Ece 382 homework determine the transfer function of the

Question

Consider a simplified cruise control system for an automobile as shown in Figure 1.

(A) Determine the transfer function of the uncompensated system and use MATLAB to obtain its root locus. Determine the dominant closed-loop poles of the uncompensated system and the time-domain specification of the uncompensated system such as the rise time t_r, the peak time t_p, the percentage maximum overshoot %M_p, and the settling time t_s due a unit step input. Show your calculations in the report.

(B) Design a lead compensator G_c^lead(s) and a PD compensator G_c^pd (s) such that the dominant closed-loop poles of the compensated system will be at s₁^∗,s₂^∗ = -3 ± j5. Determine the time-domain specification such as the rise time t_r, the peak time tp, the percentage maximum overshoot %M_p, and the settling time t_s due a unit step input. Show your calculations in the report.

(C) What are the transfer functions of the lead and PD compensators G_c(s) and their respective G_d(s)? Use MATLAB to obtain the root locus of the compensated system and obtain the open-loop gain of G_d(s). Identify and verify all the closed-loop pole locations. Attached your root-locus plot (in table form and graphical form to show the closed-loop pole locations). Use MATLAB to show the unit step response of the compensated system, and verify and discuss whether the design objectives have been met or not.

(D) Consider that the dominant closed-loop poles of the uncompensated system are satisfactory with the transient response, but its static velocity error constant K_v is not satisfactory. It is desirable to have K_v^Gd = 30 sec^-1. Design a lag compensator and a PI compensator to satisfy the design objective. What are the transfer functions G_c(s) of your respective compensators? Use MATLAB to verify your dominant closed-loop poles location of the compensated system, determine its open-loop gain, and verify that the unit step response satisfies the design objectives. If the design objectives are not satisfied, explain why they are not satisfied. Attach your MATLAB results (not all the pages of your MATLAB output; carefully include the portion of results that reflect your MATLAB work).

(E) Consider the design objectives of placing the dominant closed-loop poles of the compensated system at s₁^∗, s₂^∗ = -3± j5 and K_v^Gd = 30 sec^-1. Design a lag-lead compensator and a PID compensator for it. What are the transfer functions G_c(s) of your respective compensators? Use MATLAB to verify your dominant closed-loop poles location of the compensated system, determine its open-loop gain, and verify that the unit step response satisfies the design objectives. If the design objectives are not satisfied, explain why they are not satisfied. Attach your MATLAB results to justify your design.