Assuming the closed-loop system is stable, find the steady-state error if x,. (t) = 0 and d(t)…

1. Assuming the closed-loop system is stable, find the steady-state error if x,. (t) = 0 and d(t) = 1(t). This means that inputs are zero reference displacement and a unit step disturbance, and then calculate the steady state error e,(00)= x(w) — x,.(co) = x(co) which is caused by the disturbance signal d(t) = 1(t). If the controller co()Gs is replaced by a PD controller, i.e., Grr(s) = Kr(1+ Tas), what is the steady-state error e„(e.) with x,.(t) = 0 and d(t) = 1(t)? Explain why the steady state error is different using PD and PlD controller (2 Marks). 2. Determine the values of K,, a, T such that the closed-loop system of the inner attitude control loop, i.e., the transfer function from 61,.(s) to 13(s) exhibits a undamped natural frequency of co„ = 10 rad/sec and damping ratio at = 0.6. Next, open the Matlab script file ViS2_(12. m’, complete this file by inputting system parrameters bt, kr, J and selected K,, a, T, run the code, and plot the response of the inner attitude tracking loop with respect to a step input. Use the MATLAB ‘bode’ command to obtain the Bode plots of the compensated inner attitude loop. Find the gain margin and phase margin of the system based on the generated Bode plots. Next, check your results with the MATLAB ‘margin’ command (5.5 Marks). 3. Using the value Kr, a, T obtained in the solution of Question 2, and assuming there is no disturbance, i.e., d(t) = 0. Complete the Matlab script file `AS2_03a. m’ by inputing the values of system parrameters 65, k,„J and K,, a, T, open the Simulink file V1S2_03_Sim_EC, and then click the PID block to tune the proportional control gain of MD controller. Determine the values of Kr, Tr, and Ti of the PlD control law based on the Ziegler-Nichols’s second method. Next, complete the `AS2_(136. m’ by inputing the obtained values of K0, Tr, and Ti, and Kr, a, T, J, and then plot the system response. Given the performance of using the control parameters obtained above, discuss how to improve system performance, i.e., how to tune the values of K0, Td, and T, to improve closed-loop performance. It is noted this approach does not require any prior knowledge of the system to be controlled (3 Marks).