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.