The design and analysis of a fractional order proportional integral derivate (FOPID) controller integrated with an adaptive neuro-fuzzy inference system (ANFIS) is proposed in this study. A first order plus delay time plant model has been used to validate the ANFIS combined FOPID control scheme. In the proposed adaptive control structure, the intelligent ANFIS was designed such that it will dynamically adjust the fractional order factors (λ and µ) of the FOPID (also known as PI^{λ}D^{µ}) controller to achieve better control performance. When the plant experiences uncertainties like external load disturbances or sudden changes in the input parameters, the stability and robustness of the system can be achieved effectively with the proposed control scheme. Also, a modified structure of the FOPID controller has been used in the present system to enhance the dynamic performance of the controller. An extensive MATLAB software simulation study was made to verify the usefulness of the proposed control scheme. The study has been carried out under different operating conditions such as external disturbances and sudden changes in input parameters. The results obtained using the ANFIS-FOPID control scheme are also compared to the classical fractional order PI^{λ}D^{µ} and conventional PID control schemes to validate the advantages of the controllers. The simulation results confirm the effectiveness of the ANFIS combined FOPID controller for the chosen plant model. Also, the proposed control scheme outperformed traditional control methods in various performance metrics such as rise time, settling time and error criteria.

Conventional controllers such as PID, PI, and PD are widely used in most industrial control applications to control different processes. These controllers are selected because of their simple structure, which makes them easy to understand and implement [^{λ}D^{µ}) then the controller becomes a fractional order controller [

Many fields, particularly engineering, biology, physics, and medicine, have expanded their usage of FOCs in recent years [_{p}, K_{i}, and K_{d}, however, the FOPID controller has two extra adjustable fractional order factors (λ and µ) by which greater system design and improved control will be possible [^{λ}D^{µ} controller tuning. For the PI^{λ}D^{µ} controller design, Hwang et al. [

An intelligent ANFIS combined FOC design and its performance validation for a first order plus delay time plant is discussed in the present study. The ANFIS model has been developed to adjust the value of the FOC parameters dynamically based on system conditions. To enhance controller performance, a modified structure of the FOPID controller is proposed in this work. In the modified FOPID controller structure, the order of the integral part (λ) is used for the combined proportional and integral part (i.e., [PI]^{λ}D^{µ}). The ANFIS part of the controller block will closely monitor process dynamics by utilizing error and changes in error inputs. The designed system performance is validated under different operating conditions. The suggested controller results are compared to those of conventional PID and classical PI^{λ}D^{µ} controllers to confirm the advantages.

The following sections make up this paper: Section 2 provides an overview of fractional order control. The controller design technique is discussed in Section 3. The structure and functions of the ANFIS are detailed in Section 4. Section 5 explains how the suggested controller is implemented in the proposed system. In Section 6, the outcomes of the simulation study are discussed. The research findings are presented in Section 7 as a conclusion.

The fractional calculus is a non-integer order variation of standard differentiation and integration [

where

for (m − 1 < γ < m), where

The transfer function in the case of a commensurate order system is stated as

where

The design approach for the classical PI^{λ}D^{µ} and proposed [PI]^{λ}D^{µ} controllers are described in this section. For FOC with an integer-order pole, Luo et al. [

where K denotes the plant gain, T denotes the time constant, L denotes the delay, and α denotes the plant order. The transfer function of the classical PI^{λ}D^{µ} controller and the proposed [PI]^{λ}D^{µ} are expressed respectively, as follows

where λ is the order of the integral part and µ is the order derivative part, K_{i}, K_{p} and K_{d} are the coefficients of integral, proportional, and derivative terms, respectively. In this paper, the range of fractional order numbers (λ, µ) are chosen between 0 and 2.

Phase margin, gain crossover frequency, and robustness to changes in plant parameter are the three key characteristics that are frequently addressed for fractional order controller design [

Phase margin (ɸ_{m})

where ω and

Gain crossover frequency (ω)

In the logarithmic frequency domain, the magnitude of the open-loop transfer function should be zero at the gain crossover frequency point

Robustness to changes in the plant parameter

The phase Bode plot is flat for a certain value of the ω, as mentioned in [

Besides in the above three requirements, the controller’s noise rejection and disturbance elimination abilities are also considered during the design process to maintain system stability. The specifications for noise rejection and disturbance elimination are presented in the following formulations.

Noise rejection

In the case of noise rejection specifications, the sensitive function N can be given as

where N is the preferred reduction in noise for frequencies

Rejection of disturbance

The constraint related output disturbance rejection sensitive function D can be given as

where M is the preferred value of the sensitivity function for frequencies

The transfer function of the system under open-loop condition using the plant ^{λ}D^{µ} controller

The

where

Argument part of

The

The magnitude of

According to the robustness to plant parameters variation constraint

Using _{p}, K_{i}, K_{d}, λ, and µ of the classical PI^{λ}D^{µ} controller can be obtained.

This research proposes a modified version of the classical PI^{λ}D^{µ} controller in which the proportional and integral coefficients of the controller will use a common fractional order number(λ). The system’s open-loop transfer function employing the plant ^{λ}D^{µ} controller

In the frequency domain, the

where

Open-loop phase at ω frequency can be expressed as

where

According to the phase margin constraint

According to the gain crossover frequency constraint

where

According to the robustness to plant parameters variation constraint

where

Using _{p}, K_{i}, K_{d}, λ, and µ of the proposed [PI]^{λ}D^{µ} controller can be achieved.

The expert knowledge is expressed in terms of linguistic description in the traditional FLC, which is utilized to produce the necessary control action. The range of each input is partitioned by fuzzy membership functions with a specified boundary in traditional fuzzy system design to decide local conclusions based on fuzzy rules. However, ANFIS is a modified fuzzy inference system (FIS) architecture in which membership function ranges are adapted using neuro-adaptive learning approaches. The ANFIS combines the benefits of FLS and NN, with the FLS’ inference ability and the NN’s learning ability performing the tuning process. In NN, parameter optimization is made by back-propagation and the least-squares technique.

where σ_{ij} and n_{ij} are denotes the variance and mean of the Gaussian membership function. The nodes in the third layer represent the preconditioned part of a fuzzy rule. In this method, the layer two nodes are used to determine the degree of membership of the applicable rule. The output function of the inference node and the product (AND) operation that is commonly done in this layer are depicted as

The output from the third layer is received by normalization nodes, which are normalization layers. The fourth layer node function can be expressed as

Layer five combines the several actions advised in layer four to create a single output which is expressed as

The fractional order numbers (λ and µ) are dynamically updated based on the system parameters in the ANFIS combined modified FOC (which is designated as ANF[PI]^{λ}D^{µ}). To test the effectiveness of the suggested control approach, the model of the pressure control plant [

The initial parameters of the proposed [PI]^{λ}D^{µ} controller can be determined using _{m}) = 70°, gain margin = 10 dB, gain crossover frequency (ω) = 0.001 rad/sec., high-frequency noise rejection = 10 rad/sec., and disturbance elimination sensitivity function = 0.001 rad/sec. In addition, to acquire the optimal controller parameters, the Nelder-Mead optimization approach discussed in [^{λ}D^{µ} and a conventional PID control method are also evaluated for the same plant. The classical PI^{λ}D^{µ} controller parameters are obtained using

Controller | K_{p} |
K_{i} |
K_{d} |
λ | µ |
---|---|---|---|---|---|

ANF[PI]^{λ}D^{µ} |
1.08 | 0.19 | 0.34 | 0.91 | 0.83 |

PI^{λ}D^{µ} |
0.96 | 0.16 | 0.32 | 0.97 | 0.89 |

PID | 0.92 | 0.12 | 0.35 | – | – |

^{λ}D^{µ} control scheme in schematic form. The ANFIS component of the system is initially trained with a precise dataset linking the research model’s inputs and outputs. The Sugeno fuzzy model with three fuzzy membership functions of Gaussian type is utilized for both inputs and outputs throughout the ANFIS design process, and the hybrid (i.e., least-squares and back-propagation combined) optimization approach with an error tolerance of 0.001 is selected. The relationship between inputs and outputs is represented in the form of a surface view as shown in

The performance of the developed ANF[PI]^{λ}D^{µ}, classical PI^{λ}D^{µ}, and conventional PID controllers were examined under various operating conditions for the chosen fractional first-order plant model ^{λ}D^{µ} controller performance is comparatively better in terms of rise-time and settling time. The proposed system with ANF[PI]^{λ}D^{µ} controller reached the target level quickly than the classical PI^{λ}D^{µ} and conventional PID controller-based system. When compared to PI^{λ}D^{µ} and PID controllers, the suggested control approach has a better overall transient performance since the ANFIS modifies the controller parameters dynamically, which results in the system performing well for the unit step input.

By adjusting the system’s set-point level, the closed-loop system’s response to the input change was tested and the result is shown in ^{λ}D^{µ} controller senses input changes promptly and then modify the FOC controller’s parameters, which makes the system output to a new steady-state level within a short time. On the other hand, traditional controllers take longer to reach the new steady-state output. Therefore, the ANFIS-based adaptation technique proposed in this study improves the controller performance and thereby the system’s robustness. ^{λ}D^{µ} controller’s adaptive parameters change corresponding to the study results of

Controller | Performance indices | |||
---|---|---|---|---|

ISE | IAE | Settling time (sec.) | Rise-time (sec.) | |

ANF[PI]^{λ}D^{µ} |
2.01 | 2.32 | 3.94 | 1.45 |

PI^{λ}D^{µ} |
2.29 | 2.71 | 4.86 | 1.96 |

PID | 2.35 | 2.88 | 5.85 | 2.34 |

A modified fractional order PID controller integrated with ANFIS control scheme was investigated for a fractional first order pressure regulating plant model. The ANFIS was created for the proposed control scheme to keep track of the system parameters and thereby modify the fractional order PID controller coefficients to achieve a better control action on the system. The designed ANF[PI]^{λ}D^{µ} controller performance was studied and compared to classical PI^{λ}D^{µ} and conventional PID controllers under various operating conditions. The study’s findings indicated that when the system encounters input parameter change and external load disturbances, the designed controller performs effectively to bring back the output of the system quickly to the desired level. Compared to traditional PI^{λ}D^{µ} and PID controllers, the suggested control strategy greatly reduced the settling time and rise time of the system, which is essential for a good controller. Also, the modified fractional order PID controller integrated with ANFIS can improve the closed-loop system’s stability and robustness. The study results confirm that the present control technique will be a better option for nonlinear and complex industrial systems. Computational time and controller identification are the big challenges in this type of ANFIS combined adaptive fractional order control schemes.

^{λ}D

^{μ}controller

^{λ}D

^{μ}controller for a class of first-order delay-time systems