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°.
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
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.
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 equals 86.7 rad/sec, which
agrees well with the results from the TI-84.
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