This paper addresses improvements in fractional order (FO) system performance. Although the classical proportional–integral–derivative (PID)-like fuzzy controller can provide adequate results for both transient and steady-state responses in both linear and nonlinear systems, the FOPID fuzzy controller has been proven to provide better results. This high performance was obtained thanks to the combinative benefits of FO and fuzzy-logic techniques. This paper describes how the optimal gains and FO parameters of the FOPID controller were obtained by the use of a modern optimizer, social spider optimization, in order to improve the response of fractional dynamical systems. This group of systems had usually produced multimodal error surfaces/functions that occasionally had many variant local minima. The integral time of absolute error (ITAE) used in this study was the error function. The results showed that the strategy adopted produced superior performance regarding the lowest ITAE value. It reached a value of 88.22 while the best value obtained in previous work was 98.87. A further comparison between the current work and previous studies concerning transient-analysis factors of the model’s response showed that the strategy proposed was the only one that was able to produce fast rise time, low-percentage overshoot, and very small steady-state error. However, the other strategies were good for one factor, but not for the others.

Fuzzy-logic control (FLC), established by the use of integer orders, has been used in several engineering applications, including photovoltaic [

Control systems based on fractional calculus have recently been drawing growing attention in research, due to their extra flexibility and improved design performance [

FOPID has been demonstrated to be an effective controller in several complex nonlinear systems [

The current research aimed to determine the optimal gains and FO parameters of FOPID when using social spider optimization (SSO) to improve the response of fractional dynamic systems. This type of system usually produces a multimodal error surface that occasionally has many local minima. During the optimization process, the gains and FO parameters of PID are used as the decision variables, whereas the integral time of the absolute error (ITAE) is assigned as the objective function. The results are compared with genetic algorithms (GAs), particle-swarm optimization (PSO), harmony search (HS), gravitational search algorithms (GSA), and cuckoo search (CS).

This paper is organized as follows. In Section 2, the concept of a FOPID-like fuzzy controller is introduced. In Section 3, a brief description of the SSO algorithm is presented. Section 4 presents the discussion of the results obtained and comparative testing. Finally, in Section 5, the main findings are outlined.

In the classical PID controller, the controlling signal (control action) is calculated according to proportion of error, integral of error, and derivative of error. The constants of proportionality are the controller’s gains. They are usually named _{P}, _{I}, and _{D} for proportional, integral, and derivative gains, respectively. The control action and transfer function of the PID controller as a function of the system’s error are shown in

FOPID is an example of the use of fractional calculus in control systems. The modification in the controller’s transfer function includes the derivative and integral terms by changing the Laplace complex frequency,

where λ and α are two positive real numbers.

Since the concept’s introduction in 1965, FL has become an effective technique in industrial applications. Accordingly, FL added a new perspective to the control theory with the aim of formulating the relationship between input and output variables. Previously, this relationship had been represented mathematically. However, in the sense of FL, the relationship between inputs and outputs can be represented by a set of “if–then” rules. Every rule signifies a portion in the input–output space. Therefore, signal processing in the fuzzy controller passes through the three processing operations like a normal fuzzy system. In other words, every input should be fuzzified (converted from crisp to fuzzy) through its associated fuzzy membership functions (MFs). These fuzzy inputs are passed to the rules in the knowledge base in the inference engine to produce the rules’ fuzzy outputs. The overall fuzzy output is obtained by the aggregation (union) of the fired fuzzy rules. Finally, the defuzzification (conversion from fuzzy to crisp) operation takes place to come up with the final output value. In control systems, a Mamdani-type fuzzy rule is preferable for most systems:

IF error is NS and change-of-error is PS, THEN control action is Z

where NS is negative small, PS positive small, and Z zero MFs.

Two crucial parameters have to be set properly in the design of a fuzzy controller. The first is the controller’s inputs and their associated MFs. The second is the fuzzy rule–based list. In classical PID, the control-action value is based on information about the system’s error, integration of error, and derivative of error. However, in fuzzy control, integration of error cannot produce sufficient information to take any action based on its value. Therefore, information related to the integration term can be obtained by considering the controller’s input as a derivative term, then integrating the controller’s output. In this respect, two configurations can be adopted to implement a PID-like fuzzy controller. The first is to build both a PD and a PI fuzzy controller, and sum their outputs. The following is an example of two fuzzy rules for PD and PI controllers, respectively:

IF error is NS and change in error PS, THEN control action is Z

IF error is NS and change in error PS, THEN change in control action is Z

The PI-like fuzzy controller of the first configuration is shown in

Fortunately, the same rule base can be used for both PD and PI controllers. The second configuration of the PID-like fuzzy controller is to use a combination of PD and PI controllers, as shown in

In this study, the error (

NB | NM | NS | Z | PS | PM | PB | |
---|---|---|---|---|---|---|---|

NB | NB | NB | NB | NB | NM | NS | Z |

NM | NB | NB | NB | NM | NS | Z | PS |

NS | NB | NB | NM | NS | Z | PS | PM |

Z | NB | NM | NS | Z | PS | PM | PB |

PS | NM | NS | Z | PS | PM | PB | PB |

PM | NS | Z | PS | PM | PB | PB | PB |

PB | Z | PS | PM | PB | PB | PB | PB |

In this work, SSO, a recent and efficient optimizer, was applied to obtain optimal gains and optimal FOs for the controller. SSO simulates the cooperative behavior of spiders within a colony. It uses a population (S) of N candidate solutions, where every solution represents a spider position, whereas the general web symbolizes the search space—X. During the optimization process, every spider (_{i}) based on its best solution:

where _{i} denotes the value of the cost function of the

where _{j} denotes the spider weight and

The update process depends on the sex. For the female, these equations can be used:

where _{i, c} the vibration transferred by the closest individual (_{i, b} the vibration transferred by the best solution. For male members, this update equation can be used:

where _{f} represents the nearest female element to the individual male and _{i, f} the vibration transferred by the nearest female spider. More details about the algorithm’s mathematical modeling and physical illustration can be found in Cuevas et al. [

The objective function is used as in

To optimally track the system’s input, the tracking error should be as minimal as possible. Accordingly, the performance of a controller is best evaluated in terms of the error criterion. Most optimization techniques use the objective function in terms of system error. However, the most popular performance-assessment criteria are integral of absolute errors, integral square of errors, mean-squared error, and ITAE. In this work, ITAE was selected as the objective function [

where _{P}, _{I}, _{D}, and _{U}, as well as the controller FOs—λ and α.

Systems’ performances are usually compared by their behavior in the transient-response phase. This can be done using quantitative markers. The most important markers of a system’s response include rise time (_{r}), delay time (_{d}), steady-state time (_{s}), peak time (_{p}), percentage overshoot (%OS), and steady-state error (_{ss}). _{r}, %OS, and _{ss}.

_{r} defines the time elapsed for the system’s output to go from 10% to 90% of its final value and %OS is the percentage of the difference between output and input at the first peak, _{p}, over output at the steady state—

_{ss} is the difference between the reference value and the final value of the system’s response. For a unit-step response, it can be calculated as _{ss} = 1 –

The closed-loop system, comprising the FO system as the controlled process and the FOPID fuzzy controller, was implemented using MatLab R2020b and its Simulink toolbox. The FO part was implemented using the Fomcon toolbox. In this work, seven triangular MFs were selected as the fuzzification functions for the inputs and the output, which resulted in 49 Mamdani-type fuzzy rules. The output was defuzzified using center-of-gravity defuzzification method.

In this study, the system under consideration included a fractional dynamical system that usually produces a multimodal error surface with many local minima. The transfer function of the fractional system is presented in

The configuration of the closed-loop model of the proposed FOPID fuzzy controller that controls the fractional system is presented in

The results of the strategy proposed compared to previous work in the literature are presented in

Controller type | _{p} |
_{d} |
_{i} |
_{u} |
α | λ | ITAE | |
---|---|---|---|---|---|---|---|---|

GA | PD^{α} + I |
1.3329 | 0.6341 | 0.6130 | 5 | 0.4932 | 1 | 98.20 |

PSO | PD^{α} + I |
1.331 | 0 | 0.6937 | 5 | 5 | 1 | 311.66 |

HS | PD^{α} + I |
0.7867 | 0.8128 | 0.8271 | 3.6129 | 0.9319 | 1 | 804.16 |

GSA | PD^{α} + I |
1.0823 | 0.6463 | 0.2924 | 4.0152 | 0.5802 | 1 | 346.70 |

CS | PD^{α} + I |
1.2220 | 0.6590 | 0.6647 | 5 | 0.4232 | 1 | 105.47 |

SSO | PD^{α} + I |
1.3173 | 0.6560 | 0.5932 | 4.9797 | 0.5091 | 1 | 98.87 |

SSO | PI^{λ}^{α} |
1.4058 | 1.5646 | 1.7423 | 4.9928 | 0.5921 | 0.4854 | 88.22 |

The comparison was extended to measure transient-response factors. This analysis included calculation of system _{r}, %OS, and _{ss}. The resulting values of all factors for the strategy proposed in comparison to the other strategies are shown in _{r}, low %OS, and small _{ss}, the only strategy to accomplish this. The findings of this study confirm that the combination of FL and fractional calculus can provide a robust and efficient FOPID-like fuzzy controller that can improve the performance of complex systems, such as FO ones.

Method | Tr (ms) | %OS | |Ess| |
---|---|---|---|

GA | 405.00 | 5.24 | 0.00 |

PSO | 350.65 | 29.13 | 0.02 |

HS | 1485.24 | 0.58 | 0.17 |

GSA | 573.79 | 1.62 | 0.05 |

CS | 383.46 | 7.92 | 0.00 |

SSO | 421.67 | 4.02 | 0.00 |

SSO (FOPID) | 368.06 | 2.36 | 0.01 |

This study describes the use of a FOPID-like fuzzy controller to enhance the performance of FO systems. The properly set values of the controller played a vital role in the system’s performance. The optimal parameters of the controller relative to its gains and the FOs were obtained using SSO. The resulting gains and FOs produced the best (lowest) ITAE value: that obtained using the strategy proposed exceeded the previous best in the literature by 10.77%. Transient-response factors showed better performance in terms of fast _{r}, low %OS, and small _{ss}. Consequently, the findings of this study prove that compared to other controllers, the use of a FOPID-like fuzzy controller can produce outstanding performance in linear or nonlinear systems.