FAQ: In Section 6.4 (integral action), doesn't the value of z e depend on K?

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(Contributed by Richard Murray, 4 June 2009)

Q: Just before the example in Section 6.4, the text states

Note that the value of z_e is not specified but rather will automatically settle to the value that makes , which implies that at equilibrium the output will equal the reference value. This holds independently of the specific values of A, B and K as long as the system is stable (which can be done through appropriate choice of K and k_i).

Doesn't this mean that the result depends on K?

A: The working here is a bit confusing. The value of z_e will always converge to 0 (and hence y_e = r as long as K is chosen to be stabilizing. But it does not depend on the specific values of the state feedback, so perfect tracking is independent of the choice of gains (as long as the closed loop system is stable).