Page 182 - Modern Control of DC-Based Power Systems

146 Modern Control of DC-Based Power Systems

minimization of a quadratic cost functional. Its synthesis is thus a sub-
problem of optimal control.
A common method for the design of a state controller is the pole place-
ment. The eigenvalues of the closed loop and thus its dynamics are defined
selectively as it was performed with LSF in Section 5.1.1. The disadvan-
tages of this method are that the performances of individual states are not
placed in the foreground and actuator saturation or actuation effort can be
considered only indirectly. Both are, however, often desired in the practical
application and are enabled by the LQR controller. For this purpose, a cost
function J with the following form is assumed for a finite time horizon T.

T
1 T 1 ð T T
Ju; x 0 Þ 5 xTðÞ Sx TðÞ 1 ðxTðÞ Qx TðÞ1 utðÞ Ru tðÞÞdt (5.91)
ð
2 2
0
The states and actuating variables are factored in quadratically. With
the weighting matrices S, Q, R the end values of the states, the state tra-
jectories, and the actuating variable trajectories are prioritized. With the
diagonal elements of Q the velocity of the single states can be driven to
zero. If the system has only one input R will be a scalar. The bigger the
value of R chosen, the slower the control would be. S is used when con-
sidering a finite time horizon to minimize the final values of the states, if
the time is not sufficient to drive the states to zero. When considering an
infinite time horizon, this weighting does not apply because the states for
t-N have to trend to zero as otherwise the integral would not
converge.
The synthesis of this control is highlighted for the infinite time hori-
zon care in the following steps; for more in depth explanation the reader
is referred to [33]. A continuous time linear system, defined by:
(5.92)
_ x 5 AxðtÞ 1 BuðtÞ
with a quadratic cost function as defined in (5.93)
N
ð
T
T
Ju; x 0 Þ 5 ðxTðÞ Qx TðÞ1 utðÞ Ru tðÞÞdt (5.93)
ð
0
The feedback control law that minimizes the value of the cost is given
by:

u 52 Kx (5.94)