- The Laplace Transform
*X*_{1}(*s*) for a particular system is shown below. The damping ratio will be allowed to take on different values. The following MATLAB functions might be useful in this lab:*roots, conv, plot, subplot, linspace, residue, for*. -
For each value of shown
above, plot the poles and zero of
*X*_{1}(*s*) in the*s*-plane. Using the*residue*function in MATLAB, determine the expression for the signal*x*_{1}(*t*), and plot the time signal with a maximum time of 20 seconds. Compare the responses for the various values of . Also, compare the pole locations of*X*_{1}(*s*) for the various values of . Discuss the form of the time response (from the equations and plots) in terms of the locations of the poles and zero of*X*_{1}(*s*). Factors to consider might be the amount of overshoot of the final value, the time it takes for the output to settle close to the final value (if it does settle), and the time it takes for the output to reach the final value for the first time. - Repeat all the activities of step 2 with the new Laplace Transform shown below. Compare the results of steps 2 and 3.
- Document your work in a typed report. The important issues are the relationships between the locations of the poles and zeros in a transfer function, the coefficients in the Partial Fraction Expansions, and the forms of the time-domain responses. State what the Region of Convergence (ROC) is for each of the transforms.

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