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A mathematical model of electroslag remelting (ESR) process is established based on its technical features and dynamic characteristics. A new multivariable selftuning proportionalintegralderivative (PID) controller tuned optimally by an improved particle swarm optimization (IPSO) algorithm is proposed to control the twoinput/twooutput (TITO) ESR process. An adaptive chaotic migration mutation operator is used to tackle the particles trapped in the clustering field in order to enhance the diversity of the particles in the population, prevent premature convergence and improve the search efficiency of PSO algorithm. The simulation results show the feasibility and effectiveness of the proposed control method. The new method can overcome dynamic working conditions and coupling features of the system in a wide range, and it has strong robustness and adaptability.
The electroslag remelting (ESR) process is an advanced smelting method to make purified steels based on rudiment steel in order to reduce impurity and get the highquality steel which is defined by: uniformity, density and crystal in vertical [
From the control point of view, the ESR process is a typical complex controlled object, which has multivariable, distributed parameters, nonlinear and strong coupling features. During the early stage of research, the normal control methods include voltage swings, constant current, constant voltage and descending power. The voltage swing control method is adopted to adjust the voltage swing amplitude in order to control the electrode movements and maintain the stability of slag resistance [
The proportionalintegralderivative (PID) controller is widely used in many industrial processes because it is simple in structure, and has good robustness and high reliability. The selftuning and optimization of the PID controller parameters is a very important research field [
The paper is organized as follows. In
Electroslag furnace is a complex controlled object, whose production technique is more complex than other steelmaking methods [
Technique flowchart of the ESR furnace.
The operation flow of the ESR process is illustrated as follows: Add slag→Start arc→Melting slag→Casting→Feeding→Remelting end→Thermal insulation→Finished product. The technique of electroslag furnace may be divided into four stages: melting slag stage, exchange electrodes stage, casting stage and feeding stage.
Melting slag stage. It includes an opening stage to make slag and slag melting with low power by using the lower voltage and current. Because the slag pool has not been formed, there will be large current fluctuations. When the slag pool is formed, the current becomes stable. The slag will be melted until all slag materials have been added. From now on, slag melting will be put under a high power to melt all slag and keep a certain temperature.
Exchange electrodes. They will be finished by the movement of the robot arm, which should be finished in no more than five minutes to avoid the solidification of the melted electroslag. After the completion of the electrode exchange, because there will large heat absorption after the cold metal electrodes are inserted into the electroslag pool, the normal casting should not begin. To achieve the melting speed required by the steel casting technique, the current setpoint must be increased.
Casting stage. The electrode melting velocity in the normal casting stage is the key factor of deciding the solidification of steel ingot. So the biggest allowed electrode melting velocity of the electroslag furnace may be concluded in accord with the related formulas.
Feeding stage. The main function of the feeding stage is to remove the pit on the top of the ingot caused by the solidification.
The main circuit of the electroslag furnace consists of a power supply whose voltage and current can be adjusted, a consumable electrode, electroslag, crystallizer, ingot casting and bilge chest. The major loop and its equivalent circuit are shown in the
Main circuit and equivalent circuit of ESR process.
The relationship between the voltage and current of electroslag furnace’s major loop is described as follows [
The melting velocity of the electroslag furnace is mainly decided by the voltage and current. Its relationship is described as follows:
On the premise of the controlled object under steady working conditions, the system transfer functional is decided by the step response. By imposing the step disturbance to the secondary voltage
The PID controller is a regulator in accordance with the linear combination of the proportion, differential and integral of the error, which can be described as:
According to the dynamic characteristics of ESR process, the paper puts forward a selftuning multivariable PID control strategy optimized by the IPSO, whose structure is shown in
Configuration of selftuning PID decoupling control system.
According the above discussion, the system model can be represented as:
In order to realize the decoupling control of ESR process, the diagonal matrix decoupling method is adopted, which is described as follows:
Thus, Equations (10) and (11) are obtained based on the transfer functions of the controlled object and the diagonal matrix decoupling strategy.
Finally, the transfer functions after decoupling disposal of ESR process are calculated as follows:
So the system transfer function matrix after decoupling can be expressed as:
After decoupling, the original coupling system is decomposed into two independent, insusceptible singleinput/singleoutput (SISO) control channels (System 1 and System 2) shown in Equations (15) and (16).
The hybrid PSO algorithm is utilized to optimize the parameters of two PID controllers in order to obtain the better control performances. The design of the PID controller is actually a multidimensional function optimization problem in essence, so the improved PSO algorithm adopts real value coding pattern. The multivariable PID controller of ESR process can be directly coded as follows.
The optimization object of the PID Controller parameters is to make the system overall control deviation tend to zero, and have a fast response speed and smaller overshoot. So the four error integral criterions described as follows are used to evaluate the control effect.
Integral of squared error (ISE)
Integral of time squared error (ITSE)
Integral of absolute error (IAE)
Integral of time multiplied by absolute error (ITAE)
Particle swarm optimization algorithm [
In the course of the evolution of the flight speed
The genetic distance in GA is introduced to PSO algorithm, in which the particles diversity in the particle swarm is determined by the particle genetic distance. On the basis of this, a special mutation operation (adaptive chaotic migration mutation operator) is utilized to process the particles sinking into gathered area [
When the position of the particle
All particles, whose genetic distance is less than the average genetic distance, consist of a new particle group
Before the mutation operation, the particle swarm
Suppose the disturbance quantity of a chaotic system is Δ
The former 
The flowchart that the proposed hybrid PSO algorithm optimizes the PID controller parameters is shown in
Initialization population. Initialize the population size, random positions, particle velocities, inertia weights, learning factor, collection valve value, chaos perturbation range [−β,β], scale factor
Algorithm flowchart.
Calculate the fitness of each particle, and determine the optimal value
According to Equations (19) and (20), the particle’s position and velocity are updated. Then update the individual optimal value
Equations (21)–(23) are utilized to calculate the average distance
According to Equations (24) and (25), the chaotic optimization is processed. Extract the former 
Firstly, the simulation experiments are carried out to observe the control results before decoupling, where two input signals X_{1} and X_{2} are chosen. The coupling response curves with the input signal X_{1} = 1 and X_{2} = 0 are shown in
System response curves with input signal X_{1} = 1 and X_{2} = 0.
System response curves with input signal X_{1} = 0 and X_{2} = 1.
This paper utilizes the PID diagonal matrix decoupling controller to control the ESR process. The original coupling control system is divided into two independent control loops. The parameters of the PID controller are optimized by the proposed hybrid particle swarm optimization algorithm on two independent control channels. The paper contrasts the simulation control performances of ZN method and four different kinds of fitness functions (ISE, IAE, ITAE and ITSE). The simulation results are shown in
Output responses corresponding to different PID tuning methods. (
The optimized PID controller parameters are shown in
Tuned parameters of PID controllers.
PID Parameters  System 1  System 2  

ZN  ISE  IAE  ITAE  ITSE  ZN  ISE  IAE  ITAE  ITSE  

4.61  4.94  7.59  8.75  8.59  4.61  7.21  3.20  7.21  7.86 

5.90  7.58  9.70  9.70  9.22  5.90  1.63  3.49  8.7  1.26 

0.85  0.60  1.61  1.61  1.29  0.85  0.33  0.09  0.09  0.42 
Performance indices of PID controllers.
Performance Indices  System 1  System 2  

ZN  ISE  IAE  ITAE  ITSE  ZN  ISE  IAE  ITAE  ITSE  
Overshoot (%)  0.32  0.25  0.27  0.23  0.43  2.44  2.40  1.93  1.43  1.60 
Tr (s)  0.87  0.75  0.64  0.60  0.71  0.94  0.88  0.79  0.60  0.68 
Ts (s)  1.24  0.96  1.01  0.81  0.87  4.32  4.01  4.11  3.50  3.97 
Based on the technological characteristics, a PID decoupling control strategy based on the improved particle swarm optimization algorithm is proposed. The simulation results indicated that the proposed control strategy has many characteristics of good dynamic and steady performance, strong robustness and the adaptability of the various working conditions and antiinterference capacity.
This work is partially supported by the Program for China Postdoctoral Science Foundation (Grant No. 20110491510), the Program for Liaoning Excellent Talents in University (Grant No. LJQ2011027), the Program for Anshan Science and Technology Project (Grant No. 2011MS11) and the Program for Research Special Foundation of University of Science and Technology of Liaoning (Grant No. 2011zx10).
JieSheng Wang participated in the concept, design, interpretation and commented on the manuscript. A substantial amount of ChenXu Ning’s contribution to the draft writing and critical revision of this paper was undertaken. Yang Yang participated in the data collection, analysis and algorithm simulation. All authors read and approved the manuscript.
The authors declare no conflict of interest.