Consider a power trading system consisting of local battery storage $($for example, in individual homes$)$ referred to as node $1,$ and a central power grid storage, referred to as node $2.$ Local battery storage exchanges power with the central grid depending on which is in excess at a given time. In particular, the central grid loses power proportional to its divergence from ideal steady$-$state capacity, while local battery storage is "topped up$"$ by locally generated solar and wind power as a proportional supplement to power transferred from the grid. Consider a linearised model of this system of the following form:

$x_1^{}(t)=2x_2(t)$

$x_2^{}(t)=-x_1(t)-3x_2(t),$

Explain how the above model could describe the proposed grid mechanism, and explain the physical meaning of each term and coefficient. Then, write down updated expressions that allow a control input corresponding to the central regulator's ability to actively add or subtract power from the central grid, and an output expression that returns the total amount of power stored in the system. Write down the total system in the standard form of a linear state$-$space model.