Housekeeping

Housekeeping

close all; clear; clc; s = tf('s');

DERIVATION:

Beginning with a differential equation describing the plant to be controlled, the Laplace Transform may be used to move the equation from the time-domain into the s-domain. Because transfer functions do not account for initial conditions, this process allows us to ”separate” the input-output dynamics from the inputs and outputs themselves. A transfer function is represents the relationship of the system. Roots of the numerator are called ”zeros” (they drive the transfer function to 0) and roots of the denominator are called ”poles” (they drive the transfer function to ∞). For example, consider the equation for a simple spring-mass-damper:

m = 1;
b = 3;
k = 2;
G = (2)/(m*s^2  + b*s + k)
G =
 
        2
  -------------
  s^2 + 3 s + 2
 
Continuous-time transfer function.
Model Properties

INITIAL & FINAL VALUE THEOREMS:

Both the initial and final value theorems have a similar basis, but occur for different values of s. Recall from calculus that a system will maintain a steady value at the point where its derivative is 0. This same thought can be expanded to the s-domain, and remembering that the Laplace Transform of a function's derivative is the original transform multiplied by s such that:

. Due to the nature of s, the values used in the initial & final value theorems are the reverse of those for time such that:

Initial Value Theorem :

Final Value Theorem :

Using the Initial and Final Value Theorems, it is possible to examine some basic behavior of a system simply by multiplying by s and taking a limit.

Note: Regarding the Final Value Theorem, only stable transfer functions (poles all in the LHP) will have final values as an unstable system will tend towards infinity!

STEADY STATE RESPONSE:

Recall again the nature of transfer functions themselves as the response, such that if we wish to find the output of such a system given a particular input we can simply rearrange the equation to be:

and we can directly calculate the s-domain equation, and by the Inverse Laplace the time-domain equation for the system response. Thus by reducing the output to an equation in the s-domain, we can take advantage of the Final Value Theorem to observe the final value of our transfer function given any input!

Recall the Laplace Transforms of the Unit Step and Ramp functions:

Laplace of Unit Step Function:

Laplace of Ramp Function:

Then by application of the Final Value Theorem we have:

Unit Step Response:

Ramp Response:

steady_state_gain = dcgain(G);
fprintf('steady state gain(step) = %g', steady_state_gain)
steady state gain(step) = 1