ME 155A Homework Assignment 8 Spring ’21 Due Friday 6/4 by 11:59 pm PST via Gauchospace 1. (Critical Time Delay and the Phase Margin) (a) A pure time delay system is one where the output signal is...

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ME 155A Homework Assignment 8 Spring ’21 Due Friday 6/4 by 11:59 pm PST via Gauchospace 1. (Critical Time Delay and the Phase Margin) (a) A pure time delay system is one where the output signal is simply a copy of the input signal delayed by T time units. Mathematically this can be described as follows y(t) = u(t− T ), where u(t) is the input and y(t) is the output signal respectively. Although this system is not described by a differential equation of the type we have been considering, it is still a linear time invariant system. That means that the output Laplace transform is equal to the input Laplace transform times a transfer function, i.e. Y (s) = G(s) U(s). Using the shift property of the Laplace transform show that the transfer function of a time delay system (with delay T ) is given by G(s) = e−sT . This is an example of a system with an irrational transfer function (i.e. can not be written as the ratio of two polynomials). (b) Now let P (s) be a proper transfer function of a physical process. Let ωc denote the gain crossover frequency at which |P (jωc)| = 1. Use the Nyquist stability criterion and the definition of the phase margin γ = −π + ∠(G(jωc)) to show that the the critical delay τ that destabilizes the closed loop system is τ = γ ωc . 2. (Loop Shaping with Lead Compensators) Design a lead compensator for the open loop system P (s) = 1 s(s+1/10) using the lead compensator loop shaping method described in Lectures 7.2 and 7.3 to guarantee that the closed loop system has a phase margin γ = 42.0, a steady-state impulse response gain of 1.0. Attach the Bode plot generated using the margin function of the compensated system, as well as the uncompensated and compensated impulse response. (Hint: Consider using the [Kg,γ] = margin(·) and getGainCrossover(·) functions in MATLAB.) 3. (Loop Shaping with Lag Compensators) Design a lag compensator for the open loop system P (s) = 1 s(s+1)(s+2) using the lag compensator loop shaping method described in Lecture 7.3 to guarantee that the closed loop system has a phase margin γ = 42.0, a steady-state impulse response gain of 1.0 and a gain margin of at least 10 dB. Attach the Bode plot generated using the margin function of the compensated system, as well as the uncompensated and compensated impulse response. (Hint: You will may consider writing a helper function getPhaseCrossover(G,−180 + φM), where φM is the adjusted phase margin incorporating the design constraint and the anticipated additional phase shift from the lag compensator. Remember that the additional phase shift and the location of the zero of the lag compensator are free variables that affect downstream steps in the step-by-step process!) 1