Smart Systems for Industrial Applications. Группа авторов

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      The system we proposed uses FOPID controller instead of IPID controller for position control of pneumatic position servo system. It provides a better efficiency of the system by using FOPID controller. This system provides more accurate output compare to that of IPID controller. The power consumption of this system using FOPID controller is much lesser than the previously existing system. The robustness of this system is better than previously existing system using traditional IPID controller.

      2.4.1 Modeling of Fractional-Order PID Controller

       2.4.1.1 Fractional-Order Calculus

      Fractional-order operator, is defined

      (2.1)

      where t₁ and t₂ are the upper and lower time limits for the operator.

      The term λe is the fractional order. It is an arbitrary complex number. Real(λe) is the real part of λe.

      The Grnwald-Letniknov (GL) fractional-order derivative of the function f(t) is defined

      (2.2)

      where −1 is the rounding operation, c is the calculation step, and is the binomial coefficients defined as ϵ⁰.

      Integration and differential denoted by a uniform expression.

      (2.3)

      The fractional-order operator can be done by using the following equation [8]:

      (2.4)

      where

      (2.5)

      (2.6)

      By ignoring the very old data, an approximate fractional-order approximation is obtained by

      (2.7)

      where and L is the memory length.

       2.4.1.2 Fractional-Order PID Controller

      The equation of the IPID controller is

      (2.8)

      where K_pi, K_jj and K_di and are the proportional, integral, and differential coefficient, respectively, where e(t) = y_d (t) − y(t) is the system error, y_d (t) is the reference input, y(t) is system response, and u(t) is controlled output [6, 9]. The FOPID controller is an extension of the conventional IPID controller with the integral and the differential orders as fractional one [7].

      FOPID controller is represented as

      (2.9)

      λe indicates integral order.

      µ indicates the differential order.

      K_pf, K_if, and K_df are fractional-order controller gains.

      Laplace transfer function of the controller is given as

      (2.10)

      The FOPID has additionally more adjustable parameters, λ and µ, than IPID controller and have five control parameters (Kpf, Kif, Kdf, λe, and μ) to find a better control performance [9]. For optimization, the GA has a possibility to come with five optimum parameter space to achieve best control performance.

2.5 Genetic Algorithm

      GA is an adaptive empirical search algorithm depends on the mutative concepts of natural selection and genetics. It emphasizes the intellectual manipulation in finding solution to the optimization problems. Based on the historical information, GA searches for random variables through the best performance region of the search space. GA technique resembles the survival of the fittest principle proposed by Charles Darwin. In view of nature’s law, competition or struggle among the individuals results in the fittest predominating the inferior ones.

      Alike chromosomes in DNA, the population in every generation has certain character strings impinged from the parent. In the search space each one of the individual signifies a point and has a feasible solution. The next stage through which the individuals undergo is the evolution process. Every individual in the population strives for the best position and mates. The fittest individual competes and yields offspring, whereas the inferior individuals will not proceed to the successive process. In every generation, the offspring thus produced from the fittest parent will be more suitable for the environment.

      2.5.1 GA Optimization Methodology

GA optimization has for four major phases and requires a fitness function for optimization. The four steps are summarized as follows (Figure 2.2):

Figure 2.2 Phases in genetic algorithm.

Figure 2.3 GA initialization process.

      1 Initialization: population of chromosomes are initialized

      2 Selection: reproduce chromosomes

      3 Crossover: produce next generation of chromosomes

      4 Mutation: СКАЧАТЬ