## 4 Identification Algorithms  
As was stated earlier, the implementation results of the two algorithms (DE and NM) were compared for the LSPMSM parameter identification problem. DE is a multidimensional optimization algorithm which belongs to the stochastic optimization algorithms group. DE is the result of further development of the genetic algorithm [38]. According to the original definition, DE can be described as follows. Initially, a set of vectors called generation is created. N-dimensional points where a minimized objective function \(f(x)\) is determined are considered as vectors. The algorithm creates a new generation of randomly combined vectors from the previous generation. The number of vectors is the same for each generation, and this is one of the algorithm parameters. A new generation of vectors Is generated as follows. For each previous generation vector \(x_{i}\), three different random vectors, \(v_{1}\), \(v_{2}\), and \(v_{3}\), are selected among the previous generation except vector \(\mathrm{x_{i}}\) itself. As a result, a mutated vector (6) is generated:  
\[v=v_{1}+F_{v}\left(v_{2}-v_{3}\right), \tag{6}\]  
Figure 6: One-stage identification algorithm flow chart. where \(F_{v}\) is one of the method parameters, which is a positive real number in the range (0, 2). It is recommended to vary this value when an objective function approaches a minimum [31]. Then, the crossover procedure is implemented on mutated vector v. The procedure consists of the replacement of some of the vector's coordinates by initial vector \(x_{i}\) coordinates. Each replacement is performed with some probability, which is another parameter of the algorithm. The result after the crossover procedure vector is called a trial vector. If this vector is \(\ast\)better\(\ast\) than the initial vector \(x_{i}\), which means that an objective function value was reduced, then \(x_{i}\) is replaced by the trial vector in a new generation. Otherwise, the previous \(x_{i}\) vector remains as an initial vector. The NM is an unconstrained optimization algorithm applicable to the function of several variables [40]. The algorithm searches for a local optimum and can \(\ast\)\(\ast\)\(\ast\)\(\ast\) in one of the local optima. Different initial simplex shapes can be changed to find the better minimum if necessary. The NM is used to find the unconstrained minimum of a function of n variables defined elsewhere in \(\mathbb{R}^{n}\). The algorithm parameters are listed below:  
* Reflection coefficient \(\alpha>0\). * Contraction coefficient \(0<\beta<1\). * Expansion coefficient \(\gamma>1\). * Usually, units \(\alpha=1\), \(\beta=0.5\), and \(\gamma=2\) are selected. The algorithm contains the following steps enumerated to highlight function calls:  
1. Preparation. Initially, \(n+1\) points are selected: \(x_{i}=(x_{1}^{1},x_{i}^{2},\ldots,x_{i}^{n})\), \(i=1\ldots\) (\(n+1\)), where \(x_{1}=(x_{1}^{1},x_{1}^{2},\ldots,x_{1}^{n})\) is an initial parameter vector, \(x_{2}=(x_{1}^{1}\cdot(1+\delta)\), \(x_{1}^{2}\),..., \(x_{1}^{n})\), \(x_{3}=(x_{1}^{1}\), \(x_{1}^{2}\cdot(1+\delta)\),..., \(x_{1}^{n})\)..., \(x_{n+1}=(x_{1}^{1},x_{1}^{2},\ldots,x_{1}^{n}\cdot(1+\delta))\), \(\delta\) is the initial simplex coefficient. These vectors constitute the \(n\)-dimensional space simplex. Objective function values are calculated at the points \(f_{1}=(x_{1}),f_{2}=f(x_{2}),\ldots,\)\(f_{n+1}=f(x_{n+1})\). 2. Sorting. Three points are selected among simplex vertices: \(x_{h}\) where function value \(f_{h}\) is the greatest, \(x_{g}\) where \(f_{g}\) is the next to the greatest, and \(x_{l}\) where \(f_{l}\) is the least function value. The further steps' aim is at least the diminishing of \(f_{h}\). 3. A centroid of the aforementioned points of the simplex excluding \(x_{h}\) is determined as \(x_{c}=1/n\sum_{(i\
eq h)}(x_{i})\). Calculation of \(f_{c}=f(x_{c})\) is not mandatory. 4. Reflection.