# Optimization Procedure for Computing Sampling Time for Induction Machine Parameter Estimation
## 4 Simulation and Experiment  
### Application to the Simulated Data  
The method is first tested on the simulated data. As mentioned, the method is implemented in Matlab 2018b [24] programming language in co-simulation environment with MIDACO [25]. The motor line start is simulated with Simulink using the model described by Equation (13). Discrete solver is Runge-Kutta and the step size is 1 \(\upmu\)s.  
The simulated motor is 132 kW pump motor with parameters \(R_{s}=8.9\) m\(\Omega\), \(R_{r}=16.7\) m\(\Omega\), \(L_{ls}=0.2\) mH, \(L_{lr}=0.2\) mH, \(L_{m}=14.1\)mH, \(J=5\) kg m\({}^{2}\). The optimization variables are constrained with lower and upper boundaries determined by experience of the user and some physical consideration. They are set larger then necessary to demonstrate longest expected computation time. Selecting the boundaries of parameter space for solving Equation (24) is simple, the decimation factor is minimum of one. The maximum decimation is bounded by the maximum significant frequency of the current spectrum. It is chosen as 300 Hz giving the Nyquist frequency \(F_{ny}=600\) Hz. This corresponds to the upper bound for the decimation factor as \(d=round(T_{s,min}F_{ny})^{-1}\). The upper bound to the interval size is selected so that the entire original data set can be considered as a single interval. The lower bound is selected so that the data set can be divided into 1000 smaller intervals of 200 data points, effectively giving the smallest interval time of 0.2 ms.  
Optimal fit of the damped complex exponentials to simulated data is presented in Figure 1a, and the computed linearized model poles obtained from the damped complex exponentials and the real linearized poles in the first interval obtained from Equation (18) and Equation (19) is presented in Figure 1b. These figures show that the data segmentation and decimation by optimization procedure provides the first interval that is accurately estimated by the Prony exponential series. The estimated poles are accurate enough to be used for computing the optimal sampling time. Therefore no prior knowledge of the motor parameters is required. The results of the optimization are: decimation factor \(\hat{d}=997\) and interval size \(\hat{q}_{int}=179\),650. This reduces the the data originally sampled at \(T_{s,min}=1\) us to the interval size of 181 samples sampled at \(T_{s,fit}=0.997\) ms. Solving the Equation (26) with minimim magnitude pole obtained from the optimization yields the optimal sampling time \(T_{s,opt}=1.774\) ms and the discrete poles with optimal sampling time are \(z_{opt}=e^{\lambda T_{s,min}K}=e^{\lambda T_{s,opt}}=z^{K}\). The optimal poles, and the damped complex exponential poles are shown in Figure 2. The figure shows how using the optimal sampling time spreads the poles of the system in the \(z\)-plane. As is known, reduction of the sampling time asymptotically contracts the poles of a discrete system towards the point (1,0) on the unit circle. This in turn reduces the system robustness to computational errors. Therefore, using the optimal sampling time to spread the poles from the unit circle increases computational stability while preserving the ability to approximate continuous system with discrete versions. For the presented example, the procedure computation on Intel i7 processor with 16 GB RAM memory it takes about seven minutes to calculate the optimal sampling time. This is the longest computation time expected from this procedure, obtained by very wide bounds for the optimization variables, very fast initial sampling time of 1 us and long line start transient due to large simulated motor inertia. Using common initial sampling frequency of (1-15 kHz) dramatically reduces the computation time, as is presented with the experimental data in the following subsection. Figure 1: (**a**) Damped complex exponential fit to simulated data (\(i_{d}\)) in the 1 st interval, with \(\boldsymbol{\Gamma}=\boldsymbol{\hat{\Gamma}}\); (**b**) Poles computed from optimal fitted model (\(\times\)) and real motor poles (o).