## 1 Introduction  
The usage of adaptive control systems is suitable for the control of complex systems for which an accurate mathematical model is not available, with the system being subject to unknown parameter variations and large disturbances over time [1; 2; 3; 4; 5; 6]. Furthermore, if the operating point changes under the action of external disturbances (taking into account a wide variation range of disturbances) or due to the variation of internal parameters, the usage of classical control solutions (PI, PID controllers) becomes infeasible [5; 7]. In these cases, a minimum variance control system is a viable solution due to its adaptive nature, ensured by the self-tuning characteristic of the controller parameters [8; 9; 10]. Obviously, with these parameter estimates computed in real-time, a parameter estimator is required (as a necessary component integrated into the control system) [11; 12; 13; 14]. Based on the minimization of a cost function described as a sum of two quadratic terms, the goals of the designed control law are the minimization of the controlled output variance, simultaneously with the minimization of the controller output variance (see relation 1, where \(E\) is an expectation operator) [15; 16; 17; 18]:  
\[J=E[\text{\emph{process output variance}}]^{2}+\rho[\text{\emph{controller output variance}}]^{2}] \tag{1}\]  
The second mentioned goal (leading to a suboptimal control) is a mandatory condition to ensure the possibility of practical implementation, imposing a penalty on the control variance, so that the computed control values to be physically achievable [15; 16; 17; 18; 19]. The balance between these two objectives of the cost function is achieved through a control penalty factor (\(\rho\)), which weights the significance of the second quadratic term in the criterion function. The technical literature states that, theoretically, the control penalty factor \(\rho\in[0,1]\), but in practice it is usually set in the range \(\rho\in[0.0001,\,0.1]\), ensuring a proper weighting between the minimization of the system output variance and the control variance [1; 2; 3]. A null value of the control penalty factor (\(\rho=0\)) would hypothetically lead to an optimal control law, only theoretically feasible, its implementation in practice being impossible. A simple analysis of such an optimal control system shows that huge control values are generated, but these values are not achievable in practice. Therefore, a non-zero value for the \(\rho\) factor is a mandatory condition to ensure a practical implementation of a minimum variance control system. A smaller control penalty factor leads to a higher control variance (which can sometime destabilize the system) and, as a consequence, to more efficient minimization of the controlled output variance (ensuring good control performance). A higher \(\rho\) parameter leads to a smaller control variance and to a weaker penalization of the system output variance (which can negatively affect the control performance). However, a too low penalization of the control is the main cause that can destabilize the control system. A proper value of \(\rho\), neither too small nor too big, could be a compromise solution [12; 16; 17]. The basic idea of the proposed control strategy is that, instead of a constant value of the control penalty factor \(\rho\) (a priori set), an algorithm can be designed to on-line tune its value in accordance with the operating conditions of the control system. Even though the self-tuning feature of the minimum variance controller is implicitly ensured by using real-time estimates as parameters of the control law, the control penalty factor is the main parameter that decisively influences the performance of the control system (including the system stability).