This value falls significantly below the acceptable range of 0.1 to 0.5,
highlighting the system’s poor oscillatory behavior. In order to solve
this problem, the eigenvalues are updated by applying the strip
eigenvalue assignment where h1= − 0.05 and h2= − 0.45 resulting in
updated eigenvalues:
Λ(A) = [0.0395 ± j0.7191, − 0.1569, − 0.0173, − 0.0372 ± j0.1947, −
0.274, − 13.70]
The eigenvalues are changed by the controller’s power, with trace (𝐵 F1)
is determined to be 0.1581. After recalculating the damping ratio for
the upgraded electromechanical mode, the results demonstrate that the
system is much more stable than before, falling squarely within the
predicted range of 0.1 to 0.5.
A positive semi-definite P matrix is obtained by solving the Lyapunov
equation with the identity matrix, then solving for , in order to verify
stability. Consistent with earlier observations, the eigenvalues of this
matrix validate stability enhancements. The steady-state gain, K(t), of
the controller is found by the function K(t) contains the following
values:
103 [0.0395 ± j0.7191, − 0.1569, − 0.0173, − 0.0372
± j0.1947, − 0.274, − 13.70]
Computer simulations are employed to examine the controller’s
performance. The damping behavior is assessed across a range of
parameters:
α varies from 0 to −20.5
β varies from 0 to −10.98
the best values are determined to be α=−20.53 and 𝛽=−10.98. A minor
disturbance is applied to the system using the parameter [0, 0.02, 0,
0, 0, 0, 0]t from Fig: 3.3. Quick reduction of
low-frequency oscillations is shown by the graphical results. The
controlled system exhibits much reduced oscillations when excited with a
unit step input from Fig. 3.1 compared to the uncontrolled one.
Rotor angle modifications, extra flywheel installations, and reactance
variations are some of the system factors that can be altered to conduct
further robustness testing. The controller still shows effective
damping, as seen in the simulation results from Fig. 3.2 even when 𝑀
varies by 30%.
This research highlights the suggested controller’s impressive
flexibility and durability. In contrast to more conventional approaches,
this adaptive controller can dynamically stabilize the system in the
face of perturbations and large-scale changes to its parameters. Its
real-world effectiveness is proven by eigenvalue reassignment, Lyapunov
stability verification, and practical simulation results.