Multi-Zone Integrated Iterative-Decoupling Control of Temperature Field of Large-Scale Vertical Quenching Furnaces Based on ESRNN

Abstract

Temperature uniformity within a large vertical quenching furnace is the key factor to determine the properties of aluminum workpieces. The existing temperature control method for quenching furnaces cannot overcome the influence of multi-zone coupling issues, which lead to unstable product performance and a lack of key performance. Based on a workpiece temperature field model, a spatial-temporal dimensional extrapolation method is proposed to realize fast and accurate solving of the temperature model. In view of the over-burning and under-burning problems during the temperature rising period, a self-incentive nonparametric adaptive iterative control algorithm is presented, which realizes consistent temperature rising of multiple heating zones. Aiming at the strong coupling problem of the multi-zone heating manner during the temperature holding period, the decoupling problem of multiple control loops is converted into a multi-loop integrated control optimization problem. An eigenvector self-update recurrent neural network (ESRNN) is constructed to determine the Jacobian information and tune the control parameters of each loop controller in real time, thereby realizing the integrated intelligent decoupling control of multiple heating loops. Simulation and industrial results verify the superiority of the proposed method, which can realize high-precision and high-uniformity control of a large-scale temperature field and effectively improve the quality and performance of aluminum alloy workpieces.

1. Introduction

Large-scale vertical quenching furnaces are the main equipment employed for the thermal treatment of large-size high-quality aluminum alloy workpieces in order to improve their strength, hardness, anticorrosion and wear resistance properties. These high-strength workpieces serve as one of the pivotal constituents of high-end equipment in the aerospace and military industries. Figure 1 shows the structure of a 31 m large-scale vertical quenching furnace with a 2.8 m radius, consisting of a heating chamber and a working chamber. The aluminum alloy workpiece is hung centrally within the furnace via a hook. The electric resistance wires are emplaced surrounding the inner wall of the heating chamber, and a multi-region heating manner is employed, dividing the heating chamber wall into numerous heating regions oriented along the axis for convenient temperature control [1]. The entire thermal treatment process contains temperature rising and holding periods. To obtain the required product quality, during the temperature rising period, the thermal treatment process requires consistency of temperature rising of the multiple heating zones to avoid abnormal working conditions such as partial over-burning or partial under-burning of the workpiece. During the temperature holding period, the workpiece’s temperature uniformity distribution must be maintained at the set temperature for hours [2], within an exceedingly constricted temperature uniformity of ±3 °C. Therefore, ensuring temperature uniformity in the large-scale vertical quenching furnace during the whole process is of crucial importance to improve product quality.

However, precise homogeneity control of the temperature of the large-scale scenes is a challenging task for the following reasons: (1) Thermocouples are positioned on the working chamber wall which are 0.3 m away from the workpiece, to avoid crashing. The workpiece temperature, i.e., the most relevant process variable, cannot be measured directly. And the aluminum alloy is very sensitive to temperature change. The workpiece temperature, which serves as the most important feedback information to the temperature control system, requires a fast and accurate solving method to ensure real-time temperature control. (2) The vertical quenching furnace is a high but slim cylinder (31 m high and 2.8 m radius). The furnace temperature exhibits an intrinsic inhomogeneity feature, i.e., the temperature of the furnace top is higher than that of the furnace bottom during the temperature rising. Different heating zones are affected by different external disturbances because of their different positions. It is difficult for the temperatures of the multiple heating zones to rise up in a consistent manner. (3) The multi-zone cascaded heating manner causes strong coupling of the control effects of multi-heating zones, i.e., the temperature control of each heating zone is independent, and changing the control of any one heating zone affects the temperature distribution of the other zones. Therefore, for the vertical quenching furnace with large size, distributed parameter, nonlinearity and strong coupling characteristics, obtaining the real workpiece temperature and solving the consistent temperature rising and strong coupling issues of each heating zone to achieve cooperative, high-precision and high-uniformity temperature control are very difficult.

For the thermal processing system, the acquisition of temperature data with high accuracy is of great importance to achieve effective processing. A high degree of temperature uniformity is a crucial issue for the processing; therefore, the temperature distribution instead of the temperatures of a few single points is required. However, due to the physical limitations, temperature distributions of most of the thermal processing systems are not directly accessible. Establishing a mechanical heat transfer model is one of the methods to estimate the temperature distribution [3,4]. Combing the characteristics of the fast calculation speed of a semi-empirical model based on production data and the high accuracy of a mechanism model based on energy conservation, Yi established a dual-model temperature prediction system to realize online prediction of furnace temperature [5]. Based on the zone method to calculate radiation heat flux [6], a three-dimensional mathematical model of the transient thermal behavior in a large bloom reheating furnace was established. The study verified the feasibility and practicality of zone modeling for incorporation into a model-based furnace control system. In [7], to develop a mathematical model of a direct-fired continuous strip annealing furnace, a heat conduction problem for the furnace wall was discretized by the Galerkin method, and the zone method was utilized for the determination of the radiative heat transfer. The mechanism modeling method to obtain the temperature distribution is more explainable; however, the establishment of the model relies on many hypothetical conditions, many parameters require identification and the current solution process has low efficiency. The numerical simulation method is another method to intuitively display the temperature distribution [8,9,10,11]. A new methodology in [12] was proposed for converting the steady periodic operation of a metal–gas system into a steady-state problem for the reheating furnace, which significantly reduces the calculation time. The cyclical ephemeral reheating of the slabs and the gaseous phase combustion was computed with CFD using a proposed time-saving method and detailed maps of slab temperature can be obtained [13]. However, those methods only allowed the calculation to be solved in one steady-state or periodic transient reheating operation. CFD models are unsuitable for simulating the thermal behavior of transient steel reheating in real or near-real time. Although the numerical simulation method has relatively high accuracy, it is computationally intensive.

The control of the temperature field in the furnace has the distributed parameter characteristic. Model-based temperature control for thermal processing systems has been studied [14,15,16,17]. By constraining the heat fluxes to a piecewise linear function, a discrete-time state-space system is obtained. With the discrete-time model, a hierarchical control system that includes supervisory plant control, high-level furnace control and low-level furnace zone temperature control is designed for a pusher-type slab reheating furnace [18]. Similarly, a nonlinear model predictive controller is designed to realize user-defined desired slab temperature profile tracking [19]. In [20], the model-based temperature predictive control problem is devised by segmenting the advection equation temporally and spatially employing the method of characteristics and regionally employing linear approximations for the nonlinear constituent. These model-based temperature control methods realize temperature predictive control, feedforward control and feedback control [21,22,23,24]. However, for some industrial processes, it is very difficult to develop a precise temperature control model. Especially for the large vertical quenching furnace, which adopts the multi-zone electrical heating manner and has complex heating chambers, the thermal treatment process has a complex heat transfer mechanism and strong coupling features. In this case, intelligent decoupling is an alternative way to realize temperature control [25,26,27,28]. In [29], a decoupling predictive control method based on a convolutional neural network is proposed to solve the coupling issue between the flatness and thickness of the strip. In [30], for the combustion process of a pebble stove, a self-refreshing recurrent neural network is proposed, and a two-stage intelligent proportion-integral derivative decoupling control algorithm is developed to solve the coupling issue between the dome temperature and the exhaust gas temperature.

The principal contributions to our paper encompass the following aspects. Based on a heat conduction model of the workpiece, a spatial-temporal extrapolation method is firstly presented to realize the fast and high-accuracy acquisition of workpiece temperature. Secondly, a non-parametric adaptive iterative control based on self-incentive is designed to realize consistent temperature rising of the multi-heating zones of large vertical quenching furnace. Thirdly, an integrated intelligent decoupling control method based on ESRNN (eigenvector self-update recurrent neural network) is presented to realize high-precision and uniform temperature control within the furnace. ESRNN is devised to determine the Jacobian information of the decoupling controllers. ESRNN adopts an eigenvector-based clustering algorithm to quickly determine the clustering centers and realizes dynamic structure adjustment in accordance with on-site industrial data. The developed control strategy is tested in a real aluminum alloy processing enterprise, and the results indicate the effectiveness in improving production quality.

2. Methodology

During the temperature rising period, the furnace temperature rises continuously, and the control objectives of each heating zone change constantly. However, the change in temperature reflecting on the control effect has a time delay, and different heating zones will be affected by different external influences. These external influences primarily manifest in the heating zones at the furnace bottom, which are susceptible to the impact of inadequate furnace door sealing. In addition, the upper heating zones of the quenching furnace are close to the furnace top, where a hook hole of the overhead crane is located. Consequently, these zones are easily affected by external air from the furnace top. When the temperature in the furnace is close to the temperature set point, the top temperature is always higher than the lower zones; therefore, the workpieces are heated unevenly, resulting in abnormal working conditions such as partial over-burning and partial under-burning of the workpiece. The heated workpiece is directly scrapped. Therefore, the consistency and uniform heating of multi-heating zones is a key point in the control of the temperature rising period. As the workpiece temperature progressively approaches the set temperature, the quenching treatment process enters the temperature holding period. At the beginning of this period, the strengthening phase in the alloy slowly integrates into an aluminum matrix, and the temperature distribution in the furnace must be restricted within ±3 °C of the set temperature. During the temperature holding period, the stability and uniformity of the temperature field in the furnace are the keys to ensuring that the soluble phase in the alloy is fully dissolved to obtain a uniform alloy component, good grain structure and qualified mechanical properties. However, heat conduction and radiation coexist in the furnace, and the strong coupling effect among multiple heating loops makes it difficult to realize high-precision temperature field control in the furnace. Therefore, the framework of the multi-zone integrated iterative-decoupling control system of the vertical quenching furnace consists of four parts, as shown in Figure 2. The first is the parameter setting module (PSM). By setting the physical parameters of the to-be-heated workpiece, the PSM will match the target temperature rise curve and temperature setpoint in the module database with the corresponding type of the workpiece. The second is the temperature compensation module (TCM) for the workpiece. Fast and high-precision acquisition of the workpiece temperature is the basis to achieve high-precision and high-uniform control of the quenching furnace. With the temperature field model of the workpiece and the temporal dimensional extrapolation method, this module provides real-time and reliable feedback information of the workpiece temperature field for the control system. The third is the iterative learning control module (ILM). During the temperature rising period, the quenching process for the same type of workpiece will be repeated for multiple batches, and there will be a large amount of historical data. Based on the historical control function of each batch of these same types of workpieces, a self-incentive nonparametric adaptive iterative control method is proposed for the quenching furnace to realize consistent temperature rising of multi-heating zones. The fourth is the intelligent decoupling control module (IDM) during the temperature holding period. By transforming the decoupling control problem into a multi-zone integrated optimization control problem, using developed ESRNN, the Jacobian information matrix affected by multi-zone coupling is identified in real time, and the control parameters of each loop controller are updated synchronously, so as to achieve multi-zone integrated decoupling and high-precision control of the furnace.

3. Workpiece Temperature Compensation Based on Spatial-Temporal Finite Extrapolation Method

3.1. Workpiece Temperature Model

The workpiece is suspended in the center of the quenching furnace. The heat is provided by heat radiation of the working chamber and heat convection of the heated air. As the furnace temperature increases, the majority of the heated air gradually escapes from the bottom furnace door and the top discharge hole, leaving only a minimal amount of heated air inside. For radiative calculation, this small amount of heated air in the furnace is considered transparent and non-participating media, meaning that it does not influence the radiative heat exchange. Therefore, it is reasonable for the modeling process to only consider the radiative heat exchange between the working chamber wall and the workpiece surface. The heat conduction model of the workpiece is established as

$$\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}=\frac{\rho c}{\lambda }\frac{\partial T}{\partial t}$$

(1)

the initial condition is described as

$${\left.\frac{\partial T}{\partial z}\right|}_{z=0}=0,{\left.\frac{\partial T}{\partial z}\right|}_{z=l}=0,{\left.T\right|}_{t=0}=T_0 in \mathsf{\Omega }$$

(2)

and the boundary condition is expressed as

$$q_{surface}=q_r on \mathsf{\Gamma }$$

(3)

where $x$, $y$ and $z$ are the Cartesian coordinates, respectively; $\lambda$, $\rho$ and $c$ represent physical parameters of the workpiece, such as heat conductivity, density and specific heat capacity; $l$ and $T_0$ is the height and initial temperature value of the workpiece; $\mathsf{\Omega }$ is the computational domain; $q_{surface}$ represents the heat flux radiating on the workpiece surface, and it can be calculated through heat radiation, $q_r$; and $\mathsf{\Gamma }$ is the boundary condition, and here, it represents the workpiece surface.

The radiant heat flux $q_r$ on the working piece surface can be obtained using the zone method [31].

To solve Equation (1), according to the finite element method, the interpolation function is

$$N_{ki}\left(x_j,y_j,z_j\right)=\left\{\begin{array}{l}0 j\ne i\\ 1 j=i\end{array}\right. and {\displaystyle \sum _{i=1}^4{N}_{ki}}=1$$

(4)

The interpolation function for the temperature of the $k$th tetrahedron can be expressed as

$$T_k=N_k{T}_k={\displaystyle \sum _{i=1}^4{N}_{ki}(x,y,z)T_{ki}}, k=1,2,3,\dots ,e$$

(5)

where $e$ is the number of divided tetrahedral cells in the solution domain.

Equation (1) is changed to an ordinary differential equation as

$$C\stackrel{•}{T}+KT=P$$

(6)

where $C$, $K$ and $P$ are expressed as

$$K_{ij}={\displaystyle \sum _{k=1}^e{\displaystyle \underset{x,y,z\in {\mathsf{\Omega }}^e}{\iint }\left(\frac{\partial N_{ki}}{\partial x}\frac{\partial N_{kj}}{\partial x}+\frac{\partial N_{ki}}{\partial y}\frac{\partial N_{kj}}{\partial y}+\frac{\partial N_{ki}}{\partial z}\frac{\partial N_{kj}}{\partial z}\right)dxdydz}}$$

(7)

$$C_{ij}={\displaystyle \sum _{k=1}^e{\displaystyle \underset{x,y,z\in {\mathsf{\Omega }}^e}{\iint }\frac{\rho c}{\lambda }{N}_{ki}{N}_{kj}dxdydz}}$$

(8)

$$P_{ij}={\displaystyle \sum _{k=1}^e{\displaystyle \underset{x,y,z\in {\mathsf{\Gamma }}_k{}^e}{\iint }{q}_r{N}_{ki}dxdydz}}$$

(9)

Equation (6) is a set of dynamic ordinary differential equations that describe the heat conduction process from the surface to the interior of the workpiece.

3.2. Spatial-Temporal Finite Extrapolation Algorithm

To enhance the precision and calculation speed of the conventional finite element method, an extrapolation algorithm is presented. The temporal dimensional extrapolation is based on the finite difference method. Equation (6) is rewritten as

$$\left\{\begin{array}{l}\stackrel{·}{T}=-C^{-1}K{\rm T}+C^{-1}P\\ T_{t=0}=T_0\end{array}\right.$$

(10)

According to the Euler polyline [32], Equation (10) is rewritten as

$$\left\{\begin{array}{l}\frac{1}{\tau }\left[T^{\tau }\left(t+\tau \right)-T^{\tau }\left(t\right)\right]=-C^{-1}K{{\rm T}}^{\tau }\left(t\right)+C^{-1}P\\ T^{\tau }\left(0\right)=T_0\end{array}\right.$$

(11)

where $T^{\tau }$ is the time discretization of $T$, $t\in \left\{t=j\tau \left|j=0,1,\dots \dots ,(L/\tau -1)\right.\right\}$, where $L$ represents the upper limit of the time variant $t$. Given the initial value of Equation (10), a unique set of solutions must exist in a time interval. Therefore, the solution of $T^{\tau }\left(t\right)$ can be expressed as

$$T^{\tau }\left(t\right)=T\left(t\right)-{\displaystyle \sum _{j=0}^k{\tau }_j{V}_j\left(t\right)}$$

(12)

where $V_j\left(t\right)\in C^{\infty }\left[0,L\right]$, $T\left(t\right)$ and $T^{\tau }\left(t\right)$ are the exact and approximate solutions of Equation (10), respectively. From Equation (12), it is recognized that by performing the extrapolation method multiple times, $T^{\tau }\left(t\right)$ gradually approximates $T\left(t\right)$.

To introduce this algorithm more clearly, to give an example, if the number of the extrapolation time is $q$, let ${\tau }_i$ be the differential time step, and the extrapolation solution of each extrapolation iteration is represented by ${\mathcal{I}}_i^{(j)}\left(t\right),(j=0,1,2,\cdots ,q)$. When $j=0$, then

$${\mathcal{I}}_i^{(0)}\left(t\right)=T^{{\tau }_i}\left(t\right) \left(i=1,2,\cdots ,q+1\right)$$

(13)

where $T^{{\tau }_i}\left(t\right)$ is the approximate solution of Equation (10) under the time step ${\tau }_i$.

Refer to the Romberg algorithm [33]; the extrapolation solution is obtained after $q$ times of extrapolation conduction, and it is expressed as

$$\left\{\begin{array}{l}{\mathcal{I}}_i^{\left(1\right)}=\left({\tau }_i{\mathcal{I}}_{i+1}^{\left(0\right)}-{\tau }_{i+1}{\mathcal{I}}_i^{\left(0\right)}\right)/\left({\tau }_i-{\tau }_{i+1}\right) \left(i=1,2\cdots q\right)\\ {\mathcal{I}}_i^{\left(2\right)}=\left({\tau }_i{\mathcal{I}}_{i+1}^{(1)}-{\tau }_{i+2}{\mathcal{I}}_i^{\left(1\right)}\right)/\left({\tau }_i-{\tau }_{i+2}\right) \left(i=1,2\cdots q-1\right)\\ \cdots \cdots \cdots \cdots \cdots \\ {\mathcal{I}}_i^{\left(p\right)}=\left({\tau }_i{\mathcal{I}}_{i+1}^{\left(p-1\right)}-{\tau }_{i+p}{\mathcal{I}}_i^{\left(p-1\right)}\right)/\left({\tau }_i-{\tau }_{i+p}\right) \left(i=1,2\cdots q-p+1\right)\\ \cdots \cdots \cdots \cdots \cdots \\ {\mathcal{I}}_i^{\left(q\right)}=\left({\tau }_i{\mathcal{I}}_{i+1}^{\left(q-1\right)}-{\tau }_{i+q}{\mathcal{I}}_i^{\left(q-1\right)}\right)/\left({\tau }_i-{\tau }_{i+q}\right) \left(i=1\right)\end{array}\right.$$

(14)

The exact solution of Equation (10) is depicted as

$$T\left(t\right)={\mathcal{I}}_i^{\left(j\right)}\left(t\right)-o\left({\tau }_{i+1}{\tau }_{i+2}\cdots {\tau }_{i+j+1}\right),j=1,2\cdots ,q$$

(15)

It is discovered that the difference solution steadily converges toward the exact solution as the time of extrapolation increases. To ensure the solution accuracy of the workpiece temperature model but further reduce the computation time, a spatial dimensional finite element extrapolation based on eigenvalue is proposed. Let $\hslash$ be the partition step size of one regular tetrahedron in $\mathrm{\Omega }$, and $T_h$ represents the finite element solution under the coarse mesh $\hslash$. As shown in Figure 3b, $\frac{\hslash }{2}$ is the densified middle point of $\hslash$, and $T_{\hslash /2}$ is the finite element solution under densified mesh size $\frac{\hslash }{2}$. The spatial dimensional extrapolation solution $T_{\hslash }^{\prime }$ is obtained as

$$T_{\hslash }^{\prime }=(2^{\mathcal{R}}{T}_{\hslash /2}-T_{\hslash })/(2^R-1)$$

(16)

where $\mathcal{R}$ is a specific value. With the proposed method, the solution precision of the workpiece temperature model can be improved from $o\left({\hslash }^2\right)$ to $o\left({\hslash }^4\right)$. The specific process of the spatial-temporal extrapolation method is described below. To strike a compromise between precision and calculation time, we let $q$ = 3 and $\mathcal{R}=2$, which can be determined based on multiple experiments and ref. [34].

Step 1: Equation (17) is used to calculate the four initial matrices of the spatial finite element extrapolation:

$${\mathcal{I}}_i^{\left(0\right)}\left(t\right)=T^{{\tau }_i}\left(t\right)$$

(17)

where ${\tau }_i=\frac{\tau }{2^{i-1}}$,$(i=1,2,3,4)$.

Step 2: Conduct the first spatial extrapolation iteration using Equation (14), and obtain Equation (18):

$$\left\{\begin{array}{l}{\mathcal{I}}_1^{\left(1\right)}\left(t\right)=2{\mathcal{I}}_2^{\left(0\right)}-{\mathcal{I}}_1^{\left(0\right)}=2T^{\frac{\tau }{2}}\left(t\right)-T^{\tau }\left(t\right)\\ {\mathcal{I}}_2^{\left(1\right)}\left(t\right)=2{\mathcal{I}}_3^{\left(0\right)}-{\mathcal{I}}_2^{\left(0\right)}=2T^{\frac{\tau }{4}}\left(t\right)-T^{\frac{\tau }{2}}\left(t\right)\\ {\mathcal{I}}_3^{\left(1\right)}\left(t\right)=2{\mathcal{I}}_4^{\left(0\right)}-{\mathcal{I}}_3^{\left(0\right)}=2T^{\frac{\tau }{8}}\left(t\right)-T^{\frac{\tau }{4}}\left(t\right)\end{array}\right.$$

(18)

Step 3: Conduct the second spatial extrapolation iteration, and obtain Equation (19):

$$\left\{\begin{array}{l}{\mathcal{I}}_1^{\left(2\right)}\left(t\right)=\frac{1}{3}\left(4{\mathcal{I}}_2^{\left(1\right)}\left(t\right)-{\mathcal{I}}_1^{\left(1\right)}\left(t\right)\right)\\ {\mathcal{I}}_2^{\left(2\right)}\left(t\right)=\frac{1}{3}\left(4{\mathcal{I}}_3^{\left(1\right)}\left(t\right)-{\mathcal{I}}_2^{\left(1\right)}\left(t\right)\right)\end{array}\right.$$

(19)

Step 4: Conduct the third spatial extrapolation iteration, and obtain Equation (20):

$${\mathcal{I}}_1^{\left(3\right)}\left(t\right)=\frac{1}{7}\left(8{\mathcal{I}}_2^{\left(2\right)}\left(t\right)-{\mathcal{I}}_1^{\left(2\right)}\left(t\right)\right)$$

(20)

Step 5: According to Equation (20), $T(t)={\mathcal{I}}_1^{\left(3\right)}\left(t\right)$.

Step 6: The finite element solution $T_{\hslash }$ and $T_{\hslash /2}$ are solved on the coarse mesh $\hslash$ and refined mesh $T_{\hslash /2}$ by the spatial dimensional eigenvalue finite element extrapolation method, separately.

Step 7: According to Equation (16), the extrapolation solution is finally obtained as:

$$T_{\hslash }^{\prime }=\frac{4}{3}{T}_{\hslash /2}-\frac{1}{3}{T}_{\hslash }$$

(21)

4. Nonparametric Adaptive Iterative Control of Vertical Quenching Furnace Based on Self-Incentive

During the temperature rising period, one of the seven heating control loops is set as the $i$th control loop, $i\in \left\{1,2,\cdots 7\right\}$, and the dynamic equation of the $i$th control loop in the $k$th iteration is expressed as

$$y_k^i\left(t+1\right)=f_i(y_k^i(t),y_k^i(t-1),\cdots ,y_k^i(t-n_y), u_k^i(t),u_k^i(t-1),\cdots ,u_k^i(t-n_u))$$

(22)

where $k=0,1,2,\cdots$ represents the iteration number of the system. $f_i\left(\cdots \right)$ is an unknown nonlinear function. $y_k^i\left(t\right)$, $t\in \left\{0,1,\cdots T\right\}$ and $u_k^i\left(t\right)$, $t\in \left\{0,1,\cdots T-1\right\}$ are the output time sequence and control input time sequence of the system, respectively. $n_y$ and $n_u$ are the order of the system. For the temperature control system, the temperature change has continuous smoothness, so the following reasonable assumptions can be given:

(1)

The partial derivative of $f_i\left(\cdots \right)$ with respect to the control input $u_k^i\left(t\right)$ exists and is continuous;

(2)

For all $t\in \left\{0,1,\cdots T\right\}$ and $k=0,1,2,\cdots$, when $\left|\mathrm{\Delta }{u}_k^i\left(t\right)\right|\ne 0$, the system of Equation (1) satisfies generalized $Lipschitz$ continuity [35], that is, $\forall t\in \left\{0,1,\cdots T\right\}$ and $\forall k=0,1,2\cdots$,

$$\left|\mathrm{\Delta }{y}_k^i\left(t+1\right)\right|\le b\left|\mathrm{\Delta }{u}_k^i\left(t\right)\right|$$

(23)

where $\mathrm{\Delta }{y}_k^i\left(t+1\right)=y_k^i\left(t+1\right)-y_{k-1}^i\left(t+1\right)$, $\mathrm{\Delta }{u}_k^i\left(t+1\right)=u_k^i\left(t+1\right)-u_{k-1}^i\left(t+1\right)$ and $b$ is a constant.

Based on the above assumptions, Lemma 1 is derived as the following:

Lemma 1. 

For the nonlinear system (22), when Hypotheses (1) and (2) are satisfied, there must be a quantity ${\theta }_k^i\left(t\right)$, named as MPPD “Mimetic Pseudo Partial Derivative” [36], which makes

$$\mathrm{\Delta }{y}_k^i\left(t+1\right)=y_k^i\left(t+1\right)-y_{k-1}^i\left(t+1\right)={\theta }_k^i\left(t\right)\mathrm{\Delta }{u}_k^i\left(t\right)$$

(24)

and $\left|{\theta }_k^i\left(t\right)\right|\le b$ holds.

Proof. 

If the nonlinear system Equation (6) satisfies the assumptions (1) and (2), then we have

$$\begin{array}{ll}\mathrm{\Delta }{y}_k^i\left(t+1\right)& =f_i(y_k^i(t),y_k^i(t-1),\cdots ,y_k^i(t-n_y), u_k^i(t),u_k^i(t-1),\cdots ,u_k^i(t-n_u))\\ & -f_i(y_{k-1}^i(t),y_{k-1}^i(t-1),\cdots ,y_{k-1}^i(t-n_y), u_{k-1}^i(t),u_{k-1}^i(t-1),\cdots ,u_{k-1}^i(t-n_u))\\ & =f_i(y_k^i(t),y_k^i(t-1),\cdots ,y_k^i(t-n_y), u_k^i(t),u_k^i(t-1),\cdots ,u_k^i(t-n_u))\\ & -f_i(y_k^i(t),y_k^i(t-1),\cdots ,y_k^i(t-n_y), u_{k-1}^i(t),u_k^i(t-1),\cdots ,u_k^i(t-n_u))\\ & +f_i(y_k^i(t),y_k^i(t-1),\cdots ,y_k^i(t-n_y), u_{k-1}^i(t),u_k^i(t-1),\cdots ,u_k^i(t-n_u))\\ & -f_i(y_{k-1}^i(t),y_{k-1}^i(t-1),\cdots ,y_{k-1}^i(t-n_y), u_{k-1}^i(t),u_{k-1}^i(t-1),\cdots ,u_{k-1}^i(t-n_u))\end{array}$$

(25)

Based on assumption (1), and combined with differential mean value theorem, Equation (26) can be obtained according to Equation (25).

$$\mathrm{\Delta }{y}_k^i\left(t+1\right)=\frac{\partial f_i{}^{*}}{\partial u_{{}_k}^i\left(t\right)}\left(u_k^i\left(t\right)-u_{k-1}^i\left(t\right)\right)+{\xi }_k^i\left(t\right)$$

(26)

where $\frac{\partial f_i{}^{*}}{\partial u_{{}_k}^i\left(t\right)}$ represents a suitable partial derivative of the function $f_i\left(\cdots \right)$ with respect to $u_k^i\left(t\right)$. The expression of ${\xi }_k^i\left(t\right)$ is denoted as

$$\begin{array}{ll}{\xi }_k^i\left(t\right)& =f_i(y_k^i(t),y_k^i(t-1),\cdots ,y_k^i(t-n_y), u_k^i(t),u_k^i(t-1),\cdots ,u_k^i(t-n_u))\\ & -f_i(y_{k-1}^i(t),y_{k-1}^i(t-1),\cdots ,y_{k-1}^i(t-n_y), u_{k-1}^i(t),u_{k-1}^i(t-1),\cdots ,u_{k-1}^i(t-n_u))\end{array}$$

(27)

Consider the following Equation (28) with a variable ${\eta }_k^i\left(t\right)$,

$${\xi }_k^i\left(t\right)={\eta }_k^i\left(t\right)\mathrm{\Delta }{u}_k^i\left(t\right)$$

(28)

when $\left|\mathrm{\Delta }{u}_k^i\left(t\right)\right|\ne 0$, Equation (10) has a solution of ${\eta }_k^i\left(t\right)$, and let

$${\theta }_k^i\left(t\right)=\frac{\partial f_i{}^{*}}{\partial u_{{}_k}^i\left(t\right)}+{\eta }_k^i\left(t\right)$$

(29)

It can be directly deduced from Equation (28) that Lemma 1 holds.

Using assumption (2), it is clear that $\left|{\theta }_k^i\left(t\right)\right|\le b$. □

According to the heating rate, energy consumption and technological requirements of the large vertical quenching furnace, the expected heating curve of the heating loop is set as $y_d^i\left(t\right)$,$t\in \left\{0,1,\cdots t_{end}\right\}$, where $t_{end}$ is the end time of the temperature rising period. The error sequence of temperature rise tracking in the $i$th loop is depicted as $e_k^i\left(t+1\right)=y_d^i\left(t+1\right)-y_k^i\left(t+1\right)$, $t\in \left\{0,1,\cdots t_{end}\right\}$. Since the initial state of the system $y_k^i\left(0\right)$ is not processed by any external control input, $e_k^i\left(0\right)$ is meaningless to the system and should be neglected. To ensure the temperature rising process quickly follows the expected heating curve with high precision and strong robustness, the following control objective function is designed as

$$J(u_k^i(t))={\left|e_k^i(t+1)\right|}^2+{\lambda }_u{\left|u_k^i(t)-e^{-e_k^i(t)}{u}_{k-1}^i(t)\right|}^2$$

(30)

where ${\lambda }_u$ is a positive weight factor. $e^{-e_k^i(t)}$ is an introduced self-incentive term, which makes the designed nonparametric adaptive iterative learning controller have a weak dependence on the previous iterative control signal in the initial stage of rapid heating up; thus, the controller has a strong tracking ability and convergence ability. While in the later stage of slow heating, the anti-interference ability and robustness of the controller are improved through the dependence of the incentive control signal on the previous iterative control signal, to ensure the smoothness and stability of the heating process. Equation (30) is simplified by using Equation (24) and the definition of $e_k^i(t+1)$:

$$J(u_k^i(t))={\left|e_{k-1}^i(t+1)-{\theta }_k^i\left(t\right)\left(u_k^i(t)-u_{k-1}^i(t)\right)\right|}^2+{\lambda }_u{\left|u_k^i(t)-e^{-e_k^i(t)}{u}_{k-1}^i(t)\right|}^2$$

(31)

Using the optimal condition $\frac{1}{2}\frac{\partial J}{\partial u_k^i\left(t\right)}=0$ yields

$$u_k^i(t)=e^{-e_k^i(t)}{u}_{k-1}^i(t)+\frac{{\theta }_k^i(t)}{{\lambda }_u+{\left|{\theta }_k^i(t)\right|}^2}{e}_{k-1}^i(t+1)$$

(32)

Since the MPPD ${\theta }_k^i\left(t\right)$ is unknown, the iterative estimated value is undermined as ${\widehat{\theta }}_k^i\left(t\right)$. Based on the definition of ${\theta }_k^i\left(t\right)$, the following optimization objective function is constructed, and its best estimation value ${\widehat{\theta }}_k^i\left(t\right)$ is obtained by the optimization process.

$$J({\widehat{\theta }}_k^i(t))={\left|\mathrm{\Delta }{y}_{k-1}^i(t+1)-{\widehat{\theta }}_k^i(t)\mathrm{\Delta }{u}_{k-1}^i(t)\right|}^2+{\lambda }_{\theta }{\left|{\widehat{\theta }}_k^i(t)-{\widehat{\theta }}_{k-1}^i(t)\right|}^2$$

(33)

Using the optimal condition $\frac{1}{2}\frac{\partial J}{\partial {\widehat{\theta }}_k^i(t)}=0$ yields

$${\widehat{\theta }}_k^i(t)=\left(\frac{{\lambda }_{\theta }}{{\lambda }_{\theta }+{\left|\mathrm{\Delta }{u}_{k-1}^i(t)\right|}^2}\right){\widehat{\theta }}_{k-1}^i(t)+\frac{\mathrm{\Delta }{u}_{k-1}^i(t)\mathrm{\Delta }{y}_{k-1}^i(t+1)}{{\lambda }_{\theta }+{\left|\mathrm{\Delta }{u}_{k-1}^i(t)\right|}^2}$$

(34)

where ${\lambda }_{\theta }$ is a positive weight factor, and the iterative initial value of ${\widehat{\theta }}_k^i\left(t\right)$ is set as ${\widehat{\theta }}_0^i\left(t\right)$, $t\in \left\{0,1,\cdots t_{end}-1\right\}$, and it has an upper bound. To prevent the iterative learning process from terminating abnormally, it requires it to be ensured that $\left|\mathrm{\Delta }{u}_k^i\left(t\right)\right|\ne 0$ always holds, and the estimation ${\widehat{\theta }}_k^i\left(t\right)$ is constrained as follows:

$${\widehat{\theta }}_k^i(t)={\widehat{\theta }}_0^i(t) , {\widehat{\theta }}_k^i(t)\le \epsilon or \left|\mathrm{\Delta }{u}_{k-1}^i\left(t\right)\right|\le \epsilon$$

(35)

where $\epsilon$ is a sufficiently small positive constant. Combining Equations (32), (34) and (35), the iterative formula of the control function of the nonparametric adaptive iterative controller based on self-incentive can be obtained as

$$\left\{\begin{array}{l}{u}_k^i(t)=e^{-e_k^i(t)}{u}_{k-1}^i(t)+\frac{{\theta }_k^i(t)}{{\lambda }_u+{\left|{\theta }_k^i(t)\right|}^2}{e}_{k-1}^i(t+1)\\ {\widehat{\theta }}_k(t)={\widehat{\theta }}_0(t) , {\widehat{\theta }}_k(t)\le \epsilon or \left|u_k(t)\right|\le \epsilon \\ {\widehat{\theta }}_k^i(t)=\left(\frac{{\lambda }_{\theta }}{{\lambda }_{\theta }+{\left|\mathrm{\Delta }{u}_{k-1}^i(t)\right|}^2}\right){\widehat{\theta }}_{k-1}^i(t)+\frac{\mathrm{\Delta }{u}_{k-1}^i(t)\mathrm{\Delta }{y}_{k-1}^i(t+1)}{{\lambda }_{\theta }+{\left|\mathrm{\Delta }{u}_{k-1}^i(t)\right|}^2}\end{array}\right.$$

(36)

5. Integrated Intelligent Decoupling Control Based on ESRNN

To eliminate strong coupling among control loops and maintain the uniformity heating in the furnace, an online integrated intelligent decoupling control method based on ESRNN is developed, as shown in Figure 4. ESRNN is presented to realize intelligent identification of the Jacobian information of the multiple controllers and achieve integrated intelligent decoupling of the multi-heating loops.

5.1. Establishment of ESRNN

To thoroughly investigate the intrinsic clustering features and associations within input data, an eigenvector clustering technique is put forward to make the network better reflect the data characteristic.

The input sample $X$ is described as $X={({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_i,\dots ,{\mathit{x}}_n)}^T$ and $x_i={(x_1,x_2,\dots ,x_m)}^T$. The samples will be clustered and the clustering objective function is

$$\underset{U_K}{\mathrm{argmin}} J=Tr\left(X{X}^T\right) -Tr\left(U_{K}X{X}^T{U}_K{}^T\right)$$

(37)

where $U_K={({\mathit{u}}_1,{\mathit{u}}_2,{\mathit{u}}_3,\dots ,{\mathit{u}}_K)}^T$ is an indicator vector matrix of the cluster feature space, $K$ is the number of clusters and ${\mathit{u}}_K=(0,\dots ,0,\stackrel{n_K}{\overbrace{1,\dots ,1}},0,\dots ,0)/\sqrt{n_K}$; $n_K$ is the sample number in cluster $K$, ${\mathit{u}}_K\in {ℝ}^{1\times n}$, $U_K\in {ℝ}^{K\times n}$.

To eliminate the influence of the sample mean center on the intrinsic clustering characteristics of the samples in $X$, the input sample matrix $X$ is centralized to $\tilde{X}={({\tilde{\mathit{x}}}_1,{\tilde{\mathit{x}}}_2,\dots ,{\tilde{\mathit{x}}}_n)}^T$, where ${\tilde{\mathit{x}}}_i={\mathit{x}}_i-\overline{\mathit{x}}$ and $\overline{\mathit{x}}={\displaystyle {\sum }_{i=1}^n{\mathit{x}}_i/n}$. Substituting it into Equation (37) and yielding Equation (38):

$$\underset{U_K}{\mathrm{argmin}} J=Tr\left(\tilde{X}{\tilde{X}}^T\right) -Tr\left(Q_{K-1}^T\tilde{X}{\tilde{X}}^T{Q}_{K-1}\right)$$

(38)

where $Q_K=U_K{}^T{T}_K$, $T_K$ is a $K\times K$ orthogonal transformation matrix, and ${\mathit{t}}_K$ is the last column of $T_K$ defined as

$${\mathit{t}}_K={(\sqrt{n_1/n},\sqrt{n_2/n},\dots ,\sqrt{n_K/n})}^T, {\mathit{t}}_K\in {ℝ}^{K\times 1}$$

(39)

Therefore, the last column of $Q_K$ is expressed as

$${\mathit{q}}_K=\sqrt{1/n}\mathit{e}, \mathit{e}={(1,1,\dots ,1)}^T$$

(40)

where ${\mathit{q}}_K\in {ℝ}^{n\times 1},\mathit{e}\in {ℝ}^{n\times 1}$.

Considering that $Tr\left(\tilde{X}{\tilde{X}}^T\right)$ is determined by the input sample space as a fixed constant larger than 0, minimizing $J^k$ is equivalent to maximizing $Tr\left(Q_{K-1}^T\tilde{X}{\tilde{X}}^T{Q}_{K-1}\right)$. By utilizing the Ky Fan theorem [37], under the constraint of $Q_{K-1}^T{Q}_{K-1}=I_{K-1}$, the solution that maximizes the objective function is expressed as

$$Q_{K-1}=({\mathit{v}}_1,{\mathit{v}}_2,\dots ,{\mathit{v}}_{K-1})Z$$

(41)

where $({\mathit{v}}_1,{\mathit{v}}_2,\dots ,{\mathit{v}}_n)$ is the set of feature vectors $\tilde{X}{\tilde{X}}^T$, $Z$ is a $(K-1)\times (K-1)$ orthogonal transformation matrix and ${\mathit{v}}_x\in {ℝ}^{n\times 1}$ $1\le x\le n$. According to $Q_K=U_K{}^T{T}_K$, we can obtain $Q_{K-1}=U_{K-1}{}^T{T}_{K-1}$, and $T_{K-1}$ is a $(K-1)\times (K-1)$ orthogonal transformation matrix. Therefore, the cluster feature space indicator vector matrix $U_{K-1}^T$ can be defined as

$$U_{K-1}^T=\left({\mathit{v}}_1,{\mathit{v}}_2,\dots ,{\mathit{v}}_{K-1}\right)$$

(42)

This ensures the objective function $J^k$ achieves its minimum value, and $U_{K-1}^T{}_{}\in {ℝ}^{n\times (K-1)}$.

With $U_{K-1}^T$, all samples in set $X$ are divided into $K-1$ clusters, which can be utilized to discern if the $n$ samples belong to the cluster $j$. The discriminant equation is shown as

$$I\left(C_j\right)={\left\{x_i\left|v_{ji}>0\right.\right\}}_{i=1,2,\cdots ,n j=1,2,\cdots K-1}$$

(43)

where $I\left(C_j\right)$ represents the input samples classified to the $j$th cluster. Equation (43) can determine the distribution of the $K-1$ clusters in $X$. If a sample does not belong to the $K-1$ clusters in the input $X$, it is classified as the cluster $K$. This process is conducted using Equation (44).

$$I\left(C_K\right)=X-{\displaystyle \sum _{j=1}^{K-1}{}^{\cup }}I\left(C_j\right)$$

(44)

Equation (45) is used to calculate the center vector of each clustering.

$$c_j={{\displaystyle \sum _{x_s\in I(C_j)}{\mathit{x}}_s/\mathfrak{m}}}_{j=1,2,\cdots ,K}$$

(45)

where $𝔪$ is sample number in set $I\left(C_j\right)$.

The number of the hidden nodes of the network is equal to $K$; $c_j$ is the cluster center vector, and it can be set as the center of the basis function; and the width ${\sigma }_j$ of the basis function is calculated by Equation (46).

$$\left\{\begin{array}{l}G=\left\{\mathit{x}\left(k\right)\in I(C_j)\right\}\\ d_{\max }^j=\underset{x\left(k\right)\in G}{\max }\left\{{‖\mathit{x}(k)-c_j‖}^2\right\}\\ {\sigma }_j=\frac{2d_{\max }^j}{3} (1\le j\le K)\end{array}\right.$$

(46)

where $\mathit{x}\left(k\right)$ is the $k$th input induction data. $d_{\max }^j$ is the maximum distance between the $j$th cluster center and the sample $\mathit{x}\left(k\right)$. The model is described as follows:

$$\left\{\begin{array}{l}f\left(\mathit{x}\left(k\right)\right)=\mathit{y}\left(k\right)={\displaystyle \sum _{j=1}^K{w}_j{h}_j}\left(\mathit{x}\left(k\right)\right)\\ h_j^0\left(\mathit{x}\right)={\displaystyle \sum _{k=1}^n{\mathit{x}}_1^k{p}_{kj}}+{\displaystyle \sum _{j^{\prime }=1}^K{h}_j\left(\mathit{x}\right)z_{j^{\prime }j}}\\ h_j\left(\mathit{x}\right)=\exp \left(-\frac{{‖h_j^0\left(\mathit{x}\right)-c_j‖}^2}{{\sigma }_j{}^2}\right)\end{array}\right.$$

(47)

where $\mathit{y}\left(k\right)$ is the $k$th output, $h_j(\cdot )$ is the $j$th Gaussian basis function, $p_{kj}\left(k=1,2,\dots ,n;j=1,2,\dots ,K\right)$ is the weight between the $k$th input node and the $j$th output node, $z_{j^{\prime }j}$ is the weight between the $j^{\prime }$th hidden node and the $j$th hidden node, and $w_j$ is the output weight. The loss is defined as

$$L=\frac{1}{2}{\left(\widehat{\mathit{y}}-\mathit{y}\right)}^2$$

(48)

With the gradient descent method, each weight of the network is updated as follows:

$$\frac{\partial L}{\partial y}=\widehat{\mathit{y}}-\mathit{y}$$

(49)

$$\mathrm{\Delta }{w}_j=\eta \frac{\partial L_k}{\partial w_j}=\eta \frac{\partial L_k}{\partial {\mathit{y}}_k}\frac{\partial {\mathit{y}}_k}{\partial w_j}=\eta \frac{\partial L_k}{\partial {\mathit{y}}_k}{h}_{k,j}$$

(50)

$$\mathrm{\Delta }{z}_{j^{\prime }j}=\eta \frac{\partial L_k}{\partial z_{j^{\prime }j}}=\eta \frac{\partial L_k}{\partial h_{k,j}^0}\frac{\partial h_{k,j}^0}{\partial z_{j^{\prime }j}}=\eta \frac{\partial L_i}{\partial h_{k,j}^0}{h}_{k-1,j}$$

(51)

$$\mathrm{\Delta }{p}_{kj}=\eta \frac{\partial L_k}{\partial p_{kj}}=\eta \frac{\partial L_k}{\partial h_j^0}\frac{\partial h_j^0}{\partial p_{kj}}=\eta \frac{\partial L_k}{\partial h_{k,j}^0}{\mathit{x}}_i\left(k\right)$$

(52)

The main step of the dynamic adjustment of ESRNN is described as follows.

• Step 1: Initialize $b$ based on the characteristics of input samples and initialize the learning rate $\eta$. For the first input sample, it is set as ${\mathit{x}}_1$. Let the first cluster be $c_1={\mathit{x}}_1$ and $K=1$, the clustering center $C=\left\{c_1\right\}$ and the first hidden node $h_1$ is constructed.
• Step 2: When the $k$th input data is collected, the number of samples is $n=k$, $k=2,3,\cdots$. There have been $K$ clusters. The clustering centers $C=\left\{c_1,c_2,\cdots c_K\right\}$ and the hidden nodes are $h_p$, $p=1,2,3,\cdots ,K$. If $min{‖\mathit{x}\left(k\right)-c_j‖}_{c_j\in C}{}^2<b$, keep the current clustering result and the network structure unchanged, and go to Step 6. Otherwise, the network structure will be self-adapted. The number of the hidden-layer nodes is changed as $K+1$, then go to Step 3.
• Step 3: The input matrix is centralized to $\tilde{X}={({\tilde{\mathit{x}}}_1,{\tilde{\mathit{x}}}_2,\dots ,{\tilde{\mathit{x}}}_n)}^T$, and calculate the $K+1$ eigenvectors $v_1,\dots ,v_{K+1}$ of $\tilde{X}{\tilde{X}}^T$ to obtain the feature space indicator vector of the cluster to which each sample belongs. All samples are re-clustered using Equations (38) and (39), yielding a new cluster set $I=\left\{I\left(C_1\right),I\left(C_2\right),\cdots ,I\left(C_{K+1}\right)\right\}$.
• Step 4: Use Equation (45) to update the clustering centers and combine with Equation (41) to concurrently update the width ${\sigma }_j$.
• Step 5: Use Equations (50)–(52) to update the weight vector of the network to complete the construction and training of ESRNN.
• Step 6: Use Equation (47), and the output $\mathit{y}\left(k\right)$ is obtained as

  $$y_m(\mathit{x}\left(k\right))={\displaystyle \sum _{j=1}^c{w}_j{h}_p{}^{\prime }(\mathit{x}\left(k\right))}$$
• Step 7: End of the process until the temperature holding period is finished. Else, go to Step 2.

5.2. Integrated Intelligent Decoupling Controller

Take the $i$th control loop as an example for convenient description and the furnace’s temperature control system can be represented as:

$$\left\{\begin{array}{l}{y}_i(k)=f(y(k-1),\dots ,y(k-n_s),u(k-1),\dots ,u(k-m_s))\\ u(k)=[u_1(k),\dots ,u_n(k){]}^T\in R^n,y(k)=[y_1(k),\dots ,y_n(k){]}^T\in R^n\end{array}\right.$$

(53)

where $u(k)$ is the control inputs, and $y(k)$ is the measurement temperatures, respectively. $n_s$ and $m_s$ are the sampling time delay steps of $y(k)$ and $u(k)$. $f(\cdot )$ can be identified by ESRNN. A conventional PID controller is used, and its $K_P$, $K_I$, $K_D$ are obtained using Equation (54).

$$\left\{\begin{array}{l}\mathrm{\Delta }{k}_P^i(k)={\eta }_P{e}_i(k)\frac{\partial {\widehat{y}}_i}{\partial u_i}[e_i(k)-e_i(k-1)]\\ \mathrm{\Delta }{k}_I^i(k)={\eta }_I{e}_i(k)\frac{\partial {\widehat{y}}_i}{\partial u_i}{e}_i(k)\\ \mathrm{\Delta }{k}_D^i(k) ={\eta }_D{e}_i(k)\frac{\partial {\widehat{y}}_i}{\partial u_i}[e_i(k)-2e_i(k-1)+e_i(k-2)]\end{array}\right.$$

(54)

where ${\eta }_P$${\eta }_I$ and ${\eta }_D$ are the learning rate of the PID gains. $e$ is the error of the system and $e=\widehat{\mathit{y}}-\mathit{y}$.

$\frac{\partial {\widehat{y}}_i}{\partial u_i}$ is the Jacobian information of the controlled loop, which can be obtained by Equation (55).

$$\begin{array}{l}{\left.\frac{\partial {\widehat{y}}_i(k)}{\partial u_i(k)}\approx \frac{\partial y_{mi}(k)}{\partial {\mathit{x}}_i(k)}\right|}_{x=x_i(k)}={\left.\frac{\partial \left({\displaystyle \sum _{j=1}^K{w}_j^i{h}_p{}^{\prime }(k)}\right)}{\partial {\mathit{x}}_i(k)}\right|}_{x=x_i(k)}\\ ={\left.{\displaystyle \sum _{j=1}^K{w}_j^i\exp \left(-\frac{{‖{\mathit{x}}_i(x)-c_j^i‖}^2}{{\left({\sigma }_j^i\right)}^2}\right)}\cdot \frac{\left(c_j^i-{\mathit{x}}_i(x)\right)}{{\left({\sigma }_j^i\right)}^2}\right|}_{x=x_i(k)},\end{array}$$

(55)

where $y_{mi}$ is the ESRNN identification.

6. Simulation Results and Discussion

The experimental data sets were sampled from the quenching process of a 31 m large-scale vertical quenching furnace. The specification of the selected aluminum alloy workpiece is 360 × 16, with the alloy state 6061T6511 and the batch number CJ1552. Its corresponding temperature setpoint value is 535 °C. In this experiment, 1500 groups of data were collected, and the sampling interval was 10 s.

6.1. Model Verification

The workpiece temperature predicted by the spatial-temporal dimensional extrapolation method is shown in Figure 5 and Figure 6. Because there are seven heating zones in the quenching furnace, the temperatures of the representative upper, middle and lower zones, such as Zones 1, 4 and 7, are displayed. The measured temperature is obtained by selecting an experimental workpiece and installing thermocouples on it in the real on-site industrial process. The workpiece temperature during the temperature rising period is displayed in Figure 5. It can be seen that the predicted temperatures align well with the actual measured temperatures. The temperature prediction accuracy of Zones 1 and 4 is better than that of Zone 7. This is because Zone 7 is located at the bottom of the furnace, which is close to the furnace door. Since the furnace door cannot be completely sealed and is easily affected by the external environment, resulting in the predicted temperature values in Zone 7 being generally higher than the actual values. The temperature prediction curves of Zones 1, 4 and 7 during the temperature holding period are shown in Figure 6. Zone 1 and Zone 4 can still maintain high prediction accuracy, but at 7500 s, the furnace door accidentally falls and cold air pours into the furnace, which leads to great fluctuation of the temperature in the furnace. Zone 1 is located at the top of the furnace, which is almost unaffected. Zone 4 has a small temperature drop under the influence of cold air, but the prediction result still has a high agreement with the measured value.

Note that the temperature drops by nearly 10 °C in Zone 7; although the predicted temperature also exhibits a decreasing trend, the prediction error of Zone 7 is larger. This is because the falling of the furnace door is an unexpected working condition; the environment in the furnace suddenly changes in a short time, so the prediction error is larger, but the error is also within 5 °C. There is another temperature drop at 14,700 s. This is due to the fact that at the end of the thermal treatment process, the on-site workers will open the furnace door and then operate the overhead crane to discharge the workpiece from the furnace into the cooling well at the furnace bottom. At the moment, when the furnace door is opened, a considerable amount of cold air from outside pours into the bottom of the furnace, causing a significant decrease in the internal temperature. The process of manipulating the overhead crane to discharge the workpiece into the well will last for several minutes. During this period, the temperature control system in this paper immediately adjusts and controls the temperature after detecting the temperature drop in the furnace and raises the furnace temperature in time. This demonstrates that the proposed temperature control system exhibits strong robustness, effectively maintaining uniform quenching furnace temperature control despite being influenced by external environmental factors.

Figure 7 shows the relative error diagram of the proposed temperature prediction method. As observed from Figure 7, the temperature prediction accuracy of the temperature rising period is lower than that of the temperature holding period because the heating environment in the furnace during the temperature holding period is more stable. The first large deviation occurs in the late period of temperature rising, at 5500 s. At this time, the workpiece temperature gradually approaches the set temperature. During this period, the hot air in the furnace overflows, and the ventilators appear idle phenome. To save electricity consumption, when the fan is detected to be idling for a certain time, the fan will be set to stop working. The moment the ventilator is stopped, the environment in the furnace will fluctuate slightly. Because the ventilators are located on both sides of the furnace bottom, close to the position of Zone 7, Zone 7 has the greatest influence and has a large error. This temperature fluctuation is also reflected in the temperature curve of Zone 7 in Figure 5. However, since the workpiece temperature has not reached the set temperature during this time, the workpiece quality will not be affected. The second large deviation occurs at 7500 s when the furnace door drops; the cold air near the furnace door pours into the quenching furnace, which makes the temperature prediction of Zone 7 fluctuate. However, the relative error of the fluctuation is less than 1%, and the influence is eliminated within 500 s, so the method proposed in this paper has good robustness and accuracy. The statistical data of temperature prediction error of the proposed method are listed in

Table 1. Statistical data of temperature prediction error of spatial-temporal dimensional extrapolation method.

Index/% Zone	MRE 1	RRMSE 2
Zone 1	0.3195	0.0054
Zone 4	0.3268	0.0052
Zone 7	0.5288	0.0105

¹ Mean relative error. ² Relative root mean squared error.

. It is observed that both MRE and RRMSE values are very small. Therefore, the proposed extrapolation algorithm method can realize high-precision and fast prediction of workpiece temperature.

6.2. Validation of the Integrated Iterative-Decoupling Control

To verify the temperature control performance of the proposed control strategy in this paper, compared with the traditional PID control, the temperature curves of the whole thermal treatment process of aluminum alloy workpieces are shown in Figure 8. For the traditional temperature control, as shown in Figure 8a, it is observed that the temperature rise curves of the seven heating zones during the temperature rising period are not smooth, and the consistency of temperature rise cannot be realized. Especially in the later stage of temperature rise, when the workpiece temperature approaches the temperature set point value of 535 °C, the largest temperature disparity of the seven heating zones in the furnace reaches 46.4 °C. Even after entering the temperature holding period, there is a maximum temperature difference distribution of 7 °C among the heating zones. However, as illustrated in Figure 8b, during the temperature rising period, the seven temperature rising curves have high consistency and smooth temperature rise, and the maximum temperature difference in the furnace is only 12.3 °C, which shows that the proposed iterative learning control based on self-incentive is able to realize a uniform and consistent temperature rise in a large temperature field. During the temperature holding period, the temperature control error is confined to the range of 2 °C. Especially, when approaching 6900 s, the furnace door is closed immediately after accidentally falling off, and the temperature of the traditional temperature control appears a process of rapid drop and slow rising up, which lasts for 1000 s, as shown in Figure 8a. Other heating zones are affected by the strong coupling effect, and there are also temperature drops in different degrees. However, compared with the traditional temperature control, our proposed intelligent decoupling control, as shown in Figure 8b, exhibits its well decoupling performance. Although the temperature of Zone 7 is also affected and lower than the other zones, there is no big temperature drop fluctuation, and it is maintained at 526 °C. It is observed that the temperature of Zone 7 is quickly adjusted and continues to rise to the temperature setting value. The temperature of other heating zones is not affected by the temperature drop of Zone 7, which shows that the integrated intelligent decoupling method proposed in this paper can effectively solve the strong coupling problem among multiple heating zones and realize multi-zone consistent temperature rising and uniform temperature holding. The control performance comparison of different control methods from a statistical perspective is provided in

Table 2. Statistical table of control performance index of traditional control and proposed decoupling control based on ESRNN.

Index	Conventional	Proposed
Max Deviation (°C)	3.3000	1.1702
Average Deviation (°C)	2.5088	0.5272
Max Overshoot (%)	0.6168	0.2187
Average Overshoot (%)	0.4689	0.0985
Mean Static Error (°C)	1.1664	0.4946
Average Tuning Time (×104 s)	1.3100	1.2500
Max Steady State Error (°C)	5.9000	1.7000
Average Steady State Error (°C)	1.1375	0.4458

. All performance indices in Table 2 demonstrate that the developed control strategy is superior to the traditional control method. Note that the average tuning time refers to the time required by the temperature control system of the quenching furnace to regulate the furnace temperature to reach the set temperature before entering the temperature holding period. The obtained statistic is derived from multiple batches. A shorter average tuning time implies higher control performance, as the controller can react more quickly to accurately tune the system response.

Figure 9 is the adaptive tuning curve of control parameters of the decoupling controller during the temperature holding period. It can be seen that the parameter adjustment range is relatively large in the initial stage of the temperature holding period, because ESRNN has not been fully trained at that time, and with the continuous input of production data, the network structure and network parameters are still being adjusted and learned. However, after only 1200 s, the training of the ESRNN network is finished, and the parameters of the controller gradually converge.

7. Conclusions

During the quenching thermal treatment process of large-scale aluminum alloy workpieces, the workpiece temperature cannot be obtained directly. In this paper, a spatial-temporal dimensional extrapolation method is proposed, which can obtain workpiece temperature quickly and accurately. On this basis, a multi-zone integrated iterative-intelligent control strategy for the large-scale vertical quenching furnace is designed. The control strategy contains a nonparametric adaptive iterative control based on self-incentive and an intelligent decoupling control based on ESRNN. According to a large number of historical industrial data obtained from the repeated heating of multiple batches of the same type of workpiece, the control function of the temperature rising process of this type of workpiece can be found by self-incentive iterative learning and used to realize consistent temperature rising of multiple heating zones. Then, based on the optimization idea, an integrated intelligent decoupling control method based on ESRNN is proposed to solve the multi-zone coupling problem. By constructing an ESRNN network with a self-adjusting structure, the Jacobian information of the controller is identified in real-time, and the integrated intelligent setting of controller parameters of each heating zone is realized. Finally, the long-term stable and uniform temperature holding of multiple heating zones during the temperature holding period of the quenching furnace is realized. Compared with the traditional temperature control strategy, the simulation and industrial results indicate that the quality and key properties of large aluminum alloy workpieces using the proposed control strategy are greatly improved. The proposed control strategy can provide a meaningful reference for the thermal treatment process of metals that requires precision and high-uniformity control of temperature or for the uniformity control of large-scale temperature fields.