Optimal System for Improved Internal Model Control of Argon-Oxygen Decarburization Process Based on the Piecewise Linear Model and Time Constant of Filter Optimization

.is paper presents an improved internal model control system to raise the efficiency of refining low-carbon ferrochrome..is control system comprises of a piecewise linearized transfer function and an improved internal model controller based on optimized time constant of the filter. .e control system is mainly used to control the oxygen supply rate during the argon-oxygen refining for controlling the smelting temperature..e regulatory performance and servo of two closed-loop control schemes are compared between the improved internal model controller based on the optimized filter time 0000-0002-7606-6546and the internal model controller based on the fixed filter time constant. .e simulation analysis shows that the piecewise linearized model and the optimization of the time constant of the filter improves the response time, stability, and anti-interference ability of the controller. .en, the proposed improved internalmodel controller is used to adjust the gas supply flow in 5 tonAOD furnace to control the smelting temperature. Ten production tests performed the effectiveness of the controlling refining optimal system. .e analysis of the experimental data shows that the improved internal model control system can shorten the melting time and improve the melting efficiency. .us, the application of the improved internal model control system in low-carbon ferrochrome refining is an interesting potential direction for future research.


Introduction
With the rapid development of industrial process, the controlled object models become more and more complex. Many actual industrial processes are multivariable [1]. e complexity of the model of the controlled object increases, which leads to slow compensation and response of the system. e argon-oxygen decarburization (AOD) process is one of the most common processes for refining medium and low-carbon ferrochromium [2]. e temperature control of AOD process is influenced by many factors, such as the composition of the molten pool, the rate of oxygen and argon injection, the mixing degree of the molten pool, and the fluidity of the melting slag [3]. e refining temperature control in AOD process has the characteristics of multivariable, large lag, inaccurate mathematical model, and nonmeasurable strong disturbance [4]. e important practical considerations in control structure selection are to ensure that the selected controller does not depend on the exact model and has good disturbance resistance. Internal model control (IMC) is one of many methods to design controllers. It has been proved to be a powerful control method for multivariable systems [5], and it shows an advantage in theoretical analysis of the stability and robustness of control system, especially in the control of large time delay systems [6]. erefore, it is frequently applied in the process control with a slow response. e basic design idea of IMC is to combine an optimal controller obtained from the nominal process model with a low-pass filter to tradeoff closed-loop performance with robustness to model uncertainties. Furthermore, the IMC control structure allows for separate controller design for performance and robustness to alleviate the tradeoff between performance and robustness in the traditional feedback framework [7]. In recent years, increasingly, research has been done on the application of IMC. Ali et al. presented a general IMC structure with multiple degrees of freedom for the design of control systems with multiple objectives; the control structure is demonstrated on a simulated thin-film drying process in continuous pharmaceutical manufacturing for several multiloop control structures [8]. Jiang et al. [9] proposed an IMC method based on a damped pseudoinverse, and an analysis was made to evaluate the stability of the system. Qiu et al. [10] proposed a modified stability-constrained identification method with established boundedness and continuity properties and demonstrated the effectiveness of the proposed algorithm with an example. Although the IMC method has been widely studied and applied, the application of the IMC is restricted when strong load disturbance exists or stable control accuracy and response speed are highly required in the system. To realize the unbiased tracking of the system and eliminate the influence of the disturbance on the system, a low-pass filter is required in series with the controller. e filter time constant is one of the important factors affecting the controller's speed and robustness [11].
In this paper, an IMC method based on the piecewise identification model and particle swarm optimization filtering time parameter is developed. First, the refining process is divided into three stages according to the carbon content in the molten pool, and the piecewise linear model based on the advantages of model identification technology is obtained [12]. en, the IMC is designed with the transfer function of the real object identified. Based on the deviation of the IMC, the integrated time absolute error ITAE is used as the fitness function of the particle swarm optimization algorithm. e particle swarm optimization is used to determine the filtering time constant. e filter time constant obtained by particle swarm optimization can ensure that our estimation result is robust and less influenced by the imprecision of the model, which can maintain high efficiency. e inference for all parameters is directly derived by using existing theories. Simulation experiments are conducted to evaluate the response performance and anti-interference performance of the control method and investigate the finite sample performance of the proposed method. Finally, the effectiveness of the proposed control system is summarized.

Transfer Function of Temperature Control
System for Argon-Oxygen Refined Ferrochrome e reaction of argon-oxygen refining ferrochrome occurs in normal pressure and high temperature environment. e temperature is an important technological parameter in the refining of low-carbon ferrochrome by argon-oxygen decarburization [13]. e refining process is the dynamic equilibrium of high temperature chemical reactions. e main chemical reactions can be described with (1)-(4) [12][13][14]: where [ ] means it is dissolved melt, { } means it is gaseous phase, and ( ) means it is slag phase. In a previous work, improvements have been made on the mathematical model of the AOD refining process proposed by Guan and You [15]. e mathematic model of temperature control that gives the relationship between the carbon content and the oxygen supply rate was obtained. According to the characteristics of argon-oxygen refining process [16], the piecewise linearization treatment of the mathematic model was carried out. Temperature control models of the three refining stages are obtained by fitting to the production data. e transfer function of temperature control in three stages were developed for the AOD furnace as follows.
For the first stage, For the second stage, For the third stage,

Design and Implementation of an Internal Model Controller
3.1. IMC Structure. IMC is widely used in the process with measurable outputs and nonmeasurable disturbances [17]. e block diagram of the IMC system [18] is shown in Figure 1.
As shown in Figure 1, R(s) is the input, Y(s) is output, G c (s) is the inferential controller, G pi (s) is the process channel transfer function, G m (s) is the process channel estimation model, V(s) is the disturbance channel transfer function, Y(s) is the estimate value of Y(s), and V(s) is the estimate value of V(s).
From Figure 1, the system output is calculated as below: e temperature control system output of low-carbon ferrochrome refining process in AOD furnace can be measured, while the temperature influence of argon gas is not measurable.
us, the influence of argon gas on temperature can be regarded as the system disturbance. e control law is derived through mathematical reasoning [19]. Because the minimum phase part of the first-order inertia-hysteresis model is stable, it can meet the requirement of design the internal model controller [20]. e mathematical model of temperature control in the argonoxygen refining process with system identification is used as the transfer function of the actual object. G pi (s) is the mathematical expression.
ere is a lag component in the model with temperature as the actual control object, which is a nonminimum phase component, and then, G c (s) will contain a lead component, while the ideal controller cannot realize the dynamic compensation completely. To solve this problem, the object model can be designed as follows [21]: where G m+ (s) is the stable part of the minimum phase component, G m− (s) is the nonminimum phase component with time delay and zeros of right-half-plane.
When the system's parameters change, the model does not match the system. To overcome model mismatch, G c (s) has important roles. G c (s) is set as shown in formula (10). It can improve the stability and robustness of the system: where G f (s) is the transfer function of the filter with firstorder inertia component. G f (s) � 1/λs + 1, where λ is the time constant of the filter. en, the system output can be expressed as follows: When a large mismatch exists between the model and the plant, λ should be taken as a big value, which can avoid the oscillation of the output signal. If the degree of mismatch is small between the model and the plant, λ should be taken as a small value, which can improve the response speed of the system [22]. If λ is suitable, the control model improves the system's response time, stability, and anti-interference ability [23].

Filter Time Constant Determination.
It is a key problem for the rapidity and robustness of the controller to determine the filter time constant [24]. Eber-hart and Kennedy proposed particle swarm optimization (PSO) in 1995 [25], which is a global stochastic optimization technique for continuous methods. e ultimate purpose of this algorithm is to have whole particles find the optima in the high-dimensional data volume. To find the optima, each particle follows its previous best placement. e PSO technique will refurbish the velocity and the position of each particle. e velocity and position updating operations in PSO are as stated follows: where v id (k + 1) and v id (k) denote the velocities in the t + 1th and tth iterations, respectively, x id (k + 1) and x id (k) represent the positions of the particle i in the dth dimension in t + 1th and tth iterations, respectively, ω is the inertia weight to stabilize the local and global search, R 1id and R 2id are random vectors, and c 1 and c 2 are the acceleration coefficients. Moreover, p id (k) and g id (k) represent the personal best solution of the particle i and the global best solution in the dth dimension, respectively. Usually, c 1 + c 2 ≤ 4; in this paper, c 1 � c 2 � 2. e particle swarm optimization is updated iteratively, and the global optimal value is finally obtained.
Using particle swarm optimization algorithm to optimize the filter time constant, x � λ. e purpose of optimizing the controller parameters is to make the static difference tend to zero. erefore, the response of the system has a short regulation time and small overshoot.
If the number of particles in the initialized population is N, each particle in the swarm conserves a form of a matrix; it is shown as follows: e selection of fitness function directly affects the convergence rate of PSO. Control system design using ITAE has the advantages of small overshoot, moderate damping, and good dynamic performance, so ITAE is often used to measure system performance in the control field [26]. ITAE index is used as the fitness function of particle swarm optimization algorithm in a long-time delay system. e fitness function is shown as e steps of PSO algorithm are as follows: (1) Swarm has to be initialized by assigning a position, N � 40. d is the dimension of the particle, d � 1. ω decreases with the number of iterations, and the formula is as follows [27]: where iter is the current iteration of the algorithm and iter max is the maximum number of iterations; the PSO is allowed to continue.

Mathematical Problems in Engineering
(2) e initial value is substituted into equations (12) and (13) to get a better position and speed. (3) Fitness function is estimated for each particle.
Compare the particle's fitness value with the personal optimal value for each individual particle. (4) If the current value is better than the previous personal optimal value, then set this as the global optimal value for the current particle's position. (5) Until a sufficiently good fitness value is achieved, repeat step 2.

Filtering Time Constant of the Piecewise
Model. e structure diagram of the simulation experiment is shown in Figure 2. Firstly, the carbon content in the bath was detected by a laser-induced breakdown spectroscopy instrument [28]. e internal model controller judges the refining stage by detecting the value of carbon content. e integral of time-weighted absolute error (ITAE) performance index has the advantage of producing small overshoots and oscillations [29]. It has been chosen as the objective function applied to optimize the filter time constant. e filter time constant was optimized by particle swarm optimization algorithm according to the initial deviation of the piecewise model. e filtering time constant is shown in Table 1. According to the detected carbon content, the internal model controller selects the filtering time constant of the corresponding piecewise model from Table 1. e optimized filtering time constant can improve the rapidity and robustness of the system response.

Analysis of Rapidity and Stability.
In this section, we compare the regulatory performance as well as the servo of two closed-loop control schemes, namely, the improved internal model controller based on optimized time constant of the filter and the previous work of the internal model controller based on fixed time constant of the filter [30]. e Simulink simulation model is shown in Figure 3. It should be noted that here a piecewise model variation is imposed on the manipulated variable (i.e., oxygen rate) that is quite reasonable from the practical aspect. In the process of argonoxygen refining of low-carbon ferrochrome, the temperature is controlled at 1700-1800°C [31]. According to the change of carbon content, the refining process is divided into three stages [32]. e proportion of the mixed gas varies at each stage. When the two stages are switched, the equilibrium state of the previous stage is changed because the mixture ratio changes. is is equivalent to setting a new value. e temperature control curve is obtained through simulation; it is shown in Figure 4.
According to the refining process, the initial temperature of the chromite molten liquid in the furnace is about 1400°C, and the refining temperature is controlled at about 1700°C [33]. In the simulation study, we introduce a step change in the initial point from 1400°C to 1700°C. In this case study, the internal model controller based on fixed time constant of the filter (FTC) shows a big overshoot with a decayed oscillation (see Figure 3). On the contrary, this improved internal model controller based on optimized time constant of the filter (OTC) shows a satisfactory servo performance. To clearly see the performance indexes, the transient process of the three stages is enlarged from Figures 5-7. It is obvious from Figures 5-7 that the proposed controller is superior to the previous work of the internal model controller based on fixed time constant of the filter. is is also evident from the quantitative analysis made in Table 2 based on the ITAE values. e performance indexes of the control system are shown in Table 2.
e simulation results show that the improved internal model controller based on optimized time constant of the filter has better performance than the internal model controller based on fixed time constant of the filter in terms of speed and stability.

Antidisturbance Analysis.
In the simulation study, we introduce two consecutive step changes in the set point (a step increases from 0 to 8 at time � 0 s and thereafter, a step decreases from 0 to −8 at time � 1000 s).
ey are perturbance. e internal model controller based on fixed time constant of the filter eliminates the perturbance after 300 s, while the improved internal model controller based on optimized time constant of the filter eliminates the perturbance after 200 s. In this case study, the internal model controller based on fixed time constant of the filter shows a big overshoot with a decayed oscillation (see Figure 8). It is obvious from Figure 8 that the proposed controller is superior to the previous work of the internal model controller.
As shown in Figure 9, a sinusoidal wave disturbance signal is added. It can be seen from the amplitude that the improved internal model controller based on optimized time constant of the filter has a better effect on disturbance suppression. rough simulation analysis, both control methods are effective for disturbance signals, but the improved internal model controller based on optimized time constant of the filter shows better antidisturbance performance than the other one.

Validation Material.
e proposed improved internal model controller is used to adjust the gas supply flow in 5 ton AOD furnace to control the smelting temperature. e furnace is fitted with two bottom-blowing and a top lance. In contemporary practice, both a top lance and tow bottom lances are employed for delivering the required oxygen into the melt in combined top-blowing and bottom-blowing decarburization. Ten validation heats were conducted. In the operating practice, the combinedblowing stage is divided into steps, which differ in terms of the employed blowing mixture. In the studied heats, the last combined-blowing stage was altered so that oxygen was introduced from the bottom lance only. e computer control interface is shown in Figure 10. e validation heat site is shown in Figure 11.

Experimental Data Analysis.
e carbon content, furnace temperature, and unit refining time for the ten tests are presented in Table 3. e average unit refining time of the ten experiments was 14.51 min.
Without the use of the improved IMC method, the average unit refining time of the carbon content control was 18.43 min. From Figure 12, the average smelting time of the ten experiments with the improved IMC method reduced by 21.27% compared with that without the improved IMC         method, which resulted in an improvement of the actual production efficiency.

Conclusion
e present work proposes an improved IMC system to regulate the AOD furnace refining progress. is control system comprises of a piecewise linearized transfer function and an improved internal model controller based on optimized time constant of the filter. e simulation analysis shows that the filtering time parameter optimizer improves the controller's response time, stability, and anti-interference ability. Ten production tests using 5 ton argon-oxygen refining furnace are performed. e experimental data show that the improved IMC method can shorten the average melting time by 21.27% compared with the original control method. It can effectively improve the actual production efficiency and reduce energy consumption.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.