Problem 2
In a certain country the weather forecast takes place as follows. The percentage of sunshine per day is measured. At day \(k\) there has been \(a_{k} \%\) sunshine. The forecast for the day thereafter is made according to $$ \hat{a}_{k+1}=\left(6 a_{k}+3 a_{k-1}+a_{k-2}\right) / 10 $$ where \(\hat{a}_{k+1}\) is the forecast. Write the system in state space form for this forecast where the \% of sunshine of today is the input and where the forecast for tomorrow is the output. What is the dimension of the state?
Problem 3
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.
