# An Easily Understandable Grey Wolf Optimizer and Its Application to Fuzzy Controller Tuning
## 1 Introduction  
Fuzzy control has proven efficiency in coping with various simple and complex processes. The combinations with nature-inspired optimization algorithms, also known as metaheuristics, have a positive impact on the performance of fuzzy control systems as they contribute to the systematic design and tuning. Some recent applications of metaheuristics to the parameter tuning of fuzzy controllers dedicated to servo systems will be outlined as follows. Representative overviews on such industrial applications are offered in [1; 2; 3]. Proportional-integral-derivative (PID)-fuzzy controllers are tuned in [4; 5] using Particle Swarm Optimization (PSO)-based approaches and are tested by experiments or simulations on an electrical direct current (DC) drive benchmark and a laboratory micro air vehicle controller, respectively. A hybrid PSO and pattern search optimized PI-fuzzy controller is applied in [6] to the automatic generation control of multi-area power systems and is validated by simulation. The Gravitational Search Algorithm (GSA) is employed in [7; 8] in the optimal tuning of PI- and PID-fuzzy controllers for DC servo systems and load frequency control in power systems and is validated by simulation and experimental results. Charged System Search algorithms are appliedin [9] to the optimal tuning of PI-fuzzy controllers for DC servo systems. Ant Colony Optimization is applied in [10; 11] to the optimal tuning of fuzzy controllers for robots and ball and bam systems. The Grey Wolf Optimizer (GWO) algorithm is proposed in [12] on the basis of modeling grey wolf social hierarchy and hunting habits towards finding prey, represented by the solution to the optimization problem. The social hierarchy is simulated by categorizing the population of search agents into four types of individuals, i.e., alpha, beta, delta, and omega, based on their fitness. The search process is modeled with the aim of mimicking the hunting behavior of grey wolfs making use of three stages, searching, encircling, and attacking the prey. The first two stages are dedicated to exploration and the last one covers the exploitation. The reduced number of search parameters is an important advantage of GWO algorithms reflected in various applications, which include blackout risk prevention in smart grids [13], training multi-layer perceptrons [14], optimization of reactive power dispatch [15], solutions to benchmarks generally used to test optimization algorithms [16], hyperspectral band selection [17], maximum power point tracking [18], and optimal tuning of PI- and PID-fuzzy controllers [19; 20; 21]. This work builds upon the authors' previous results on the optimal tuning of Takagi-Sugeno proportional-integral fuzzy controllers (T-S PI-FCs) and fuzzy models by means of various standard nature-inspired algorithms like: Gravitational Search Algorithm [7; 22], Charged System Search [9], adaptive variations [23], or applications of multiple optimization algorithms [3; 24]. This paper proposes an easily understandable presentation of GWO. Based on this, GWO is later applied in an innovative method for the optimal tuning of T-S PI-FCs to ensure fuzzy control systems with a reduced process parametric sensitivity by the inclusion of sensitivity functions in the objective functions, which are minimized by the GWO in appropriately defined optimization problems. The presentation is focused on a family of servo systems considered as controlled processes, which are modeled by second-order dynamics with an integral component, variable parameters, along with a saturation and dead zone static nonlinearity. The results are confirmed through comparisons with other metaheuristics that have the same unsupervised characteristics. This paper is organized as follows: the optimization problem is set in the next section. GWO is presented and analyzed in Section 3 along with the tuning approach for T-S PI-FCs dedicated to the considered family of servo systems.