Page 183 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 147

The optimal gain is determined by the optimality principle, stating
that, if the closed-loop control is optimal over an interval, it will be opti-
mal over any subinterval within it. Where K is defined by:

21 T
K 5 R B PðtÞ (5.95)
And P is found by solving the continuous algebraic Riccati equation.
T 21 T
PA 5 A P 2 PBR B P 1 Q 5 0 (5.96)
The performance index is therefore minimized for the infinite time
horizon, thus obtaining a constant optimal state feedback gain, by assum-
ing that the controller will operate for a longer period that the transient
time of the optimal gains [33].

5.4.1.2 Kalman Filter
The ISPS is an application where possible transients happen in a very
short time and it is desirable to estimate those transients exactly, therefore
it is not reasonable to neglect them. Thereby applying a steady-state
Kalman filter with constant gain will not lead to optimal results, but it
yields to sufficient estimation performance. The advantage of the steady-
state Kalman filter in comparison with its more advanced versions [42,43]
is that it provides fast computation, as the covariance matrixes and the
Kalman gain are not updated. According to [33] it brings savings in filter
complexity and computational effort while sacrificing only small amounts
of performance.
The Kalman filter fulfills in this context the role of a linear optimal
dynamic state estimator. The Kalman filter has two main steps: the pre-
diction step, where the next state of the system is projected given the pre-
vious measurements, and the update step, where the current state of the
system is estimated given the measurement at that time step [44,45].
A key structural result that allows optimal state estimation to be con-
sidered as a dual to optimal control is the separation principle [46]. The
paper shows that the optimal feedback control of a process whose state is
not available may be separated, without loss of optimality, into two linked
tasks:
• Estimating optimally the state x by a state estimator to produce a best
estimate ^x.
• Providing optimal state feedback based on the estimated state ^x, rather
than the unavailable true state x.