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Q1 (60 p.) : Consider the digital control system above with G(s) = 0.6 e^{-T_d s} / (0.3 s + 1) where the time-delay is T_d = 20 ms and the sampling period is T = 20 ms. Then, answer the following questions. a) Find the discrete-time open loop transfer function D(z)G?(z) where D(z) = K and Z {G_{ZOH}(s)G(s)} = G?(z). ( Hint: Use Z {0.6 / (0.3s+1) s} = 0.6 z / (z-1) - 0.6 z / (z-0.935) for T = 20 ms. ) b) Sketch the change of the location of the closed loop poles of the system (root locus plot) in z-plane with respect to K for 0 < K < ?. c) Find the critical gain value K_{cr} such that the closed-loop system is unstable for K > K_{cr}. Use the root locus plot you found in (b). d) Design a digital controller D(z) which satisfies the closed-loop response without overshoot and makes the steady-state error zero to step inputs.

          Q1 (60 p.) : Consider the digital control system above with G(s) = 0.6 e^{-T_d s} / (0.3 s + 1) where the time-delay is T_d = 20 ms and the sampling period is T = 20 ms. Then, answer the following questions.

a) Find the discrete-time open loop transfer function D(z)G?(z) where D(z) = K and Z {G_{ZOH}(s)G(s)} = G?(z). ( Hint: Use Z {0.6 / (0.3s+1) s} = 0.6 z / (z-1) - 0.6 z / (z-0.935) for T = 20 ms. )

b) Sketch the change of the location of the closed loop poles of the system (root locus plot) in z-plane with respect to K for 0 < K < ?.

c) Find the critical gain value K_{cr} such that the closed-loop system is unstable for K > K_{cr}. Use the root locus plot you found in (b).

d) Design a digital controller D(z) which satisfies the closed-loop response without overshoot and makes the steady-state error zero to step inputs.

Q1 (60 p.) : Consider the digital control system above with G(s) = 0.6 e^-Td s / (0.3 s + 1) where the time-delay is Td = 20 ms and the sampling period is T = 20 ms. Then, answer the following questions.

a) Find the discrete-time open loop transfer function D(z)G?(z) where D(z) = K and Z GZOH(s)G(s) = G?(z). ( Hint: Use Z 0.6 / (0.3s+1) s = 0.6 z / (z-1) - 0.6 z / (z-0.935) for T = 20 ms. )

b) Sketch the change of the location of the closed loop poles of the system (root locus plot) in z-plane with respect to K for 0 < K < ?.

c) Find the critical gain value Kcr such that the closed-loop system is unstable for K > Kcr. Use the root locus plot you found in (b).

d) Design a digital controller D(z) which satisfies the closed-loop response without overshoot and makes the steady-state error zero to step inputs.

Added by John P.

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