# Considerations about Parameters Estimation into a Minimum Variance Control System
## 2 Analytical Demonstration Based on Minimum Variance Control Law Design  
The general structure of a minimum variance control system (particularized for the considered process-an induction generator integrated into a wind energy conversion system), is depicted in Figure 1a. In addition, the nonlinear process operating in open-loop is illustrated in Figure 1b. As already mentioned, the starting point of a minimum variance control law design is a linear model described by a discrete difference equation (Equation (2)):  
\[A\Big{(}z^{-1}\Big{)}y_{t}=z^{-1}B\Big{(}z^{-1}\Big{)}u_{t} \tag{2}\]  
where: \(A\big{(}z^{-1}\big{)}\) and \(B\big{(}z^{-1}\big{)}\) are polynomials in \(z^{-1}\) shift operator, \(y_{t}\)-process output at discrete time \(t\) (terminal voltage for the considered process), \(u_{t}\)-process input (excitation voltage), \(z^{-1}\)-shift operator (with one sampling time, therefore \(y_{t}=z^{-1}y_{t+1},\ y_{t-1}=z^{-1}y_{t}\), and so on). This discrete linear model (Equation (2)) is valid around an operating point, for a proper set of parameters values. Another operating point means another set of parameters values. The control law design involves minimizing a criterion function described by Equation (3) [2,8,9,20]:  
\[J=E\Big{\{}\big{[}y_{t+1}-w_{t}\big{]}^{2}+\rho\big{[}u_{t}-u_{t}^{*}\big{]}^{ 2}\Big{\}} \tag{3}\]  
where, in the context of a control system: \(y_{t+1}\)-controlled output at discrete time \(t+1\) (next sampling time); \(u_{t}\)-controller output and \(u_{t}^{*}\) -steady state controller output; \(w_{t}\)-set point (reference), \(\rho\)-control penalty factor (properly set); \(E\)_[. ]_-mean operator (expectation). Considering a steady state regime, therefore \(z^{-1}=1\) and \(u_{t}^{*}=\frac{A(1)}{B(1)}y_{t}\) (through \(\hat{\ }\) being noted the estimates of polynomials from Equation (2)), by minimizing the criterion function (Equation (3)) and solving the equation \(\frac{dI}{du_{t}}=0\), the following control law is obtained [9,13,20]:\[u_{t}=\frac{-\mathrm{z}\big{[}1-\hat{A}\big{(}z^{-1}\big{)}\big{]}}{\hat{B}(z^{-1} )+\rho}y_{t}+\frac{1+\rho\frac{\hat{A}(1)}{\hat{B}(1)}}{\hat{B}(z^{-1})+\rho}w_{t} \tag{4}\]  
It should be noted that \(\hat{A}\big{(}z^{-1}\big{)}\) and \(\hat{B}\big{(}z^{-1}\big{)}\) are the estimated polynomials (in closed-loop) which approximates the real polynomials \(A\big{(}z^{-1}\big{)}\) and \(B\big{(}z^{-1}\big{)}\) (identified in open-loop). Based on linear model described by Equation (2), \(y_{t}=z^{-1}\frac{B\big{(}z^{-1}\big{)}}{A\big{(}z^{-1}\big{)}}u_{t}\) and in the steady state (\(z^{-1}=1\)), under the control system constraints, the reference and the controlled output are equal: \(w_{t}=y_{t}\). Taking into account those mentioned and substituting \(w_{t}\) and \(y_{t}\) in Equation (4), it results:  
\[u_{t}=\frac{-\big{[}1-\hat{A}(1)\big{]}+1+\rho\frac{\hat{A}(1)}{\hat{B}(1)}}{ \hat{B}(1)+\rho}\frac{B(1)}{A(1)}u_{t} \tag{5}\]  
By simplifying with \(u_{t}\), Equation (5) becomes:  
\[A(1)\hat{B}(1)+\rho A(1)=\hat{A}(1)B(1)+\rho\frac{\hat{A}(1)}{\hat{B}(1)}B(1) \tag{6}\]  
Multiplying Equation (6) with \(\hat{B}(1)\) and grouping on the right member the terms depending on \(\rho\), it is obtained:  
\[\hat{B}(1)\underbrace{\big{[}A(1)\hat{B}(1)-\hat{A}(1)B(1)\big{]}}_{0}=-\rho \underbrace{\big{[}A(1)\hat{B}(1)-\hat{A}(1)B(1)\big{]}}_{0} \tag{7}\]  
The equality Equation (7) (valid in steady-state) can be fulfilled only if both right member and left member are zero, leading to the following equality relation (practically the same, fulfilled by both members):  
\[\frac{\hat{B}(1)}{\hat{A}(1)}=\frac{B(1)}{A(1)} \tag{8}\]  
Supposing that \(A(1)\hat{B}(1)-\hat{A}(1)B(1)\
eq 0\) and simplifying the equality (7) with \(A(1)\hat{B}(1)-\hat{A}(1)B(1)\), for the fulfillment of this equality would result: \(\hat{B}(1)=-\rho\). However, the control penalty factor \(\rho\) is a chosen constant and \(\hat{B}(1)\) is a variable estimate (which changes with the operating point), so the condition \(\hat{B}(1)=-\rho\) cannot be met.