A Fuzzy Gravitational Search Algorithm to Design Optimal IIR Filters

Danilo Pelusi, Raffaele Mascella, Luca Tallini
2018 Energies  
The goodness of Infinite Impulse Response (IIR) digital filters design depends on pass band ripple, stop band ripple and transition band values. The main problem is defining a suitable error fitness function that depends on these parameters. This fitness function can be optimized by search algorithms such as evolutionary algorithms. This paper proposes an intelligent algorithm for the design of optimal 8th order IIR filters. The main contribution is the design of Fuzzy Inference Systems able to
more » ... nce Systems able to tune key parameters of a revisited version of the Gravitational Search Algorithm (GSA). In this way, a Fuzzy Gravitational Search Algorithm (FGSA) is designed. The optimization performances of FGSA are compared with those of Differential Evolution (DE) and GSA. The results show that FGSA is the algorithm that gives the best compromise between goodness, robustness and convergence rate for the design of 8th order IIR filters. Moreover, FGSA assures a good stability of the designed filters. Energies 2018, 11, 736 2 of 18 possibility to specify a maximum radius for the poles of the designed rational transfer function. A computationally low intensive method for designing IIR multi-Notch filters was proposed by Duarte et al. [11] . The design of IIR filters may be oriented on magnitude and delay together: by combining the root-mean-square error function of variable frequency response and a suitable stability constrained function, the stability problem is overcome [12] . Generally, the problem of designing IIR filters is formulated as a nonlinear optimization problem. Moreover, the traditional methods based on gradient search can easily be stuck at local minima of error surface. In order to solve this problem, some methods based on metaheuristic approaches have been proposed. Due to their fast convergence property, Differential Evolution (DE) algorithms [13] have been applied to design robustly stable IIR filters [14] [15] [16] . Karaboga [17] proposed a technique to design IIR filters through DE. A seeker-optimization-algorithm based on evolutionary methods has been proposed for digital IIR filter design [18] . Other evolutionary algorithms such as Particle Swarm Optimization (PSO) [19] have been used for the design of IIR filters to reconstruct missing segments of multidimensional data [20] . A multi-swarm PSO with particle reallocation strategy is applied to design IIR filters with null constraint and specified error in the stop band [21] . Wang and Chen [22] proposed the use of multi-objective optimization evolutionary algorithms with the aim of minimizing magnitude response error, phase response error and order of IIR filters. An improved Immune Algorithm (IA) was proposed by Tsai and Chou to solve the problem of designing optimal IIR filters [23] . The process of IIR filters' design optimization is difficult because some constraints should be satisfied: (i) the determination of the lowest filter order; (ii) the filter stability; and (iii) the minimum value of passband and stopband ripple magnitudes. Because the Genetic Algorithms (GA) [24] are able to optimize complex and discontinuous functions that are difficult to analyze mathematically, some research [25] [26] [27] [28] [29] proposed different methods based on GA to solve the digital IIR filter design problems. A multi-crossover approach to design optimal GA-aided IIR filters was proposed by Chang [27] . Robust D-Stable IIR filters was designed by using GA where the stability criterion is embedded in the evolution of each generation [29] . Yu and Xinjie [28] proposed a coevolutionary GA that evolves coordinately as two different species: the control species and the coefficient species. A multi-parameter and multi-criterion optimization method based on a quantum genetic algorithm was proposed by Zhang et al. [25] . Stable IIR filters have been designed with the application of GA [26] . IIR filters' designing problems can be formulated as a multi-modal optimization problem with multiple decision variables. The Gravitational Search Algorithm (GSA) is a search method based on a law of gravity [30] able to optimize multi-modal functions. Saha et al. [31, 32] proposed a simple GSA and a GSA with Wavelet Mutation for the optimization of 8-th order IIR filter design. On the other hand, GSA has been combined with fuzzy logic for various applications [33] [34] [35] [36] . A fuzzy logic-based adaptive gravitational search algorithm dedicated to the optimal tuning of fuzzy controllers for servo systems was proposed by Precup et al. [33] . GSA and fuzzy logic have been combined to design optimal Proportional Integral (PI) controllers for a class of servo systems characterized by saturation and dead zone static nonlinearities [34] . The idea of enhancing GSA using fuzzy logic is inspired from the exploration and exploitation principle in meta-heuristics. The fuzzy regulation of GSA parameters assures this principle. Fuzzy Gravitational Search Algorithms (FGSA) with dynamic alpha parameter value adaptation for the optimization of modular neural networks in echocardiogram and pattern recognition have been proposed [35, 36] . Moreover, other versions of GSA with a fuzzy dynamic parameters adaptation have been proposed [37] [38] [39] [40] [41] [42] [43] . The improvements of GSA are based on the dynamic regulation of suitable parameters during the search procedure. This paper aims to design optimal IIR filters with the help of a revised GSA and the design of suitable Fuzzy Inference Systems (FIS). The first contribution of the work is the re-definition of a parameter of GSA able to improve the search performances. The second one is the design of two FIS's for GSA parameters adjustment. Both the approaches give rise to a Fuzzy Gravitational Search
doi:10.3390/en11040736 fatcat:f35fyxi5cvhehg4qzicwgcb34u