Discussion
The minimum number of states required to describe the two degree differential equation :
Answer: B
The minimum number of states is 2.
An nth-order differential equation needs n state variables, one for each independent initial condition. A second-order equation such as \(\ddot{y} + a\dot{y} + by = u\) needs two, usually \(x_{1} = y\) and \(x_{2} = \dot{y}\), giving
\[ \begin{aligned} \dot{x}_{1} &= x_{2} \\ \dot{x}_{2} &= -b x_{1} - a x_{2} + u \end{aligned} \]
One state cannot capture both the value and the rate of change. Three or four states would be more than the minimum, a non-minimal realisation.
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