# Control System Problem

Problem 1:

Consider the differential equation: d2y/dt2 + (3)dy/dt +2y = u

where y(0) = dy(0)/dt = 0 and u(t) is a unit step.

1. Determine the solution y(t) analytically.

I have the following(using Laplace transform):

[s2Y(s) - sy(0)] + 3[sY(s) - y(0)] + 2Y(s) = 1/(s)

s2Y(s) + 3sY(s) + 2Y(s) = 1/(s)

Y(s)[s2 + 3s + 2] = 1/(s) -> Y(s) = 1/(s)[s2 + 3s + 2]

roots = 1, -2, -1 thus

y(t)1 = 1/[(s+2)(s+1)] |s=0 -> y(t)1 = ½

y(t)2 = 1/[(s)(s+1)] |s=-2 -> y(t)2 = ½

y(t)3 = 1/[(s)(s+2)] |s=0 -> y(t)3 = -1

y(t) = 1/2e-2t - e-t = ½

Need to verify this and the next two porblems

2. Verify with MATLAB by co-plotting the analytic solution and the step response obtained with the step function.

Problem 2:

Consider the block diagram shown in Figure 1.

Figure 1

1. Use MATLAB to reduce the block diagram in Figure 1, and compute the closed-loop transfer function.

2. Generate a pole-zero map of the closed-loop transfer function in graphical form using the pzmap function.

3. Generate the list of poles and zeros.

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#### Solution Summary

The solution solves a control system by using MATLAB. The closed-loop transfer function is calculated.