# Optimization Procedure for Computing Sampling Time for Induction Machine Parameter Estimation
## 3 Optimal Induction Machine Poles and Sampling Time
### Linearized Induction Machine Model  
For any nonlinear state space model operating at the stationary point \(\mathbf{x}_{0}\) with the stationary input \(\mathbf{u}_{0}\), the response for small change in input variables \(\Delta\mathbf{u}\) can be approximated using the linearized form of the model. The linearized form is represented in Equation (16).  
\[\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\Delta\mathbf{x}& =\mathbf{A}\Delta\mathbf{x}+\mathbf{B}\Delta\mathbf{u}\\ \Delta\mathbf{y}&=\mathbf{C}\Delta\mathbf{x}+\mathbf{D}\Delta\mathbf{u}\end{split} \tag{16}\]  
With:  
\[\begin{split}\mathbf{A}&=\left.\frac{\partial}{\partial\mathbf{x}} \mathbf{f}\right|_{\begin{subarray}{c}\mathbf{x}=\mathbf{x}_{0}\\ \mathbf{u}=\mathbf{u}_{0}\end{subarray}}&\mathbf{B}&=\left.\frac{\partial}{\partial\mathbf{u }}\mathbf{f}\right|_{\begin{subarray}{c}\mathbf{x}=\mathbf{x}_{0}\\ \mathbf{u}=\mathbf{u}_{0}\end{subarray}}\\ \mathbf{C}&=\left.\frac{\partial}{\partial\mathbf{x}}\mathbf{g}\right|_{ \begin{subarray}{c}\mathbf{x}=\mathbf{x}_{0}\\ \mathbf{u}=\mathbf{u}_{0}\end{subarray}}&\mathbf{D}&=\left.\frac{\partial}{\partial\mathbf{u }}\mathbf{g}\right|_{\begin{subarray}{c}\mathbf{x}=\mathbf{x}_{0}\\ \mathbf{u}=\mathbf{u}_{0}\end{subarray}}\end{split} \tag{17}\]Particular interest of this paper is in the matrix \(A\) of the linearized induction machine model. If the known simplification (\(\frac{d}{dt}\omega=0\)) found in [23] is used, the right hand side of the rotational motion equation of the model in Equation (13) becomes zero. Using this assumption, which is valid if the current transients in the model are much faster than the speed transients, the linearized matrix \(A\) of the induction machine model in Equation (13) is:  
\[A=\begin{bmatrix}-R_{s}G_{s}&\omega_{k}&-R_{s}G_{m}&0&0\\ -\omega_{k}&-R_{s}G_{s}&0&-R_{s}G_{m}&0\\ -R_{r}G_{m}&0&-R_{r}G_{r}&\omega_{k}-p\omega_{0}&-\psi_{rq0}\\ 0&-R_{r}G_{m}&p\omega_{0}-\omega_{k}&-R_{r}G_{r}&\psi_{rq0}\\ 0&0&0&0&0\end{bmatrix} \tag{18}\]  
The poles \(\lambda\) of the linearized system described by Equation (16) are the eigenvalues of the matrix \(A\). The \(k\)-th discrete pole (\(z_{k}\)) of the system, used for discrete and recursive computation, is calculated using the \(k\)-th continuous eigenvalue (\(\lambda_{k}\)) and discrete sampling time \(T_{s}\) as:  
\[z_{k}=\exp\left(\lambda_{k}T_{s}\right) \tag{19}\]  
Since the IM model is nonlinear, the only discussion about the poles location can be in context of linearized models. In order to apply any method for selection of optimal sampling time based on poles location, one must know poles a-priori. If poles are known from the system model, the parameter estimation is not necessary. Therefore the problem of finding optimal sampling time for parameter estimation as described by [15] can be rephrased as a problem of determining the system poles without the knowledge of the system and the optimal sampling time.  
The procedure for finding the system poles by Prony approximation is described, but it is not straightforward applicable to the measurements of the IM transient. Some considerations need to be made: the IM model is nonlinear, the measured transient is not expected to be an impulse response.  
In this paper these issues are overcome by applying the procedure to the measurements of the direct line start of an IM. This is effectively a step response of a nonlinear system with \(x(0)=\mathbf{0}\). The response around the initial condition can be represented by the linearized model, but as time goes on, the linearized response and real nonlinear response becomes very different. For observing the small interval around the initial condition point, the step function can be thought of as a finite impulse.  
The question that arises is how long is the initial interval, and how many points, sampled at what rate are needed to achieve the accurate pole estimation via Equation (7). This paper proposes the use of an optimization technique, described in the following subsection, to determine these factors.