An Adaptive Neuro-Fuzzy Inference System to Improve Fractional Order Controller Performance

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 PID) 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 ANFISFOPID control scheme are also compared to the classical fractional order PID 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.


Introduction
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 [1][2][3]. The performance of traditional controllers is satisfactory for systems that are simple and have a linear input-output relationship [4]. However, real-world industrial processes are highly dynamic, nonlinear, and more complex. Conventional controllers are not good enough to provide better control performance for those industrial processes due to the various constraints [5]. Besides, conventional controller-based closed-loop systems are more sensitive to system parameter change and load disturbances [6]. Hence, most recent researchers are focusing on the fractional order control (FOC) as an alternative, the integral and derivative terms of the conventional PID controller are replaced by fractional order numbers (PI λ D µ ) then the controller becomes a fractional order controller [7]. In the past, several research studies have confirmed that FOCs can exhibit better performance than classical controllers [8][9][10][11]. If the integral and derivate parts order of a fractional order controller is equal to unity (i.e., λ = µ = 1) then the FOPID controller will work as a conventional PID controller. The FOC technique can be used for integer order or fractional order plant models in a particular control application [12].
Many fields, particularly engineering, biology, physics, and medicine, have expanded their usage of FOCs in recent years [13][14][15]. The accurate model of a real-time system can be developed with the help of FOC, which is necessary to achieve precise and reliable control action. The traditional PID controller only has three adjustable parameters: K 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 [16]. Unfortunately, parameter tuning in fractional order controllers is a difficult part of system design, and there is currently no easy and effective tuning solution. Caponetto et al. [17] suggested a design strategy for the PI λ D µ controller tuning. For the PI λ D µ controller design, Hwang et al. [18] used the minimal error criterion method. Intelligent algorithms such as fuzzy logic control (FLC), neural network (NN), ANFIS etc. have increased the flexibility of controller tuning and system design. Recent studies show that intelligent algorithms combined with fractional order controllers provide a better outcome in closed-loop control [19][20][21][22]. Controller tuning is an important component of the system design, and it becomes difficult as the number of controller parameters increases. On the other hand, intelligent algorithm combined with closed-loop control schemes will reduce the challenges in system design. In the past, only a few investigations on controller adaptation employing intelligent techniques were documented in connection with the FOC. Nooshin and Hossein introduced an adaptive intelligent robust controller for stabilizing uncertain fractional order chaotic systems in [23], demonstrating that the controller is more robust against external disturbances and process uncertainties. Gholamreza et al. [24] used a fuzzyneural controller to fine-tune the settings of a PID controller for controlling the speed of a DC motor; improved results were reported in terms of settling time and rising time.
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 Grunwald-Letnikov (GL) and Riemann Liouville (RL) methods [26][27][28]. As per the GL definition, the fundamental fractional operator is expressed as where p and x are the limits, γ is the order of operation, while in RL definition the fractional operator is expressed as for (m − 1 < γ < m), where À Á ð Þ Euler's gamma function. The Laplace transform of the GL and RL fractional derivative/integral is provided as under a zero initial condition which is expressed as The transfer function in the case of a commensurate order system is stated as where g q ¼ qg; b q ¼ qg and g 2 0; 1 ð Þ; 8q 2 Z þ

Controller Design Procedure
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. [29] have proposed the controller design method.
Malek et al. [30] have combined this fractional order pole controller design method with a time-delay mechanism. A fractional first order plus delay time plant model general transfer function is given as 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 [30][31][32]. The fractional order controller must meet the following parameters based on the fundamental definitions of gain crossover frequency and phase margin.
where ω and f m are the required gain crossover frequency and phase margin, respectively.
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 [9], indicating that the system is more robust to changes in the parameter. As a result, the controller must meet this requirement, 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 x ! x r rad/sec.

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 x ! x d rad/sec.

Classical PI λ D µ Controller Design
The transfer function of the system under open-loop condition using the plant Eq. (5) and the classical PI λ D µ controller Eq. (6) is given as The Eq. (13) can be written as where Argument part of Eq. (14) is The Eq. (15) can be rewritten as The magnitude of Eq. (14) yields According to the robustness to plant parameters variation constraint Eq. (10), the open-loop phase is expressed as Using Eqs. (16)-(18) the parameters K p , K i , K d , λ, and µ of the classical PI λ D µ controller can be obtained.

[PI] λ D µ Controller Design
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 Eq. (5) and the suggested [PI] λ D µ controller Eq. (7) is written as In the frequency domain, the Eq. (19) can be represented in the following form by replacing s with jω where C mfoc jx and G jx arg G jx ð Þ j j¼ Àtan À1 Open-loop phase at ω frequency can be expressed as where A ¼ 1 þ Tx a cos ap 2 ; B ¼ Tx a sin ap 2 According to the phase margin constraint Eq. (8), the open-loop phase could satisfy the following expression According to the gain crossover frequency constraint Eq. (9), the open-loop gain at the gain crossover frequency satisfies the following equation x a cos lp 2 According to the robustness to plant parameters variation constraint Eq. (10), the open-loop phase is expressed as where F ¼ aTx aÀ1 sin ap 2 1 þ T 2 x 2a þ 2T x a cos ap 2 À L: Using Eqs. (28), (30) and (31) the parameters K p , K i , K d , λ, and µ of the proposed [PI] λ D µ controller can be achieved.

Adaptive Neuro-Fuzzy Inference System
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. Fig. 1 depicts the internal design of a typical ANFIS. The structure consists of five layers; the input layer is followed by a membership layer, the rule layer, the normalization layer, and the output layer. The first layer, which is made up of nodes for each input, will just transfer data to the second layer and will not do any calculations. For the second layer, the type of membership function and the number of input membership functions are specified in advance by the designer. The membership value to a specific fuzzy set is expressed as 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

Implementation of the Proposed Control Configuration
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 [32] was chosen and is given in Eq. (36), G s ð Þ ¼ 2:15 4:803s 1:2 þ 1 e À1:4s : The initial parameters of the proposed [PI] λ D µ controller can be determined using Eqs. (28), (30), and (31). Using the controller tuning procedure in [33], the controller parameters' value has been determined with the following specifications: phase margin (ɸ 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 [34] and integral absolute error (IAE) performance metrics are used. To ensure the applicability of the identified system, the system stability and controller design specifications described in Eqs. (8)- (12) are validated using the MATLAB software toolbox. To assess the usefulness of the suggested control strategy, a classical PI 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 Eqs. (16)- (18). Tab. 1 lists the parameters of the three distinct controllers used in this research. Fig. 2 depicts the suggested ANF[PI] 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 backpropagation 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 Fig. 3. The trained ANFIS will adjust λ and µ parameters to accomplish adaptability. The fuzzy rules for creating the outputs λ and µ have their own ANFIS block, as shown in Fig. 2. The process error e(t) and  change in error Δe(t) are used as inputs by both ANFIS parts of the controller to fix the value of fractional factors λ and µ in the range of 0 to 2.

Results and Discussion from the Simulation Study
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 (36). In the initial part of the study, the effectiveness of the controllers has been validated for the unit step input. From the study results shown in Fig. 4, one can understand that the designed ANF[PI] 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. Fig. 5 depicts the time response of the proposed controller's adaptive settings for the unit step input study corresponds to Fig. 4. This result illustrates that the controller's ANFIS rightly changes the controller's parameters and ensures the system responds quickly without any oscillation or overshoot. Therefore, the designed FOC with modified structure exhibit better transient performance than traditional control methods. External load disturbances and changes in input test have been performed to confirm the robustness of the proposed controller. When the system is at a steadystate level, load disturbances have been applied. The external load disturbances at 10 and 20 s and the associated system output using three different control schemes are shown in Fig. 6. When compared to the traditional controllers, the present fractional order controller with ANFIS control scheme outperforms for the load disturbances. The ANFIS section of the proposed controller immediately detects load disturbances     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 Fig. 8. The input change test has been carried out using three distinct controllers for the input to the system reduced by 20% from its normal level after 10 s and then boosted by 40% after 20 s. The system with ANF[PI] 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. Fig. 9 depicts the ANF[PI] λ D µ controller's adaptive parameters change corresponding to the study results of Fig. 8. Tab. 2 shows a numerical comparison of the performance of three different controllers using various metrics such as rise time, settling time, IAE and integral square error (ISE). In all these performance metrics, the adaptive mechanism using ANFIS for the fractional order controller outperforms when compared to other controllers. Therefore, the proposed ANFIS-based adaptive control scheme will be a better option to enhance controller action for non-linear and complex systems.

Conclusions
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.

Conflicts of Interest:
The author declares that he has no conflicts of interest to report regarding the present study.