This problem is intended to give you more insight into controllability and observability. Consider the circuit in Fig. 7.92, with an input voltage source u(t) and an output current y(t).
(a) Using the capacitor voltage and inductor current as state-variables, write state and output equations for the system.
(b) Find the conditions relating R 1, R 2, C, and L that render the system uncontrollable. Find a similar set of conditions that result in an unobservable system.
(c) Interpret the conditions found in part (b) physically in terms of the time constants of the system.
(d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the conditions derived in part (b) (that is, when the system is uncontrollable or unobservable).
