# Grey Wolf Optimizer-Based Optimal Controller Tuning Method for Unstable Cascade Processes with Time Delay

## Abstract

**:**

## 1. Introduction

_{2}optimum performance. For time-delayed unstable cascade processes, Chandran et al. [16] presented a simple controller design method based on fractional order calculations. They designed fractional-order PI and fractional-order PD designs in the primary loop and controller designs based on the IMC principle in the secondary loop according to the synthesis method. An unstable cascade process with a time delay was controlled by a generalized predictive control scheme in another study [17]. In their study, they proposed a control scheme in which they used an IMC-based controller in the inner loop and a set-point achievement controller in the outer loop. The technique proposed by Kalim and Ali [18] improves the servo response of the system in the outer loop while performing disturbance rejection in the inner loop in serial cascade systems. An enhanced series cascaded control approach with Smith predictor was designed in this work [19] by combining some techniques. Based on the direct synthesis approach and the pole placement method, the study designs controllers in cascade controller structure for some processes with dead time [20]. One of the recent studies on the cascade control of time-delayed unstable processes is presented by Mukherjee et al. [4]. Two of the three controllers used in the proposed method are fractional order controllers designed using the IMC approach. The other controller used as a stabilizer is in the PD structure. The controller parameters are determined by an artificial bee colony algorithm using a multi-objective objective function. The authors emphasized that the presented technique provides good improvement in closed-loop system responses according to some studies.

- The article improves both the set-point tracking and disturbance rejection performances in the control of unstable cascade systems by presenting a novel control scheme given in Section 2.
- The Grey wolf optimizer algorithm, which uses the Euclidean distance function as the objective function, is introduced to the literature as an effective method for determining the controller parameters in the proposed control scheme.

## 2. Fundamentals of Cascade Control and Proposed Cascade Control Scheme

_{p1}and G

_{p2}are system transfer functions of processes belonging to primary (outer) and secondary (inner) systems, respectively. G

_{c1}and G

_{c2}indicate controllers of primary and secondary systems. In addition, d

_{1}and d

_{2}represent disturbance inputs affecting the outer and inner loops. In the cascade control block diagram, the controller of the secondary system is used to provide the stability in the inner loop, while the controller of the primary system is used to provide the stability in the outer loop.

_{1}and d

_{2}illustrate disturbance inputs for the inner and outer loop, respectively.

## 3. Tuning of Controller Parameters Based on the GWO Algorithm

- Tracking, chasing, and approaching the prey
- Pursuing, encircling, and harassing the prey until it stops moving
- Attack toward the hunt

_{t1}, x

_{t2}and x

_{t3}represent target settling time, steady-state error, and maximum percent overshoot, respectively. Similarly, y

_{a1}, y

_{a2}, and y

_{a3}show the actual settling time, steady-state error, and maximum percent overshoot.

## 4. Simulation Study

**Example 1.**

_{R}(s) = 1/(6.821s + 1). In addition, the primary and secondary process controllers are G

_{c1}(s) = 0.4780 [1 + (1/0.99s) + 0.2130s] [(6.8210s + 1)/(0.5670s + 1)] and G

_{c2}(s) = (2.07s + 1)/(0.3s + 1). In Yin’s method, the designed controllers (G

_{CS}, and G

_{CD}) are as G

_{CS}(s) = 4.6456[1 + (1/4.2549s) + 0.2798s][(0.4695s + 1)/(0.23s + 1)]. In addition, the secondary process controller and set-point filter are G

_{c2}(s) = (2.07s + 1)/(0.3s + 1) and F(s) = 1/(1.1904s

^{2}+ 4.2549s + 1), respectively.

**Example 2.**

_{c1}(s) = 0.438 × [1 + (1/6.71s) + 0.85s] × [(39.76s + 1)/(4.21s + 1)], G

_{c2}(s) = (20s + 1)/(2s + 2) and F

_{R}(s) = 1/(39.76s + 1), respectively.

_{CS}and G

_{CD}controllers are the same values, and G

_{CS}(s) = 4.6456 × [1 + (1/4.2549s) + 0.2798s] × [(0.4695s + 1)/(0.23s + 1)]. The secondary process controller and set-point filter are G

_{c2}(s) = (2.07s + 1)/(0.3s + 1) and F(s) = 1/(33.51s

^{2}+ 34.31s + 1), respectively.

**Example 3.**

_{c}= 0.6495 and T

_{i}= 2.2043 (for PI), K

_{c}= 2.5245, T

_{i}= 23.5919 and T

_{d}= 1.6901 (for PID).

_{c1}(s) = 0.1223 × [1 + (1/5.50s) + 1.7280s] × [(72.25s + 1)/(0.8856s + 1)] and G

_{c2}(s) = (s + 1)/(6s + 2). The set-point filter is as F

_{R}(s) = 1/(72.25s + 1).

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Summary of some meta-heuristic algorithm [24].

**Figure 4.**Position updating in the GWO algorithm [27].

**Figure 5.**GWO algorithm flowchart [25].

**Figure 9.**Closed-loop responses (

**top**) and control action (

**bottom**) of the perturbed system for Example 1.

Primary Process Controllers | Secondary Process Controllers | |||||||
---|---|---|---|---|---|---|---|---|

K_{p1} | K_{i1} | K_{p2} | K_{d1} | K_{p3} | K_{i2} | K_{p4} | K_{d2} | |

Ex 1 | 3.3961 | 4.6422 | 15.00 | 3.0507 | 1.8294 | 0.0010 | 4.2725 | 1.0915 |

Ex 2 | 0.6585 | 0.1877 | 4.9138 | 9.2398 | 4.0649 | 0.0014 | 2.2291 | 0.0171 |

Ex 3 | 0.0010 | 0.0564 | 1.9446 | 1.2782 | 0.0012 | 0.4900 | 0.1502 | 0.1070 |

Tuning Method | Nominal System | Perturbed System | |||
---|---|---|---|---|---|

IAE | TV | IAE | TV | ||

Ex. 1 | Proposed | 3.483 | 30.17 | 3.604 | 58.83 |

Dasari’s method | 7.902 | 10.97 | 7.948 | 16.61 | |

Yin’s method | 4.95 | 18.68 | 7.019 | 87.79 | |

Ex. 2 | Proposed | 27.76 | 18.8 | 27.8 | 32.82 |

Dasari’s method | 44.1 | 20.93 | 44.96 | 15.65 | |

Yin’s method | 38.16 | 69 | 39.07 | 12.53 | |

Ex. 3 | Proposed | 28.7 | 4.307 | 40.49 | 7.241 |

Dasari’s method | 63.62 | 2.89 | 65.26 | 3.899 | |

Kaya’s method | 30.46 | 12.66 | 77.3 | 86.47 |

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Dogruer, T.
Grey Wolf Optimizer-Based Optimal Controller Tuning Method for Unstable Cascade Processes with Time Delay. *Symmetry* **2023**, *15*, 54.
https://doi.org/10.3390/sym15010054

**AMA Style**

Dogruer T.
Grey Wolf Optimizer-Based Optimal Controller Tuning Method for Unstable Cascade Processes with Time Delay. *Symmetry*. 2023; 15(1):54.
https://doi.org/10.3390/sym15010054

**Chicago/Turabian Style**

Dogruer, Tufan.
2023. "Grey Wolf Optimizer-Based Optimal Controller Tuning Method for Unstable Cascade Processes with Time Delay" *Symmetry* 15, no. 1: 54.
https://doi.org/10.3390/sym15010054