Hi,
Thanks for all the responses. I am aware that both a Nyquist plot and Bode plot provide insight into the response of a system after the transient effects have died out. More to the point, I am curious as to where this magical complex analysis using this magical s = a + j w comes from.
Let me elaborate using my original example, just modified slightly.
x_dot_dot + 1 x_dot + 2x = u
Back in my differential equations class we first solved equations such as the one above as follows:
Assume the steady state response is x = A e^(L t), then x_dot = A L e^(L t) and x_dot_dot = A L^2 e^(L t). We will sub these in:
A L^2 e^(L t) + 1 A L e^(L t) + 2 A e^(L t) = u
We will first solve the homogeneous solutions, so set u = 0 for the time being.
A L^2 e^(L t) + 1 A L e^(L t) + 2 A e^(L t) = 0 ( L^2 + 1 L + 2 ) A e^(L t) = 0 (L + 1/2 + (7/4)^0.5j)(L + 1/2 - (7/4)^0.5j) = 0
Thus the eigenvalues of the above equation are L = -1/2 +/- (7/4)^0.5j = a +/- j w . Then, using our assumed solution:
x = A e^ (a t +/- j w t) x = e^(a t) (A e^( j w t) + B e^( j w t ))
Next you can solve for the particular solution if u is known and augment it with the homogeneous solution. If u is not known the convolution integral must be employed.
SO, when we look at the asymptotic response we are assuming the e^(a t) effect has died out, and we are just looking at A e^( j w t) + B e^( j w t ) system. Okay, I buy that. To me however, there is a suspicious similarity to what I define as L (i.e. L = a +/- j w ) and the Laplace variable s.
I am looking for the connection between what I have been taught in controls classes (using Laplace transforms and the like) and what I have learned in DE classes. I all ready know the answer is in both classes we are being taught the same thing, just a different way (i.e many ways of of skinning a cat), but still, I want to know the connection or the thrust to always use the Laplace domain. I am not looking for an answer like "replace differential equations in the time domain, with an equivalent algebraic representation in the Laplace domain" either. I think one of the reasons we use Laplace stuff it enables us to better understand signals in the frequency domain, but evidently I don't know if this is true, of if it is true why we do it.
I find while I am using these tools (Laplace transforms, bode plots, Nyquist diagrams etc.) I am "faking it", or "going through the motions". I am executing what I have been taught over and over, but I feel I don't understand why we are doing things the way we do them. I no longer want to "fake it", but rather really understand the history and background as to why we use all this complex math.
JRF