Question 4: In the following figure the force u is applied to the mass m. The spring with the stiffoess coefficient k and the damper with damping coeffieient b model the coupling between the mass and the wall. a) Write the equations of motion of this system with input u and outpur y (Take m=1, k=100 and b=0.01 ). b) Find the state ipace representation of this system. (Find F,G,H,J matrices). c) Is this syatem controllable? Why? d) Design a linear state feedback regulator for this system ruch that the closed loop poles are it -2 and -5 . (Find the gain K.) e) Simulate the elosed loop system with a Simullink modet, - Model be plant with Simulink inteprators and Simulink gains - Use following initial conditions: x2 anitial =1. - Model the the state feodback controtler with a Simulink gain. - Add "time" and "to workapace" blocks. - Simslate the sysiem and plot x1,x2,y and u verius time. - Discuss simulation recults. 9) Is this system observable? Why? e) Design an obierver such that the observer poles are at -10 and -15 . h) Simulate the closed loop system with the observer. - Model the obscrver with a Simulink "State-Space" block. - Simulate the syetem and plot x3,x2,x^1,s^3,y and u versus time. - Discuss simulation resulte. 1) Add a step reference input with compensator in the foedforward path - Usea slep of 1 uaits. - Simulate the system and plot x1,x2,P1,x2,y and u versus time - Discuss simulation realts. Is there a steady state error? m) Now we want to avoid the stcady stale crror. For this purpose, use the compensator in the feedback path itructure. N1 : Feedforward gain to avoid steady state error Nx - State reference gain to convert the reference for y into a reference for x [NxNu]=[FNGJ]1[01]N4+KNx - Use a step of 1 units - Snulate the syicm and plot x1,x2,x1x2,y and u versus time - Discus simulation results. Is there a steady stale erron