Answer Happy

Problem 2: (65 Points) In this problem, all parts may be done independently and in any order except part (d) which needs parts (a) and (c). An LTI SISO system has internal representation: X = 4X+bu y = cx with; 2 3. b= -R C=(-32) For this system a- Find, using any method you prefer the state feedback gain k so that the resulting system has eigenvalues la =-1+1 12 =-1-j. Is the resulting system capable of regulation? Prove your assertion. Is the resulting system capable of tracking a constant input other than zero? Prove your assertion b- Design an observer for this system, using any method you prefer, with eigenvalues iz =-3.1. =-3. Are you able to use Lyapunov equation to do this design? If yes, how? If no, why? C- Design a first order, (reduced order), observer with eigenvalue is =-3. d- Draw and interconnect the system and the observer of part (c) and use output of the observer with the feedback gain of part (a) to obtain a closed loop system e State the condition for, and check the system to see whether you can have a robust design so that constant disturbances would be rejected and parameter variations would not affect steady state system output. If possible design a robust scheme that will put the eigenvalues of the system at l =-1+) 1 =-1-j1=-1. f Design a minimum order compensator so that the overall system has poles at P. =-1+1, P, =-1-j.P; =-1. Calculate the transfer function of the resulting system. Does it have the assigned poles? If yes, show it. If no. Why? Is the resulting system capable of regulation? Prove your assertion. Is the resulting system capable of tracking a constant input other than zero? Prove your assertion