A New Methodology for Tuning PID‐Type Fuzzy Logic Controllers Scaling Factors Using Genetic Algorithm of a Discrete‐Time System

In this chapter, a proportional-integral derivative (PID)-type fuzzy logic controller (FLC) is proposed for a discrete-time system in order to track a desired trajectory generated using the flatness property. In order to improve the performance of the proposed con- troller, genetic algorithm (GA) based on minimizing the integral of the squared error (ISE) is used for tuning the input and output PID-type FLC scaling factors online. The considered controller is applied to an electronic throttle valve (ETV). GA tuning shows a better and robust performance compared to Simulink design optimization (SDO) algo- rithm in terms of tracking a desired trajectory with disturbances rejection.


Introduction
Fuzzy logic control (FLC) has been widely used in many successful industrial applications. The first FLC algorithm was implemented by Mamdani in 1974. Unlike conventional control, which is based on mathematical model of a plant, an FLC usually embeds the intuition and experience of a human operator and may provide a nonlinear relationship induced by membership functions, rules and defuzzification. In that respect, FLC has been reported to be successfully used for a number of complex and nonlinear systems and are proved to be more robust and their performances are less sensitive to parametric variations than conventional controllers.
In the literature, various types such as proportional integral (PI), proportional derivative (PD) and proportional-integral derivative (PID) of FLCs have been proposed. For example, PI-type FLCs have been successfully implemented in many physical applications such the control of the temperature and pressure of a steam engine and control the steering and speed of an automobile. However, performance of PI-type FLCs for higher order systems and nonlinear systems may be poor due to the large overshoot and the excessive oscillation. PD-type FLCs are suitable for a limited class of systems and they are not recommendable in the presence of measurement noise and sudden load disturbances. Theoretically, PID-type FLCs provide a good performance. However, there are difficulties associated with the generation of an efficient rule base and the tuning of parameters.
In the proposed PID-type FLC, the design of parameters within two groups: structural parameters and tuning parameters. Basically, structural parameters include input/output (I/O) variables to fuzzy inference, fuzzy linguistic sets, membership functions, fuzzy rules, inference mechanism and defuzzification mechanism, which are usually determined during offline design. Tuning parameters include I/O scaling factors (SF) and parameters of membership functions (MF), which can be calculated during online adjustments of the controller in order to enhance the process performance [1].
The appropriate selection of input and output scaling factors is very important because they have significant effects on the dynamic of fuzzy controller. This leads researchers to explore the best method in searching optimum PID-type FLC parameters. Various strategies or methods have been used up to now. In Ref. [2], Qiao and Mizumoto proposed a peak observer mechanism-based method to adjust the PID-type FLC parameters. This self-tuning mechanism decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time. Furthermore, Woo et al. [3] developed a method based on two empirical functions evolved with the system's error information. In Ref. [1], Guzelkaya et al. proposed a technique that adjusts the scaling factors, corresponding to the derivative and integral components, using a fuzzy inference mechanism. However, the major disadvantages of all these PID-type FLC tuning method are the difficult choice of their relative parameters and mechanisms. To overcome these difficulties, differential search algorithm (DSA) meta-heuristic technique is proposed for systematically tuning the scaling factors of the PID-type FLC in Ref. [4]. The fuzzy control design is formulated as a constrained optimization problem, which is efficiently solved based on an improved DSA. In this proposed technique, different optimization criteria such as integral square error (ISE) and maximum overshoot are considered in order to guarantee more robustness and performance control objectives.
In this chapter, a genetic algorithm (GA)-based heuristic optimization technique has been implemented to obtain better performance compared to the Simulink design optimization (SDO) technique. GA is based upon minimizing the error between the output system and the desired trajectory starting from a flat output variable generated using the flatness property. Various performance indices can be used. In this study, the integral of squared error (ISE) index has been used in order to minimize the error between the output and the desired flat trajectory. The methods are applied in a discrete-time framework to an electronic throttle valve as a case of study.

PID-type fuzzy logic controller description
In this study, we will deal with fuzzy PID-type controllers formed using one PD-type FLC with an integrator at the output.
The PID-type fuzzy logic controller structure is shown in Figure 1 [5], where and are the input scaling factors are α and β the output scaling factors.
The inputs variables, well known as the error e k between the desired trajectory and the measure , as well as the error variation Δe k given by Eqs. (1) and (2) where T e is the sampling period.

ð1Þ ð2Þ
The output variable of such a controller is the variation of the control signal which can be defined as Eq. (3).

ð3Þ
The output of the PID-type fuzzy is given by Eq. (4) [5] ð4Þ Thus, the equivalent control components of the PID-type FLC are such that Proportional gain: Integral gain: Derivative gain: where the terms and are given by Eqs. (5) and (6) [2].

ð5Þ ð6Þ
The fuzzy controllers with a product-sum inference method, centroid defuzzification method and triangular uniformly distributed membership functions for the inputs and a crisp output proposed in Refs. [2,6] are used in our case of study. Table 1 gives the linguistic levels, assigned to the variables e k , Δe k and Δu k , as follows: NL: negative large; N: negative; ZR: zero; P: positive; PL: positive large.

Scaling factors tuning using genetic algorithm
In this work, a new method is proposed for tuning the coefficients of PID-type FLCs. This method adjusts the input scaling factor corresponding to the derivative coefficient and the output scaling factor corresponding to the integral coefficient of the PID-type FLC using genetic algorithm, as shown in Figure 2. The integral of squared error (ISE) index has been used in order to minimize the error between the output and the desired flat trajectory.

Genetic algorithm
Genetic algorithm was first proposed by Holland [7]. It is a heuristic optimization technique inspired by the mechanism of natural selection. It is used in order to solve highly complex problems. GA starts with an initial population containing a number of parameters, where each one is regarded as the genes of a chromosome and can be structured by a string of concatenated values. Each chromosome represents a solution of the problem and its performance is evaluated based on fitness function.
In the beginning, an initial chromosome population is randomly generated. crossover and mutation are applied in order to create the next population solution. The above steps are repeated until a specific termination criterion is found.
• Reproduction: Create a part of the new population by simply copying without changing the selected individuals from the present population. Also, new population has the possibility of selection by already developed solutions [8].
• Crossover: Create new individuals as offspring of two parents. It is a recombination operator that combines selected subparts called crossover points of two parent chromosomes. The individuals resulting in this way are the offspring [8].
• Mutation: Create a new individual for the new population by randomly mutating a selected individual. The modifications can consist of changing one or more values in the representation or adding/deleting parts of the representation [8].
To compute the fitness of each chromosome, the objective functions are used. Many authors use integral of time multiplied by absolute error (ITAE), mean of the squared error (MSE), integral of absolute error (IAE) and integral of the squared error (ISE) as performance index [9,10].
In this chapter, the method of tuning PID-type FLC parameters using GA consists in finding the optimal I/O scaling factors, which minimize the defined objective function, chosen as the ISE in order to specify more performance in terms of tracking a desired trajectory.
If is the desired trajectory and is the output trajectory, then error e(t) is ð7Þ and the ISE can be defined by Fitness function is taken as inverse of error, i.e., performance index.

Tuning procedure
The overall flowchart for optimization using GA is shown in Figure 3. Initially, a number of populations N have been generated for the scaling factors α and β. Each individual of these N sets in the current population is evaluated using the objective function ISE. Based on the values of the objective function, out of these N possible solutions, the good solutions are retained and the others are eliminated. A new population is formed by applying the genetic operators (reproduction, crossover and mutation) to these selected individuals. This process of production of a new generation and its evaluation is performed repetitively. The algorithm continues until the population converges to the stop criterion.

Flatness and trajectory planning
The flat property has been introduced in Ref. [11] for continuous-time nonlinear systems. It can be stated in a discrete-time version which leads for the design of a control which ensures a tracking of a desired trajectory. One major property of differential flatness is that the state and the input variables can be directly expressed, without integrating any differential equation, in terms of the flat output and a finite number of its derivatives. The flatness approach will be used in this chapter in a discrete-time framework.
The studied dynamic linear discrete system is described by Eq. (10). where the parameters and are constants, i=0,1,…,nÀ1. The partial state of such a dynamic system can be considered as a discrete flat output which can be expressed as a function of input and output signals as following ð13Þ ð14Þ Often, the real output signal to be controlled is not a flat output. Then, it is necessary to plan a desired trajectory for the flat output [11] and to consider thereafter the relation (14).
The open loop control law can be determined by the following relations [12].

ð15Þ ð16Þ
where and g are the vectorial functions. Then, it is sufficient to find a desired continuous flat trajectory that must to be differentiable at the order.
The polynomial interpolation technique is used in order to plan the desired flat trajectory .
Let consider the state vector containing the desired continuous flat output and its successive derivatives. The expression of can be given in Eq. (17); and are the two moments known in advance.

ð17Þ
where and are such as [12] ð18Þ ð19Þ and the vectors and defined by ð20Þ ð21Þ Then, the output desired trajectory is defined. In the discrete-time framework, the real output has asymptotically to track this such as Eq. (22).

ð22Þ
In the following section of this chapter, the efficiency of the proposed methodology for tuning PID-type FLC scaling factors has been validated on a discrete-time system: an electronic throttle valve for a defined desired trajectory generated using the flatness property and compared to the Simulink design optimization (SDO) technique.

Case of study: electronic throttle valve (ETV)
Throttle valve is one of the most important devices in the engine management system. In conventional engine, the amount of airflow into the combustion system has been adjusted by the throttle valve, which is connected mechanically to an accelerator pedal [13]. The electronic throttle body (ETB) regulates air inflow into the car engine. Compared to the mechanical throttle, a well-controlled ETB can reduce fuel consumption.

System modelling
The case of the ETV is described in Figure 4.

The electrical part is modelled by Eq. (23)
where L is the inductance R is the resistance u(t) and i(t) are the voltage and the armature current, respectively, is an electromotive force constant and is the motor rotational speed.
The mechanical part of the throttle is modelled by a gear reducer characterized by its reduction ratio such as Eq. (24) ð24Þ where is the load torque and C g is the gear torque. The mechanical part is modelled according to Eq. (25), such that [14,15]. where is the throttle plate angle, J is the overall moment of inertia, is the electrical torque where K e is a constant, C f is the torque caused by mechanical friction, C r is the spring resistive torque and is the torque generated by the airflow. The electronic throttle valve involves two complex nonlinearities due to the nonlinear spring torque and the friction torque . They are given by their static characteristics [16]. The static characteristic of the nonlinear spring torque C r is defined by For , is the spring constant, is a constant, is the default position and sgn(.) is the following signum function The friction torque function of the angular velocity of the throttle plate can be expressed as where f v and f c are two constants. By substituting in Eq. (25) the expressions of C g , C f and C r and by neglecting the torque generated by the airflow C a , the two nonlinearities and and the two constants and f v the transfer function of the linear model becomes (30) [15].

Simulation results
The identified parameters of are given in Table 2 at 25 C temperature [15].
The corresponding discrete-time transfer function is given by Eq. (31) for the sample time = 0.002s. The desired trajectory is then given in Figure 5.
The obtained optimal I/O scaling factors for Simulink design optimization technique and GA are summarized in Table 3.
The obtained results are given in Figures 6-9.  Table 3. PID-type fuzzy scaling factors values.

Conclusion
In this chapter, an optimization technique was introduced to tune the parameters of PID-type fuzzy logic controller (FLC). The idea is to use the genetic algorithm (GA)-based heuristic optimization technique in order to solve highly complex problems. In order to specify more robustness and performance of the proposed GA-tuned PID-type FLC, optimization criteria such as integral square error (ISE) is considered.
The proposed controller is applied to an electronic throttle valve (ETV) in the discrete-time framework in order to track a desired trajectory starting from a flat output generated using flatness property. The performance comparison with the Simulink design optimization (SDO) technique shows the efficiency of the proposed GA-tuned approach in terms of tracking a desired trajectory with disturbances rejection.