Problem 1.1 Consider again the storage tank example (1.4), but now with a slightly modified effect of the input on the output, such that bu(t) flows into the system. On the basis of this a priori knowledge of the tank system, different types of representation will be investigated. (a) Give some physical conditions under which b = 1. (b) Give the differential equation describing this system in terms of the relationship between u(t) and y(t) and provide the solution, under zero initial conditions, for a unit input, such that u(t) = 1 for all t. (c) Represent the system in terms of an impulse response or convolution model and give the continuous-time impulse response. (d) Represent the system in state-space form. Are there other state-space forms that lead to the same input–output behavior? If there is, give an example. If not, motivate your answer. (e) Is this system linear, time-invariant, and causal? Justify your answer.