# Optimization Procedure for Computing Sampling Time for Induction Machine Parameter Estimation
## 2 Parameter Estimation Problem and System Poles
### Determination of the Linear System Poles  
The system described by Equation (1) with \(f(\mathbf{x},\mathbf{u})=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}\) and \(g(\mathbf{x},\mathbf{u})=\mathbf{C}\mathbf{x}\), where \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are matrices of appropriate dimensions is a linear state space system. For the specific case with the initial condition \(\mathbf{x}(0)=\mathbf{0}\) and for systems with single input that is a Dirac delta function \(u=\delta\), the linear system response can be expressed in terms of the state transition matrix \(\mathbf{\Phi}=e^{\mathbf{A}t}\) in the Equation (4). \[\mathbf{x}(t)=\int_{0}^{t}\mathbf{\Phi}(t-\tau)\mathbf{B}\delta(\tau)\mathrm{d}\tau=\mathbf{ \Phi}(t)\mathbf{B} \tag{4}\]  
The state transition matrix \(\mathbf{\Phi}\) can be computed based on modal decomposition of the state matrix \(\mathbf{A}\) such that:  
\[\mathbf{\Phi}=\mathbf{W}e^{\mathbf{A}t}\mathbf{W}^{-1} \tag{5}\]  
where \(e^{\mathbf{A}t}=\mathrm{diag}\left[e^{\lambda_{1}t}\dots e^{\lambda_{n}t}\right]\) is the matrix exponential of the \(\mathbf{A}\) matrix eigenvalues \(\lambda_{i}\), and \(\mathbf{W}\) is the matrix of corresponding eigenvectors. Since the system output is a linear combination of states and since the state transition matrix is defined using the matrix exponential and system eigenvalues (for further details see [18]), it can be concluded that the \(i\)-th system output under given conditions is the sum of modal exponentials:  
\[y_{i}=\sum_{j=1}^{k}G_{ij}\exp(\lambda_{i}t) \tag{6}\]  
where \(G_{ij}\) are constants determined from the initial conditions of the output variables and \(\lambda_{i}\) is the system \(i\)-th eigenvalue. The Prony method approximates a measured signal with the sum of damped complex exponentials. It can be used for determining the system eigenvalues directly from the measured impulse response. The general discrete \(n\)-th order Prony approximation \(\hat{y}[p]\) of the measured signal \(y[p]\) is represented in Equation (7)  
\[\hat{y}[p]=\sum_{i=1}^{n}R_{i}\exp(j\phi_{i}+\lambda_{i}pTs)=\sum_{i=1}^{n}R_{ i}e^{j\phi_{i}}z_{i}^{p} \tag{7}\]  
If it is assumed that \(G_{ij}=R_{ij}e^{j\phi_{i}}\) is constant the Equation (6) becomes the classical expression of the Prony approximation of a measured signal. An efficient and computationally stable method for determining the poles of the damped complex exponential approximation is to use the matrix pencil method introduced by [PERSON] and [PERSON] [19; 20]. This method turns the problem of finding the eigenvalues into problem of solving for the matrix pencil eigenvalues. The method is explained in mentioned references and only its application is presented here. The matrix pencil can be formed using the Hankel matrix \(\mathbf{H}\) for the \(i\)-th system output measurement set \(\mathbf{y}_{i}=[y_{i}[1]\ \dots\ y_{i}[n]\). \[\mathbf{H}= \begin{bmatrix}y_{i}[1]&\dots&y_{i}[n]&y_{i}[n+1]\\ y_{i}[2]&\dots&y_{i}[n+1]&y_{i}[n+2]\\ \vdots&\vdots&\vdots&\vdots\\ y_{i}[q-n]&\dots&y_{i}[q-1]&y_{i}[q]\end{bmatrix}  \tag{8}\]  
Written in terms of collumn vectors \(\mathbf{h}\) the matrix is:  
\[\mathbf{H}= \begin{bmatrix}\mathbf{h}_{1}&\dots&\mathbf{h}_{n}&\mathbf{h}_{n+1}\end{bmatrix}  \tag{9}\]  
a matrix pencil \(\mathbf{P}\) is formed as:  
\[\mathbf{P}=\mathbf{H}_{l}+z\mathbf{H}_{r} \tag{10}\]  
where:  
\[\mathbf{H}_{l}= \begin{bmatrix}\mathbf{h}_{1}\dots\mathbf{h}_{n}\end{bmatrix}  \tag{11}\]  
With defined matrix pencil \(\mathbf{P}\), the equation for discrete eigenvalue computation is formed as:  
\[\mathbf{P}=\mathbf{H}_{l}+z\mathbf{H}_{r}=0 \tag{12}\]  
Solving Equation (12) for all \(z\) yields the discrete system eigenvalues. In order to compute the eigenvalues one must solve Equation (12) by means of singular value decomposition since the matrices \(\mathbf{H}_{l}\) and \(\mathbf{H}_{r}\) are rank deficient. Therefore the eigenvalues are just an approximation of the real system eigenvalues.