## 2 Design of the Minimum Variance Control Law  
It is common that the design of the minimum variance control law starts with the assumption of a linearized model of the controlled plant, which describes its functionality around an operating point accurately enough [1,3]. The behavior of a nonlinear plant can be described in the vicinity of an operating point by such a linearized model of a certain order, with parameters computed by a parameter estimator. A change of the operating point has as consequence a change of the linear model's parameters and the real-time tracking of these can be done by such a parameter estimation algorithm (in this case, the recursive least square estimator-RLS) [12, 13, 14]. For the case of an induction generator connected to a power system, its complete functionality is described by a nonlinear mathematical model of the seventh order (the \(d\)-\(q\) model represented by the Park equations) [10,22,23]. The complete model (and the parameter values) of the considered process has already been presented in extenso in one of the author's recent paper [10] and therefore it is not detailed in the present paper (being mentioned only in an Appendix A at the end of the paper). The goal of the proposed control system is to maintain constant the voltage at generator terminals through the excitation voltage control, considering the plant perturbed by various external disturbances (electrical consumer load/unload, mechanical torque variations). The technical literature shows that this type of relation between the excitation voltage (considered as the plant input variable) and the terminal voltage (as plant output variable) can be enough accurately described by a 4 th order linear mathematical with time-varying parameters [12,22,24,25]. This reduced order (4 th) is based on several simplifying assumptions regarding the electrical phenomenology, which does not affect the accuracy of the model near an operating point [12,15,26]. For maximum accuracy of the simulation results, the complete 7 th order nonlinear model of the controlled process was used to test the proposed control strategy  
Minimizing the criterion function (4) (by solving the equation: \(\frac{dI}{du_{t}}=0\)) and taking into account that, in steady state regime: \(q^{-1}=1\), \(u_{t}^{*}=\frac{\hat{A}(1)}{\hat{B}(1)}w_{t}\), respectively \(w_{t}=y_{t}\), the following control law results [17]:  
\[u_{t}=\frac{-q[1-\hat{A}\big{(}q^{-1}\big{)}]}{\hat{B}(q^{-1})+\rho}y_{t}+ \frac{1+\rho\frac{\hat{A}(1)}{\hat{B}(1)}}{\hat{B}(q^{-1})+\rho}w_{t} \tag{5}\]  
where \(\hat{A}\) and \(\hat{B}\) are the estimates of \(A\) and \(B\) polynomials (calculated by the RLS estimator). The control law described by relation (5) will be used for all performed tests to validate the proposed control strategy. The adaptive feature of such a controller is highlighted by the presence of parameter estimates (time-varying) in the control law (5). In case of RLS estimator, these estimates depend on the forgetting factor \(\lambda\). However, the influence of this factor over the control system perturbed by external disturbances is quite low [18; 25; 27; 28]. The influence of this tuning parameter (\(\lambda\)) is much more significant in the case of process with time-varying internal parameters [10; 12; 18; 29]. Therefore, a constant value (off-line properly set) is used during the entire operating regime of the process. The plant is considered disturbed by a stochastic noise (with zero mean and variance \(\sigma^{2}=0.01\)), as a mandatory condition for a proper functioning of the RLS estimator (ensuring the numerical stability) [16; 27; 30; 31; 32]. Also, a constant value \(\lambda\)=0.995 is identified and set for the forgetting factor, providing the best results. Instead, the control penalty factor (\(\rho\)) is the tuning parameter that has a much greater influence over the control system performance and stability [12; 16; 32].