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Question: question 1 review of signals and systems you are given...

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Question 1: Review of Signals and Systems You are given a dynamic system described by the following differential equation. The initial conditions are zero. Given that Oh-2 rs and ζ-0.707-1/V2, determine the transfer function x(s/us) a) b) Compute by hand the time response x() as a function of time only. Use inverse Laplace transforms. Use Matlab to plot the time response. Is Matlab consistent with your closed form solution. The input u(), is the unit step Now let ζ-0.005. Draw the bode diagram for the system. Clearly show the low and high frequency asymptotes. Use hand calculations to compute the magnitude and phase of the system at 5 different input frequencies. Specifically, compute the magnitude and phase at input frequencies a-1.0 rad/sec and ω-10rad/sec. c) d) For input u(t)-sin(t), plot the steady state output for several cycles. Then plot the response again to an input of frequency u(t)-sin(10t). Use Matlab to compute and plot the time response. How does the time response relate to your computations in part c) above.
Question 2: Review of Discrete Time Systems You are given a dynamic system described by the following differential equation. The nitial conditions are zero. a) Write the equation in state space matrix fornm b) Given that 2 rs and 0.707-1/2 and there is a sampling rate ofT 0.05 sec. Write out the discrete time state space equations. Show your hand calculations. Assume that there is a zero order hold on the input signalu c) Compute the Z transform of the discrete time state space equations. d) Convert the Z transform back into the time domain and write out the second order difference equation that describes the sampled version of the given system
Question 3: Least Squares Solutions Simulate the following system y(k)-1.5y(k-1)+0.7y(k-2) u(k-1)+0.Su(k-2) +e(k) Where e(k) is a white noise. Create data sets where e(k) has a variance of 4.0 and 9.0 The input( is a random binary signal with uk 10. Create the data sets with N-400 points. a) Compute the least squares estimate of the parameters given the uncorrupted data b) Compute the least squares estimate of the parameters using the two corrupted data c) Use first, second and third order models to estimate the parameters. Compare the set. That is for a data set of (y, u) and c(k)-0. sets. What is the standard deviation of the parameter estimates? parameter estimates, the prediction errors and the cost function for different model orders. Plot on the same graph the actual and predicted outputs. Do this for both sets of corrupted data.
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