Bode Plot for Controller Design

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1 Bode Plot for Controller Design Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by

2 This Lecture Contains Bode Plot for Controller Design Example Bandwidth and it s significance Assignment Joint Initiative of IITs and IISc - Funded by

3 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.

4 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

5 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 From the phase plot find out at which frequency the phase plot will be about This will happen at omega = 2.03 rad/sec. The corresponding Gain is db 5

6 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: x = This corresponds to K = 130.8/10 = The corresponding root-locus is shown below: 6

7 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 s + 2). The Bode plot for this function is provided in the next slide. 7

8 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

9 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

10 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

11 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

Course Outline. Time vs. Freq. Domain Analysis. Frequency Response. Amme 3500 : System Dynamics & Control. Design via Frequency Response

Course Outline. Time vs. Freq. Domain Analysis. Frequency Response. Amme 3500 : System Dynamics & Control. Design via Frequency Response Course Outline Amme 35 : System Dynamics & Control Design via Frequency Response Week Date Content Assignment Notes Mar Introduction 2 8 Mar Frequency Domain Modelling 3 5 Mar Transient Performance and