## A feedback control system is shown in Figure The controller and process transfer functions are given by $ + 40 Ge(s) = K and G(s) s(s + 10) and the feedback transfer function is H(s) =1/ (s + 20). (a) Determine the limiting value of gain K for a stable system. (b) For the gain that results in mar- ginal stability, determine the magnitude of the imagi- nary roots. (c) Reduce the gain to half the magnitude of the marginal value and determine the relative stabil- ity of the system (1) by shifting the axis and using the Routh-Hurwitz criterion and (2) by determining the root locations. Show the roots are between – 1 and -2. Controller Process E.(S) ROS) G (S) Gís) Y(S) Sensor His)

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### ANS:

The image you sent shows a block diagram of a simple feedback control system. The system has a proportional controller with gain *K* and a feedback signal with gain 1/*D*. The input to the system is the torque *T*(*s*) and the output is the angular velocity *θ*2(*s*).

The transfer function of the system is given by:

```
G(s) = \frac{\theta_2(s)}{T(s)} = \frac{K}{1 + DKs}
```

This transfer function can also be written in the following form:

```
G(s) = \frac{\omega_2(s)}{T(s)} = \frac{1}{\tau s + 1}
```

where *τ*=*D*/*K* is the time constant of the system.

**Answer:**

The transfer function of the system shown in the image is:

```
G(s) = \frac{\omega_2(s)}{T(s)} = \frac{1}{\tau s + 1}
```

where *τ*=*D*/*K* is the time constant of the system.