ABSTRACT
Using the continuous system transfer function, find the initial and final values using the discrete forms of the IVT and FVT. Assume a unit step input and a sample time equal to 0.1 seconds.
And apply the ZOH:
Now we can use the tables in Appendix B where a= 2 and b = 4, and the transform for the portion inside the brackets is
A= 1-e-aT cos(bT)- ~e-aT sin(bT) B = e-2aT + l!..e-aT sin(bT)- e-aT cos(bT)
Recognizing that [z/(z-1)] cancels with portion of the ZOH outside of the transform, including the 6/20 factor, and substituting in for a, b, and T, results in
This is now the discrete transfer function approximation of the continuous system transfer function. To find the initial and final values, we can apply the discrete forms of the IVT and FVT. For both cases we need to add the step input since we have only derived the discrete transfer function, G(z), not the system output Y(z). In discrete form the step input is simply
To get the initial value, multiply G(z) by the step input and let z approach infinity:
(O) _ _ 1' Y( ) _ 1' 0.026z2 + 0.023z _ O y -Yo - lzl~oo z - lzl~oo (z-l)(z2 - 1.605z + 0.670)- With the discrete FVT the step input and the term included with the theorem
cancel, as they did with the continuous form of the theorem. For a unit step input only, we can thus simply let z approach unity in the discrete transfer function to solve for the final value of the system.
