Consider the circuit that we have been using for homework below, where I have called the input, u, and the output, y. Do the following.
 

(a) Find a system of dynamical equations for this circuit, using our "typical choice" for state variables.

(b) Let R1 = R2 = 1, L = 1, C = 1/4. Find the state transition matrix, (t).

(c) Assuming that IL(0) = 0, Vc(0) = 1, find the function y(t) (you don't need to evaluate the convolution integral, but reduce its integrand to the simplest possible form.

(d) Find H(s) = Y (s)/ U(s). I'll think more highly of your answer if you find it two ways.

Solution:

Now all we need to do is divide the capacitor current by the capacitor value and the inductor voltage by the inductor value and we get the state equations. Finally, we need to recognize that the output is simply equal to the capacitor voltage. This immediately allows us to write down the dynamical equations as follows:

(b) Let R1 = R2 = 1, L = 1, C = 1/4. To find the state transition matrix, (t), we will first need to get the eigenvalues of the state matrix, A. I have done this below.

Having found the eigenvalues, we can find the time functions in the matrix polynomial expansion of the state transition matrix we got using the Cayley-Hamilton theorem. Specifically,

Recalling the polynomial expansion and using the above results, we have,

This defines the state transition matrix.
 

(c) Assuming that IL(0) = 0, Vc(0) = 1, we can find the solution to the state equations using the state transition matrix. This is done as follows:

Next, we put this result into the input-output equation to find the output, y. This gives us,

(d) Now let's compute the transfer function using the state space method.

Now let's compare this to the result we got from homework #1. There we found the transfer function using other methods.

Looks like it all agrees.