Selectable Fractional-order Controller for Industrial Control Designs

. Differential Equation (DE) of fractional-order specifically gives clear view of fractional-order systems. Since genuine processes are typically or most anticipated to be fractional, employing fractional-order’s concept might be results to take us closer to the actual world. A lot of recent publications concentrate on employing fractional-order dynamics is to describe actual physical processes. In this paper, fractional calculus is applied in the field of control systems. Fractional-order controller also known as FOC has been proposed in numerous studies. The fundamental benefit of a FOC is that it gives the control mechanism of greater flexibility of time and frequency responses, enabling better and more reliable functioning of the system. The industrialization of fractional-order control has practical benefits of better solutions for control problems. The industrial controller has the requirement of different gains and orders of fractional-order controllers. The selectable improved design is proposed, and an optimal and efficient controller is suggested with fractional-order approach. The results show that the best controller is selected from different controllers for water tank and bio-reactor systems. It is found in the results that the Mod FPID controller has the least overshoot of 4.31% and the fastest settling time of 76.5 s for water tank system and fractional-order controller (PID) n is selected for bio-reactor control systems.


Introduction
Fractional calculus was used in mathematics for the last three centuries, but recently, it has been used in the fields of engineering, control theory, material science, biomedicine, signal processing, robotics, and new applications where the overlapping of several disciplines occur.Thus, fractional calculus has been applied as an interdisciplinary tool [1].The list of abbreviations is given in Table 1.
Table 1 List of Abbreviations used in this paper The study of derivatives and integrals of arbitrary orders forms the foundation of fractional calculus.The differentiation (D) and integration (J) for fractional calculus are shown in equations 1 -2.
() = ∫ ()  0 (2) The Grunwald-Letnikov (GL) and the Riemann-Liouville (RL) definitions are frequently used in fractional calculus.The Caputo derivative, another form of the Riemann-Liouville differential explains better physical phenomena.Caputo defined a fractional derivative by the equation 3.
The fractional calculus takes the history of all previous events, and they provide intrinsically a memory for dynamic fractional-order systems.Podlubny introduced the concept of fractional-order PID (FOPID) as a PI λ D µ controller with λ as order of integrator and a µ as an order of differentiator [2].It provides flexibility in the process of design with fine tuning parameters, but with more complexity due to settings in tuning of parameters.These tuning methods include Ziegler-Nichols tuning method and other methods in [3][4][5][6].
The theory belonging to the integers along with derivatives formulated from arbitrary numbers covered by fractional calculus.Additionally, it expands on the formalism for n-fold integration and integer orders.Both short-term and longterm memories are displayed in fractional calculus.Long-term memory results in the absence of a specific timeframe, while short-term memory relates to the dispersion of time constants [7].A potent tool for the explanation of short and long memory is provided by the fractional-order derivatives and integrals.Let U () and Y () represent the input and output signals of Laplace transformations, and assume that all initial conditions are equal to zero.Equation 4 can therefore be used to represent the fractional-order transfer function of the aforementioned system.
The differential-order integral and differential plants are described in equations 5-6.
The latest innovations and trends of fractional calculus by researchers in the different areas of research are: 1. Control systems: Fractional calculus is applied in the field of the design of control systems as fractional-order controllers.These controllers are used by researchers to improve the systems like to improve the performances in robotics, automation in industries, aerospace etc. 2. Signal processing: fractional calculus is used by researchers in the analysis of fractal or non-stationary signals and also in image and audio processing applications [8]. 3. Solution of fractional differential equations: New techniques are developed for the solution of fractional differential equations as these describes the physical phenomena.The applications are in the modelling of materials and heat conduction.The materials based on acrylic-epoxy nanocomposite coatings containing varying amounts of graphene oxide (GO) nanoparticles used in [9] can be described by fractional differential equations.4. Model and predict of financial mathematics: Fractional calculus is used in the modelling and prediction of complex financial systems.The model for 3-pillar with environmental, social, and ethical pillars and their imtegration by financial economic pillar is discussed in [10].The model can further be designed for the sustainability with the fractional calculus.5. Artificial intelligence: Fractional calculus is used to model complex and non linear data.The applications in the food industry for different AI technologies are reviewed in [11].The fractional calculus can do better in food industry to get better results for the intelligent systems.6. Machine learning: Machine learning algorithms are improved by fractional calculus [12].7. Anomalous diffusion: Fractional calculus is used to model anomalous diffusion which is used in the field of physics, biology and chemistry.
An approach to an integer order transfer function in a system is analogous to the concept of the fractional-order transfer function (FOTF).If all initial conditions zero, the frequency characteristic is shown in equation 7 for fractional-order systems(FOS).
G(s) = 1/s α (7) The fractional-order signal processing is used in the analysis of fractional-order linear system and filter theory, estimation and identification of fractional system etc.The relationship among input and output are described by mathematical models.These models reflect the overall behaviours of the system.Differential Equation (DE) of fractional-order specifically gives clear view of fractional-order systems.Since genuine processes are typically or most anticipated to be fractional, employing fractional-order's concept might be results to take us closer to the actual world.The relationship between voltage and current of a RC line with a semi-infinite loss [13] or the heat diffusion into a semi-infinite material is typical examples of FOS [14].
Recent developments have been achieved in the analysis of FODEs to model the dynamic systems.For instance, integrator and differentiator of fractional-order concepts were used to modify PID controllers or Proportional-Integral-Derivative controllers, which usually used to handle industrial control scenarios.The additional level of flexibility by the fractional-order integrator differentiator enables the efficiency of conventional PID controllers to be further improved [15][16].A controlled system's asymptotic phase curve can be changed by a certain amount by using the FOC or fractional-order control approach, changing the time response.Theoretically, control systems can comprise both the fractional-order controller and the dynamic system that is being regulated.However, it is more typical in practices of control system to take the fractional-order controller into account [17][18].And this is because of the possibility that the plant model has been developed being an integer order model in the conventional sense.
A lot of recent publications concentrate on employing fractional-order dynamics to describe actual physical processes.For example; slope fraction on the log-log as discussed in [19][20].The Bode plot is used to describe a specific category of physical events and is also known as the fractal system or sometime called as the fractional power pole or zero.An approach based on the singularity function, that compose of cascaded branches poles-zero pair's number values, is suggested to depict and examine its dynamical behaviours.Additionally, the system's distributing spectrum is likewise simple to build, and the correctness of the distributions relies on an initial prescribed mistake.Following this, the fractional-order differentiators s 1 and integrators s -l (1 is a real positive number) would be widely utilized to approximate the fractional-order transfer functions for a specific frequency range [21][22].A syatematic approach to designing the controllers for the fractional-order plants offered in [23].
A fractional-order controller provides better results for time delays in [24].An investigation with order between 1 and 2 for the stability in fractional-order systems is shown in [25].The fractional-order controller has more stable and robust characteristics, as demonstrated by the suggested method.A practical realization and theoretical implementation of fractional-order controllers using memristive systems was proposed in [26].The memory effect of such device is used for an analogue implementation.The digital FOC in form of IIR filter and analogue realizations realization based on analogue circuits or fractance can be used for realizations.Based on the approximation methods, an analog implementation technique was created developed in [27].
There is a wide scope of systems and control engineering in the perspective of industries due to the requirements of nonlinear, robust or optimal controls which can meet the performance required in the design of controllers.The examples of such controllers or systems are in the field of manufacturing of sensors or transducers, chemical process control systems, robot-based control, and energy transmission industries [28][29].FOC outperform the integer order controllers as for linear controllers, FOC provide flexibility due to a graeter number of controlled parameters and the stability region of proportional, derivative and integral terms of FOC for guaranteed stability for specific phase margin as compared to IOC.Efficient constraints of FOPID controllers tuning due to a greater number of controlled parameters and the tuning of FOPID controllers is easy as compared to high order IOPID controllers for same performances.FOPID controllers are more robust against uncertainties than IOPID controllers.FOPID decreases 20% power consumption of dc motor than IOPID.Further, FOC are more proper than IOC, hence used for modelling as compared to IOPID.FOC increases the precision by decreasing static distortion of the systems.
The innovations and ongoing developments in the field of fractional-order controllers are discussed as: 1. Fractional-order PID controllers: These controllers are developed and optimized for specific applications.
These controllers improve the performance as compared to integer order controllers [30].The system composed of an Arduino Uno, a conveyor motor, a gear motor, and an inductive proxy sensor is shown in [31].The fractional-order controller can be used with such hardware control systems.

Adaptive fractional-order controllers:
The adaptive control strategies adapt the orders in real time on the basis of system dynamics.Thus, the control performances are control in changing conditions [32].
3. Fractional-order sliding mode controller: The robust control can be obtained due to disturbances in dynamic systems by the combination of fractional calculus and sliding mode control [33][34].
4. Fractional-order control in energy managements: Fractional-order controllers are used in renewable and battery management systems.It can improve the stability and efficiency on the systems.The data-based field model is proposed for reducing human energy consumption involved in the plastic injection moulding process with optimized variables [35].To get more stability fractional-order controller can be used in plastic industry for energy managements.
5. Non-linear fractional-order controllers: Integer-order controllers are not efficient for non linear systems and thus fractional-order controllers are used such as in robotics [36].
6. Fractional-order controllers in water systems: Fractional-order controller is used in optimization of water treatment systems and resource management.
7. Real-time implementation: Hardware and software efforts to implement fractional-order controller efficiently and use in industries.
The limitations and lack of research in the area of fractional-order controllers are: 1. Complex design: Fractional-order controllers are more complex and appropriate fractional-order and tuning of controller parameters are time consuming.

Limited analytical tools:
The analysis and tools for these controllers are not sufficient to analyze the system behavior.The custom software support is required for these designs.

No standard approach:
The design of fractional-order controller is not a standardized approach and thus cannot apply them consistently.
4. The realization of FO controllers is complex due to the approximation required in these designs.
5.Not applicable for all systems: Fractional-order controllers are used for non-integer dynamic systems.
Despite these shortcomings, fractional-order controllers are having unique characteristics with significant advantages.The ongoing research is to overcome these limitations and it should be carefully applied for specific applications with particular constraints and requirements.In this research paper mentioned limitations are addressed and a design is proposed for selectable or easy selection of order and tuning of controller with optimum control.

Materials and Methods
The controller performance is degraded due to process dynamics, high delay time, set point variations and external disturbances.Different parameters with the configuration of controller is shown in Fig. 1.

Fig.1 Configuration of controller
The PID controller has three algorithms with proportional, integral and derivative.The PID provides the simple structure and easy implementation.It is used position and speed control due to simplicity and easy maintenance.It is used in manufacturing industries as well.Overall, 90 -95 % of the control applications use PID form as a controller.
Integral term corrects the controlled output for reducing the offset of the process variable by cumulating the error term whereas; derivative term corrects the controlled output for unusual variations.
The purpose of controllers is to match a set point by feedback, such as a thermostat that forces the heating and cooling units to turn on/ off based on a set temperature.It is recommended for systems to automatically compensate and mass, the set point, or the amount of energy available change frequently and the controller is expected to automatically adjust.The very often utilized closed loop controller in industrial processes is the PID controller.It has been used for more than 50 years with great success [37].Because of the simplicity of the feedback mechanism, it is most common in industries [38].It is a combination of proportionate, integral, and derivative which offers a number of benefits.
Where Kp, Ki and Kd are the controls as proportional, integral and derivative gains.The controller is a simple structure and has tuned gains parameters.The closed loop response of the system is expressed as equation 10.
The controller transfer function C(s) maintain output Y(s) based on input R(s) and eliminate the input disturbance D(s) as shown in Fig. 2 and the simple PID control mechanism is presented in Fig. 3.In Fig. 3, input is known as the control signal, which is setup product of three different factors.Furthermore, the error represents the tracking error, which is a factor of each of the following parameters.The Kp represents a symbol denoting a proportionality Furthermore, the two symbols Ki/s and Kds are known as integral and derivative terms, respectively.The phrases function "independently" of each other.Considering that Kp = Kd = 0, it left with equation 11.

𝑢(𝑡) = 𝐾 𝑖 ∫ 𝑒(⋋)𝑑 ⋋
0 (11) This integral is added to create the open loop forward path of Type I.Because of this, in case the system remains stable, there will be no steady-state error in response to a step input.It can be considered as an instance of the internal model concept in action.If e (t) is non-zero for any amount of time, such as when it is positive, the control signal increases over time.In the situation where the plant's output starts to fluctuate, it is compelled to respond.The integral term can be seen as an accumulation of the previous values of the error.Gain and integral gain are frequently connected by equation 12.

𝑢(𝑡) = 𝐾 𝑑 𝑑𝑒(𝑡)
(13) The control relies on how quickly the error is changing.A true differentiator typically not present and this is because of absence of the proper differentiation is a "wide-band" with the gain of this term increase linearly as frequency increases.It doesn't follow a causal chain.It is feasible to move constantly in the PID plane as opposed to hopping between four xed places in the plane.In general, fractional-orders between 0 and 2 are taken into consideration.The function of controller based on the selected value of ⋋ and μ are as follows in Table 2.Both processes either I or D are typically of fractional-order in controller.Finding the best combination of Kp, KI, and KD values is necessary to satisfy the complete identification for a particular process, and this requires parameter improvement in two-dimensional hyperspace.Industrial control applications have been using proportional integral derivative (PID) controllers since many years.The wide-ranging appeal can be attributed to their straightforward design and amazing performance, which includes minimal percentage overrun and a short settling time for sluggish process plants [39].An addition to the traditional PID controller is a fractional PID controller.A fractional-order controller can quickly acquire the iso-damping feature [40][41].
For fractional PID controllers a generalized function has been described below equation 14.
Where C(s) for the outcome of the control system, U(s) representing the control signal, E(s) representing the error signal, Kp representing the proportional constant gain, KI representing the integration constant gain and Kd representing the derivative constant gain.This is also for the order of integration, and also for differentiator.All conventional controllers are special examples of fractional controllers wherein ⋋ and μ equal to 1.

Selectable Optimized Controllers
The fractional-order differentiator/integrator (FOE) will be used for FOC for the adjustment of the gain and order of the proportional and integrative/derivative controllers.The mentioned variable FOE will be used with selection of fractional-order controller from different controllers for better system performance is proposed.
In 1995, Kennedy and Eberhart proposed the swarm intelligence based optimization technique called as particle swarm optimization (PSO).The simulation is taken of the social behavior including the cooperation and communication in bird and fish schooling.PSO algorithm is applied in real world complex, non-linear and non-differentiable problems as mention in [42].PSO is an optimization technique which is based on population and quality results are produced in specified time with stable convergence.On iteration of the main loops, it updates the each particle based on local information and global information of swarm.After this update, the position of particle is update with the new information as in equation 15.
The optimized fractional order controller is shown in Fig. 4. It shows the optimization technique and the parameters which are taken by the controller and then the plant is being controlled.The algorithm as a pseudo code required for the selection of controller is shown in Algorithm 1.

Results and Discussion
In Industries, many real-time processing units like spherical tank system and biochemical reactor are non-linear systems.Due to their complexity, the tuning and disturbance rejection is a difficult task to handle, and they can exhibit stable and unstable steady states respectively for mentioned units.The tuning of controller is important for all industrial control systems and different controller structures can be used to stabilize the dynamic systems.
Chemical, oil and waste-treatment industries are widely used level control of a tank by control the liquid or water level at the desired state [51].A water tank controller is used to controls automatically the water level in a tank.To maintain the optimal value in the tank, it senses the level and switches on or off accordingly.Fully automatic controllers detect the level in the tanks and control accordingly with sensors in tanks with prevention of dry running of the pump.Simple water level controllers sense the undesired low and high levels in a tank and work accordingly to maintain the optimum water content in the tank.These controllers help in water conservation with prevention of water overflows and help in saving of energy with switching on and off of the motor.These are easy to use and also cost effective.Thus, water level controllers are having many advantages and applications.An optimal and efficient controller is required for industries and a fractional-order control system is designed to fulfil mentioned requirements.
A bio-reactor control system is used to regulate the biochemical or other processes for stable environment.Different factors and parameters are controlled to maintain the optimal process conditions in these controllers.Mostly, industries are using classical control approaches and due to the complexity in monitoring and control, there is a need of new better control systems.An optimal control system is suggested in this research paper for better responses.

Water Tank Control System
The industrial controllers are used in many industries, and they are designed for technicians/engineers to maintain the required performance.The modules of controllers are used in forward and feedback loops.The commercial controllers are improved by fractional dynamics as FOPID controllers.The FOPID contains amplifier (proportional), fractionalorder differentiator, and fractional-order integrator.The variable fractional-order controller as an improved controller is presented.Then an algorithm is applied to select the fractional-order controller from different controllers for improved system performance.Mass balance is used for mathematical modeling of the system for the tank system as shown in Fig. 5.
The mass balance for the mentioned SISO tank system is Mass in -Mass out = accumulation of mass as in equations 26 -27.
The output of the flow rate (lit/hr) follows nonlinear relationships i.e. square root of the height as in equation 28.
Also, we have equations for inlet water flow (lit/hr) as in equations 29-30.
After Laplace for both sides of equations 31 -32 as : The transfer function model is considered for first-order stable process of tank level after using values of the parameters as in equation 33.The research problem is to study the existing controller designs.The simulations of MATLAB are performed for approximation and realization techniques required for fractional-order element.The fractional-order integrator and differentiator are designed.The fractional-order controller designs with fractional-order element are proposed.The results will be analyzed and compared with the existing designs.The design and analysis of the performance of the fractional-order controllers based on fractional-order elements are analysed and compared with the proposed algorithm.From the Table 2, it can be seen, the Mod FPID controller has the settling time of 76.5 seconds and the least amount of overshoot, at 4.31%.The rise time is better as 9.14 seconds for the (PID) n controller.Therefore, the fractional-order controller Mod FPID can be chosen based on the information provided above.

Bio-reactor Control System
The transfer function model is considered for an unstable bioreactor process [53][54] is in equation 34.
The step response of the system mentioned for different systems using MATLAB are shown in Fig. 7.The numerical assessment of open-loop and closed-loop systems and for different controllers is mentioned in Table 4.
(g) Fig. 7 Step and bode responses of Table 4 Performance assessment of Bio-reactor system A control system uses a controller to manipulate the input of a device or system (the plant) in order to achieve a desired behavior at the output of the system.In automatic control systems, feedback that is based on observations of process output through sensing elements is important factor.The controller's goal is to maintain zero error or reduce it at a sufficient rate.On the other hand, tracking systems aim to track a reference input that varies over time.Table 4 shows that the Mod FPID controller has the least overshoot (12.1%), but it takes 114 seconds to settle.Nonetheless, a controller is optimal for (PID) n (settling time and rise time).Therefore, a fractional-order controller (PID) n can be chosen based on the information.

Conclusion
If the desired performances in complex systems are not fulfilled by the tradional controllers systems, fractional-order control laws may be able to help and fulfill those performance requirements.Firstly, the basics of feedback control system and fractional calculus focusing on its applications in different fields are discussed.A brief insight in the importance of fractional-order controllers is given.Then the motivating factors to carry out the research work pertaining to fractional-order controller.Fractional-order (FO) system modeling and fractional calculus are becoming more and more popular in many scientific and engineering domains.The analog circuit realizations of approximate FO derivative models are used in different systems.The fractional-order element is used for the design of electronic circuits.The approximation is done for these fractional elements and further the networks are emulated.The analogue implementation techniques are discussed with approximation and realization techniques in [56][57].
The industry grade FOPID controller is required so that the gains of proportional, differentiator, and integrator controllers and also the orders of differentiator/integrator controllers can be varied.Thus the selectable type of controller for controlling the gains and orders are required for industrial use.The research work is centered around to find an accurate fractional-order controlller corresponding to real systems using a new algorithm.The fractional-order controller is designed for different dynamic systems and then the analysis is done for the selection of better fractionalorder controller.The validation is done for the proposed study with the MATLAB software.The selectable fractionalorder controller is proposed for an optimum and efficient control design.The suggestions are for the automation with the use of microcontroller for the selection of gain and the orders and then the implementation of the selectable fractional-order controllers.Thus the variable FO differentiator as an improved fractional-order elements can be used and these analog circuits are recommended as a future scope for creating the FO-PID controller.

Fig. 2
Fig.2 Closed loop system with controller transfer function (gain) = 3.6215 sec/cm 2 and  ( ) = 330.46minutes.The Oustaloup approximation is used for fractional-order controller design with N = 5, wlow = 10^-3 and whigh = 10^3.Other parameters used for the design are λ = α = 0.98, µ = 0.75, n= 0.05.γ = 0.1 *kd, Ti = Kp/Ki, Td = Kd/Kp and Kc = Kp/Ti.The controller design process to find the optimal values of controller parameters with PSO algorithm based PID controller tuning are shown in[52] as Kp = 68.115,Ki = 1.0042, and Kd = 1.9016.The design of different Integer and fractional-order controllers like PID, PI, FOPID, (PID) n , IMC-FOPID, and Mod FPID are performed after putting different parameters.The step response of the system mentioned for open-loop and closed-loop systems are shown in Fig.6.The numerical assessment of open-loop and closed-loop systems and for different controllers is mentioned in Table3.

Table 2
Controller function classification

Table 3
Performance assessment of water level system