Nonlinear Systems Stability Analysis

Appendix A1: Application of VLF in Nonlinear Power system Stabilization

Consider the following smooth nonlinear large-scale system (Σ_N), which consists of N affine nonlinear subsystems such as:

$\begin{array}{l}{\text{Σ}}_N:{\dot{x}}_i=f_i(x_i)+g_i(x_i)u_i+h_i(x)+k_i(x_i)d_i,\\ y_i=z_i(x_i).\end{array}$

(A1.1)

where x_i ∈ Rⁿu_i ∈ R^md_i ∈ R^q and y ∈ R^p x_iu_i and y_i are the state vector, the input, and the output of each subsystem, respectively. f_i(x_i), g_i(x_i), Z_i(x_i), and h_i(x) are smooth functions with appropriate dimensions and d_iis the disturbance. It is assumed that f_i(x_0i) = 0, h_i(x₀) = 0 and z_i(x_0i) = 0.

Defining the system states, inputs, outputs, and disturbances as follows:

$x=\left[\begin{array}{l}{x}_1\\ x_2\\ ⋮\\ x_N\end{array}\right]u=\left[\begin{array}{l}{u}_1\\ u_2\\ ⋮\\ u_N\end{array}\right],y=\left[\begin{array}{l}{y}_1\\ y_2\\ ⋮\\ y_N\end{array}\right]\text{and}d=\left[\begin{array}{l}{d}_1\\ d_2\\ ⋮\\ d_N\end{array}\right];$

(A1.2)

the augmented large-scale system model is given by: