Confinement Phenomenon Effect on the CO2 ... - ACS Publications

7248

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 65, NO. 9, SEPTEMBER 2018

Nonlinear Control Tools for Fused Magnesium Furnaces: Design and Implementation Zhiwei Wu, Tengfei Liu

, Member, IEEE, Zhong-Ping Jiang , Fellow, IEEE, Tianyou Chai, Fellow, IEEE, and Lina Zhang

Abstract—This paper employs the recently developed nonlinear control tools to solve the engineering problem of stable current control of fused magnesium furnaces (FMFs). By fully taking into account the nonlinearity, uncertainty, and couplings in FMFs, this paper first develops a dynamic model for controller design. A mild assumption for controller design is verified by using experimental data. Based on the dynamic model, this paper proposes a class of stable control rules for the disturbance-free model. Then, the result is extended to handle both measurement errors and external disturbances. With the proposed control algorithm, it is proved that the closed-loop system is robustly stable, and the three-phase currents ultimately converge to neighborhoods of the setpoints. If the system is disturbance-free, then asymptotic stability can be guaranteed. It is shown that the proposed control algorithm can be readily implemented as rule-based control. The notion of input-to-state stability and the nonlinear small-gain theorem are employed as tools for the designs. An industrial application is demonstrated to show the validity of the proposed design in solving practical engineering problems. Index Terms—Fused magnesium furnace (FMF), robustness, rule-based control, small-gain control, stabilization.

I. INTRODUCTION HIS paper studies the control problem of fused magnesium furnaces (FMFs). Fused magnesia is an important refractory for many industries, such as metallurgical, chemical, electric apparatus, and aerospace industry. Due to the complex characteristics of the magnesite material, high-purity fused magnesia is normally produced by the unique three-phase ac FMF. In engineering practice, appropriate control of the three-phase current values has been recognized to be vital to guarantee the

T

Manuscript received May 9, 2017; revised September 2, 2017; accepted September 21, 2017. Date of publication November 1, 2017; date of current version May 1, 2018. This work was supported in part by the NSFC under Grant 61503066, Grant 61522305, Grant 61633007, Grant 61374042 and Grant 61533007; in part by the NSF under Grant ECCS-1501044; and in part by the State Key Laboratory of Intelligent Control and Decision of Complex Systems at BIT. (Corresponding author: Tengfei Liu.) Z. Wu, T. Liu, T. Chai, and L. Zhang are with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110004 China (e-mail: [email protected]. cn; [email protected]; [email protected]; [email protected]. edu.cn). Z.-P. Jiang is with the Tandon School of Engineering, New York University, Brooklyn, NY 11201 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2017.2767545

product quality and, at the same time, minimize the electricity consumption per ton. During the smelting process, given specific three-phase voltages, the relation between the positions of the electrodes and the three-phase current values depends on various (and maybe time-varying) factors of the raw materials including melting point, resistivity, and impurity constituent, which leads to strong nonlinearity and large uncertainty. For different phases (e.g., normal and abnormal phases) with various working stages/conditions (e.g., heating and melting, feeding, and exhausting stages) of the smelting process, the factors may be significantly changed. Also, due to the inherent physical couplings, the position adjustment of each electrode affects not only its corresponding current value but the current values of the other two phases as well. The complexity mentioned above causes the major difficulty in designing control laws for FMFs. Moreover, the actuator subsystems which drive the three-phase electrodes are normally based on AC asynchronous motors without speed regulation mechanisms, which means that continuous control laws cannot be implemented, and discontinuous control is expected. Stability is the fundamental requirement for industrial control systems. Although some essential control strategies such as rulebased control have been tested in some of the FMF facilities, it is frequently questioned how stability can be guaranteed. It should be noted that the well-known stability tools such as Lyapunov theory [1] have not been rigorously used to analyze the existing control designs for FMFs. Today, most of the FMFs in China are manually controlled by skilled operators, mainly due to the stability issues. In the past decade, quite a few theoretical and experimental studies were carried out to handle the complexity of the smelting process of FMFs [2]–[4]. In particular, some intelligent control strategies, e.g., fuzzy control in [5], rule-based control in [6], and case-based control in [7], have been tested for the control of FMFs without theoretically rigorous stability analysis. Other results that related to the control of electric arc furnaces include [8]–[17], which cannot be directly applied to the control of FMFs due to the specific system complexity of FMFs. Based on the recently developed nonlinear control tools, this paper proposes a systematic design for stable rule-based control of FMFs. By fully taking into account the nonlinearity, uncertainty, and couplings in FMFs, this paper first develops a dynamic model for controller design. The designs in this paper are based on the mild assumption on the relationship between the positions of electrodes and the corresponding currents, the

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WU et al.: NONLINEAR CONTROL TOOLS FOR FUSED MAGNESIUM FURNACES: DESIGN AND IMPLEMENTATION

validity of which is verified by using experimental data. Then, to clarify the basic idea of the design, this paper first considers the disturbance-free case, and proposes a class of current control algorithms. In this case, it is proved that the three-phase currents asymptotically converge to the current setpoints. A more general case with both measurement errors and external disturbances is then considered, and the closed-loop system is proved to be robustly stable. This paper employs the notion of inputto-state stability (ISS) to characterize the robust stability of the closed-loop system; see the original paper on ISS [18] and a nice tutorial [19]. The basic idea is to consider the current control system as an interconnection of three subsystems corresponding to the three-phase electrodes. A nonlinear control law is developed for each subsystem, such that each controlled subsystem is ISS and the closed-loop system is transformed into a dynamical network composed of three ISS subsystems. Then, a nonlinear small-gain synthesis is proposed to fine tune the controller parameters and to guarantee the robust stability of the closed-loop system; see [20] for the original reference on the nonlinear ISS small-gain theorem, and [21]–[23] for recent extensions of the small-gain theorem to nonlinear dynamical networks. It should be noted that the small-gain design makes it possible to take into account not only nonlinearity and uncertainty in system dynamics but also discontinuity in the actuation system. One advantage of the small-gain design lies in that the proposed control law can be readily implemented with rule-based control algorithms, which is easily programmable for PLCs. The effectiveness of the proposed algorithm has been verified by practical applications. The rest of the paper is organized as follows. Section II formulates the control problem, which contains a brief description of the process and modeling for control design. In Section III, we introduce the basic idea of small-gain design for current control of FMFs for the disturbance-free case. Section IV extends the result in Section III by using gains to describe the influence of measurement errors and external disturbances. Then, in Section V, a stable rule-based control algorithm is developed based on the method proposed in Section IV. An industrial application is introduced in Section VI. Section VII contains some concluding remarks and future research directions. II. FORMULATION OF THE CONTROL PROBLEM In this section, we first give a brief description of the magnesia smelting process and its control system, and then propose a control-oriented dynamic model based on experimental data. A. Description of the Magnesia Smelting Process and its Control System The physical and chemical changes during smelting process of FMFs are shown as follows: 640 d e g C−800 d e g C

MgCO3 −−−−−−−−−−−→ CO2 + MgO (solid state) 2800 d e g C

MgO (solid state) −−−−−→ MgO (liquid state) cooling

MgO (liquid state) −−−→ MgO (crystal).

(1) (2) (3)

Fig. 1.

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Structure of the manual control system for FMFs.

The raw material for fused magnesia production is magnesite with main ingredient MgCO3 . However, magnesite normally contains impurities such as CaO, SiO2 , Fe2 O3 , Al2 O3 , etc. When MgCO3 is heated from the room temperature to the temperature range of 640 °C–800 °C, it is decomposed into gas CO2 and solid-state MgO. Such solid-state MgO is normally called caustic-burned magnesia, and there exist drawbacks in the lattice of caustic-burned magnesia such as loose texture and great chemical activity. During the smelting process, solid MgO continues to be heated up to 2800 °C–2850 °C , which turns it into molten (liquid) state. After natural cooling process, the liquid becomes crystal gradually. In the entire process of heating and cooling, the lattice drawbacks of MgO in caustic-burned magnesia are corrected, crystal particles gradually grow up, and the organization structure becomes compact. Nowadays, the smelting process is normally controlled manually in industrial fields as shown in Fig. 1. Before smelting, a small amount of raw material is fed into the furnace from the storage silo above the FMF. Then, it is required to determine the positions of the three-phase electrodes for preparation at the beginning of the smelting process. After moving the AC motor-driven electrodes to proper positions, AC power is applied to the electrodes to generate electric arcs between the electrodes and the raw material, and the smelting process starts. As more raw material is feed into the furnace, a molten pool is formed with the raw material melted by electric arc heating. The level of the molten pool goes up as more raw material is feed and melted. Accordingly, the positions of the three-phase electrodes should be adjusted to keep a certain gap between the electrodes and the surface of the molten pool, and to control the current values of the electrodes to track specific setpoints. The main purpose of current control is to maintain the temperature. The operator manually controls the up-and-down movement of the motors by observing the current values of the three-phase electrodes to continuously adjust the positions of the electrodes, insuring that the currents track the current setpoint. The smelting process ends when the surface of the molten pool reaches the top of the furnace. The furnace body is then removed from the smelting rack for cooling. During the cooling off period, the liquid becomes crystal gradually. Finally, the fused magnesia meeting the desired product grade is produced by breaking the crystal MgO into certain granularity.

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Fig. 2.


Couplings between the three-phase currents.

Fig. 3.

B. Modelling for Current Control To maintain the temperature within the desired range, the current values of the three-phase electrodes can be adjusted by controlling the resistance and intensity of the electrical arcs, which can be achieved by adjusting the gaps between the three-phase electrodes and the surface of the molten pool. The complexity lies in the uncertain resistivity of the raw material and the nonlinear couplings between the three-phase electrodes. During the smelting process, the granule size and impurity constituents (mainly SiO2 , Al2 O3 , Fe2 O3 ) are frequently changed, and such unmeasurable changes cause fluctuations in resistivity of the raw materials. The temperature cannot be maintained within the desired range without proper control of the three-phase current values. Moreover, the control action on the electrode of each phase influences the currents of the other two phases. Fig. 2 shows the trajectories of the three-phase currents when Phase A electrode is manually controlled to make up-and-down movement and the positions of the other two electrodes are fixed. As shown with the experimental data, the movement of Phase A electrode changes not only the current value of Phase A but also the current values of Phase B and Phase C, due to strong couplings. In the presence of uncertainty, nonlinearity and strong couplings, it is not easy to develop a precise model for current control. In this section, the three-phase currents are intuitively considered as functions of the positions of three-phase electrodes. Some specific characteristics of the model, which are useful for controller design, will be revealed by using practically measured data. For i = 1, 2, 3, we use Ii to represent the three-phase current values, and use Ii∗ to represent the corresponding setting points. Normally, the three-phase currents should be “balanced”, i.e., I1∗ = I2∗ = I3∗ . We use pi to represent the positions of the threephase electrodes. Since the three-phase currents are considered to be determined by the positions of the electrodes, there exist functions fi : R3 → R such that Ii = fi (p1 , p2 , p3 ), i = 1, 2, 3.

(4)

Corresponding to the three-phase current settings, if we use p∗i with i = 1, 2, 3 to represent the ideal positions of the electrodes, then Ii∗ = fi (p∗1 , p∗2 , p∗3 ).

(5)

Historical data used for identification.

To develop a control error model, define I˜i = Ii − Ii∗ and p˜i = pi − p∗i . Define f˜i (˜ p1 , p˜2 , p˜3 ) = fi (˜ p1 + p∗1 , p˜2 + p∗2 , p˜3 + p∗3 ) − fi (p∗1 , p∗2 , p∗3 ) = fi (p1 , p2 , p3 ) − fi (p∗1 , p∗2 , p∗3 ).

(6)

Then, from (4) and (5), direct calculation yields I˜i = f˜i (˜ p1 , p˜2 , p˜3 ).

(7)

C. Characteristics of the Dynamic Model: Analysis With Experimental Data If fi in (7) with i = 1, 2, 3 is totally unknown, then it seems impossible to directly use the existing control theory to solve the current control problem, and some complex learning algorithm might be expected. However, the computing capacities of FMF controllers, e.g., programmable logic controller (PLC), are usually very low and cannot afford large computing tasks. In this section, we will identify some characteristics of model (7) by using practically measured data. To figure out the strength of the interaction between the electrodes, around the working point, we consider the linear approximation of model (7) as I˜i = ai1 p˜1 + ai2 p˜2 + ai3 p˜3

(8)

for i = 1, 2, 3, where Aij with j = 1, 2, 3 are unknown coefficients of the linearized model. Clearly, Aij represents the strength of influence from position change of electrode j to value change of current i. Then, we use the historically measured data to identify Aij with i, j = 1, 2, 3. The historical data that is used for the identification is shown in Fig. 3, and the identified model is as follows: ⎡˜ ⎤ ⎡ ⎤⎡ ⎤ I1 360.83 −50.89 −60.39 p˜1 ⎢˜ ⎥ ⎢ ⎥⎢ ⎥ (9) ⎣ I2 ⎦ = ⎣ −61.47 291.11 36.29 ⎦ ⎣ p˜2 ⎦ p˜3 −10.67 62.98 249.74 I˜3 From (9), it can be easily checked that |aij | aii >

(10)

j = i

for each i = 1, 2, 3. This means that, although the interaction between the electrodes cannot be ignored, the current of each


phase is mainly determined by the position of its corresponding electrode. The observation of (10) is reasonable, as the skilled operators use similar ideas for manual control. Based on (10), by also taking into account uncertainty and nonlinearity, we make the following assumption on model (7). Assumption 1: For i, j = 1, 2, 3, there exist functions Lij : R3 → R such that each fi can be written in the form of p1 , p˜2 , p˜3 ) = Lij (˜ p1 , p˜2 , p˜3 )˜ pj . (11) fi (˜ j =1,2,3

Moreover, there exist constants δi > 0 such that Lii (˜ p1 , p˜2 , p˜3 ) ≥ (1 + δi ) |Lij (˜ p1 , p˜2 , p˜3 )|.

(12)

i= j

Besides the nonlinearity, uncertainty, and couplings, the following challenging issues should also carefully addressed: 1) The position p∗i , which is basically the gap between the electrode and the surface of the molten pool, is not measurable, and thus, p˜i = pi − p∗i cannot be directly considered as a control input. 2) The velocities of the electrodes may not be continuously adjusted, due to the predesigned actuation mechanism. For each i = 1, 2, 3, we use ui to represent the velocity of electrode i. Then, by taking the derivative of p˜i , we have p˜˙i = ui .

(13)

D. Objective of Controller Design The objective of this paper is to design a control law ui = κi (I˜i )

(14)

to adjust the positions of the electrodes, such that lim I˜i (t) = 0, i = 1, 2, 3.

t→∞

(15)

Remark 1: Clearly, control objective (15) is achieved if limt→∞ p˜i (t) = 0, which can be easily achieved if p˜i is available for feedback. However, in our problem setting, p˜i is not directly measurable, and I˜i is used instead for feedback. The difficulty lies in the fact that I˜i depends not only on p˜i but also on p˜j (j = i). For the ith electrode, p˜j (j = i) can be considered as measurement disturbances. In the literature of nonlinear control, it is well-known that, small (even finitely converging) measurement disturbance may destroy the global asymptotic stability of a nonlinear control system [24]. Remark 2: Modern control systems are normally based on digital computers and sensors. Digitalized implementation of analog (continuous-time) controllers is one fundamental topic with computer control system designs [25]. The design in this paper is based on the basic idea that the validity of a continuoustime control law can be guaranteed if the sampling period is small enough. III. DESIGN AND IMPLEMENTATION OF RULE-BASED CONTROL ALGORITHM: A NONLINEAR SMALL-GAIN APPROACH In this section, we propose a class of control laws for current control of FMFs. In particular, we consider a class of control

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laws in the form of (14) with κi : R → R satisfying κi (I˜i ) = −ρi (|I˜i |)I˜i

(16)

where ρi : R+ → R+ is a positive function to be designed later. Remark 3: Intuitively, the control algorithm (16) can be considered to be a variable-coefficient proportional (P) control algorithm with ρi representing the updating law of the P coefficient. Clearly, the control algorithm (16) can be easily implemented in industrial control systems with limited computing capabilities. Substituting (14) into (13) and using (7) and (11) yield ⎛ p˜˙i = κi ⎝

⎞ Lij (˜ p1 , p˜2 , p˜3 )˜ pj ⎠

j =1,2,3

⎞ ⎛

⎝ Lij (˜ p1 , p˜2 , p˜3 )pj

⎠ = −ρi

j =1,2,3

× Lij (˜ p1 , p˜2 , p˜3 )˜ pj .

(17)

j =1,2,3

Then, the closed-loop system can be considered as an interconnection of three p˜i -subsystems. If limt→∞ p˜i = 0, then the objective of (15) is achieved. Clearly, if the closed-loop system with (˜ p1 , p˜2 , p˜3 ) as the state is asymptotically stable at the origin, then the objective defined by (15) is achievable. However, due to the interaction between the electrodes, even if each p˜i -subsystem is stabilized by designing κi , the stability of the closed-loop system is still questionable. Theorem 1 shows the validity of the proposed control law. Theorem 1: Consider the closed-loop system composed of (7) and (13) with control law (14). Under Assumption 1, the closed-loop system (17) is asymptotically stable at the origin, and the control objective (15) is achievable. Proof: We first prove that each p˜i -subsystem is ISS, and then use the nonlinear small-gain theorem to prove stability of the closed-loop system. For each p˜i -subsystem (i = 1, 2, 3), define the following Lyapunov function candidate pi ) = Vi (˜

1 2 p˜ . 2 i

(18)

For system (17), following the standard approach of Lyapunov-based ISS analysis (see, e.g., [19]), we calculate ∇Vi (˜ pi )p˜˙i

⎞ ⎛

= −˜ pi ρi ⎝

Lij (˜ p1 , p˜2 , p˜3 )˜ pj

⎠

j =1,2,3

× Lij (˜ p1 , p˜2 , p˜3 )˜ pj j =1,2,3

7252


⎞ ⎛

⎝ ≤ ρi Lij (˜ p1 , p˜2 , p˜3 )˜ pj

⎠

j =1,2,3

⎛ ⎞ × ⎝−Lii (˜ p1 , p˜2 , p˜3 )˜ p2i + |Lij (˜ p1 , p˜2 , p˜3 )| |˜ pj | |˜ pi |⎠ . j = i

(19) Fig. 4.

To analyze the ISS property of the closed-loop system, we choose constant i satisfying 0 < i < δi and consider the case of |˜ pi | ≥

1 max {|˜ pj |} . 1 + i j = i

(20)

Then, we have ∇Vi (˜ pi )p˜˙i

⎞ ⎛

⎝ ≤ ρi Lij (˜ p1 , p˜2 , p˜3 )˜ pj

⎠

j =1,2,3

⎛ ⎞ × ⎝−Lii (˜ p1 , p˜2 , p˜3 )˜ p2i + (1 + i ) |Lij (˜ p1 , p˜2 , p˜3 )| p˜2i ⎠ j = i

⎛ =

p1 , p˜2 , p˜3 )˜ p2i − ⎝Lii (˜

− (1 + i )

Interconnection gains of the closed-loop system.

By using the definition of Vi in (18), we have Vi (˜ pi ) ≥

1 max {Vj (˜ pi )} (1 + i )2 j = i

⇒ ∇Vi (˜ pi )p˜˙i ≤ −αi (˜ p1 , p˜2 , p˜3 )˜ p2i

(24)

which guarantees ISS of each p˜i -subsystem. Use γij (i = j) to represent the ISS gain from the p˜j subsystem to the p˜i -subsystem. Then, we have γij (s) =

1 s (1 + i )2

(25)

for all s ∈ R+ . The interconnections of the closed-loop system with the ISS gains are shown in Fig. 4. Since all the ISS gains are less than the identity function, the following small-gain condition is readily satisfied [22]:

⎞

γ12 ◦ γ21 < Id

(26)

|Lij (˜ p1 , p˜2 , p˜3 )|⎠

γ23 ◦ γ32 < Id

(27)

γ31 ◦ γ13 < Id

(28)

γ12 ◦ γ23 ◦ γ31 < Id

(29)

γ13 ◦ γ32 ◦ γ21 < Id

(30)

j = i

⎞ ⎛

⎝ × ρi Lij (˜ p1 , p˜2 , p˜3 )

⎠ p˜2i .

j =1,2,3

(21)

Substituting (12) into (21) yields ∇Vi (˜ pi )p˜˙i 1 + i ≤ − Lii (˜ p1 , p˜2 , p˜3 ) − Lii (˜ p1 , p˜2 , p˜3 ) 1 + δi

⎞ ⎛

× ρi ⎝

Lij (˜ p1 , p˜2 , p˜3 )

⎠ p˜2i

j =1,2,3

⎞ ⎛

δi − i

⎝ = − Lii (˜ p1 , p˜2 , p˜3 )ρi Lij (˜ p1 , p˜2 , p˜3 )

⎠ p˜2i 1 + δi

j =1,2,3

=: − αi (˜ p1 , p˜2 , p˜3 )˜ p2i .

(22)

Clearly, αi is a positive function. Recall that the validity of (22) is based on (20). Thus, we have

IV. ROBUSTNESS WITH RESPECT TO UNCERTAINTIES AND DISTURBANCES Control algorithms in the form of (16) designed in Section III satisfy the following sector-bound property: −ρi (|I˜i |)I˜i ≤ κi (I˜i ) ≤ −ρi (|I˜i |)I˜i

1 max {|˜ pj |} |˜ pi | ≥ 1 + i j = i ⇒ ∇Vi (˜ pi )p˜˙i ≤ −αi (˜ p1 , p˜2 , p˜3 )˜ p2i .

where “◦” represents composition of functions, and “Id” represents the identity function. Condition (26)–(30) means that the composition of the gain functions along each simple cycle in the interconnected system is less than the identity function. The stability property of the closed-loop system is proved by directly using the small-gain theorem. Remark 4: It can be observed that the current control objective can be achieved as long as the control law can be transformed into the form of (14) with κi satisfying (16). One advantage of the small-gain design is that function κi in (14) is not required to be continuous. Thus, the proposed design makes it possible to implement discontinuous control actions.

(23)

(31)

where ρi , ρi : R+ → R+ are continuous and positive functions satisfying ρi (s) ≤ ρi (s) ≤ ρi (s) for all s ≥ 0. This means that such control algorithms cannot take into account the deadzone


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properties around the origin, which is desired for practical implementation. Moreover, although Assumption 1 can cover a large class of dynamics, external disturbances and the measurement noise cannot be directly handled with the design in Subsection III. The objective of this subsection is to extend the design in Subsection III by proposing a robust control result for FMFs.

Remark 5: It should be noted that wi can also be used to represent the deadzone property in the feedback of I˜i . Specifically, if we ignore the measurement error, then the deadzone can be represented by defining wi as −I˜i , if |I˜i | ≤ I˜Δ (38) wi = 0, otherwise.

A. Problem Formulation of Robustness Analysis

Here, I˜Δ represents the size of the deadzone. If the measurement errors cannot be ignored, then one may redefine wi by adding the measurement errors. If κi in (37) is still designed as defined by (16), then by substituting (37) into (36) and using (32) and (33), we have ⎛ ⎞ p˜˙i = κi ⎝ Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + ν i ⎠

If the current values of the electrodes depend not only on the positions of the electrodes but also on other variables, then the model described by (7) is not valid. We use di ∈ Rm with i = 1, 2, 3 to represent the other variables that influence the currents of the three-phase electrodes, and modify model (7) as I˜i = f˜i (˜ p1 , p˜2 , p˜3 , di ).

(32)

j =1,2,3

⎞ ⎛

⎝ = −ρi Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + νi

⎠

j =1,2,3

⎛ ⎞ ×⎝ Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + ν i ⎠

To represent the influence of the external disturbance, we modify Assumption 1 as follows. Assumption 2: For i, j = 1, 2, 3, there exist functions Lij : R4 → R and Lid : R4 → R such that each fi can be written in the form of p1 , p˜2 , p˜3 , di ) = Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj fi (˜ j =1,2,3

+ Lid (˜ p1 , p˜2 , p˜3 , di ).

(33)

where νi = Lid (˜ p1 , p˜2 , p˜3 , di ) + wi .

Moreover, there exist constants δi , Lbii > 0 such that p1 , p˜2 , p˜3 , di ) Lii (˜ ⎧ ⎫ ⎨ ⎬ ≥ max (1 + δi ) |Lij (˜ p1 , p˜2 , p˜3 , di )|, Lbii ⎩ ⎭ i= j

(34) for i, j = 1, 2, 3, and σi ∈ K such that p1 , p˜2 , p˜3 , di )| ≤ σi (|di |) |Lid (˜

(36)

In the presence of external disturbances and measurement errors, the objective of this paper is to design a control law to adjust the positions of the electrodes according to the current errors such that the closed-loop current control system is stabilized and the control errors of the currents ultimately converge to some neighborhoods of the origin with the sizes of the neighborhoods depending on the magnitudes of the external disturbances and the measurement errors. In this case, the designed control law would be ui = κi (I˜i + wi ) where wi ∈ R represents the additive measurement error.

(40)

Different from (17), the closed-loop system (39) is subject to the external disturbance di and the measurement error wi . For such system, asymptotic convergence may not be achievable, and some practical stability result would be reasonable. For practical implementation, it is also expected to get an estimate on the strength of the influence of di and wi . B. Robustness Analysis

(35)

for i = 1, 2, 3. Here, K represents the class of functions defined on [0, ∞) such that if γ ∈ K then γ is strictly increasing and γ(0) = 0. In this case, the relation between velocity and position of the electrodes, which is represented by (13), still holds. For completeness of this section, we rewrite it as p˜˙i = ui .

(39)

j =1,2,3

(37)

By refining the small-gain analysis in Section III, we propose a robust stability result for system (39). In particular, the robust stability property is formulated by the notion of ISS, and the influences of the external disturbance di and the measurement error wi are represented by ISS gains. Under Assumption 2, Theorem 2 gives a robust stability result on the closed-loop system. Theorem 2: Consider system composed of (32) and (36) with control law (37). Under Assumption 2, the closed-loop system (39) is ISS with di and wi (i = 1, 2, 3) as the inputs. Proof: For each p˜i -subsystem, define an ISS-Lyapunov function candidate as (18). Then, we have ∇Vi (˜ pi )p˜˙i

⎞ ⎛

⎝ = −˜ pi ρi Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + νi

⎠

j =1,2,3

⎛ ⎞ ×⎝ Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + ν i ⎠ j =1,2,3

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⎞ ⎛

pi | ≤ ρi ⎝

Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + νi

⎠ |˜

j =1,2,3

×

By substituting (46) into (41), we have

−Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi |

+

|Lij (˜ p1 , p˜2 , p˜3 , di )||˜ pj | + |νi | .

(41)

j = i

Note that ρi is a positive function. From the last inequalp1 , p˜2 , p˜3 , di )|˜ pi | > ity in (41), it can be observed that if Lii (˜ ˙ |L (˜ p , p ˜ , p ˜ , d )||˜ p | + |ν |, then ∇V (˜ p ) p ˜ is negaij 1 2 3 i j i i i i j = i tive, and thus Vi is decreasing along the trajectories of the p˜i subsystem. To analyze the ISS property of the closed-loop system, we consider the case of 1 + μi 1 + δi max |˜ |νi | pj |, |˜ pi | ≥ 1 + i j = i μi Lbii

(42)

|Lij (˜ p1 , p˜2 , p˜3 , di )||˜ pj | 1 + i Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi | (1 + μi )(1 + δi )

≤

≤ (1 + μi ) max

j = i

(44)

1 |Lij (˜ p1 , p˜2 , p˜3 , di )||˜ pj |, |νi | . μi

With properties (43) and (44), it also holds that |Lij (˜ p1 , p˜2 , p˜3 , di )||˜ pj | + |νi |

j = i

≤

1 + i Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi |. 1 + δi

j = i

(49)

|νi | ≤ |Lid (˜ p1 , p˜2 , p˜3 , di )| + |wi | ≤ σi (|di |) + |wi | 1 (50) ≤ max (1 + ηi )σi (|di |), 1 + |wi | . ηi To explicitly characterize the gains from di and wi , define γid (s) = γiν ((1 + ηi )σi (s)) and γiw (s) = γiν ((1 + η1i )s). Then, based on the discussions above, we have

(45)

(48)

where γij (s) = (1 + μi )2 s/(1 + i )2 , and γiν (s) = (1 + δi )2 (1 + μi )2 s2 /μ2i (Lbii )2 (1 + i )2 . Recall the definition of νi in (40) and the property of p1 , p˜2 , p˜3 , di ) in (35). For any specific ηi > 0, we have Lid (˜

|Lij (˜ p1 , p˜2 , p˜3 , di )||˜ pj | + |νi |

j = i

⎞ ⎛

δi − i ⎝

= ρi Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + νi

⎠ 1 + δi

j =1,2,3

⇒ ∇Vi (˜ pi )p˜˙i ≤ −αi (˜ p1 , p˜2 , p˜3 , di , νi )Vi (˜ pi )

Note that for any specific μi > 0, it holds that

αi (˜ p1 , p˜2 , p˜3 , di , νi )

Vi (˜ pi ) ≥ max {γij (Vj (˜ pj )), γiν (|νi |)}

μi Lbii (1 + i ) |˜ pi | (1 + μi )(1 + δi ) μi (1 + i ) Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi |. (1 + μi )(1 + δi )

(47)

Define

(43)

and |νi | ≤

=: −αi (˜ p1 , p˜2 , p˜3 , di , νi )˜ p2i .

Clearly, αi is a positive function. Recall that the validity of (47) is based on condition (42). By also using the definition of Vi in (18), we have

j = i

≤

× Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi |2

× Lii (˜ p1 , p˜2 , p˜3 , di ).

where μi and i are constants satisfying 0 < μi < i < δi . In this case, by also using Assumption 2, we have

pi )p˜˙i ∇Vi (˜

⎞ ⎛

pi | ≤ ρi ⎝

Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + νi

⎠ |˜

j =1,2,3

1 + i × −Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi | + Lii (˜ p1 , p˜2 , p˜3 , di )|˜ pi | 1 + δi

⎞ ⎛

δi − i ⎝

= − ρi Lij (˜ p1 , p˜2 , p˜3 , di )˜ pj + νi

⎠ 1 + δi

j =1,2,3

(46)

Vi (˜ pi ) ≥ max {γij (Vj (˜ pj )), γid (|di |), γiw (|wi |)} j = i

⇒ ∇Vi (˜ pi )p˜˙i ≤ −αi (˜ p1 , p˜2 , p˜3 , di , νi )Vi (˜ pi ).

(51)

It can be easily checked that the nonlinear small-gain condition (26)–(30) still holds in the case with measurement errors and external disturbances. The ISS property of the closed-loop system can thus be proved. Remark 6: Control of nonlinear systems in the presence of measurement error is one of the fundamental and challenging problems in the literature of robust nonlinear control. Different from linear systems, even very small measurement errors that converge to zero in finite time may destroy the asymptotic


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stability of a nonlinear system. For practical control applications, the control algorithm should be designed to be robust or less sensitive to disturbances/uncertainties. In this paper, such problem is solved by considering the closed-loop system as an interconnection of three subsystems, and applying the recently developed nonlinear small-gain theorem. Moreover, the influence of the measurement errors and the external disturbances can be explicitly described by ISS gains, which can be tuned by appropriately designing the control algorithm. Remark 7: From the proof of Theorem 2, it can be observed that function ρi of the control law determines αi , which represents the convergence rate of the closed-loop system; see (47). Intuitively, increasing ρi means improved convergence. However, it should be noted that larger ρi requires a faster actuation system, while the study in this paper neglects dynamics of the actuation system. V. RULE-BASED IMPLEMENTATION OF THE CONTROL ALGORITHM

Fig. 5.

Logic-based controllers (e.g., PLCs) are quite popular, and accordingly rule-based control is used widely in practical industrial control systems. Based on the design proposed in Sections III and IV, in this section, we fully take into consideration the discontinuity of the actuation mechanism, and design a class of rule-based control algorithm for practical implementation. In particular, by appropriately choosing κi and wi , one can implement control laws in the form of (37) with rule-based control. For i = 1, 2, 3, define u¯ for 0 ≤ s ≤ I˜iΔ ˜Δ , (52) ρi (s) = Iu¯i , for s > I˜Δ i

s

and

wi =

−I˜i , 0,

for − I˜iΔ ≤ I˜i ≤ I˜iΔ otherwise.

Fused magnesium furnace.

Fig. 6. Comparison of the current trajectories with manual control and rule-based control.

10 h. Siemens S7-300 control system is employed for low-level loop control. (53)

Then, the control law (37) proposed in this paper can be implemented as ⎧ u, if I˜i > I˜iΔ ; ⎪ ⎨ −¯ ui = κi (I˜i + wi ) = (54) 0, if − I˜iΔ ≤ I˜i ≤ I˜iΔ ⎪ ⎩ Δ ˜ ˜ u ¯, if Ii < I . i

VI. INDUSTRIAL APPLICATION A. Implementation of the Control System The proposed method has been applied to a real fused magnesia factory located in Liaoning Province of China. This factory has ten fused magnesium furnaces, and annually produces around 100 000 tons of fused magnesia. Fig. 5 shows the scene of a FMF during smelting process. The main parameters of the manufacturing process are as follows: the electrode diameter is 350 mm, the rated voltage is 110 V, the shell diameter is 3.5 m, and the rated smelting time is

B. Operational Performance Analysis Before the automation system proposed in this paper is implemented, the whole smelting process is manually controlled based on human experience. The rule-based controller proposed in this paper is applied to the smelting process with parameters I˜iΔ = 1500 A and u ¯ = 5 Hz. Fig. 6 shows the current trajectories with setpoint value being 13 500 A. From Fig. 6, it can be observed that the three-phase currents can only be restricted to within a large neighborhood of the setpoint value [7500 A, 19 500 A]. Specifically, during the time period of 22:06–22:18, the maximum tracking error of the currents is up to 6717 A, which significantly exceeds the acceptable range of the tracking error. It can be easily checked that, the proposed rule-based control algorithm significantly reduces the range of the currents. A comparison between manual control and the proposed rulebased control is also made by using control error probability distributions (EPD). The EPDs of manual control and rule-based control are shown in Fig. 7. The current control error with manual control is within the range of [−1500 A, 1500 A] for

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REFERENCES

Fig. 7. Probability distributions of the control errors with manual control and rule-based control. TABLE I COMPARISON OF THE CURRENT CONTROL PERFORMANCES

Manual control Tracking control Decrease

IEA

MSE

1 721 875 1 156 813 32.8%

2 323 721 1 454 951 37.4%

62.5% of time, and the current control error with rule-based control is within the range of [−1500 A, 1500 A] for 83.1% of time. In addition, the advantage of the proposed rule-based control algorithm is also verified by using the integration of absolute error index and mean squared error (MSE) [26], which are listed in Table I. Since 2014, the proposed current control system has been safely and reliably controlling ten furnaces, and has shown significantly improved performance in application. VII. CONCLUSION Based on the recent development of nonlinear control theory, this paper proposed a new design of the stable control rules for fused magnesium furnaces. By fully taking into account the nonlinearity, uncertainty, and couplings in FMFs, a nonlinear multivariable dynamic model was developed and verified with experimental data. Then, a class of stable control rules were developed. By considering the FMF as an interconnectoin of three subsystems, each of which corresponds to one electrode, this paper first proved the stability of each controlled subsystem and then analyzed the stability of the closed-loop multivariable system by applying the nonlinear small-gain theorem. The robustness of the design with respect to model uncertainties and external disturbances was also analyzed in the nonlinear smallgain framework. Interestingly, it is shown that the proposed control algorithm can be readily implemented as rule-based control, and practically applied for current control of FMFs. An industrial application verifies the validity of the design in solving the practical engineering problem. ACKNOWLEDGMENT The authors would like to thank the anonymous AE and reviewers for their invaluable comments for the improvement of the paper.

[1] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 2002. [2] G. Liu, “Analysis of work about smelting furnace making MgO,” Ind. Heating, vol. 32, pp. 41–43, 2003. [3] G. Liu and Y. Liu, “Analysis of couples in the three-phase current about smelting furnace making MgO,” Ind. Heating, vol. 33, pp. 55–58, 2004. [4] Y. P. Tong, X. Zhang, and H. G. Zhang, “Analysis of optimal operation about purifying magnesium oxide three-phase ac electric smelting furnace,” Control Eng. China, vol. 44, pp. 205–211, 2007. [5] D. Bin, Z. Li, Y. J. Wu, J. Feng, and T. Y. Chai, “The fuzzy control research on electrodes of electrical-fused magnesia furnace,” in Proc. Chin. Control Decision Conf., 2008, pp. 216–220. [6] Z. W. Wu, Y. J. Wu, T. Y. Chai, and L. Zhang, “Intelligent control system of fused magnesia production via rule-based reasoning,” J. Northeastern Univ. (Natural Sci.), vol. 30, pp. 1526–1529, 2009. [7] Y. J. Wu, L. Zhang, H. Yue, and T. Y. Chai, “The intelligent optimal control based on CBR for the fused magnesia production,” J. Chem. Ind. Eng., vol. 59, pp. 1686–1690, 2008. [8] S. Srdic, M. N. Nedeljkovic, S. N. Vukosavic, and Z. Radakovic, “Fast and robust predictive current controller for flicker reduction in DC arc furnaces,” IEEE Trans. Ind. Electron., vol. 63, no. 7, pp. 4628–4640, Jul. 2016. [9] P. Ladoux, G. Postiglione, H. Foch, and J. Nuns, “A comparative study of AC/DC converters for high-power DC arc furnace,” IEEE Trans. Ind. Electron., vol. 52, no. 3, pp. 747–757, Jun. 2005. [10] Z. W. Wu, Y. J. Wu, T. Y. Chai, and J. Sun, “Data-driven abnormal condition identification and self-healing control system for fused magnesium furnace,” IEEE Trans. Ind. Electron., vol. 62, no. 3, pp. 1703–1715, Mar. 2015. [11] R. D. Zhang and J. L. Tao, “Data-driven modeling using improved multiobjective optimization based neural network for coke furnace system,” IEEE Trans. Ind. Electron., vol. 64, no. 4, pp. 3147–3155, Apr. 2017. [12] H. P. Ren and X. Guo, “Robust adaptive control of a CACZVS three-phase PFC converter for power supply of silicon growth furnace,” IEEE Trans. Ind. Electron., vol. 63, no. 2, pp. 903–912, Feb. 2016. [13] A. K. Fard, A. Khosravi, and S. Nahavandi, “Reactive power compensation in electric arc furnaces using prediction intervals,” IEEE Trans. Ind. Electron., vol. 64, no. 7, pp. 5295–5304, Jul. 2017. [14] J. Ling and C. H. Gao, “Binary coding SVMs for the multiclass problem of blast furnace system,” IEEE Trans. Ind. Electron., vol. 60, no. 9, pp. 3846– 3856, Sep. 2013. [15] E. U. Logoglu, O. Salor, and M. Ermis, “Online characterization of interharmonics and harmonics of AC electric arc furnaces by multiple synchronous reference frame analysis,” IEEE Trans. Ind. Appl., vol. 52, no. 3, pp. 2673–2683, May/Jun. 2016. [16] E. A. C. Plata, A. J. U. Farfan, and O. J. S. Marin, “Electric arc furnace model in distribution systems,” IEEE Trans. Ind. Appl., vol. 51, no. 5, pp. 4313–4320, Sep./Oct. 2015. [17] G. W. Chang, M. F. Shih, Y. Y. Chen, and Y. J. Liang, “A hybrid wavelet transform and neural-network-based approach for modelling dynamic voltage-current characteristics of electric arc furnace,” IEEE Trans. Power Del., vol. 29, no. 2, pp. 815–824, Apr. 2014. [18] E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Autom. Control, vol. 34, no. 4, pp. 435–443, Apr. 1989. [19] E. D. Sontag, “Input to state stability: Basic concepts and results,” in Nonlinear and Optimal Control Theory, P. Nistri and G. Stefani, Eds. Berlin, Germany: Springer-Verlag, 2007, pp. 163–220. [20] Z. P. Jiang, A. R. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” Math. Control, Signals Syst., vol. 7, pp. 95–120, 1994. [21] Z. P. Jiang and Y. Wang, “A generalization of the nonlinear small-gain theorem for large-scale complex systems,” in Proc. 7th World Congr. Intell. Control Autom., 2008, pp. 1188–1193. [22] T. Liu, D. J. Hill, and Z. P. Jiang, “Lyapunov formulation of ISS cyclicsmall-gain in continuous-time dynamical networks,” Automatica, vol. 47, pp. 2088–2093, 2011. [23] T. Liu, Z. P. Jiang, and D. J. Hill, Nonlinear Control of Dynamic Networks. Boca Raton, FL, USA: CRC, 2014. [24] R. A. Freeman and P. V. Kokotovi´c, Robust Nonlinear Control Design: State-space and Lyapunov Techniques. Boston, MA, USA: Birkh¨auser, 1996. ˚ om and B. Wittenmark, Computer-Controlled Systems: Theory [25] K. J. Astr¨ and Design, 3rd ed. Englewood Cliffs, NJ, USA: Prentice Hall, 1996. [26] T. Hagglund, “A control-loop performance monitor,” Control Eng. Practice, vol. 3, pp. 1543–1551, 1995.


Zhiwei Wu received the B.S. degree in electronic and information engineering from Dalian Nationalities University, DaLian, China, in 2004; the M.S. degree in control theory and engineering from Shenyang University of Chemical Technology, Shenyang, China, in 2007; and the Ph.D. degree in control theory and engineering from Northeastern University, Shenyang, in 2015. He is currently a Lecturer with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang. His current research interests include operational control for complex industry process and industrial embedded control systems. Tengfei Liu (M’12) received the B.E. degree in automation and the M.E. degree in control theory and control engineering from South China University of Technology, Guangzhou, China, in 2005 and 2007, respectively, and the Ph.D. degree in engineering from the Australian National University, Canberra, ACT, Australia, in 2011. He was a Visiting Assistant Professor at Polytechnic Institute of New York University (now Tandon School of Engineering at New York University), New York, NY, USA, from 2011 to 2013. Since 2014, he has been a Professor in Northeastern University, Shenyan, China. His research interests include stability theory, robust nonlinear control, quantized control, distributed control and their applications in mechanical systems, power systems, and transportation systems. Dr. Liu received the “Guan Zhao-Zhi” Best Paper Award at the 2011 Chinese Control Conference. Zhong-Ping Jiang (M’94–SM’02–F’08) received the B.Sc. degree in mathematics from the University of Wuhan, Wuhan, China, in 1988, the M.Sc. degree in statistics from the University of Paris XI, Orsay, France, in 1989; and the Ph.D. degree in automatic control and mathematics from the Ecole des Mines de Paris (now called ParisTech-Mines), Paris, France, in 1993, under the direction of Prof. L. Praly. He is currently a Professor of Electrical and Computer Engineering at the Tandon School of Engineering, New York University, New York, NY, USA. He is a coauthor of the books Stability and Stabilization of Nonlinear Systems (with Dr. I. Karafyllis, Springer, 2011), Nonlinear Control of Dynamic Networks (with Drs. T. Liu and D.J. Hill, Taylor & Francis, 2014), and Robust Adaptive Dynamic Programming (with Y. Jiang, Wiley-IEEE Press, 2017). He also is the coauthor of 14 book chapters, 195 published/accepted journal papers, and numerous conference papers. His work has received more than 17 400 citations, with an h-index of 67, from Google Scholar. His main research interests include stability theory, robust/adaptive/distributed nonlinear control, adaptive dynamic programming and their applications to information, and mechanical and biological systems. Dr. Jiang is a Deputy Co-Editor-in-Chief of the Journal of Control and Decision, a Senior Editor for the IEEE CONTROL SYSTEMS LETTERS, and an Editor for the International Journal of Robust and Nonlinear Control and has served as an Associate Editor for several journals, including Mathematics of Control, Signals and Systems, Systems and Control Letters, the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the European Journal of Control, and Science China: Information Sciences. Dr. Jiang received the prestigious Queen Elizabeth II Fellowship Award from the Australian Research Council in 1998, the CAREER Award from the U.S. National Science Foundation in 2001, the JSPS Invitation Fellowship from the Japan Society for the Promotion of Science in 2005, the Distinguished Overseas Chinese Scholar Award from the NSF of China in 2007, and the Chair Professorship by the Ministry of Education of China in 2009. His recent awards include the Best Theory Paper Award (with Y. Wang) at the 2008 WCICA and the Guan Zhao Zhi Best Paper Award (with T. Liu and D. Hill) at the 2011 CCC, the Shimemura Young Author Prize (with his student Yu Jiang) at the 2013 Asian Control Conference in Istanbul, Turkey, and the Steve and Rosalind Hsia Best Biomedical Paper Award at the 2016 World Congress on Intelligent Control and Automation in Guilin, China. He is a Fellow of the IFAC.

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Tianyou Chai (M’90–SM’97–F’08) received the Ph.D. degree in control theory and engineering in 1985 from Northeastern University, Shenyang, China. In 1988, he has been a Professor with Northeastern University. He is the Founder and the Director of the Center of Automation, which became a National Engineering and Technology Research Center and a State Key Laboratory. He is the Director of the Department of Information Science, National Natural Science Foundation of China. He has published 180 peer reviewed international journal papers. His current research interests include modeling, control, optimization, and integrated automation of complex industrial processes. Prof. Chai is a member of the Chinese Academy of Engineering and a Fellow of IFAC. His paper titled “Hybrid intelligent control for optimal operation of shaft furnace roasting process” was selected as one of three best papers for the Control Engineering Practice Paper Prize for 2011–2013. He has developed control technologies with applications to various industrial processes. For his contributions, he has won four prestigious awards from the National Science and Technology Progress and the National Technological Innovation and the 2007 Industry Award for Excellence in Transitional Control Research from the IEEE Multipleconference on Systems and Control.

Lina Zhang received the B.E. degree in geographic information systems from Inner Mongolia Normal University, Hohhot, China, in 2006 and the M.E. degree in environmental science from Shandong Normal University, Jinan, China, in 2009. He is currently an Enginneer with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China. Her current research interests include the design and implementation of industrial embedded control systems.

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