# Considerations about Parameters Estimation into a Minimum Variance Control System
## 1 Introduction  
Based on adaptive feature, the minimum variance control systems represent viable solutions for nonlinear process control, for which classical control solutions are difficult to implement due to nonlinearities [1; 2; 3; 4; 5; 6]. This adaptive feature of such control system is assured by the integration in its structure of an on-line parameters estimator for a linearized model, which describes the nonlinear process functionality around a functioning point. Thus, for the design of a minimum variance control law, the starting point is the identification of a linearized model for the considered process. In the present work, the recursive least squares (RLS) estimator was considered. The control law, designed for an operating point, is obtained by minimizing a criterion function that contains a sum of two quadratic terms (see Equation (1), where \(E\) is an expectation operator). The first quadratic term assures the minimization of the controlled output variance, respectively the second term (weighted by a control penalty factor) allows a minimization of the control variance, absolutely necessary in order to obtain physically achievable control values and implicit stability of the control system [7; 8; 9]. The technical literature states that in practice the control penalty factor is usually set in the range \(\rho=0.0001\dots\ 0.1\), ensuring a proper control variance penalization [2; 5; 8]. \[J=E\Big{\{}[\text{output}\ \ \text{variance}]^{2}+\rho[\text{control}\ \ \text{variance}]^{2}\Big{\}} \tag{1}\]  
Although designed based on a linearized model, the real-time estimation and the continuous tracking of the estimates changes leads to a nonlinear control law. In the present paper, it will be proved analytically and validated through a numerical simulation (for a particular chosen case) that, in the closed-loop (due to the constraints imposed by the minimum variance control law), the linear model parameters estimates are not identical with the real ones estimated when the process operates in open-loop. In spite of this fact, in the steady-state (and based on different estimates for the open-loop, respectively the closed-loop regime), the process gain computation for both cases leads to similar values. In other words, in the closed-loop, the parameter estimator allows the computation of exact values of the steady-state process gain and not of the real parameters estimates (taking into account that the real values are the ones obtained in open-loop). The demonstration of the above mentioned was performed analytically and validated through simulations for the case of an induction generator integrated into a wind energy conversion system (a strong nonlinear process). Even if the complete functionality of an induction generator is described by a seventh order nonlinear model (the classical Park's equations [10; 11]), the technical literature shows that its functionality can be approximated by a fourth order linearized model, accurately enough around an operating point [11; 12; 13]. This simplified linearized model is used only for the design of the minimum variance control law and, obviously, the accurate seventh order nonlinear model is used to simulate the process. For the tuning of the minimum variance control system (an issue that is not subject of this paper, but is well described by technical literature [6; 8; 14]), appropriate values have been set for the control penalty factor (\(\rho\)) and for the estimator's forgetting factor (\(\lambda\)). In addition, the plant will be disturbed by a stochastic noise of zero mean and \(\sigma^{2}=0.01\) variance (ensuring the numerical stability of the RLS estimator) [15; 16; 17; 18; 19]. The main objective of the control system is to maintain a constant terminal voltage in two operating regimes, under the action of external disturbances such as: mechanical torque variation (due to wind speed changes), respectively load/unload disturbance (by connecting/disconnecting electrical consumers at induction generator terminals).