Problem $2.$ Block Diagram Reduction. $(30$ points$)$

Consider a cascade control system that has a block diagram shown below.

Figure $2\mathrm{.}1$ Block diagram of a cascade control system

In the inner loop, a flowmeter is used to measure the flow and transmit the measurement

to a flow controller that manipulates the valve opening. The set point for this inner loop is

supplied by an outer loop where the liquid level is measured and transmitted to a level

controller. In figure $2\mathrm{.}1,$ the subscript i denotes the various inner loop operations while the

subscript o denotes the various outer loop operations.

By using block reduction operations, show that the above block diagram can be reduced

to the one shown in Figure $2\mathrm{.}2.$ Show all the intermediate steps that lead to this result.

Figure $2\mathrm{.}2.$ Simplified block

diagram of the cascade control

system after applying block

reduction operations