Bode Plot for Controller Design Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by
This Lecture Contains Bode Plot for Controller Design Example Bandwidth and it s significance Assignment Joint Initiative of IITs and IISc - Funded by
Bode Plot for Controller Design From the last lecture we have noted how we can use the Bode-plot for obtaining the frequency response of a system. This is often needed for the identification of the system. For example, a system showing -20db/decade decay indicates the presence of a simple pole, for -40db/decade decay indication of a second order system with two poles becomes evident. Bode plot can also be used effectively for Gain control (controller design) of a SISO system. In this lecture, we will illustrate this with the help of an example.
Example: A Position Control System R(s) + Power Pre amplification Amplification DC Motor Integrator Y(s) Position K 10 1 ( s 10) s 6 1 s For the position control system shown above, find out the controller gain K such that the system will show only 9.5% overshoot corresponding to a step input. 4
Steps for Design Choose an initial control gain of K=0.8065, so that at 0.1 rad/s, the gain will be 0dB. This will help in normalization. An overshoot of 9.5% is equivalent to a damping ratio of 0.6 which you may obtain by following the earlier lecture on specifications. Now, if you remember the approximate relationship between Phase Margin and ζ is given by: PM 100 In our case, the Phase Margin will be about 60 0. From the phase plot find out at which frequency the phase plot will be about 120 0. This will happen at omega = 2.03 rad/sec. The corresponding Gain is -44.2 db 5
Steps for Design [contd..] We need to increase the gain by 44.2 db such that it becomes zero at the phase margin. This means the gain required will be: 0.8065 x 162.18 = 130.8. This corresponds to K = 130.8/10 = 13.08. The corresponding root-locus is shown below: 6
Bandwidth of a System from Bode Plot Bode Plot can be effectively used to obtain the system bandwidth. The bandwidth of a system refers to the range of frequency beyond which h the magnitude of the closed loop response exceeds -3dB. By using a second order system approximation, one can also identify bandwidth of a system from the open loop frequency response. Assuming that the phase of the system would be between -135 to -225 degree. Consider a closed loop transfer function to be 1/(s 2 + 0.5s + 2). The Bode plot for this function is provided in the next slide. 7
Bode Plot and the Band-width The plot shows that the Magnitude falls below 3dB at approximately 1.76 rad/sec. Hence, the bandwidth of the system is 1.76 rad/sec orabout100 Hz. 8
Frequency responses below and above the bandwidth In order to realize the significance of bandwidth, let us consider two excitation frequencies one below the bandwidth (Case A:1 rad/sec) and one above the bandwidth (Case B: 3 rad/sec) and show the frequency response for the two. Case A when the excitation freq is well within the band Case B when the excitation freq is beyond the band 9
Assignment 1. A System to be controlled has the following transfer function: T ( s) 2 s( s K 10 s 8 4s 40) Use Bode Plot to select a rate feedback compensator. 2. Design gain for the following plant such that at least 60 0 Phase Margin is available. T s 3 s) ( s 1)( s 4s 5) ( 2 10
Special References for this lecture Feedback Control of Dynamic Systems, Frankline, Powell and Emami, Pearson Control Systems Engineering Norman S Nise, John Wiley & Sons Design of Feedback Control Systems Stefani, Shahian, Savant, Hostetter Oxford 11