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Unformatted text preview: Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Venkata Sonti * Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 560012 This draft: March 12, 2008 1 Introduction:Frequency domain design With lecture 4, we have completed what is considered the time domain method of system design. The reason being that time domain parameters, rise time, settling time, overshoot and steady state error were directly addressed by the PID controller which differentiates and integrates the time domain error. PID modifies effectively the pole positions and we can use the Root Locus method or Ziegler Nichols for finding the PID gains. The paramaters under the engineer’s control are Kp, Ki and Kd, which using the method of Root Locus modify the pole positions and influence the final objective, viz., t r ,t s ,M p andSSE . Let us say we model a system and then build it. We may find that the model differs quite a bit from reality. And so we have to upgrade the model and say we converge. Then we measure the t r ,t s ,M p etc. and we are not satisfied. Then we specify what we want and proceed with PID design. It seems simple enough. But, it is difficult to measure time domain characteristics. We need to give a step input which can be light and no energy input occurs. Or the input can be strong, then the system can become nonlinear. If the system is stiff, then the response can be very quick and die away. Measuring it is difficult. On the other hand the system may oscillate with a lot of low frequency dynamics. For every kind of system we cannot design a step inputu system. Locally, step inputs create nonlinear responses and deformations. On the other hand, the frequency response of a system can be very easily measured in the lab. Either a hammer impact is given or a sinusoid/random input is given and this method has been standardized. Now, does such a frequency response which captures sustained oscillation behavior capture information about transient response is the question. We will see that the frequency response is characterized by the resonance peak Mp, resonance frequency ω n , BW, Phase and gain margin. Indirectly, these indicate the time domain response parameters t r , etc. Here, the final requirement is still the time response. The tools here are compensators which use bode diagram and nyquist diagram methods. * 1 Nyquist Criterion Nyquist Criterion: For a closed loop system to be stable, the Nyquist plot of G ( s ) H ( s ) must encircle the (- 1 ,j 0) point as many times as the number of poles of G(s)H(s) that are in the right half of s-plane, and the encirclement, if any, must be made in the closkwise direction. (Read from Benjamin Kuo) 2 Nyquist or polar plots The characteristic equation is 1 + GH = 0 or GH =- 1 so that when the system is just unstable amplitude of GH = 1 and phase = 180 The Nyquist plot is the polar (amplitude phase) plot of a GH transfer function on the complex plane as...
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