Problem 5. (State equations for interconnections of linear systems) Let $H_i(s)$, i = 1,2, be two transfer matrices with state-space realizations ($A_i$, $B_i$, $C_i$, $D_i$), i = 1, 2, respectively. Suppose the dimensions of $H_i(s)$ are such that all the expressions in the following are well defined.
1. Serial connection: A block diagram that implements the system with the transfer function $H_{series} = H_2(s)H_1(s)$ is given below. Write the space-space model (A, B, C, D) of it. Note that A, B, C, D should be in terms of subsystem matrices $A_i$, $B_i$, $C_i$, $D_i$, i = 1, 2.
2. Parallel connection: Draw a block diagram on your own of a system that is the parallel connection of the two subsystems with the transfer function $H_{parallel} = H_1(s) + H_2(s)$, and find its state-space model (A, B, C, D) in terms of subsystem matrices $A_i$, $B_i$, $C_i$, $D_i$, i = 1,2.
3. Negative feedback connection: Draw a block diagram of a system that is the negative feedback connection of the two subsystems with $H_1(s)$ in the feedforward path and $H_2(s)$ in the negative feedback path, so that the overall system has the transfer function $H_{feedback} = (I + H_1(s)H_2(s))^{-1}H_1(s)$. Find its state-space model (A, B, C, D) in terms of subsystem matrices $A_i$, $B_i$, $C_i$, $D_i$, i = 1,2.
