# Considerations about Parameters Estimation into a Minimum Variance Control System
## 3 Case Studies for Validation
### Case 1. Initialization of the Minimum Variance Control System  
The starting of a minimum variance control system is a complex process, requiring several steps [8; 9].  
The first step is to start the open-loop process, for computation and initialization of the linearized model parameters. The considered operating point is defined by the following conditions: step input excitation voltage: 440 V, zero initial values of the estimates, process input disturbed by a zero-mean stochastic noise and variance \(\sigma^{2}\) = 0.01. In this case, the estimates of the model parameters are presented in Figure 2a (practically a zoom of the initial time interval (\(t=[0\;6]\;s\)) from Figure 2b presenting the estimates for the complete procedure of starting and initializing the control system, as will be shown below). The same colors used in Figure 2a (see the figure legend) to represent the parameters estimates of polynomials described by Equation (11) are used for all next case studies which will present parameters estimates (\(a_{1..4},b_{0..3}\) in open-loop and \(\hat{a}_{1..4},\hat{b}_{0..3}\) in closed-loop).  
In the second step, the loop is closed (performing a synchronization between the controller and the plant) and an initial tuning of the control penalty factor to a high value \(\rho=0.1\) is performed to ensure the control system stability and an efficient control penalization [8; 9]. The last step requires a decrease of the control penalty (\(\rho=0.01\)), allowing a higher control and much better performances [7; 8; 9; 22; 23; 24; 25; 26]. All these operating stages are highlighted and can be observed in Figure 2b-d. Figure 2c presents the controller output (the control loop is closed after time moment t = 6 s and the control penalty factor is changed at time moment t = 10 s from \(\rho=0.1\) to \(\rho=0.01\)). Figure 2d shows the controlled output. It can be noticed that the closing of the control loop leads to a decrease of the controlled output variance after time moment t = 6 s, according to the control goal.  
The purpose of this case study is to demonstrate that, into a well-established operating point, by closing the control loop, the parameter estimates of the linearized model are different from those obtained in the open-loop. Thus, by analyzing the estimates presented in Figure 2b, different evolutions of the estimates for each stage can be observed. As already stated, the initial estimates until the moment t = 6 s, are those from Figure 2a (open-loop). Immediately after closing the loop (at time moment t = 6 s), the time evolution of the estimates changes under the action of the constraints imposed by the control system.  
Figure 2: (**a**) Open-loop process parameters estimates (polynomials \(A(z^{-1})\) and \(B(z^{-1})\)); (**b**) open-loop and closed-loop process parameters estimates (polynomials \(A(z^{-1})\), \(B(z^{-1})\) in open-loop, \(\hat{A}(z^{-1})\), \(\hat{B}(z^{-1})\) in closed-loop); (**c**) controller output (excitation voltage); (**d**) controlled output (terminal voltage).  
Moreover, without changing the operating point of the controlled process, just a simple retuning of the minimum variance controller (by decreasing the control penalty factor at time t = 10 s) affects the parameters estimates (see the same Figure 2b).  
The resulted conclusion is that the constraints imposed by the control law in closed-loop affect the estimates.  
The following studies present disturbed regimes and perform an analysis regarding the parameters estimates, respectively, the process gain in the two considered operating cases: open-loop and closed-loop.