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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.

Closed loop system including the I-controller

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). 

transfer function of PI-controller
formula to calulate the integrating factor of the PI-control

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°.

Analysis open loop including PI controller
Bode diagram results for the optimized PI-controller

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

Open loop Bode diagram of drive system with PI control

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

Bode diagram of the closed loop system with some overshoot

                                                                     -3dB @ X=1.94                                            X=1.95 equals 86.9 rad/sec .

Bode plot of closed loop system by Simulink

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.

For questions or a request for the program, BODEHOC.8XP mail to ti84.org@gmail.com and the program will be sent. Don't forget to provide your email address!!

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