Problem 55: (26 points) A linear time-invariant system can be represented by an ordinary differential equation, an impul

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Problem 55 26 Points A Linear Time Invariant System Can Be Represented By An Ordinary Differential Equation An Impul 1 (229.27 KiB) Viewed 32 times

Problem 55: (26 points) A linear time-invariant system can be represented by an ordinary differential equation, an impulse response function, a frequency response function, or a transfer function. From one form of representation, you must be able to derive the others. As an example, a certain linear time invariant system has a frequency response function whose exact magnitude and phase plots are shown in Figure 4. Bode Diagram 40 20 Magnitude (dB) 0 -20 -40 180 135 Phase (deg) 90 45 0 101 103 102 Frequency (rad/s) Figure 4: Exact magnitude and phase response of a LTI system. 1. (4 points) Can the system be described by a first-order differential equation? Justify your answer in one or two sentences to receive credit. 2. (8 points) Based on the magnitude and phase plots shown in Figure 4, determine the transfer function H(s) of the system. Let num and den be the the polynomial representations of the numerator and denominator of H(s). Check your answer by using the MATLAB command bode(num,den) to generate the exact magnitude and phase plots; verify your results by comparing them to key values from Figure 4. Attach a copy of your exact magnitude and phase plot obtained using the bode command, and add your name and section to the plot using the gtext command. 3. (4 points) Based on your result in part 2, find an ordinary differential equation that describes the response y(t) of the system to the input f(t). 4. (5 points) Based on your result in part 2, do you expect the unit-step response of the system to be underdamped, critically damped, or overdamped? What is the steady-state response of the system to a unit-step input? Verify your answers by using the MATLAB command step(num,den) to generate the unit-step response of the system, where num and den are the polynomial representation of the transfer function obtained in part 2. Attach a copy of the simulation output obtained using the step(num,den) command, and add your name and section to the plot using the gtext command. 5. (5 points) Suppose that the input to the system is given by the trigonometric Fourier series f(t) = 10 sin(10 t) + 10 sin(100 t) + 10 sin(1000 t). Using the magnitude and phase plots in Figure 4, determine the compact trigonometric Fourier series repre- senting the steady state response y(t).