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LECTURE 18 An Example of Bode-Based Control System Design
The block diagram of an antenna angular position command control system is shown below.
motor:
antenna:
controller: . position sensor: .
CLOSED LOOP SPECIFICATIONS:
(S1): unit ramp steady state error no greater than 0.025o.
(S2): closedat a gain crossover frequency .
(a)Assuming that the closed loop system is stable, then it is a Type-1 system. Because it is a unity feedback system, the error constant is: .
Hence, for a unit ramp, , this requires . So we must have
It follows that our controller must have the form where
(b)Design a unity static gain lead controller to obtain open loop phase of 120o at
The Bode plot for is at right.
At , we currently have ~3.9dB and -183o.
We will center the lead controller here, with : & .
and . The augmenting lead controller is:
(c)With the inclusion of the lead controller, at the OL phase exactly 120o, and the magnitude is now 3.9+12.35 = 16.25 dB (= 6.494). Hence, we need a unity static gain lag controller must have high frequency attenuation . We must choose the upper frequency to be sufficiently less than , so that the lag controller phase is essentially back to zero at this frequency. I will set . So. Then. Hence,
(d) The final controller is:
We see have CL at .
The PM is slightly low due to the fact that the lag compensator phase
at is not exactly back to zero. In fact, it is -2.42o.
(e)The plots below validate the design. The left plot is the closed loop response to a unit step command. The right plot is the error response to a unit ramp command. They are compared to those obtained using the root locus pole placement method. Using that method, the specifications, in addition to (S1), were: (S2): All time constants no greater than 0.5 seconds, (S3) Dominant complex pole damping ratio at least 0.85.
We see that the frequency domain design is better in many respects. The rise time is faster, as is the settling time. The ramp response is ~4 times as fast. And the initial ramp response peak error is lower. The frequency domain also has two more advantages:
(A1) It only requires an OL Bode plot; not a transfer function model.
(A2) It can accommodate time delays far easier that the root locus method.
Explain why our design began by addressing the steady state error, as opposed to addressing the PM.