QUESTION $1($Test$1-$SEM$2-2020-$QUEST$-$III$)$

$1\mathrm{.}1$ Speed of response, stability and accuracy are usually considered as common specifications to be achieved on closed$-$loop control systems. Among the three, stability can be thought of as the most important specification. Please justify this proposition by giving a practical example.

$1\mathrm{.}2$ State and then clearly explain at least, three methods you can use to study the stability of closed$-$loop control systems.

QUESTION $2($Test$1-$SEM$2-2020-$QUEST$-$III$)$

FIGURE $1$ shows a closed$-$loop control system of an industrial process, where the parameter $K$ is the qain of the proportional controller.

FIGURE $1$: Closed$-$loop control system.

$2\mathrm{.}1$ Reduce this closed$-$loop system to a single block diagram.

$2\mathrm{.}2$ Construct the Routh table of this system.

$2\mathrm{.}2$ Briefly discuss the stability of this control system. Your discussion should include the following:

a$)$ The range of the parameter $K$ that stabilizes this closed$-$loop system.

b$)$ The range of the parameter $K$ that will make this closed$-$loop system to be unstable.

c$)$ The condition of the parameter $K$ that will make this closed$-$loop system ${?}^{}$

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marginally stable.