Coordination Control of Multiple Electrical Excited Synchronous Motors and Its Application in High-Power Metal-Rolling Systems

Abstract

This study focuses on the coordination control problem of multiple electrical excited synchronous motor systems. A robust coordination controller is designed on the basis of cross coupling and an interval matrix. The proposed control strategy can deal with load uncertainty. In addition, the proposed control strategy is applied to a high-power metal-rolling system. Simulation and experiment results demonstrate that the proposed control strategy achieves good dynamic and static performance. It also shows better coordination performance than traditional proportional-integral controllers.

I. INTRODUCTION

The use of coupling multi-motor systems (CMMSs) in industry, agriculture, and transportation is steadily increasing. Coupling multi-motor systems require good coordination to achieve good production quality, reduce contouring errors, and improve the operation security of systems. In recent decades, many coordination control strategies have been designed for multi-motor systems. The work in [1] developed a chaotic speed synchronization controller for multiple induction motors using stator flux regulation. In [2], a cross-coupled intelligent complementary sliding mode control for dual linear motor servo systems was proposed on the basis of a Takagi–Sugeno–Kang-type fuzzy neural network estimator. The work in [3] simplified the control structure by presenting an adjacent cross-coupling control architecture incorporating sliding mode control for multiple induction motors. In [4], precision motion control was achieved by combining individual axis iterative learning control (ILC) and cross-coupled ILC into a single control input. To track different desired trajectories, the work in [5] exploited a generalized synchronization controller for multi-axis motion systems by integrating cross-coupling technology into optimal control architecture. In [6], an optimal synchronization for high-precision motion system was designed by introducing coupling and synchronization factors into the synchronization error. Many other control strategies have been adopted in this field, and they include adaptive control [7]-[9], fuzzy control [10], [11], electronic line-shafting control [12], [13], robust control [14]-[19], neural network control [20], gain-scheduled control [21], and model-free control [22].

Compared with asynchronous induction motors, electrical excited synchronous motors (EESMs) achieve higher efficiency, power density, and so on. EESMs are applied in many fields, such as in mine-hoisting, metallurgy, metal-rolling, and marine propulsion, particularly in the field of high-power industrial drives. The present work explores the coordination control of multiple EESMs.

The dynamic model of driving devices is seldom or never considered in CMMSs when controllers are designed [2], [5], [12], [13], [16]–[21]. If the dynamic model of a driving device is fully considered, the dynamic performance of a system significantly improves. Therefore, a multiple externally excited synchronous motor system is studied in the current work. A dynamic model of an EESM is considered to improve the dynamic performance of the studied system.

When a parameter is uncertain or changes with time, an interval matrix is normally introduced [23], [24]. Confined by supply voltage and protection equipment, the current and speed of an EESM are limited to a rated range. Although many coordination control strategies have been developed, cross coupling remains widely adopted in coordination controller design [2]. Interval matrix and the cross-coupling concept are introduced to design the controller of a multiple EESM system.

This paper is organized as follows. The mathematical model of a multiple EESM system is presented in Section II. The coordination controller design is presented in Section III. The application of the proposed design in a high-power metal-rolling system is described in Section IV. Conclusions are given in Section V.

 

II. MATHEMATICAL MODEL

The following notations are used throughout the paper. I is the identity matrix with an appropriate dimension, and ei is the ith column of matrix I . 0ij denotes the zero matrix with i×j dimensions.

If matrices for all 1≤i≤n and 1≤j≤n , then can be defined. Assume A∈Rn×n and A∈[Am AM] ; then, A is an interval matrix.

Lemma 1 [29]. For a given interval matrix A∈[Am AM ] and A∈Rn×n , A can be rewritten as

where . The elements in matrices Am and AM consist of the lower bound and upper bound of the elements in matrix A , respectively. Each element in matrix H is nonnegative.

Lemma 2 [24]. For any given interval matrix A∈[Am AM ] , matrix , A can be written as , where |fij|≤hij.

Lemma 3 [30] For any scalar ε > 0 and real matrices X and Y with appropriate dimensions, the following inequality is established.

According to [31], the mathematical model of EESMs can be described as follows.

where Lsd and Lsq are the d- and q-axis self-inductions, respectively; Lmd and Lmq are the d- and q-axis inductances, respectively; Lf is the excitation winding self-induction; is the leakage coefficient; Rf is the excitation winding resistance; Rs is the stator resistance; np is the number of pole pairs; B is the frictional coefficient; J is the moment inertia; if is the excitation current; isd and isp are the d- and q-axis currents, respectively; ω is the rotor angular speed; uf is the excitation voltage; usd and usq are the d- and q-axis input voltages, respectively; and Tl is the load torque.

Remark 1. Damper winding is not considered in the aforementioned model. That is, damping winding is assumed to be constant in this work.

If model error and disturbance are considered, the ith subsystem (i.e., motor i ) is described as

where Lsdi and Lsqi are the d- and q-axis self-inductions of motor i , respectively; Lmdi and Lmqi are the d- and q-axis inductances of motor i , respectively; Lfi is the excitation winding self-induction of motor i ; is the leakage coefficient of motor i ; Rfi is the excitation winding resistance of motor i ; Rsi is the stator resistance of motor i ; npi is the number of pole pairs of motor i ; Bi is the frictional coefficient of motor i ; Ji is the moment inertia of motor i ; ifi is the excitation current of motor i ; isdi and isqi are the d- and q-axis currents of motor i , respectively; ωi is the rotor angular speed of motor i ; ufi is the excitation voltage of motor i ; usdi and usqi are the d- and q-axis input voltages of motor i , respectively; Tli is the load torque of motor i ; dfi , ddi , dqi , and dωi present the model error and/or disturbance; and subscript i = 1,⋯,n .

As the changes in disturbance and load are usually slow, the following condition is often satisfied, similar to that in [25], [26].

The current and speed of each EESM are constrained by supply power, rated parameter, protection equipment, and so on. Thus, they are restricted to a certain range or “rated” range. Hence, the following assumptions are reasonable.

The state variables and control inputs of subsystem i are chosen as

where is the desired speed of the ith subsystem and is a constant.

Tracking error is defined as

In coordination control, the desired speed of each motor may be different. may not equal when i≠j . Scalars should be introduced to describe the coordination error of the ith motor and jth motor. The coordination error between the ith motor and the jth motor can be defined as

The mathematical model of subsystem i is described as follows , according to (2) and (5).

where

According to (8), the model of a multiple three-level EESM system is expressed as

where

 

III. COORDINATION CONTROLLER DESIGN

A coordination controller should be designed to ensure that the system achieves a good coordination performance and that each motor can track its own desired speed. The control structure of the system is described in Fig. 1.

Fig. 1.Control structure of the system.

The auxiliary matrices are defined as

where,

Theorem 1. For system (9), each EESM can track its own desired speed and maintain a good coordination with other motors under controller U = KX if there exist a symmetric and positive-definite matrix P1∈R6n×6n , matrix K∈R3n×6n , and positive scalar λ such that the following BMI holds:

Proof:

As robust control law is designed as U = KX , (9) is rewritten as

Let

is continuously differentiable with regard to hence, the equation offers a unique solution under condition (4). Assume that X0 is the solution of equation Let

Then, Z = 0 when X = X0 . According to (11) and (12), the following equation is obtained.

where

A Lyapunov function is chosen as

According to Lemma 1, can be rewritten as

where

The following equality is obtained via (13), (14), and (15).

According to Lemma 3 and ΔTΔ≤I , inequation (16) is obtained.

If

the time derivative of the Lyapunov function Vc1 is negative. This result indicates that the system can be stabilized by the controller, i.e., each EESM can follow its desired speed, and the system remains coordinated. According to Schur complement theory, (17) can be rewritten as (10). This completes the proof of Theorem 1.

Remark 1. The proposed controller is U = KX , where k∈R3n×6n ; thus, matrix K can be expressed as

In this expression, kij is the feedback gain of motor j to motor i and kii is the self-feedback gain of motor i . As each motor must track its own desired speed, kii≠0 for i = 1,2,...n.

If kij=0 , then the signals of the state variables of motor j are not sent to motor i , in which case motor i is not influenced by motor j . If kij=0 when i≠j , each motor only tracks its own desired speed without coordinating with the other motors. If every motor tracks its own desired speed and considers its coordination with other motors, then kij≠0 for all i, j = 1,2,...,n . If motor i only considers the coordination with motor j , then let kij≠0 and kil=0 with l≠i, j .

Take three motors as an example. Motors 1 and 2 ,motors 2 and 3 should achieve good coordination performance. Thus, we should let k13 = 0 , k31 = 0 , k12 ≠ 0 , k21 ≠ 0 , k23 ≠ 0 , k32 ≠ 0 , and kii ≠ 0 with i = 1,2,3 .

If the cross feedback signals are enhanced in the proposed controller, the coordination performance consequently improves. As the controller design is based on the dynamic model of a drive motor and the current signal is used for cross feedback, the dynamic performance of the system is improved. A detailed description of the controller is presented in Fig. 2.

Fig. 2.Controller structure.

Remark 2. The model error and disturbance are considered in the system model, thus giving the proposed control strategy strong robustness to the environment disturbance and perturbation parameters. The load information of two motors is not required in the proposed controller, which can thus deal with load uncertainty.

Ten changeable elements make up Qi . According to Lemma 2, Qi can be expressed as

where

Equation (19) is obtained by combining and (11).

Theorem 2. For system (9), each EESM can track its desired speed and maintain coordination performance with other motors if there exist a symmetric and positive-definite matrix P2∈R5n×5n , matrix K =∈R3n×6n , and real scalar , such that the following BMI holds:

where

Proof: The Lyapunov function is chosen as

On the basis of (13), (19), and (21), the following equation is obtained.

The following inequality is obtained via Lemma 3 and

where

If M < 0 , then the time derivative of the Lyapunov function Vc2 is negative. This outcome indicates that each EESM can follow its desired speed and that the system can remain coordinated. M < 0 can be rewritten as (20) via Schur complement theory. Doing so completes the proof of Theorem 2.

 

IV. APPLICATION IN HIGH-POWER METAL-ROLLING SYSTEM

The proposed coordination controller is utilized for an aluminum-rolling line in a Chinese factory. Fig. 3 shows the structure of the metal-rolling system. This system comprises an inverter and an rectifier, both of which make up a typical three-level NPC topology. The transformer converts the 10 kV AC into 3 kV AC, and DC voltage is obtained via the PWM rectifier. The converter for field winding is used to control the field winding current of the EMSM. Two inverters arranged in parallel are used to control two 5 MW EESMs (Fig. 4). The parameters of the EESMs are shown in Table I. Each EESM can achieve up to 10 MW in a short time.

Fig. 3.Structure of metal-rolling system.

Fig. 4.Metal-rolling system with two EESMs.

TABLE IPARAMETERS OF EESM

EESMs 1 and 2 respectively drive the up roller and the down roller to realize metal rolling. In the working process, the loads of the two EESMs often change rapidly when the metal touches the roller. The loads of the two EESMs are also different for numerous reasons, one of which is the shape of the metal, which further results in different output torques between the two motors. To achieve high-precision metal-rolling, the speeds of the two EESMs must be the same, and the speed drop should be minimal. These goals are feasible if the proposed coordination controller is adopted.

The circuit of a three-level diode clamp inverter based on IGCT is shown in Fig. 5. The output voltage state is defined in Table II. Considering the motor parameters, we can obtain the following control parameters by solving the inequality (10) via MATLAB.

Fig. 5.Main circuit topology for three-level diode clamp inverter.

TABLE IISWITCH STATES FOR THREE-LEVEL IGCT

LAB soft

Next, a simulation and an experiment are conducted to verify the effectiveness of the proposed control algorithm.

A. Simulation

The simulation is conducted in MATLAB with varying loads ( Tl ) and parameters ( Lmd ) for the two motors.

The desired speed of the two motors is 27 r/m. The load is added to the system at 2 s and removed at 3.5 s.

A comparison between the proposed method and the traditional proportional–integral (PI) controller is conducted to further demonstrate the control performance of the proposed controller. The PI speed controller is used in the outer loop, whereas the PI current controller is used in the inner loop of each motor. The PI law can be presented as

where The control gains of all controllers shown below are elaborately selected to obtain optimal synchronization and tracking performance.

The control performance of a traditional PI controller is depicted in Fig. 6, which shows the speed error at maximal (0.36 r/m) when the load is changed. Fig. 7 shows the results achieved with the proposed controller. The maximal speed error is 0.11 r/m, which proves the effectiveness of the proposed method when the parameters and loads of the two motors are different (25). This result also indicates that the proposed controller exhibits a superior performance relative to the traditional PI controller. Note that the load information of two motors is not required in the proposed controller and that the proposed controller can deal with load uncertainty.

Fig. 6.Performance of traditional PI controller.

Fig. 7.Performance of proposed controller.

B. Experiment

The proposed coordination controller is then used in the metal-rolling system of Guangxi Liuzhou Yinhai Aluminium Co., Ltd., and the experimental data are recorded with a specific software program. Fig. 8 shows the results achieved with the proposed algorithm without load. The results indicate that regardless of whether the system is in steady state or dynamic state, the speed error between the motors is nearly zero.

Fig. 8.Start process in no-load condition.

Fig. 9 shows the results obtained with the proposed algorithm under the rolling condition. When the roller touches the metal, the maximum speed error between the motors is less than 0.7 rpm, and the error converges to nearly zero in a short time. This result demonstrates the good control performance of the proposed controller. Similarly, the load information of two motors is not required in the proposed controller, which can thus deal with load uncertainty. Fig. 10 shows the production of an aluminum strip and reveals the successful use of the proposed controller in the metal-rolling system.

Fig. 9.Process under the rolling condition.

Fig. 10.Aluminum strip produced with the designed system.

 

V. CONCLUSIONS

This paper presents a coordination control strategy for multiple EESM systems via cross coupling and an interval matrix. The proposed controller is successfully applied in a high-power aluminum rolling system. It can deal with load uncertainty, i.e., the load information is not required in the controller. The simulation and experimental results verify the good dynamical and static performance of the proposed control strategy and its superior coordination performance relative to traditional PI controllers.