It is desirable to use well-designed controllers to maintain building temperature with solar collector space-heating systems. One solar heating system can be described by [10]
dx1/dt = 3x1 + u1 + u2
and
dx2/dt = 2x2 + u2 + d
where x1 = temperature deviation from desired equilibrium, and x2 = temperature of the storage material (such as a water tank). Also, u1 and u2 are the respective flow rates of conventional and solar heat, where the transport medium is forced air. A solar disturbance on the storage temperature (such as overcast skies) is represented by d. Write the matrix equations and solve for the system response when u1 = 0, u2 = l, and d = 1, with zero initial conditions.