71
oo
J = / [x^(t)QxA(t) + u' (t)Rtf(t)]dt
(4-51)
where
and
x(t) xss(t)
n(-t) riss(t)
(4-52)
u(t) = Cu(t) uss(t)] = Cu(t) + KlXss(t) + K2nss.(t)] (4-53)
The above observation simply points out that the transients behave in
some optimal fashion.
Now consider the state trajectory [x$s(t), nss(t)] which is unique
for a specific feedback gain K = [K-^, K2]. By assumption, this gain has
been selected by optimal control techniques. If this is not the case,
it is quite possible that the steady-state trajectory [xss(t), nss(t)]
as well as the input uss(t) might be improved in some respect. For
example, a different choice of gains might actually lessen the average
power required to maintain tracking of a certain reference signal.
Increased Degree of Stability Using the
Optimal Control Approach
In the previous section we discussed using the optimal control
approach to obtain stabilizing feedback gains for the linear servo
mechanism problem. The method can also be employed in the nonlinear
servomechanism problem when the linearized equations are used; however,
the control can no longer be considered optimal. It is sometimes
advantageous to make a slight modification on the performance index to
give a higher degree of stability to the linearized model. It was
previously indicated that when the linearization of NCT about the origin