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# ps9 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.060: Principles of Automatic Control Fall 2003 PROBLEM SET 9 Due: 12/04/2003 Problem 1: Representing a Pure Delay in the Frequency Domain One situation in which frequency response techniques are useful is when a transfer function can not be represented easily in terms of poles and zeros. A very common example is a pure delay, which is always present in any control system implemented on a digital computer (e.g. Simulink). For a pure delay of time τ , the output y ( t ) in terms of the input u ( t ) is given by: y ( t ) = u ( t τ ) 1. Using the definition of the Laplace transform, find the transfer function G ( s ) = Y ( s ) U ( s ) corresponding to a pure delay. 2. Draw the polar plot for G ( s ) for a time delay of 1 second. (The polar plot is the part of the Nyquist plot corresponding to 0 < ω < .) Problem 2: Nyquist Plots with Negative Gain For each of the following open-loop transfer functions, draw one Nyquist plot for K > 0, and then another Nyquist plot for K < 0. In

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