Problem 2b -- Does this mean exactly or at least fifty percent?

Question: In question 2b it says “design a controller that settles 50% percent faster than the default controller”. Does this mean exactly 50% or at least 50%? For example, if my settling time for 2a was, say, 30 seconds, would you want us to design a new controller that settles in exactly 15 seconds or could it settle anywhere from 0 to 15 seconds?

Answer: you can find any controller that allows a settling time at least 50% faster than the default. Finding the controller that gives you precisely the 50% faster settling would require mathematical manipulations that you do not know yet. Moreover, keep in mind that in the real world one always has to deal with uncertainty. Therefore as far as settling times and performances you will be more likely to encounter requirements specified by inequalities rather than exact equalities.