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MIT16_30F10_rec11 - Recitation 11 Time delays Emilio...

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Recitation 11: Time delays Emilio Frazzoli Laboratory for Information and Decision Systems Massachusetts Institute of Technology November 22, 2010. 1 Introduction and motivation 1. Delays are incurred when the controller is implemented on a computer, which needs some time to compute the appropriate control input, given a certain error. More in general, the evaluation of sensory information aimed at deciding the best course of action, will require a finite computation time. 2. In some systems, delays may also be part of the physical plant. For example, on an airplane, the effect of lift variation at the main wing are felt on the tail when the vortices shed by the wing reach the tail plane. A rather extreme example is remote tele-operation: communication with a deep-space spacecraft or planetary rover may require several minutes. 2 The frequency response of a time delay A time delay is an operator that transforms an input signal t �→ u ( t ) into a delayed output signal t �→ y ( t ), with y ( t ) = u ( t T ), where T 0 is the amount of the delay. Clearly, this is a linear operator: you can check easily that the delayed version of a linear combination of signals is equal to the linear combination of the delayed signals. Let us indicate this operator with Delay T . In order to compute the transfer function of this linear operator, let us compute the Laplace transform of the output as a function of the Laplace transform of the input. We get Y ( s ) = L [ y ( t )] = y ( t ) e st dt = u ( t T ) e st dt = u ( τ ) e sT = e sT U ( s ) , 0 0 0
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from which we conclude that Delay T ( s ) = e sT . What is the frequency response of the time delay? We need to study the function ω �→ Delay T ( ) = e jωT . Recall that the polar representation of a complex number z C is z = | z | e j ( z ) , from which we deduce immediately that | e jωT | = 1 , e jωT = ωT. This is not unexpected, since the time-delayed version of a sinusoid of unit amplitude and zero phase u ( t ) = sin ( ωt ) is another sinusoid, y = sin ( ω ( t T )), with unit magnitude and a phase delayed by ωT . (The peak of the input is reached at times t p = π/ (2 ω ) + 2 πn , where the phase of the input is equal to π/ 2; at those times, the phase of the output is π/ 2 ωT .) See Figure 1 . -1 -0.5 0 0.5 1 0 5 10 15 20 y(t) t Figure 1: Time-delayed sinusoid. The blue line represents the signal y ( t ) = sin(2 π/ 10 t ) (for t 0); the red line is the same signal delayed by 2.5 time units, i.e., Delay 2 . 5 [ y ]. Since ω = 2 π/ 10 rad/s, and T = 2 . 5 s , ωT = π/ 2: the phase of the time-delayed signal is lagging that of the original signal by exactly π/ 2 radians, or 90 degrees.
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