Question

You will look at both 1^st and 2^nd order approaches as you perform your analysis. Use math lab and Simulink as needed.

Please show all calculations and any other means of analysis / simulation for full credit.

Question three is the one needed to be solved, but 2 is needed to solve 3.

Question 2) (25 points) Suppose instead of the cooling chamber modeled as a 1t order system, that the chamber was modeled as a 2nd system due to the closed in nature and is described by the closed loop diagram Plant Command input Output G(s) error H(s) Where K1(a) s(s+a) G(s)where a-6.78, K1-6.78, and the controller block K-2 a) Find the closed loop equation for T(S) which can be rewritten as T(s) in terms of a plant gain, damping ratio and natural frequency b) Discuss the advantages and disadvantages of the relationships between the natural frequency and the damping coefficient in terms of the performance criteria for %os, settling time, and the peak time c) What type of response is the system, i.e. Overdamped, Critically-damped, Underdamped and what are the pole locations for T(s)? Justify your answer d) Provide a simulation to match with your analysis in parts a-c. Question 3) (30 points) Using the analysis from Question 2 Plant Command input Output G(s) error H(s) a) For this system, for a step input from 0 to 1 where 1 is the 'on' value, is there steady state error? If so how much? b) using the controller block K find a gain that results in a having a %OS of 20% and a settling time within 3 seconds? Explain your answer c) For your answer in part b) suppose the input was a ramp input, is there steady error now, and if so how much? Justify your answer. unity ramp analogous to a step.) (Note: In Simulink, the ramp function is call 'ramp' and is a d) Modify the controller block to remove the steady state error to be less than for the ramp input as used in part c. e) Further modify the controller block such that the output approaches the input ramp such that the steady state error is within 5%, and that it occurs within 1 second. Explain your approach and why you used it. f For parts a-e provide a simulation to compare with your analysis.

Answer 1

SI s+6チろ 2 x45. 9684 2 stb. 78 s-t 2245-9684- 6.18 556 1517

AIDO 了wワ 3-14 8-963