
$$
The conditions in terms of matrices \(A, B, C\) and \(D\) for controllability and
observability are the same as in the time-continuous case. This will be shown
in the next theorem for controllability. Sometimes one distinguishes null
controllability \(\left(x_{1}=0\right)\) and reachability \(\left(x_{0}=0\right)
.\) It can be shown that "standard" controllability, i.e. with arbitrary
\(x_{0}\) and \(x_{1}\), is as strong as reachability (see also the proof of next
theorem); the condition is:
$$
\operatorname{rank}\left[B A B \ldots A^{n-1} B\right]=n
$$
wheras null controllability is not as strong as "standard" controllability.
