# Considerations about Parameters Estimation into a Minimum Variance Control System
## 3 Case Studies for Validation
### Case 2. Mechanical Torque Variation  
In the following case studies, the process is considered disturbed (both in open and closed-loop) by a successively variation of the mechanical torque at time moments t = 2 s and t = 12 s (Figure 3a). Figure 3: _Cont._  
Case 2a: First, the closed-loop case is analyzed. Figure 3b,c show the terminal voltage (the controlled output), respectively, as well as the excitation voltage (the controller output) (small overshoots, small settling times, good control penalization). In Figure 3d is presented the time evolution of the parameter estimates and, based on them, in Figure 3e is computed and presented the closed-loop process gain estimate. By \(P_{0}\) and \(P_{1}\) are noted the steady-state operating points and very good performances can be noticed of the controlled process (induction generator): \(P_{0}\)-before time moment t = 2 s, \(P_{1}\)-between time moments t = 2 s and t = 12 s, and again \(P_{0}\)-after time moment t = 12 s.  
Regarding these operating points, Figure 4 presents the operational characteristics of the induction generator disturbed by mechanical torque variations. The initial operating point is \(P_{0}\) (placed on the curve corresponding to a mechanical torque \(M_{0}\)). In the closed loop, under the action of the control system that maintains constant terminal voltage (\(y_{0}\)) and under the action of a disturbance produced by the variation of the mechanical torque (\(M_{1}=M_{0}+\Delta M\)), the operating point horizontally slides on another operating curve (corresponding to the new mechanical torque \(M_{1}\)) and moves to position \(P_{1}\). In Figure 3e can be noticed the steady-state values \(k_{0}\) and \(k_{1}\) of the process gain corresponding to the two closed-loop operating points \(P_{0}(u_{0},y_{0},M_{0})\) and \(P_{1}(u_{1},y_{0},M_{1})\) depicted in Figure 4. This representation of the induction generator operational characteristics is necessary to explain the setting of similar operating conditions in both the open and the closed-loop. Case 2b: The next case study considers the open-loop process (perturbed by the same stochastic noise). The first temptation is to consider (as process input) a constant excitation voltage \(U=U_{0}\), respectively at time moments t = 2 s and t = 12 s, a change of the mechanical torque (\(\Delta M=\pm 5\%\), as in the previous closed-loop case). However, these operating conditions lead the process from the initial operating point \(P_{0}\) to the operating point \(P_{2}(u_{0},y_{1},M_{1})\) (other than the desired point \(P_{1}\)), and respectively back to \(P_{0}\) (see Figure 4). For the process transition between these operating points \(P_{0}\)-\(>P_{2}\)-\(>P_{0}\), the parameters estimates are presented in Figure 5a and, based on them, the calculated gain is presented in Figure 5b. Comparing the results from Figures 3d and 5a), a different variation of the parameters estimates can be observed. However, more important is the fact that, by comparing the results from Figures 3e and 5b, it can be noticed the value inconsistency regarding the median time zone (the time interval between t = 2 s and t = 12 s) of the computed gain factor in closed-loop (\(k_{1}\approx 0.84\)), respectively, in open-loop (\(k_{2}\approx 0.86\)). This value discrepancy can be explained by the fact that the open loop operating conditions took into account only the disturbances action (mechanical torque variation), which lead the process to the operating point P\({}_{2}\) and not to the desired target point P\({}_{1}\) (see Figure 4). Obviously, for different operating points, the gain values are different. Therefore, it is necessary to correct the open-loop operating conditions by considering an appropriate variation of the process input (excitation voltage: \(u_{1}=u_{0}+\Delta u\)), which maintains a constant terminal voltage (\(y_{0}\)) and thus the operating point follows the desired transition P\({}_{0}\)-> P\({}_{1}\)-> P\({}_{0}\) (see Figure 4).