# Optimization Procedure for Computing Sampling Time for Induction Machine Parameter Estimation
## 3 Optimal Induction Machine Poles and Sampling Time
### Optimal Poles of Induction Machine  
The application of the method for obtaining system poles from the IM line start is formed as an optimization problem. The general optimization problem is formulated as finding the vector variable \(x\) that minimizes the scalar objective function \(f(x)\), subject to equality and inequality constraints presented using vector functions \(g(x)\) and \(h(x)\) that define the set \(\mathbb{S}\) of all admissible values for \(x\). The general optimization problem is presented in Equation (20). \[ \begin{split}& f(x)\rightarrow\min.\\ &\hat{x}=\operatorname*{argmin}_{x\in\mathbb{S}}f(x)\\ &\text{subject to constraints}\\ & g(x)=\mathbf{0}\\ & h(x)\leq\mathbf{0}\end{split}  \tag{20}\]The optimization problem for obtaining the IM poles by fitting Prony complex exponential series to the measured line start transient is derived in the following paragraphs. The observed value for computing the Prony complex exponential approximation is the \(d\)-axis current, \(y=i_{d}\), in the voltage oriented reference frame. Since the machine model given by Equation (13) is fifth order, the total number of elements in the sum and the order of the estimated model by Equation (7) is \(n=5\). Therefore, the 5 eigenvalues of the linearized model can be approximated. Given the limited order of the estimated damped exponentials, it is not expected that the entire transient can accurately estimated by Equation (7). Therefore, the transient is divided into shorter intervals, and the aim is that each interval is short enough for the accurate estimation with the fifth order model. This criteria gives the interval duration where the nonlinear transient can be approximated with the fifth order damped complex exponential sum. The total number of observations in the interval is noted \(q_{int}\), with the reminder that the total number of observations in the measured set is \(q\). The total number of intervals the data are separated into is described by Equation (21) (rounded to the nearest integer). \[N_{int}=\frac{q}{q_{int}} \tag{21}\]  
Secondly, the transient is sampled at the sampling rate \(T_{s,min}\), the fastest sampling rate possible. When fitting the data with model from Equation (7) it is assumed that less data is needed. Therefore a decimation factor \(d\in\mathbb{N}\) is introduced such that the decimated sampling time is \(T_{s,fit}=dT_{s,min}\). Now selecting every \(d\)-th element of the original measured set (down-sampling) can provide the data reduction. The original measured set \(y[p]\) is decimated to \(y_{d}[p]=y[dp]\), and with decimated data, the new interval size is \(q_{dt}=q_{int}/d\). With interval size \(q_{int}\) and decimation factor \(d\) introduced, the optimization problem can be formed. The optimization objective function is presented in Equation (22)  
\[J_{est}=\sum_{w=1}^{N_{int}}MSE_{w} \tag{22}\]  
where the \(MSE_{w}\) is the mean sum of squared residuals for \(w\)-th interval:  
\[MSE_{w}=\frac{1}{q_{d}}\sum_{r=(w-1)q_{d}}^{wq_{d}}\left(y_{d}[r]-\hat{y}_{d}[ r]\right)^{2} \tag{23}\]  
and where \(\hat{y}_{d}[r]\) is computed by fitting the decimated data \(y_{d}[r]\) from the interval \((w-1)q_{d}<r<wq_{d}\) with damped complex exponentials from Equation (7) by the procedure described in Section 2.2. The decision variables of the optimization are \(\boldsymbol{\Gamma}=[d\quad q_{int}]\) and the optimization problem is to find the optimal \(\hat{\boldsymbol{\Gamma}}\) that minimizes the objective function, Equation (22), subject to constraints \(\boldsymbol{\Gamma}\in\mathbb{S}\subset\mathbb{N}^{2}\). The set \(\mathbb{S}\) is the set of all permissible integer values of the decision variables, bounded by the lower and the upper bound for each decision variable. The total optimization problem is presented in Equation (24). \[