Fuzzy rules‐base.

## Abstract

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 controller, 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) algorithm in terms of tracking a desired trajectory with disturbances rejection.

### Keywords

- PID‐type FLC
- scaling factors
- genetic algorithm
- integral of the squared error
- Simulink design optimization technique
- flatness
- electronic throttle valve

## 1. 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.

## 2. 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 *α* and *β*

The inputs variables, well known as the error *ek* between the desired trajectory *ek* given by Eqs. (1) and (2) where *Te* is the sampling period.

The output variable

The output of the PID‐type fuzzy is given by Eq. (4) [5]

Thus, the equivalent control components of the PID‐type FLC are such that

Proportional gain:

Integral gain:

Derivative gain:

where the terms

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 *ek*, Δ*ek* and Δ*uk*, as follows: *NL*: negative large; *N*: negative; *ZR*: zero; *P*: positive; *PL*: positive large.

e_{k}\Δe_{k} | N | ZR | P |
---|---|---|---|

N | NL | N | ZR |

ZR | N | ZR | P |

P | ZR | P | PL |

## 3. 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.

### 3.1. 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. The chromosomes are candidate solutions to the problem. Then, the fitness values of all chromosomes are evaluated by calculating the objective function. So, a group of the best chromosomes is selected based on the fitness of each individual. In this ‘surviving’ population, the genetic operators of 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 *e(t)* is

and the ISE can be defined by

Fitness function is taken as inverse of error, i.e., performance index.

### 3.2. 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.

## 4. 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 *q* is the forward operator,

where the parameters *i=0,1,…,n−1*. The partial state of such a dynamic system can be considered as a discrete flat output

Often, the real output signal

The open loop control law can be determined by the following relations [12].

where *g* are the vectorial functions. Then, it is sufficient to find a desired continuous flat trajectory

The polynomial interpolation technique is used in order to plan the desired flat trajectory

where

and the vectors

Then, the output desired trajectory

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.

## 5. 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.

### 5.1. 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,

The mechanical part of the throttle is modelled by a gear reducer characterized by its reduction ratio

where *Cg* is the gear torque. The mechanical part is modelled according to Eq. (25), such that [14, 15].

and

where *J* is the overall moment of inertia, *K*_{e} is a constant, *Cf* is the torque caused by mechanical friction, *Cr* is the spring resistive torque and *Cr* is defined by

For

The friction torque function

where *fv* and *fc* are two constants. By substituting in Eq. (25) the expressions of *Cg*, *Cf* and *Cr* and by neglecting the torque generated by the airflow *Ca*, the two nonlinearities *fv* the transfer function of the linear model becomes (30) [15].

with *ks* = (180 / *π / γ2*)*kr* and *q* as the Laplace operators.

### 5.2. Simulation results

The identified parameters of

Parameters | Values |
---|---|

R(Ω) | 2.8 |

L(H) | 0.0011 |

ke (N.m/A) | 0.0183 |

kv (v/rad/s) | 0.0183 |

J (kg.m^{2}) | 4 × 10^{−6} |

γ | 16.95 |

The corresponding discrete‐time transfer function is given by Eq. (31) for the sample time

The desired continuous time flat trajectory

where

The desired trajectory is then given in Figure 5.

The obtained optimal I/O scaling factors

SF\Method | SDO | GA |
---|---|---|

Ke | 0.1144 | 0.0086 |

Kd | 1.5997 | 0.4612 |

α | 2.6239 | 0.1108 |

β | 0.0001 | 0.1934 |

The obtained results are given in Figures 6–9.

Figures 8 and 9 show the responses with Simulink design optimization technique and GA tuning using ISE criterion. Based on a comparative analysis, better results were there obtained with the GA tuning method.

All results, for obtained scaling factors values, are acceptable and show the effectiveness of the proposed GA tuning method in terms of the tracking desired trajectory with disturbances rejection in comparison with the SDO technique.

## 6. 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.