# Optimal tuning of FOPID controller based on PSO algorithm with reference model for a single conical tank system

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## Abstract

The fractional order proportional integral derivative controller (FOPID) replaces the conventional PID controller for its immense merits such as its simple structure, better set point tracking, high disturbance rejection, higher capability of handling model uncertainties in nonlinear and real time applications. This paper addresses the effectiveness of the FOPID controller on a level control of single conical system in real time. It is one of the classical nonlinear control problem and its shape not only contributes the proper mixing of liquids and also guarantee to drainage of solid wastes. This study, a dual loop controller is proposed and constructed with a master process and slave process for a level control of a single conical tank system. In the slave process, the output response obtained by the traditional PID controller combined with the conical tank reference model and compared with the output response of the master process. The obtained error signal is used to dynamically correct the parameters of the FOPID controller in the master process. The tuning of FOPID controller is a challenging task due to its presence of extra parameters and it efficiently carried out by particle swarm optimization (PSO) algorithm. The obtained result reveals that the Proposed PSO optimized FOPID controller with reference model demonstrated the better set point tracking, smooth controller response than other conventional integer order controllers.

## Keywords

FOPID controller Single conical tank system Reference model PSO algorithm## 1 Introduction

Fractional order calculus is a famous mathematical area with over 300 years of antiquity that generalizes a conventional integer calculus into arbitrary orders. In the initial theory of fractional order derivative was developed in the seventeenth epoch assumption between L’Hospital and Leibniz [1]. Mostly, real time problems can be described, defined, modeled and controlled more precisely in fractional order methods than the integral order methods. Several engineering fields and science applications utilize fractional calculus due to well theoretical explanation and development of the computing area in the last two decades. Subsequently, new design tools makes fractional-order integro-differential equations more effortlessly. Moreover, the applications of fractional-order differential equations has got in different prospects in control systems. Accordingly, various research work has been completed in the fractional order control (FOC) in the last decades due to its extra flexibility to meet control applications more specifically.

Conventional PID is the utmost typical controller employed in the process industries in last six decades, because of simplicity, ease of implementation, availability of many tuning methodologies and minimum knowledge of the process enough to control. However, there is no guarantee of dynamic responses, structural complexity, nonlinearities, and large time delays. The uncertainties in some realistically constrained cases in modern industries is makes increasing the attention for an improvement of PID controller. Podlubny et al. [2] has stretched integer order PID controller theory to fractional order PID controller and suggested in fractional order formats of PI and PID controller i.e. \(PI^{\lambda }\) and \(PI^{\lambda } D^{\mu } .\) These controllers have an integrator order \(\lambda\) and differentiator order \(\mu\) and this two extra parameter provides an added degree of freedom in the performance of controller that makes FOPID controller performance better than conventional PID controller. Then Podlubny’s works related to fractional order theory and fractional order controller has given lot of new ideas to researches and industrial real time application. Some of the very important literatures are: Roy et al. [3] developed adaptive fractional order PI controller for a variable quadruple tank process with feed forward controller and compared with decentralized PI and sliding mode controller. Padula et al. [12] proved advantages of FOPID controller than other integer order controller with large number of experiment under set point following and load disturbance rejection. This paper also minimize the integrated absolute error subject to a constraint to the maximum sensitivity. Ranganayakula et al. [4] studied the different tuning rules for a FOPI and FOPID controllers and validated using different stable first order time delay processes. Sharma et al. [5] developed fuzzy FOPID controller for a robotic manipulator. This paper shows advantages of FOPID controller under trajectory tracking, model uncertainty, disturbance rejection and noise rejection compared to the other integer order controller. Azarmi et al. [6] demonstrated and proved FOPID controller for a CE 150 type laboratory helicopter model is better than PID and FOPI controller under change in disturbance. Moreover, the flexibility of handling uncertainties, robustness, sinking undesired oscillations and fast change of control signal makes requirement of fractional order controller concepts into many advanced control strategies such as phase lead lag compensator [7, 8], sliding mode control based FOC [9, 10], quadratic regulator based FOC [11, 12], smith predictor based FOC [13], internal model based FOC [14, 15], \(H\infty\) norm based FOC [16, 17], set-point weighted FOC [18], FOC with pre-filter [19] and Loop shaping method [20]. From the above literature clearly shows FOPID controller is better than other integer order controller. Also, currently many researchers chosen FOPID controller due to extra parameters makes system more robust and effective for different applications.

The fine tuning of fractional order PID control is challenging compared to conventional PID control due to the existence of extra two parameters and to meet some special constraints like gain margin, phase margin, gain crossover frequency, and sensitivity conditions. Nevertheless, evident from a literature shows that the development of the meta-heuristic methods are created the tuning of constraints very ease [21, 22, 23] in last few years such as genetic algorithm [24, 25], Big bang big crunch algorithm [26], particle swarm optimization [27], bacterial foraging optimization algorithm [22, 23, 28], artificial bee colony algorithm [29, 30], stochastic multi- parameters divergence optimization [31], multi objective optimization design [32], cuckoo search algorithm [33, 34], bacterial foraging chemotaxis gravitational search algorithm [35], differential evolution [36], chaotic ant swarm optimization [37], gases brownian motion optimization [38], bat algorithm [39], tabu search algorithm [40] than analytical approach.

Even though development of many natural inspired algorithms, PSO has got some constant place among researchers due to its simplicity, simple calculation, lots of freedom given to modify the structure of algorithm and a substitute solution to the non-linear complex optimization problem. Its base idea was stimulated from the social activities of animals such as fish schooling, bird clustering, etc. PSO depends on the natural activity of communication between the teams and to share individual information when a team of birds or insects searching food or throughout migration. All the birds don’t know wherever the best position, however from the social behavior, if anyone of the members within the group can find the desired path to travel and therefore remainder of the particle track quickly. In this article, to optimize the FOPID controller parameters PSO algorithm is concentrated.

As far as design for tuning of PID and FOPID controller, performance index such as integral square error (ISE), integral absolute error (IAE), and integral time absolute error (ITAE) only used in an objective function. IAE and ISE produce comparatively minor overshoot but a lengthy settling time. Though ITAE surpassing this drawback, but produces problem in the stability margin. In this article includes time domains performance criteria such as settling time, rise time, and peak overshoot with weighting values in the objective function to improve the optimization performance. Recently many researchers are giving importance for tuning of PID controller with reference model and this idea was originate from the concept of model reference adaptive control. A reference model is designed based on the expected output from the plant model [41]. Recently, many researchers given importance to tuning the controller with reference model. Barbosa et al. [42] and Li et al. [43] utilized Bode’s ideal transfer function for tuning PID controller. Neural network based reference model is proposed for positioning control system by [41] and [44]. Wang et al. [45] developed reference model based sliding mode control for helicopter system. Jeng [46] addressed different specifications for a reference model and analyses the control strategy with reference model for a different stable and unstable plants. In this article, time domain criteria such as settling time and peak overshoot values are utilized for conical system reference model implementation. In the proposed dual loop structure, closed loop reference model with PID controller setup is known as slave process and plant model with FOPID controller is known as a master process. The difference between the response from the reference model and plant response is considered as an error which is used as an objective function with some time domain specifications for tuning of FOPID controller parameters using PSO algorithm. This difference of error value improves the efficiency of the controller to perform control in the plant.

The proposed dual loop control structure is validated in real time using highly nonlinear single conical tank system. It is one of the classical nonlinear control problem whose objective is to control the level of the tank. This shape not only contributes the proper mixing of liquids and also guarantee to drainage of solid wastes [47]. This type structure widely used in food factories, petroleum industries and chemical industries. So controlling of conical tank is important and challenging task in process control industry. If the level of the tank increases, then overflow of the valuable liquid. Similarly decreases in the level produces bad significances in the sequential operation. Hence dedicated and efficient control algorithm is required for efficient operation. Vijayalakshmi et al. [48] developed linear parameter varying model and then controlled using adaptive PI controller. Ramanathan et al. [47] presented reinforcement learning technique based on Q-learning algorithm for a conical tank level control. Ravi et al. [49] demonstrated regime-based multi-model adaptive control strategy for decoupling-based decentralized PI controller for a conical tank system. Tamilselven et al. [50] estimated the plant parameters and control using online kalman filter based fuzzy logic controller. Similarly, adaptive passivity based adaptive control algorithm with Taylor polynomial approximation for a level regulation of a conical tank system is developed by [51]. Lot of control algorithm is proposed for control of conical tank system but still, it is an open issue in current and future. In this article proposed FOPID controller with reference model is proposed for level control of a single conical tank system.

- 1.
Better controller performance and smooth control signal is achieved using dual loop FOPID control structure with reference model for a single conical tank system.

- 2.
The difference between the reference model and plant model, dynamically adjust the FOPID controller parameter by using PSO algorithm with multi-objective function.

This paper is prepared as, a detailed review of fractional order calculus and FOPID controller is described in Sect. 2. Section 3 comprises the detail description of single conical tank system and first order plus dead time approximation. Overview of PSO is explained in Sect. 4. A detail block diagram arrangement of the proposed technique and reference model formation of a single conical tank system with multi-objective function is given in Sect. 5. A practical result of the FOPID controller based PSO algorithm with reference model for a single conical tank system is described in Sect. 6. Lastly Sect. 7 concludes article.

## 2 Review of fractional calculus and FOPID controller

The first study of fractional calculus was described by Liouville (1832), who was principally involved by deep components of Laplace and Fourier. Later M. Axtell discussed and recommended an extensive usage of fractional calculus in control system in 1990, first time. The detailed review of fractional order calculus and FOC is available in [52, 53, 54].

### 2.1 Fractional order calculus and approximation

*t*are limits of an operator. The integro-differential operator in continuous domain is defined as

*t*are limits of an operator. Some definitions available for fractional order calculus in literature but the commonly used are as follows.

### **Definition 1**

*b*is the integer which fulfills the stipulation \(b - 1 \le \chi < b\).

\(\varGamma \left( \cdot \right)\) is the gamma function.

### **Definition 2**

### **Definition 3**

Fractional order differential equation is not a simpler in numerical simulation like a standard differential equation. Therefore, Laplace transforms tool is frequently utilized by engineers, especially control engineering applications.

Hence, the fractional integer operator *β* embodied by the transfer function \(L \, f\left( t \right) = F(s) = {1 \mathord{\left/ {\vphantom {1 {s^{\chi } }}} \right. \kern-0pt} {s^{\chi } }}\) in the frequency domain.

### 2.2 Integer-order approximation of fractional derivative

### 2.3 Fractional order proportional integral derivative controller

*μ*= 1, a conventional PID controller is develop again. Similarly other combination is represented in the Fig. 2.

## 3 Description of a single conical tank system

Adjusting the level of a conical tank system is a very interesting task, owing to their variations in the control area, uncertainty, time varying parameters and dead time on input and output variables. Nonlinear systems like conical tanks used in wide choice of applications such as gas plants, food industries, petrochemical plants because of merits like better disposal of solids and easy mixing. In this section real time setup, mathematical modeling and identification of conical tank system is described.

### 3.1 Real time setup and technical specifications of single conical tank system

Technical specifications of a single conical tank system

Part name | Technical specifications |
---|---|

Conical tank | Stainless steel body |

Top radius (R): 240 mm | |

Bottom radius (r): 8 mm | |

Height (H): 50 cm | |

Inflow: 0–1000 lph | |

Differential pressure transmitter | Capacitive type Range : 2.5–250 mbar Output : 4–20 mA |

Pump type | Centrifugal 0.5 HP Single phase AC motor |

Control valve | Size : ¼” Equal percentage valve Air–open type Input : 3–15 psi |

Rotameter | Range: 0–18 lpm |

Air regulator | Size: ¼” BSP Range: 0–2.2 bar |

I/P converter | Input: 4–20 mA Output: 0.2–1 bar |

Pressure gauge | Range: 0–30 psi |

Compressor | 20 psi |

### 3.2 Modeling of a single conical tank system

The objective is to control a level of the tank, which can be achieved by controlling the input flow of the conical tank by using a valve at an inlet. At steady state, both the inflow (*F*_{i}) and outflow (*F*_{o}) rates remain the same. At each height of the conical tank, the radius will vary due to the non-linear nature which is due to the shape of the tank. The difference between the inflow and the outflow rate will be based upon the cross section area of the tank and level of the tank with respect to time. The flow and the level of the tank can be regulated by properly modeling the tank.

*θ*is the angle difference which relates the current height of fluid to the total height of the tank (Degrees), R is the maximum radius of the tank and H is the maximum height of the tank.

*F*

_{o}) can be written as,

*dh*/

*dt*) can be written as,

### 3.3 FOPDT approximation of a single conical tank system

*K*is process gain,

*τ*is dead time and

*T*is time constant of process.

*t*

_{1}and

*t*

_{2}when the open loop reaches 35.3% and 85.3% of its last steady state value respectively. This method is very simple and effective compared to other techniques in the literature. The open loop response of single conical tank system as shown in Fig. 5. By using estimated

*t*

_{1}and

*t*

_{2}values dead time calculated as follows:

## 4 Particle swarm optimization

This section already explain in [41], and is repeated in this article for a sake of completeness. PSO is a unique population based stochastic search algorithm proposed by Kennedy and Eberhart [65]. The original idea of PSO algorithm is inspired from a model of the communal performance of animals such as fish schooling and bird gathering. This algorithm is based on communication of individual knowledge and natural learning when birds or insects hunt for food or migration in the search space. Popularity of PSO in the last decades due to its simple structure and only few parameters needed to adjust the optimization of any kind of problems.

*m*> 1. The set of particle is in a colony (m) is flown in the D dimensional search planetary. This location of every particle is represented as \(X_{i} (t) = \left( {x_{i1} (t),x_{i2} (t), \ldots ,x_{iD} (t)} \right)\). The D dimensional space is created by \(X_{i} (t) \in \left[ {l_{d} ,u_{d} } \right]\) where \(l_{d} {\text{ and }}u_{d}\) are the limits. The velocity vector also represents like a position vector \(V_{i} (t) = \left( {v_{i1} (t),v_{i2} (t), \ldots ,v_{iD} (t)} \right)\). Each particle tracks the personal best particle position from its own experience and particle fitness value calculated from objective function, which is represented as \(P_{{b_{i} }} (t) = \left( {p_{{b_{i} 1}} (t),p_{{b_{i} 2}} (t), \ldots ,p_{{b_{i} D}} (t)} \right)\). An optimal value among the \(P_{{b_{i} }}\) is called \(G_{b}\) and it is represented as \(G_{b} (t) = \left( {g_{b1} (t),g_{b2} (t), \ldots ,g_{bD} (t)} \right)\). A PSO algorithm is worked based on (30–33).

*i*;

*x*

_{i}denotes the current location of particle

*i*with objective value fitness; \(P_{{b_{i} }}\) is the best past position of particle

*i*itself; \(g_{{b_{i} }}\) is the global best location among the group. \(rand_{1} \left( t \right) \, ,rand_{1} \left( t \right) \in \left( {0,1} \right)\) are the homogeneously distributed random numbers.

*c*

_{1}and

*c*

_{2}are the positive accelerated constraints. The constants \(c_{1} {\text{ and }}c_{2}\) express the confidence of the particle has in itself and in its neighbor and these parameters improve fast convergence of the optimal solution. The selection of values of

*c*

_{1}and

*c*

_{2}are depends on the problem, in the first version of PSO \(c_{1} \, = c_{2} = 2\), for better performance on real time problems recent work propose that \(c_{1} \, + \, c_{2} \le 4\).

*w*) component in velocity updating rule. Clerc [67] proposed constriction factor \(\left( \varPsi \right)\) for ensuring convergence of optimal solution and control of the magnitude of particle velocity. This constriction model also helpful to value selection of \(w,c_{1}\) and

*c*

_{2}. This proposed constriction model with inertia weight is (34) and the pseudo code of PSO algorithm is given in Table 2.

Pseudo code for a PSO algorithm

## 5 Tuning of FOPID controller for a single conical tank system using PSO algorithm with reference model

*ω*

_{n}is the natural frequency of a system. Thus the transfer function of a reference model is specified in (38)

*e*(

*t*)is the error (difference between the reference model and real model),

*M*

_{p}is maximum peak overshoot,

*t*

_{r}is rise time and

*t*

_{s}is settling time. The \(\omega_{1} , \ldots ,\omega_{5}\) are the weighting values of objective function which vary from 0 to 20 in this case. This proposed objective function is applied to the PSO algorithm for optimal tuning of FOPID controller parameters \((K_{p} ,K_{i,} K_{d} ,\lambda \, ,\mu )\) for the conical tank system model. It is essential to control the reference model and is done by PID controller with Z-N tuning method.

## 6 Simulation results and discussion

In this segment, the advantage of FOPID controller with reference model is proved through the real time level control of a single conical tank system and compared with PI controller, PID controller and FOPID controller. This work is carried using MATLAB R2013a (Version 8.1.0.604) platform and implemented in a PC with Intel Core2 Duo Processor with 2.27 GHz speed and 2.00 GB RAM. National Instruments conical tank system module is used for real time analysis of the proposed controller. The Simulink program is directly interfaced with the real time process system through DAQ module. It is enabled with National instruments VISA serial communication interface. The module supports ASCII data format with a sampling time of 0.1 s and a baud rate of 38400. With this, monitoring and control of the real-time process can be easily established with MATLAB software.

Z–N tuning result of slave PID controller for reference model

Controller parameters | \(K_{p}\) | \(K_{i}\) | \(K_{d}\) |
---|---|---|---|

PID | 6.544 | 3.094 | 5.189 |

Values of PSO

Parameters | PSO |
---|---|

Dimension | 5 |

Number of particles | 50 |

Number of iteration | 100 |

\(c_{1}\) | 0.8 |

\(c_{2}\) | 1.3 |

Inertia weight \((w)\) | 0.9 |

Optimized controller parameters using PSO algorithm

Controller | \(K_{p}\) | \(K_{i}\) | \(K_{d}\) | \(\lambda\) | \(\mu\) | J | |
---|---|---|---|---|---|---|---|

PI | Best | 29.136 | 1.898 | – | – | – | 0.9955 |

Worst | 29.152 | 1.861 | – | – | – | 0.9985 | |

Mean | 29.121 | 1.826 | – | – | – | 0.9923 | |

SD | 00.023 | 0.048 | – | – | – | 0.0026 | |

PID | Best | 28.655 | 4.965 | 8.944 | – | – | 0.9165 |

Worst | 28.685 | 4.956 | 8.956 | – | – | 0.9189 | |

Mean | 28.625 | 4.925 | 8.925 | – | – | 0.9145 | |

SD | 00.056 | 0.066 | 0.056 | – | – | 0.0016 | |

FOPID | Best | 24.852 | 3.466 | 6.473 | 0.469 | 0.656 | 0.7923 |

Worst | 24.865 | 3.468 | 6.486 | 0.459 | 0.668 | 0.7985 | |

Mean | 24.898 | 3.462 | 6.476 | 0.486 | 0.685 | 0.7953 | |

SD | 00.045 | 0.012 | 0.028 | 0.056 | 0.049 | 0.0023 |

Performance analysis of different control methodologies of single conical tank system

Controller | ISE | \({\text{M}}_{\text{p}}\) | \(t_{r} ( {\text{s)}}\) | \(t_{s} ( {\text{s)}}\) |
---|---|---|---|---|

PI | 196 | 5.35 | 42.82 | 146.86 |

PID | 102 | 4.86 | 36.40 | 86.85 |

FOPID | 75 | 1.34 | 21.36 | 29.34 |

FOPID with reference model | 41 | 0.93 | 16.79 | 19.45 |

## 7 Conclusion

The novel FOPID controller with the reference model has been offered for a single conical tank system in this article. A mathematical model of a single conical tank has been identified and approximated as a first order plus dead time model. Closed loop reference model with PID controller is proposed with FOPID controller to tune the highly nonlinear single conical tank model effectively. The PSO algorithm with multi-objective function is used for dynamically adjust the FOPID controller parameters depends on the error difference between the reference model and plant model. The effectiveness of proposed dual loop controller over other controllers are proved in real time under change in setpoint. This critical analysis of the obtained result reveals that the proposed FOPID controller based PSO algorithm with reference model is provided smooth control signal, improved stability and better qualitative performance compared to integer order controllers. Robustness and stability of the controller also analyzed for a single conical tank system model. We believe that the proposed methodology for control can benefit for control practitioners in lot of ways. One of the important benefit and key point of this proposed controller is that we can able to obtained required output response by formulating reference model. This work will serve as a new control scheme in process control and several other applications.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interests regarding publications of this article.

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