(Solved): 9. Given the unity feedback system of Figure P9.1 with G(s)=(s+2)(s+4)(s+6)(s+8)K Find the tran ...

Expert Answer

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To design a lag-lead compensator, we need to first determine the desired closed-loop poles of the compensated system, which will give us the desired settling time and damping ratio. Since the settling time of the compensated system is to be 0.5 seconds less than that of the uncompensated system, we can set the desired settling time as follows:
ts_compensated = ts_uncompensated - 0.5 = 4 - 0.5 = 3.5 seconds

Next, we need to determine the damping ratio of the compensated system, which we have been given as 0.5. We can use the following formula to relate the damping ratio to the closed-loop poles:
? = -ln(?(1-?2)) / ?(1-?2)
Solving for ?, we get:
? = 0.5
Using the desired settling time and damping ratio, we can calculate the desired closed-loop poles of the compensated system using the following formula:
s = -??n ± j?n?(1-?2)

where ?n = 4 / ts_compensated = 4 / 3.5 = 1.143 rad/s
Plugging in the values, we get:
s = -0.571 ± j1.032
We need to place the compensator zero at -5 to improve the steady-state error by a factor of 30. To achieve this, we can use a lag-lead compensator with the following transfer function:
Gc(s) = Kc (s+1/?1)/(s+1/??1)
where ?1 is the time constant of the lag network, ? is the ratio of the time constant of the lead network to that of the lag network, and Kc is the compensator gain.