69
Equation (4-46) is a linear dynamic equation modeling the transient part
of the state trajectory of the system LC. Figure 4-5, shows a typical
illustration of the actual, steady-state, and transient trajectories
which might result when a sinusoidal signal is being tracked. It is
important to note that neither w(t) nor r(t) appear in (4-46) so that
these exogenous signals play no role in the transient trajectory or, in
otherwords, how fast tracking occurs. In addition, without the exo
genous signals, (4-46) is of a form which makes possible the interpre
tation of using optimal control techniques for the selection of the
feedback gains.
Now let us assume that the stabilizing gain K := [K^, 1^] is found
by solving the algebraic Riccati equation for optimal control. That is,
the positive semidefinite P satisfying
Q + PFa + F^P PGaR_16P = 0 (4-48)
is obtained and K is selected as
K = R_1G^P
(4-49)
Here, Q > 0 is a symmetric matrix of dimension n+pr x n+pr, R > 0 is a
symmetric matrix of dimension m x m, and
FA =
F
-BH
0
A
9
G
A
G
0
(4-50)
If the reference signal r(t) and the disturbance w(t) are applied at
time t=0 then the following quadratic performance index is minimized