Algorithmic design and application of feedback control for coiling temperature in hot strip mill

Abstract

Microstructure and mechanical properties of hot-rolled strips are directly influenced by coiling temperature, and feedback control is one of the most significant approaches to improve coiling temperature control precision. Laminar cooling control system was introduced, and the novel feedback control algorithm based on proportional–integral controller and Smith predictor was designed, respectively, and the online adaptive algorithm was used to optimize parameters of proportional–integral controller. This control system has been successfully applied in a hot strip mill, and the record curves indicate that the novel feedback control system can get better dynamic characteristic. The coiling temperature can be precisely controlled with a high accuracy of ±15°C over 98.3%.

Keywords

Hot strip mill coiling temperature feedback control Smith predictor online adaptive algorithm

Introduction

As an important elementary industry of national economy, the iron and steel enterprises are facing serious resource and energy shortages with the rapid development of society and the increasing environmental pressure. With the increasing of customers’ requirements on the quality of hot-rolled strip, the operating performance of laminar cooling processes becomes more and more important in hot strip manufactory.^1–3 Cooling process is a complex heat transfer process; therefore, the cooling control system is a complex time-varying nonlinear system. In practical production, the calculation error exists as the existing mathematical model is very difficult to accurately describe the impact of various factors on the cooling process,^4–6 so the advanced control algorithm for temperature control is quite significant for product quality. Many scholars have made constant efforts to improve the control precision of temperature in recent 20 years. Zhang et al.⁷ brought Smith predictor into laminar cooling system to eliminate the effect of delay. Liu et al.⁸ designed integration controller based on Smith predictor to improve the coiling temperature control precision. Dong et al.⁹ proposed weighted multiple models adaptive controller to deal with the problem that the thickness of the plate differs largely from the real one.

With the development of the intelligence control, its application is increasingly attracting people’s attention. And intelligent control scheme for hot strip rolling based on the fuzzy control was proposed to conduct the dynamic stochastic adaptive control of coiling temperature.^10,11 In order to obtain optimal fuzzy proportional–integral–derivative (PID) controller parameters of active automobile control system, the particle swarm optimization (PSO) reinforcement evolutionary algorithm and the improved chaotic ant swarm algorithm were used to achieve better match the model.^12,13 In addition, the graphical tuning method, the direct synthesis method, and the support vector machine were used to tune PID controller to obtain the optimal performance of control system.^14–16

However, there still exist some problems in coiling temperature control research. For instance, although the study on the feedback control algorithm and intelligent PID algorithm has made relatively abundant achievements on offline models, the study on the online control algorithm is not enough.

In recent years, the traditional feedback control system was used in a hot strip mill; the control system, control models, and application results of novel feedback control system will be described in this article.

Feedback control algorithm

Structure of coiling temperature control system

The cooling equipments can be divided into two functional parts: the main cooling section with 16 cooling banks and the trimming section with 2 cooling banks. Each main cooling bank is made up by four spray headers. Each trimming cooling bank comprises eight spray headers. The coiling temperature control system consists of basic automation and process automation. Process automation is responsible for cooling schedule setting which consists of model setup calculation and coil temperature self-learning. Taking advantage of the high speed of basic automation, the material tracking, the feed forward control, the feedback control, and adaptive control are realized in basic automation, as illustrated in Figure 1. The feedback control system which plays a very important role in coiling temperature control system because it related to product quality directly is one of the critical paths to improve product quality.

Figure 1.

Structure of coiling temperature control system.

Feedback control for coiling temperature with Smith predictor

For increasing measurement accuracy of the finish cooling temperature, the coiling pyrometer is usually installed about 1000–5000 mm far from the exit of laminar cooling equipment, as illustrated in Figure 2. Taking response time of valves into account, the delay time of the system can be expressed by equation (1)

τ = τ_{1} + τ_{2} = \frac{L}{v} + τ_{2}

(1)

where $τ$ is delay time, s; $τ_{2}$ is response time of valves, s; $L$ is the distance between coiling pyrometer and the adjustable header, m; $v$ is rolling speed, $m / s$ .

Figure 2.

Schematic illustration of feedback control system.

The delay length changes when spray headers are adjusted as rolling speed $v$ may increase or decrease, so the delay time of the system varies in real time. The delay length is divided into a number of segments with a certain length, which changes when the delay time varies. The segment length $L_{g}$ can be defined by equation (2)

L_{g} = \frac{L + v τ_{2}}{n}

(2)

where $n$ is integer and $n \geq 1$ ; $(L + v τ_{2})$ is delay length, m.

Segment time can be expressed by equation (3)

T_{s} (i) = \frac{L_{g}}{v (i)}

(3)

The strip temperature drop can be considered proportional to numbers of spray headers, and coefficient $K$ is the temperature drop of a trimming header, which can be calculated by cooling models, °C. In addition, there is a response time when coiling pyrometer works, which can be considered as first-order inertial, and the transfer function can be expressed by equation (4)

G_{p} (s) = \frac{1}{T_{0} s + 1}

(4)

where $T_{0}$ is inertia time constant of coiling pyrometer, s.

The feedback control is shown in Figure 3. $T^{*} (s)$ is temperature reference, $Δ N (s)$ is adjusting water flow value related to control algorithm, $T (s)$ is real temperature measured by the pyrometer, $Δ T (s)$ is difference between setting temperature and feedback temperature.

Figure 3.

Block diagram of feedback control system.

The object function can be expressed by equation (5)

G (s) = G_{p} (s) \cdot e^{- τ s} = \frac{K}{T_{0} s + 1} \cdot e^{- τ s}

(5)

Traditional feedback control based on Smith predictor

For the pure delay characteristics of the temperature feedback control system in the hot strip, Smith predictor is used to eliminate the effect of delay, and the $G_{c} (s)$ was designed with typical second-order optimality, as illustrated in Figure 4.

Figure 4.

Block diagram of traditional feedback control system based on Smith predictor.

The Smith Predictor function is expressed by equation (6)

G_{τ} (s) = \frac{K}{T_{0} s + 1} (1 - e^{- τ s})

(6)

The transfer function after compensation can be expressed by equation (7)

\begin{array}{l} G^{'} (s) = G (s) + G_{τ} (s) = \frac{K}{T_{0} s + 1} \cdot e^{- τ s} \\ + \frac{K}{T_{0} s + 1} (1 - e^{- τ s}) = \frac{K}{T_{0} s + 1} \end{array}

(7)

From Figure 4, the explicit expression of control rate can be expressed by equation (8)

\begin{matrix} Δ n (i) = \frac{a (i) + 2 R {(i)}^{2} \frac{v (i - 1)}{v (i)} - 1}{a (i)} Δ n (i - 1) - \frac{2 R {(i)}^{2} \frac{v (i - 1)}{v (i)}}{a (i)} Δ n (i - 2) \\ + \frac{1}{a (i)} Δ n (i - m - 1) + \frac{R (i) + 1}{a (i) \cdot K} Δ t (i) - \frac{R (i)}{a (i) \cdot K} Δ t (i - 1) \end{matrix}

(8)

where

R (i) = \frac{T_{0}}{T_{s} (i)}

(9)

a (i) = 2 R (i)^{2} + 2 R (i) + 1

(10)

The control rate of segment i is relative to temperature difference $Δ t (i)$ of segment i, temperature difference $Δ t (i - 1)$ and control rate $Δ n (i - 1)$ of segment $(i - 1)$ , control rate $Δ n (i - 2)$ of segment $(i - 2)$ and control rate $Δ n (i - m - 1)$ of segment $(i - m - 1)$ .

Feedback control based on PI with Smith predictor

Considering the response speed and the steady-state precision, the novel feedback control system based on PI controller was designed with Smith predictor, the method of variable segment length was used, and then the optimal control strategy based on the variable segment length method was derived. Figure 5 shows the novel feedback control system.

Figure 5.

Block diagram of novel feedback control system.

From Figure 6, equations (11) and (12) can be obtained

\begin{array}{l} Δ N (s) = Δ N_{τ} (s) k_{p} (1 + \frac{k_{i}}{s}) or Δ N_{τ} (s) \\ = \frac{Δ N (s)}{k_{p} (1 + \frac{k_{i}}{s})} = \frac{Δ N (s) \cdot s}{k_{p} (k_{i} + s)} \end{array}

(11)

\begin{array}{l} Δ N_{τ} (s) = T^{*} (S) - T (S) - T_{τ} (S) \\ = Δ T (s) - \frac{K}{T_{0} s + 1} Δ N (s) + (\frac{K}{T_{0} s + 1} e^{- τ s}) Δ N (s) \end{array}

(12)

Figure 6.

Flowchart of novel feedback control.

Combining equations (11)–(13) can be obtained

\begin{array}{l} \frac{Δ N (s) \cdot s}{k_{p} (k_{i} + s)} = Δ T (s) - \frac{K}{T_{0} s + 1} Δ N (s) \\ + (\frac{K}{T_{0} s + 1} e^{- τ s}) Δ N (s) \end{array}

(13)

Discretizing equation (13) using the sampling time of segment i is $T_{s} (i)$ , and the explicit expression of control rate can be expressed by equation (14)

\begin{matrix} Δ n (i) = \frac{R (i) + R (i - 1) + K \cdot k_{p} + 1}{a (i)} Δ n (i - 1) - \frac{R (i - 1)}{a (i)} Δ n (i - 2) \\ + \frac{T_{s} (i) (u (i - τ) \cdot k_{p} + K \cdot k_{p} \cdot k_{i})}{a (i)} Δ n (i - τ) - \frac{T_{s} (i) (u (i - τ) \cdot k_{p}}{a (i)} Δ n (i - τ - 1) \\ + \frac{k_{p} (R (i) + T_{0} \cdot k_{i} + T_{s} (i) \cdot k_{i} + 1)}{a (i)} Δ t (i) - \frac{k_{p} (R (i) + R (i - 1) + T_{0} \cdot k_{i} + 1)}{a (i)} Δ t (i - 1) \\ + \frac{k_{p} \cdot R (i - 1)}{a (i)} Δ t (i - 2) \end{matrix}

(14)

where

\begin{array}{l} R (i) = \frac{T_{0}}{T_{s} (i)}, u (i) = \frac{K}{T_{s} (i)}, a (i) = R (i) \\ + T_{s} (i) K \cdot k_{p} \cdot k_{i} + K \cdot k_{p} + 1 \end{array}

(15)

Equation (15) shows that the control rate of segment i is relative to temperature difference $Δ t (i)$ of segment i, temperature difference $Δ t (i - 1)$ and control rate $Δ n (i - 1)$ of segment $(i - 1)$ , temperature difference $Δ t (i - 2)$ and control rate $Δ n (i - 2)$ of segment $(i - 2)$ , control rate $Δ n (i - τ)$ of segment $(i - τ)$ and control rate $Δ n (i - τ - 1)$ of segment $(i - τ - 1)$ .

PI controller online adaptive algorithm

The parameters of PI controller need to be adjusted to fit the Smith model in order to keep the stability of the system. The expression of conventional PI controller can be expressed by equation (16)

Δ N (i) = k_{p} \cdot Δ t (i) + k_{i} \cdot \sum_{j = 1}^{i} Δ t (j)

(16)

The control algorithm is based on incremental, thus equation (16) can be written as equation (17)

Δ N (i) = Δ N (i - 1) + Δ n (i)

(17)

The equation (18) can be acquired

{\begin{matrix} Δ n (i) = k_{p} (k) Δ t_{1} (i) + k_{i} (k) Δ t_{2} (i) \\ Δ t_{1} (i) = Δ t (i) - Δ t (i - 1) \\ Δ t_{2} (i) = Δ t (i) \end{matrix}

(18)

The index of PI control is selected as equation (19)

y (i) = \frac{1}{2} (Δ t (i))^{2} = \frac{1}{2} (T^{*} (i) - T (i))^{2}

(19)

The online adaptive algorithm for the PI controller can be formulated as equation (20)

{\begin{matrix} Δ k_{p} (k) = - η_{p} \frac{\partial y (i)}{\partial T (i)} \frac{\partial T (i)}{\partial N (i - 1)} \frac{\partial N (i - 1)}{\partial k_{p} (k - 1)} = η_{p} Δ t (i) \frac{\partial T (i)}{\partial N (i - 1)} Δ t {(i - 1)}_{1} \\ Δ k_{i} (k) = - η_{i} \frac{\partial y (i)}{\partial T (i)} \frac{\partial T (i)}{\partial N (i - 1)} \frac{\partial N (i - 1)}{\partial k_{i} (i - 1)} = η_{i} Δ t (i) \frac{\partial T (i)}{\partial N (i - 1)} Δ t {(i - 1)}_{2} \end{matrix}

(20)

where $(\partial T (i)) / (\partial N (i - 1))$ is the Jacobian information of object which can be expressed by equation (21)

\frac{\partial T (i)}{\partial N (i - 1)} = \frac{K}{T_{0} s + 1}

(21)

The change of the parameters can be expressed by equation (22)

{\begin{matrix} Δ k_{p} (k) = η_{p} Δ t (i) Δ t {(i - 1)}_{1} \frac{K}{T_{0} s + 1} \\ Δ k_{i} (k) = η_{i} Δ t (i) Δ t {(i - 1)}_{2} \frac{K}{T_{0} s + 1} \end{matrix}

(22)

According to equation (22), the proportional gain $k_{p}$ and integral gain $k_{i}$ of PI controller can be obtained as equation (23)

{\begin{matrix} k_{p} (k) = k_{p} (k - 1) + Δ k_{p} \\ k_{i} (k) = k_{i} (k - 1) + Δ k_{i} \end{matrix}

(23)

The parameters of PI controller such as the proportional gain $k_{p}$ and integral gain $k_{i}$ can be automatically readjusted online to maintain the system error $y (i) = 0$ .

Program realization and results of feedback control

Program realization of feedback control

The feedback control system is the critical system affecting the microstructure and mechanical properties which is designed as a subsystem of laminar cooling control system. Feedback control subsystem gets schedule setting from process automation and gets real-time data (such as coiling temperature, strip speed, and actual header configuration) from measuring instrument. The adjustment calculation of feedback control is done for each segment when the strip passes the coiling pyrometer. Flowchart of feedback control based on Smith predictor is shown as Figure 6.

Results and discussions

After a long time of debugging, the novel feedback control system has been applied to the production of various specifications, comparing with the traditional feedback control system based on typical second-order optimality and Smith predictor, the novel feedback control system has faster response speed and higher steady-state precision. The temperature curves of traditional feedback control system and novel feedback control system were shown in Figures 7 and 8.

Figure 7.

Temperature curve of traditional feedback control system.

Figure 8.

Temperature curve of novel feedback control system: (a) Q345, thickness4.25 mm, coiling temperature 620°C–640°C and (b) SPHC, thickness 3.2 mm, coiling temperature 670°C.

The temperature curve of the traditional feedback control system and the novel feedback control system were analyzed, and the statistical result is illustrated in Table 1.

Table 1.

Temperature control effect statistic.

Control strategy	Control precision (<15°C)	Response speed (ms)
Traditional feedback control system	92.6%	260
Novel feedback control system	98.3%	90

Statistical data show that after applying the novel feedback control system, the strip temperature deviation can be controlled within the target tolerances ±15°C over 98.3%, and the respond time could be controlled within 90 ms. The control precision was improved significantly, and the response speed of the system is significantly accelerated.

Conclusion

On the basis of cooling theory, the self-tuning PI-Smith feedback control system was developed and the explicit expression was derived, which provided theory basis for field application.

The online adaptive algorithm was used to readjust the PI parameters such as the proportional gain $k_{p}$ and integral gain $k_{i}$ automatically to enhance the stability and accelerate the response speed of the system.

The novel feedback control system has been applied in a hot strip mill. The coiling temperature can be precisely controlled with a high accuracy of ±15°C over 98.3% and the respond time within 90 ms. The control precision was improved significantly, and the response speed of the system is significantly accelerated.

Footnotes

Academic Editor: Baozhen Yao

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully appreciate the financial support by the National Natural Science Foundation of China (Grant no. 51404159), Natural Science Foundation of Shanxi Province, China (Grant No. 201601D202027), and doctoral promoter of Taiyuan University of Science and Technology of China (Grant no. 20152013).

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