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.
Consider the block diagram shown in 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.
The solution solves a control system by using MATLAB. The closed-loop transfer function is calculated.