Q: Obtain the transfer function Y(s)/R(s) for the system shown in figure below G4(s) G-(s) 2 R(s) Y(s) G,(s) - G2(s) G3(s) G,(8) G,(s)

Q4) An LTI system is characterized by the differential equation below: d² = y(t) + 5 = y(t)+6y(t) = x(t) dt2 dt a) Write the characteristic equation. b) Find the characteristic roots. c) Write the form of the zero-input response. d) Write the final zero-input response if yzı(0)=2 and dyzi(0)/dt=-1. e) Plot the final zero-input response in Matlab/Octave and select the correct graph on the side: f) What are the characteristic modes for this system? 1.5 1 0.5 0 -0.5 1 2 3 4 (1) 0.5 0 -0.5 -1 -1.5 (2) 4 01234 (3) 1.5 0.5 234 (4)

4. Find the equivalent single transfer function G(s) = C(s) for the system shown below using Mason's R(S) gain formula. R(s) G₁(s) G₂(s) G₁(s) G5(s) G3(s) G6(s) H₁(s) H3(s) G7(s) H₂(s) ) C(s)

2. Let {A1, B1,C1, D1} and {A2, B2, C2, D2} be realizations of the transfer functions G1(s) and G2(s), re- spectively. Write down the realizations corresponding to the following transfer functions: (a) G(s) = G1(s) + G2(s). (b) G(s) = G1(s)G2(s). %3D (c) G(s) = [G:(s) G2(s)]. [G:(s)] G2(s)] 8- (d) G(s) =