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This paper proposes a novel method to tune the I-PD controller structure for the time-delayed unstable process (TDUP) using Bacterial Foraging Optimization (BFO) algorithm. The tuning process is focussed to search the optimal controller parameters (

Proportional + Integral + Derivative (PID) controllers are widely used in various industrial applications in which setpoint tracking and disturbance rejection are necessary. This controller provides an optimal and robust performance for a wide range of operating conditions for stable, unstable and nonlinear processes. Based on the controller configuration (position of P, I, and D), the PID is classified as ideal PID, series PID, and parallel PID.

Since an ideal PID controller has practical difficulties due to its unrealizable nature, it is largely considered in academic studies. Parallel PID controllers are widely used in industries due to its easy accomplishment in analog or digital form. The major drawbacks of the basic parallel PID controllers are the effects of proportional and derivative kick. In order to minimize these effects, modified forms of parallel controller structures such as ID-P and I-PD are widely considered [

Time Delayed Unstable Processes (TDUP) considered in this work are widely observed in chemical process industries (exothermic stirred reactors with back mixing, pump with liquid storage tank, combined feed/effluent heat exchanger with adiabatic exothermic reaction, bioreactor, polymerization reactor, jacketed CSTR) [

In control literature, many efforts have been attempted to design optimal and robust controllers for TDUP. Panda has proposed a synthesis method to design an Internal Model Controller-based PID (IMC-PID) controller for a class of time-delayed unstable process [

In recent years, evolutionary approach-based controller autotuning methods has attracted the control engineers and the researchers due to it is nonmodel-based approach, simplicity, high computational efficiency, easy implementation, and stable convergence [

Recently, the author has attempted BFO-based PID and I-PD tuning for a class of TDUP [

In this work, a multiple-objective function-based BFO algorithm has been proposed for the controller parameter tuning for TDUP. Further, an attempt has been made by considering a TDUP with a zero. A comparative study on various cost functions such as ISE, IAE, ITSE, and ITAE, has been attempted. To evaluate the performance of the proposed method, a simulation study is carried out using a class of unstable system models.

The remaining part of the paper is organized as follows: an overview of bacterial foraging optimization algorithm is provided in Section

Bacteria Foraging Optimization (BFO) algorithm is a new class of biologically inspired stochastic global search technique based on mimicking the foraging (methods for locating, handling, and ingesting food) behavior of

Flow chart for bacterial foraging algorithm.

The working principle for the bacterial foraging optimization algorithm can be defined as shown in Figure

In process industries, PID controller is used to improve both the steady state as well as the transient response of the plant. Consider the closed loop control system as shown in Figure

Block diagram of the closed loop control system.

For practical applications, the term “

(i) Let,

Structure of parallel PID control system.

Mathematical representation of parallel form of PID controller is given in (

The output signal from the controller is

(ii) The noninteracting form of I-PD controller structure is shown in Figure

Structure of parallel I-PD control system.

The output signal from the I-PD controller is

A generalized closed loop response of a system is shown in Figure

“

“

“

Closed loop response of the system.

The Cost Function (CF) guides the algorithm to search the optimised controller parameters. Equation (

The performance criterion in the proposed method is expressed as,

The performance criterion presented in (

The controller tuning process is to find the optimal values for

BFO-based I-PD controller tuning.

Prior to the optimization search, it is necessary to assign the following algorithm parameters.

Dimension of search space is three; number of bacteria is chosen as ten; number of chemotaxis step is set to five; number of reproduction steps and length of a swim is considered as four; number of elimination-dispersal events is two; number of bacteria reproduction is assigned as five; probability for elimination dispersal has a value of 0.2.

In the literature, there is no guide line to allot the tuning parameters for the BFO algorithm.

In this study, before proceeding with the BFO-based I-PD controller tuning, the following values are assigned.

The three dimensional search space is defined as:

If the search does not converge with an optimal

The maximum overshoot (

The steady state error

There is no guideline to specify the values for CF and settling time

For each process example, five trials with a particular CF are carried out and the finest set of values among the trials are selected as the optimized controller parameter set.

A unity reference signal is considered for all the process models, (that is,

To study the closed loop performance of the TDUP with BFO tuned I-PD controller, mathematical model of the processes from literature are considered.

The first order system with the following transfer function model is considered:

The BFO-based I-PD controller tuning is proposed for the system as in Figure

Optimised controller values.

Method | Iteration number | |||
---|---|---|---|---|

ISE I-PD | 35 | 1.7726 | 0.2809 | 0.1339 |

IAE I-PD | 41 | 2.2530 | 0.4752 | 0.4628 |

ITAE I-PD | 57 | 1.8399 | 0.3722 | 0.3194 |

ITSE I-PD | 62 | 1.8874 | 0.5199 | 0.3836 |

Final convergence of controller parameters with (a) ISE-, (b) IAE-, (c) ITAE-, and (d) ITSE-guided BFO algorithm.

The process model (

PID controller performance for Example

An enlarged view of the controller output “

The effect of proportional and derivative kick in

The given process model is then proceeded with an I-PD controller (Figure

I-PD controller performance, (a) process output, (b) controller output.

The regulatory response is then studied with a load disturbance of 0.1 (10% of setpoint) introduced at 30 sec. From Figure

Load disturbance rejection for Example

Table

Performance comparisons for Example

Method | Reference tracking | Disturbance rejection | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

ISE | IAE | ITAE | ITSE | ISE | IAE | ITAE | ITSE | |||||

ISE I-PD | 1.959 | 2.750 | 4.989 | 3.582 | 4.50 | 0 | 8.20 | 0 | 2.051 | 3.234 | 20.36 | 19.62 |

IAE I-PD | 1.813 | 2.637 | 4.687 | 3.225 | 4.60 | 0 | 8.25 | 0 | 1.872 | 3.938 | 46.79 | 37.85 |

ITAE I-PD | 1.793 | 2.441 | 3.889 | 2.923 | 3.65 | 0.0323 | 9.55 | 0 | 1.926 | 3.151 | 19.44 | 17.03 |

ITSE I-PD | 1.565 | 2.230 | 3.675 | 2.681 | 2.85 | 0.1039 | 8.25 | 0 | 1.700 | 2.935 | 17.92 | 15.92 |

The second order TDUP with the following transfer function is considered. It has one unstable pole and a stable pole:

For this model, the BFO-based I-PD is proposed with a bacteria size of 18 and the other values as given in Section

Optimised

Method | Iteration number | |||
---|---|---|---|---|

ISE I-PD | 174 | 1.3394 | 0.0479 | 0.6195 |

IAE I-PD | 189 | 1.9037 | 0.1173 | 0.9032 |

ITAE I-PD | 238 | 1.5518 | 0.0968 | 0.8471 |

ITSE I-PD | 251 | 1.5892 | 0.1204 | 0.8949 |

Figure

Performance evaluation for Example

Method | Reference tracking | |||||||

ISE | IAE | ITAE | ITSE | |||||

ISE I-PD | 5.775 | 7.485 | 33.73 | 24.71 | 11.1 | 0.0343 | 27.9 | 0 |

IAE I-PD | 4.852 | 7.706 | 47.50 | 39.42 | 13.7 | 0.0003 | 40.2 | 0 |

ITAE I-PD | 4.541 | 5.702 | 18.62 | 17.55 | 8.59 | 0.0002 | 19.1 | 0 |

ITSE I-PD | 4.184 | 5.239 | 10.00 | 9.681 | 7.33 | 0.0009 | 18.5 | 0 |

Servo response for Example

The robustness of the I-PD controller is analysed by changing the delay time of the process model. The controller values by ISE are employed to test the controller performance.

−50% change is applied in the delay

The above model has been studied by the researchers and classically tuned controller settings are existing in the literature [

+25% change in the delay

BFO-based I-PD controller tuning.

BFO-based PID controller tuning.

From Figures

A third order unstable process with delay studied by Chen et al. [

Optimised controller parameters.

Method | Iteration number | |||
---|---|---|---|---|

ISE I-PD | 153 | 1.5309 | 0.0349 | 1.1747 |

IAE I-PD | 169 | 1.7045 | 0.0596 | 2.1298 |

ITAE I-PD | 378 | 1.8996 | 0.0807 | 1.5813 |

ITSE I-PD | 391 | 2.0109 | 0.1052 | 1.8934 |

Figure

Reference tracking performance.

Method | Reference tracking | |||||||

ISE | IAE | ITAE | ITSE | |||||

ISE I-PD | 11.70 | 15.42 | 147.9 | 162.8 | 23.2 | 0.0134 | 67.8 | 0 |

IAE I-PD | 9.899 | 13.48 | 118.1 | 137.5 | 19.0 | 0.0189 | 62.5 | 0 |

ITAE I-PD | 8.591 | 11.58 | 95.06 | 124.9 | 15.9 | 0.0121 | 68.8 | 0 |

ITSE I-PD | 7.841 | 10.85 | 87.12 | 115.2 | 14.0 | 0.0184 | 75.5 | 0 |

Servo response for Example

Continuous Stirred Tank Reactor (CSTR) with nonideal mixing considered by Liou and Yu-Shu [

The optimization search is initiated with the following algorithm parameters.

Number of bacteria is chosen as 25; number of chemotactic steps is set to ten; number of reproduction steps and length of a swim is considered as ten; number of elimination-dispersal events is five; number of bacteria reproduction is assigned as ten; probability for elimination dispersal has a value of 0.3.

In the performance criterion (

The BFO I-PD parameters converge at 439th iteration for ISE and 558th iteration for ITSE. The controller gains and the performance measure are presented in Tables

Optimised parameters for CSTR model.

Method | Iteration number | |||
---|---|---|---|---|

ISE I-PD | 439 | 1.913 | 0.0412 | 0.1094 |

IAE I-PD | 493 | 2.4571 | 0.0509 | 0.2109 |

ITAE I-PD | 526 | 1.8662 | 0.0258 | 0.2168 |

ITSE I-PD | 558 | 1.6984 | 0.0195 | 0.1107 |

Performance index for the CSTR model.

Method | Reference tracking | |||||||

ISE | IAE | ITAE | ITSE | |||||

ISE I-PD | 137.8 | 279.3 | 82112.9 | 67391.4 | 59.4 | 0.1409 | 287 | 0 |

IAE I-PD | 135.3 | 275.2 | 81416.4 | 66942.7 | 53.7 | 0.0681 | 396 | 0 |

ITAE I-PD | 113.3 | 253.0 | 80461.3 | 59448.0 | 94.7 | 0.0431 | 380 | 0 |

ITSE I-PD | 106.9 | 244.2 | 79848.5 | 57489.3 | 111.3 | 0.0407 | 320 | 0 |

Reference tracking performance of CSTR model.

In this work, an attempt has been made for tuning an I-PD controller structure for a class of unstable process models using Bacterial Foraging Optimization (BFO) algorithm with minimizing the multiple objective performance criterion. A comparative study with Integral of Squared Error (ISE), Integral of Absolute Error (IAE), Integral of Time weighted Squared Error (ITSE), and Integral of Time weighted Absolute Error (ITAE) have been discussed. The ISE based method provides the optimized value with a small iteration time than the IAE, ITAE, and ITSE. Hence, ISE-based controller tuning can be employed for unstable systems to obtain optimal controller settings compared to other methods. The I-PD structure provides an enhanced setpoint tracking performance with minimal cost function. It also provides improved time domain specifications and robust performance for the system with time delay uncertainty.

Reference/input signal for the closed loop system

Process output

Controller output

External disturbance

Laplace inverse of

Proportional gain

Integral gain

Derivative gain

Rise time

Settling time

Over shoot

Steady state error

CSTR constants.

Constants

Derivative

Integral

Proportional

Feed.