Recursive design of nonlinear \(H_\infty\) excitation controller
A robust model of a one-machine infinite-bus system with excitation control is under consideration. The motion equations are reduced to the form: \[ \begin{cases} \dot x_1 = x_2, \\ \dot x_2 = d_1 x_2 + d_2 x_3 + d_3 (x_3 + x_{30}) (\sin \delta_0 - \sin (\delta_0 + x_1)) + \varepsilon_1 ,\\ \dot x_3 = d_4 x_3 + d_5 \cos (\delta_0 + x) - d_5 \cos \delta_0 + \varepsilon_2 + d_6 u, \\ z = (q_1 x_1, q_2 x_2) . \end{cases} \tag{1} \] The problem of nonlinear \( H_{\infty} \) control design for system (1) is formulated as follows: for any given small enough positive number \( \gamma \) find a state feedback \( u = a (x_1 , x_2, x_3) \) such that for the disturbance \( \varepsilon = [\varepsilon_1, \varepsilon_2]^T \) \[ \int\limits_0^T \|z \|^2 dt \leq \gamma^2 \int\limits_0^T \|\varepsilon \|^2 dt + E (x_0), \quad T > 0 \] and when \( \varepsilon_1 = 0 \) and \( \varepsilon_2 = 0 \), the corresponding closed system is asymptotically stable at the equilibrium point. Here \( T > 0 \) is a given constant, and \( E (x_0) \) is the initial value of a storage function to be determined. This problem is solved and a nonlinear \( L_2 \) gain disturbance attenuation controller is constructed. Simulation results show the effectiveness of the design method.
scientific article; zbMATH DE number 1570538