#
Design of a Robust H_{2} State Feedback Temperature Controller for a Steel Slab Reheating Furnace

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## Abstract

**:**

_{2}state feedback controller (H

_{2}-SFC) for the effective temperature control in the heating zone of the steel slab reheating furnace. Based on the available field data and system identification procedures, a mathematical model of the heating zone is derived, which presents autoregressive-moving average with exogenous input (ARMAX) structure and fourth order. The design of an H

_{2}-SFC controller for the effective control of the heating zone temperature of the slab reheating furnace under study is developed. The simulation results of the designed control system showed its high effectiveness compared to the conventional PID control.

## 1. Introduction

_{2}control is an effective control technique that has greatly contributed to practical industry applications minimizing the effect of disturbances [21,22,23]. The robust H

_{2}control is based on minimizing the quadratic norm of the transfer function between the input disturbance signal and the plant output signal or H

_{2}norm [24,25,26].

_{2}state feedback controller (H

_{2}-SFC) for the effective temperature control in the heating zone of a steel slab reheating furnace.

_{2}-SFC controller uses a mathematical model of the industrial process to be controlled to calculate an optimal sequence of actions that are projected to design the dynamics of the system through state feedback based on the minimization of the quadratic norm of the transfer function between the input disturbance signal and the plant output signal.

_{2}-SFC controller is the possibility of taking advantage of the representation in states space of the transfer function taking advantage of the continuous feedback of the information on the states and the output of the system in order to stabilize the said function minimizing its H

_{2}norm [26].

_{2}norm.

_{2}-SFC controller has a greater tolerance or insensitivity to changes or disturbances that may occur in the operation conditions of the industrial process [26].

_{2}state feedback controller (H

_{2}-SFC) for the effective temperature control in the heating zone of a steel slab reheating furnace is designed.

_{2}-SFC controller is developed. Section 4 shows the discussions of the obtained results. Finally, Section 5 gives some conclusions.

## 2. Identification of the Dynamic Behavior of the Steel Slab Reheating Furnace

_{2}-SFC.

## 3. Design of a H_{2}-SFC Controller

_{2}-SFC controller that generates a better behavior than a conventional PI/PID controller against the effect of complex dynamic behavior influenced by external disturbances. For this reason, this work raises the design of a robust H

_{2}-SFC controller.

_{2}controller, conformed by the controller K, the linear dynamic process P, the output of the system y, the control signal u, the output signal z, and the disturbance of input d with which we have $\left(\begin{array}{c}z\\ y\end{array}\right)=\left(\begin{array}{cc}{P}_{11}& {P}_{12}\\ {P}_{21}& {P}_{22}\end{array}\right)\left(\begin{array}{c}d\\ u\end{array}\right)$ and $u=Ky$. The realization of P is equal to $\left[\begin{array}{ccc}A& {B}_{1}& {B}_{2}\\ {C}_{1}& {D}_{11}& {D}_{12}\\ {C}_{2}& {D}_{21}& {D}_{22}\end{array}\right]$.

- (i)
- $\left(A,{B}_{2}\right)$ is stabilizable and $\left({C}_{2},A\right)$ is detectable.
- (ii)
- ${D}_{12}$ and ${D}_{21}$ have full rank.
- (iii)
- $\left[\begin{array}{cc}A-jwI& {B}_{2}\\ {C}_{1}& {D}_{12}\end{array}\right]$ has full rank.
- (iv)
- $\left[\begin{array}{cc}A-jwI& {B}_{1}\\ {C}_{2}& {D}_{21}\end{array}\right]$ has full rank.

_{2}controller is linear and invariant in time and dimensional finite (nonsingular case). Assumptions (iii) and (iv) guarantee the solvability of algebraic Riccati equations. For ${D}_{11}=0$ and ${D}_{22}=0$. The assumption ${D}_{22}=0$ is made to simplify algebra. With the assumption ${D}_{11}=0,{P}_{11}$ becomes strictly proper.

_{2}is to find a stabilizer controller K that minimizes the $\Vert H{\Vert}_{2}$ norm of the states and the action of the controller given the realization of the state space [42,43,44]. The tracking gain F allows the system output to reach the R reference by solving the nonzero reference tracking problem of the system being the error $e=R-y$. Here, it should be fulfilled:

_{2}-SFC controller, it is necessary to consider the model of the plant under study in Equation (5) for which we assume that the new input $v\left(k\right)=0$. The transfer matrix from $d\left(k\right)$ to ${z}_{2}\left(k\right)$ is as follows:

_{2}is to find K such that $\left(A-{B}_{2}K\right)$ is Hurwitz and $\Vert {G}_{d,{z}_{2}}{\Vert}_{2}<\gamma $ for the minimum value of $\gamma >0$ that meets the following linear matrix inequality, with X and Q being symmetric matrices [41]:

_{2}-SFC controller are: Delay time: 8 s, rise time: 40 s, peak time: 40 s, overshoot: 0%, and settling time: 40 s.

## 4. Analysis and Discussion of Results

_{2}-SFC controller, some simulations of the control system of the furnace heating zone temperature under study using the equivalent model in discrete time (6) were carried out for the conventional PI/PID controller, (which was designed in the frequency domain using the second Ziegler-Nichols method). Moreover, using the equivalent state space model (7) for the controller H

_{2}-SFC, both were designed under similar time response conditions.

_{2}-SFC controllers were tested. The results of the control system responses with both controllers are shown in Figure 6. The time response with the H

_{2}-SFC controller does not present overshoot and is much faster with respect to the response of the conventional PI/PID controller. In Figure 7 the controller efforts are shown when the reference presents step characteristics.

_{2}-SFC controllers, these were tested in the presence of internal and external disturbances in the feedback loop. Figure 8 shows the results of the time responses of the control systems with both controllers in the presence of an external step disturbance d(k) of 1 L/h at the fuel flow inlet that feeds the burner that causes an increase in heating temperature from 1250 to 1260 °C. It can be seen that the H

_{2}-SFC controller is more robust than the PI/PID controller by completely rejecting this negative effect almost immediately while the PI/PID controller rejects it in a period of 150 s.

_{2}-SFC controller is more robust than the PI/PID controller by completely rejecting this negative effect almost immediately while the PI/PID controller rejects it in a considerable period of time.

_{2}-SFC controllers.

_{2}-SFC controllers.

_{2}-SFC controller has a lower ISE value and a lower IAE value, which means it has a lower position error, reaching the desired temperature conditions faster with respect to the PI/PID controller. This implies that the H

_{2}-SFC controller has greater robustness and better performance indexes.

_{2}-SFC controller under the influence of the external disturbance d(k) with phase margin (Фm) equal to 72.8° and with gain margin (Am) equal to 35.9 dB.

_{2}-SFC controller due the fact that it shows a greater phase margin and a greater gain margin under the influence of the external step disturbance d(k) of 1 L/h at the fuel flow inlet that feeds the burner.

_{2}-SFC controller makes it possible to obtain a significant increase in the effective control of the heating zone temperature of the slab reheating furnace under study, which implies producing better quality steel with a reduction in mechanical equipment deterioration, and with a decrease in the emission of toxic gases to the environment.

## 5. Conclusions

_{2}-SFC controller for the effective control of the heating temperature of the slab reheating furnace under study was carried out. The results of the control system designed with conventional PI/PID controllers vs. H

_{2}-SFC controller showed that the second one has a better performance guaranteeing a faster response.

_{2}-SFC controller for the effective control of the heating zone temperature in a slab reheating furnace is presented. To obtain better results on the effectiveness of the designed controller, it is necessary to implement the said controller in the heating zone of the slab reheating furnace under study and develop experiments with respect to its operation in real time.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram of the experimental arrangement for the collection of output/input data of the heating zone of the slab reheating furnace under study. PLC: Programmable Logic Controller, PC: Personal Computer.

**Figure 2.**Results of the experiment with a pseudorandom binary sequence (PRBS) signal in the furnace heating zone.

**Figure 5.**Block diagram of the heating zone temperature control system of the slab reheating furnace under study with the designed H

_{2}state feedback controller (H

_{2}-SFC) controller.

**Figure 6.**Time responses of the conventional (PID) and H

_{2}-SFC control system with internal step reference.

**Figure 8.**Time responses with H

_{2}-SFC and PID controllers in the presence of an external step disturbance.

**Figure 9.**Time responses with H

_{2}-SFC and PID controllers in the presence of an internal disturbance.

**Figure 10.**Phase margin of the PI/PID controller against variations in the reference signal and external disturbances.

**Figure 11.**Phase margin of the H

_{2}-SFC controller against variations in the reference signal and external disturbances.

STRUCTURE | na | nb | nc | nd | nf | nk |
---|---|---|---|---|---|---|

ARMAX | 4 | 4 | 15 | X | X | 1 |

BJ | X | 4 | 4 | 4 | 8 | 1 |

OE | X | 4 | X | X | 5 | 1 |

ARX | 20 | 20 | X | X | X | 1 |

**Table 2.**First performance index (ISE) in the presence of external disturbances of the PI/PID and H

_{2}-SFC controllers.

ISE | ISE | |
---|---|---|

(NOMINAL) | (DISTURBANCE) | |

PI/PID | 1449 | 165 |

H_{2}-SFC | 145 | 1 |

**Table 3.**Second performance index, integral absolute error (IAE) in the presence of external disturbances of the PI/PID and H

_{2}-SFC controllers.

IAE | IAE | |
---|---|---|

(NOMINAL) | (DISTURBANCE) | |

PI/PID | 85 | 31 |

H_{2}-SFC | 13 | 1 |

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**MDPI and ACS Style**

Salcedo-Hernández, J.; Rivas-Perez, R.; Sotomayor-Moriano, J. Design of a Robust H_{2} State Feedback Temperature Controller for a Steel Slab Reheating Furnace. *Appl. Sci.* **2020**, *10*, 1731.
https://doi.org/10.3390/app10051731

**AMA Style**

Salcedo-Hernández J, Rivas-Perez R, Sotomayor-Moriano J. Design of a Robust H_{2} State Feedback Temperature Controller for a Steel Slab Reheating Furnace. *Applied Sciences*. 2020; 10(5):1731.
https://doi.org/10.3390/app10051731

**Chicago/Turabian Style**

Salcedo-Hernández, José, Raul Rivas-Perez, and Javier Sotomayor-Moriano. 2020. "Design of a Robust H_{2} State Feedback Temperature Controller for a Steel Slab Reheating Furnace" *Applied Sciences* 10, no. 5: 1731.
https://doi.org/10.3390/app10051731