Page 35 - IJEEE-2023-Vol19-ISSUE-1

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