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Proportional + integral + derivative (PID) controllers are widely used in industrial applications to provide optimal and robust performance for stable, unstable, and nonlinear processes. In this paper, particle swarm optimization (PSO) algorithm is proposed to tune and retune the PID controller parameter for a class of time-delayed unstable systems. The proposal is to search the optimal controller parameters like

Poportional integral derivative (PID) controllers are widely used in industrial applications to provide optimal and robust performance for stable, unstable, and nonlinear processes. It can be easily implementable in analog or digital form. Further, it supports tuning and online retuning based on the performance requirement of the process to be controlled.

Open-loop unstable systems are mostly observed in chemical process industries and for economical and/or safety reasons, the chemical process loops to be operated in unstable steady state [

Optimization is a powerful tool for finding the controller parameters. Soft computing-based PID controller parameter optimization is widely addressed by the researchers. Wang et al. have proposed a PSO-based autotuning of PID controller on a stable system [

In this work, the PID controller parameter tuning is proposed for unstable system using particle swarm optimization (PSO) algorithm introduced by Kennedy and Eberhart [

This paper is organized as follows: principle of PSO algorithm is discussed in Section

Particle swarm optimization (PSO) algorithm is a population-based evolutionary computation technique developed by the inspiration of the social behavior in bird flocking or fish schooling. It attempts to mimic the natural process of group communication of individual knowledge, to achieve some optimum property. In this method, a population of swarm is initialized with random positions

The updated velocity of each particle can be calculated using the present velocity and the distances from

% Assign values for the PSO parameters %

Initialize: swarm (

% Initialize random values and current fitness %

% Initialize Swarm Velocity and Position %

Current position = 10 * (rand (

current velocity = 0.5 * rand (

Evaluate the objective function of every particle and record each particle’s

Compare the fitness of particle with its

for

If current fitness (

Then local best fitness = current fitness;

local best position = current position (

end

% Same operation to be performed for

Change the current velocity and position of the particle group according to (

Steps 2–5 are repeated until the predefined value of the function or the number of iterations has been reached. Record the optimized

Perform closed-loop test with the optimised values of controller parameters and calculate the time domain specification for the system.

If the values are within the allowable limit, consider the current

Otherwise perform the retuning operation for

Industrial PID controllers are usually available as a packaged form, and to perform well with the industrial process problems, the PID controller requires optimal tuning. Figure

Block diagram of a closed-loop control system.

Closed-loop response of the system with setpoint

The final steady state response of the system for the setpoint tracking and the load disturbance rejection is given in (

To achieve a satisfactory

In this study, a noninteracting form of PID (

The PID structures are defined as the following:

Block diagram for PID parameters tuning and retuning using PSO.

In this work, ISE (

To study the closed-loop performance of the unstable process with PSO-tuned PID controller, practical examples from literature are considered.

The first order plus delayed time (FOPDT) unstable process with the following transfer function model is considered:

In this study, the optimization algorithm is initiated with the following values. Dimension of search space is three (i.e.,

PSO-based PID tuning is proposed with the method as in Figure

Classical and optimally tuned PID parameters for Process 1.

Method | |||
---|---|---|---|

HC | 0.565 | 0.046025 | 0.34352 |

SC | 0.548 | 0.049294 | 0.56115 |

Visioli | 0.624 | 0.054021 | 0.72446 |

PSO PID | 0.5338 | 0.0307 | 0.4931 |

PIDr1 | 0.5338 | 0.0211 | 0.4931 |

PIDr2 | 0.5338 | 0.0379 | 0.4931 |

Convergence of

Figure

Performance comparison for Process 1.

Method | Over Shoot | Settling Time, s | ||||
---|---|---|---|---|---|---|

HC | 1.7803 | 34.95 | 13.47 | 11.70 | 13.93 | 13.83 |

SC | 1.3233 | 41.55 | 10.11 | 11.87 | 10.42 | 14.02 |

Visioli | 1.3031 | 62.85 | 8.324 | 10.97 | 8.621 | 13.03 |

PSO PID | 1.2632 | 41.83 | 11.25 | 13.76 | 11.62 | 16.30 |

PIDr1 | 1.1815 | 59.80 | 12.34 | 17.45 | 12.77 | 20.73 |

PIDr2 | 1.3229 | 36.26 | 11.04 | 12.61 | 11.39 | 14.91 |

Decrease in cost function with optimized controller parameter.

The process is then controlled with the optimized values of controller parameters, and the performance of the controller is tested in terms of the overshoot, settling time and the error criterion. From Table

Retuned value of integral controller gain.

Servo response for Process 1 with PSO-tuned PID.

Figure

Reference tracking for Process 1 with conventional PID parameters.

Servo and regulatory response for Process 1.

The robustness of the PSO-based PID controller is then tested with the measurement noise introduced in the feedback path. A band-limited white noise with a noise power of 0.001 is introduced along with the feedback signal and from the result it is observed that the proposed controller can perform well even in the noisy environment. Figure

Reference tracking performance with measurement noise.

Isothermal continuous stirred tank reactor (CSTR) considered by Liou and Chien [

The values of the operating conditions are given by flow rate

For the above transfer function model, PSO-based PID tuning is proposed with the method as in Figure

Servo response for Process 2 with PID and PIDr1.

The regulatory response value is recorded with a disturbance value of 0.2 (20% of setpoint) introduced at 500 s. From the performance measure values in Table

Cost function for Process 2.

Method | Over Shoot | Settling Time, s | ||||
---|---|---|---|---|---|---|

PSO PID | 1.201 | 360.7 | 77.64 | 99.89 | 80.72 | 119.7 |

PIDr1 | 0.985 | 239.6 | 59.45 | 78.77 | 61.79 | 94.36 |

The robustness of the PID controller is then tested with a band-limited white noise with a noise power of 0.001and is introduced along with the feedback signal. From Figure

Servo response with measurement noise.

The bioreactor plays a major role in most of the biotechnological and chemical industry. It can be defined as a reactor tank to execute a number of biological reactions to create a large amount of intermediate and final products. In recent years, biosynthesis process is widely utilised to convert the living cells (biomass) into marketable chemical, pharmaceutical, food, and beverage products. In biosynthesis, the biomass consumes nutrients from the substrate (feed) to cultivate and produce more cells and important products. During this operation the bioreactor is kept under a controlled environment with constant pH, temperature, agitation rate, and dissolved oxygen tension to attain better growth of microbes.

In this work, a small scale bioreactor widely analysed by the researchers [

Schematic diagram of a bioreactor.

where “

The stoichiometry for biomass activity is very complex since it varies with environmental conditions microorganism and nutrient in the system. Due to these reasons, unstructured models are mainly considered for analysis purpose. The following mathematical equations can describe a variety of industrial bioreactors. Equations (

The steady state solutions and the mathematical model of the system are depicted in Tables

Steady state solutions of the bioreactor.

Biomass concentration ( | Substrate concentration ( | System condition | | Initial arbitrary value for modelling | |

0 | 4.0 | Stable | Trivial | 0 | 1 |

0.9951 | 1.5122 | Unstable | Nontrivial | 0.125 | 1 |

1.5302 | 0.1746 | Stable | Nontrivial | 1 | 1 |

Mathematical model of the bioreactor.

Operating region | State-space model | Transfer function model |
---|---|---|

Growth phase | ||

Stationary phase |

The unstable bioreactor model is a benchmark problem in the unstable system study.

For substrate inhibition model, the following parameters are considered (Table

PID parameters considered for Process 3.

Method | |||
---|---|---|---|

Linearised (VC) | −0.9513 | 5.5363 | 0.5235 |

Vivek and Chidambaram (VC1) | −0.9524 | 5.6228 | 0.5304 |

Majhi and Atherton | −1.1855 | 4.7255 | 0.4377 |

PSO PID | −0.8722 | 5.4073 | 0.3645 |

^{−1}, ^{−1} (the residence time is 3.33 h) and the feed substrate concentration is ^{−1}. For the unstable operating region (equilibrium 2, nontrivial) biomass concentration

For the unstable operating point, the local linearized model for the unstable bioreactor is

In this work, the PSO-based PID tuning is attempted for second order model (

The closed-loop response of the unstable bioreactor system is tested with the first order model (

Figure

Servo response with classical and PSO-tuned PID.

The actual bioreactor model constructed using the nonlinear equation ((

Servo response of the unstable bioreactor model.

Regulatory response of the bioreactor model.

From the result, it is observed that the PSO algorithm based PID controller provides a nonoscillatory response with minimized overshoot and settling for both the setpoint tracking and disturbance rejection applications.

The robustness of the proposed control scheme is then tested by introducing a measurement noise (noise power of 0.001; sampling time of 0.1 sec). The nonlinear model is considered for the simulation study.

Figure

Variation of Biomass concentration with respect to noise.

Variation of substrate concentration and dilution rate with respect to noise.

Most of the industrial process loops use conventional or modified structure PID controllers. Tuning the controller parameter for time-delayed unstable system is a challenging work if the system model is other than a first order plus dead time. In this work, design of optimization-based model independent controller tuning for unstable process models has been attempted. In this work, a PSO-based PID controller tuning and retuning is presented in detail for a class of unstable systems. The design of controller is formulated as an optimization problem using ISE as the performance index. This is a model free, online tuning method which can identify the optimal controller parameters effectively. PSO can be practically used as an alternative to obtain the controller parameters with an algorithm-based PID controller. The result obtained from the computer simulation shows that the proposed method improves the performance of the process in terms of time domain specification, setpoint tracking, disturbance rejection, error minimization, and measurement noise attenuation.

Positive constants (0–2)

Feed concentration

R concentration

Load disturbance

Error

Global best position

Controller model

Process model

Integrated absolute error

Integral squared error

Integral time absolute error

Iteration

Proportional gain

Integral gain

Derivative gain

Filter constant

Measurement noise

Local best position

Process variable

Inlet flow rate

Random number (0-1)

Reference input

Position of particle

Setpoint

_{i}

Integral time constant

Derivative time constant

Velocity of particle

Inertia weight of particle

Process output

Process output + noise.

Iteration number

Updated iteration number.

Load disturbance

Servo response.