Bode diagram program with the Ti-84 Plus (CE) page 2
PI-control
In Electrical engineering, PI control is a frequently used control principle. The Ti-84 Basic program can help you to gain a better understanding of this method of control. It is known that if only a proportional gain is used, a difference between the actual speed and the desired speed in a motion control system will occur if disturbances are present. Adding a parallel integrator with a time constant of τi to the proportional gain solves this problem for static and low-frequency disturbances.
Although the PI controller compensates low-frequency disturbances, it decreases the phase margin (stability) at 0dB. It can be shown that if taui follows the rule below, the phase margin decreases with about 5 degrees. Kp is the proportional gain for a 60° phase margin (see page 1 of the Bode diagram).
taui=1/(0.1*0.52*51.12)=0.38 sec
Drawing the Bode diagram of an open loop with PI controller (with Kp=0.52 and taui=0.38 sec} should give a phase margin of approximately 55° (60° - 5°). This results in a phase of -125°.
Inserting the PI controller, Hr(s) with Kp=0.52, TAUI=0.38 sec., gives as results a phase margin of 180deg-125.7deg=54.3 deg at 51.18 rad/sec, which is in exact agreement with the Matlab results below.
0dB @ x=1.71=51.18 rad/sec Phase margin 180-10*12.57= 54.3deg x=1.71 equals to 51.18 rad/sec
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Bode plot with Matlab
Closed loop analysis with the program BODEHOC.8XP
At the end, we notice that it is also possible to create a Bode plot of the closed loop system with Kp=0.52 and taui=0.38 sec. The closed loop results :Hc=HrHs/(1+HrHs) are given:
Bandwidth : -3 dB at 86.79 rad sec. In agreement with Matlab
-3dB @ X=1.94 X=1.95 equals 86.9 rad/sec .
A Bode plot with Matlab showed that at -3 dB, the frequency was 86.7 rad/sec, which
agrees well with the results from the TI-84.
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