All the problems in this set are represented with the following diagram, without considering the measurement noise (that is N (s) = 0 ).

1. Consider the problem of controlling the position of a system with loop transfer function G(s) = 1/s(s + 5)

with a proportional controller Gc (s) = K . The transfer function of the sensor

is H (s) = 1/(s/2 + 1)

- Select the gain K to meet the following design requirements based on its response to a step input: (a) overshoot is less than 25% and (b) settling time (with a 2% criterion) is less than 5 seconds. In case the two specifications cannot be met simultaneously, you could compromise by relaxing them by the same percentage, and then select the gain K (K > 0) accordingly.

- Once the value of gain K is determined, check the resulted steady state error due to a step input, or a step disturbance.

- Suggest an approach to reduce or eliminate the steady state error caused by a step disturbance by using a step input.

2. Consider the problem of controlling an unstable system, whose transfer function is obtained as G(s) = 0.1/(s² +2s -8).

A controller in the form of G_c(s) = K(s +6)/(s +10) with a unity feedback H (s) = 1 has been selected for the problem. One of the design objectives is to maintain the output y(t) ≈ 0 when r(t) = 0 in the presence of disturbance. The design specifications, based on a step disturbance, are as follows:

- steady-state tracking error less than 1% in the presence of the disturbance
- settling time (with a 2% criterion) less than 4 seconds, and
- percent overshoot less than 30%.

Select the value of K (K > 0) to meet all the specifications and describe your approach used in your selection process. Plot the results based on your final selection of K value.