Multi-objective evolutionary framework for non-linear system identification: A comprehensive investigation

Abstract

The present study proposes a multi-objective framework for structure selection of nonlinear systems which are represented by polynomial NARX models. This framework integrates the key components of Multi-Criteria Decision Making (MCDM) which include preference handling, Multi-Objective Evolutionary Algorithms (MOEAs) and a posteriori selection. To this end, three well-known MOEAs such as NSGA-II, SPEA-II and MOEA/D are thoroughly investigated to determine if there exists any significant difference in their search performance. The sensitivity of all these MOEAs to various qualitative and quantitative parameters, such as the choice of recombination mechanism, crossover and mutation probabilities, is also studied. These issues are critically analyzed considering seven discretetime and a continuous-time benchmark nonlinear system as well as a practical case study of non-linear wave-force modeling. The results of this investigation demonstrate that MOEAs can be tailored to determine the correct structure of nonlinear systems. Further, it has been established through frequency domain analysis that it is possible to identify multiple valid discrete-time models for continuous-time systems. A rigorous statistical analysis of MOEAs via performance sweet spots in the parameter space convincingly demonstrates that these algorithms are robust over a wide range of control parameters.of correct terms which captures the system dynamics is one of the major challenges of the system identification [6,7], and, therefore, is the main focus of this investigation.The exhaustive search to solve the structure selection problem is intractable even for moderate number of terms 'n', as this would require an evaluation of 2 n term subsets/structures. The structure selection problem is often 'NP-Hard' [13] and therefore requires an efficient search strategy. Since the search for system structure is primarily dependent on the limited number of observed data, it involves bias-variance trade-off [2, 3, 7], i.e., too sparse a structure may yield a high bias error (under-fitting) whereas a very complex structure may yield a high variance error (over-fitting). Further, earlier investigations [14,15] show that the over-fitted structures tend to induce 'ghost' dynamics which are not present in the actual system. Pursuing these arguments, it is clear that determination of the size of the identified structure (i.e., number of terms or 'cardinality') is a critical aspect of the structure selection process. However, this issue has received relatively less attention in the existing investigations.Among the existing structure selection approaches, the Orthogonal Forward Regression (OFR) proposed by Billings and Korenberg [16] has extensively been studied. It is a sequential model building approach in which the most significant term is added in each step, which is determined on the basis of a statistical criterion. Over the years, this notion of the original OFR approach has been refined to further improve the search performance. These include OFR approaches...

MultiObjective Evolutionary Approach to Grey-Box Identification of Buck Converter

“…For such a scenario, the required static data could be obtained experimentally, by the steadystate input-output measurements. Since the static behavior of the candidate structure can be quantified using (18), it is now possible to integrate static behavior as one of the search objectives, as follows:…”

“…This can be accomplished by any multi-objective evolutionary algorithm such as NSGA-II [15], SPEA-II [16], MOEA/D [17] and others. The comparative analysis of these algorithms on the structure selection problem in [18] indicates that NSGA-II often yields an improved Pareto front in comparison to SPEA-II and MOEA/D. Hence, in this study, NSGA-II is selected to solve the structure selection problem given in (19).…”

“…Drawing on the recommendations in [18], the qualitative and quantitative control parameters of NSGA-II are set as follows: Population Size: 50; Selection: crowded tournament if pc > rand then 5 for j = 1 to n do 6 if 0.5 > rand then 7βp,j ← βq,j,βq,j ← βp,j Evaluate the fitness ofβi as per Algorithm 1 19 end 'CTS' denotes the Crowded Tournament Selection, see [15] selection [15]; Recombination: uniform crossover; crossover rate (p c ): 0.9; Mutation: flip-bit mutation; and mutation rate (p m ): 0.006. The reproduction operators being used in this study are shown in Algorithm 2, where 'β i ' and 'β i ' respectively denote the i th parent and the corresponding offspring.…”

“…Note that the other search components of NSGA-II such as non-dominated sorting and determination of crowding distance are omitted here for sake of brevity. Further implementation details about the NSGA-II can be found in [15], [18].…”

Identificação de um Sistema de Bombeamento Hidráulico com Algoritmo Evolucionário Multi-objetivo

“…Em Retes e Aguirre (2019) foi utilizado o algoritmo randomizado para seleção da estrutura do modelo (RaMSS), sendo este método classificado como tendo configurações probabilísticas. Outro trabalho relevante, desenvolvido por Hafiz et al (2019a), propõe um quadro multiobjetivo para a seleção de estruturas de sistemas nãolineares representados por modelos polinomiais NARX. Os Algoritmos Evolucionários Multiobjetivos (MOEAs) utilizados foram: Nondominated Sorting Genetic Algorithm-II (NSGA-II) (Deb et al, 2002), Strength Pareto Evolutionary Algorithm-II (SPEA-II) (Zitzler et al, 2001) e Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D) (Zhang e Li, 2007).É demonstrado que os algoritmos MOEAs podem ser adaptados para determinar a estrutura correta de um sistema não-linear.…”