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Chapter 10 - Digital Control Systems - Section 2.4: Final Value Theorem

By Harold Rothbart and Thomas H. Brown
From Mechanical Design Handbook 2nd Edition

The final-value theorem for continuous systems

 (10.13)

is often used to find steady-state system errors and/or steady-state gains of portions of
a control system. The analog for discrete systems is obtained by noting that a continuous
steady response is denoted by X(s) = A/s and leads to the multiplication by s in
Eq. (10.13). Therefore, since the steady response for discrete systems is X(z) = A/(1 - z-1),
the discrete final-value theorem is

 (10.14)

For example, to find the dc gain of the transfer function,

  

let u(n) = 1 for n ≥ 0, so that

  

Applying the final-value theorem yields

  

and therefore the dc gain of G(z) is unity. In general, we see that to find the dc gain of
any transfer function, we simply substitute z = 1 and compute the resulting gain.

Since the gain of a system does not change whether represented continuously or
discretely, this calculation is an excellent check on the calculations associated with
determining the discrete model of a system.

 

© 2009 McGraw-Hill Companies, Inc. (The)
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