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 exogenous signals, (4-46) is of a form which makes possible the interpretation of using optimal control techniques for the selection of the feedback gains.
Now let us assume that the stabilizing gain K := [K1, K2] is found by solving the algebraic Riccati equation for optimal control. That is, the positive semidefinite P satisfying
Q + PFA + F P - PGAR- G P = 0 (4-48)
is obtained and K is selected as
K = R1G'P (4-49)
A
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
F 0 G
FA [ H , GA [ (4-50)
If the reference signal r(t) and the disturbance w(t) are applied at time t=O then the following quadratic performance index is minimized