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Basim T. Kadhem | 31
problem is the challenge of analyzing and creating an accurate By adjusting the filter gain, Kf, the Target Feedback Loop
control system. (TFL), GKF, has to be constructed in this design step. This
approach is frequently known as the (LQG) approach. It is
For the robust control problem, numerous strategies have important to note that this filter also avoids torsional
been created, and others are currently being investigated. interference, which prevents negative damping of torsional
However, the Linear Quadratic Gaussian with Loop Transfer oscillations [18].
Recovery (LQG/LTR) methodology is especially alluring
because of how effectively it handles plant uncertainties in a B. Controller Design
systematic and easy-to-understand manner [20–24].
An optimal control problem exists in this stage. In order to
A block schematic of the system with the robust controller
is shown in Fig. 5. All of the subsystems seen in Figs. 1, 2, 3, restore the TFL, we must use the optimal control technique to
and 4 are represented by the block referred to as "system." The
objective is to build a reliable controller that generates three solve for the full state feedback regulator gains Kc. The metric
control signals to account for differences in speed (USf, USr and
Usc). While the signal Usr and Usc are intended to help the for performance is provided by (7)
SVC and TCSC by damping torsional modes, the signal Usf is 9 = ?-,[;)(<) + &(6+&] =>
intended to help the power system stabilizer (PSS) by damping
electromechanical hunting modes [23]. This setup is appealing Where Qc and Rc are positive definite matrices that penalize
because it is simple and affordable to implement with actual
systems, which should be the end goal, and needs minimal the states and controls, respectively, and q > 0 is a scalar design
information (just the output).
parameter. In equation (7), y is the system's output and is the
input vector [Usf, Usr, Usc]. The best control law can be found
in:
Usf & = -.+# (8)
.+ = 6+)*%(5+ (9)
Usr
?refg -S LQG / LTR Usc where Pc satisfies another algebraic Riccati equation:
Controller "(5+ + 5+" - 5+%6+)*%(5+ + ;*(7+* = 0
System ?g (10)
If one is able to create a Kc and tweak Kf so that GKF(S)
has the desired loop shape over the frequency range relevant to
our performance and robustness concerns, then K(s) is a robust
Fig. 5. System with robust controller. compensator.
There are two steps in the design process. A filter design Given that the plant is stabilizable, detectable, has a
makes up the first, while a controller design makes up the
second. The LQG/fundamental LTR's tenet is to treat the minimum phase, and has fewer outputs than inputs, there is
unstructured uncertainties on the plant as noises in the
processes and measurements. In this scenario, a "fictitious" such a compensator, and the closed-loop system is internally
filter created to exclude these disturbances effectively
eliminates the consequences of uncertainties. The Kalman stable [12]. Fig. 6 depicts the configuration of the dynamic
filter is used to create a target feedback loop (TFL) with the
required loop shape in the first design step. This loop acts as robust controller K(s) that was created through the
the point of convergence for the controlled system. In the
second step, the Linear Quadratic Regulator (LQR) is mostly aforementioned procedures.
used to regain the target loop shape.
A. Kalman Filter Design
A linear model's state space representation is: Fig. 6. Dynamics of the LQG/LTR controller.
!"
!# = "# + %& + '( (1) IV. MODEL ORDER REDUCTION
(2)
) = *# + +& + , The control design techniques, like LQG or H¥ , result in
controllers that are at least as ordered as the plant, and typically
where w and v are zero-mean white-noise Gaussian processes higher due to the incorporation of the necessary additional
weights. To reduce the complexity of the final controller and
with Qf and Rf, respectively, covariance’s. Here, will be used as simplify the design process, model order reduction is
necessary. For proper control design, the reduced plant that
design parameters in the LQG/LTR technique to create a was employed in the design had to be a close approximation of
the full order counterpart. Thus, the following is the main issue
compensator that satisfies the required requirements. The state raised.
estimation, error, and gain equations for the Kalman filter are Determine a low-order approximation Gr(s) such that the
!"$ infinity norm of their difference ||G – Gr||8 is sufficiently small
!# = "#- + .%[) - *#-] + %& (3) given a high-order linear model G(s).
!& = 2" - .%*34 + '( + .%' (4)
!#
.% = 5%*(6%)*
(5)
where Kf stands for gain Kalman filter and e is the error in state
estimation. The Riccati equation's positive-semidefinite
solution is Pf. (6)
5%"( + "5%-5%*(6%)**5% + '7%'( = 0