Saturation Constraint One-step Optimal Control of Electrode Current for the Fused Magnesia Smelting Process
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摘要: 电熔镁砂熔炼过程通过电极电流熔化物料, 采用埋弧方式, 边熔化边加料, 其被控对象是以转动方向与频率为输入, 以电极电流为输出的三相电机. 本文通过引入中间变量并转化控制目标, 将电熔镁砂熔炼过程三相电极电流的复杂非线性控制问题简化为线性控制问题, 提出了一种简化的电极电流饱和约束一步最优控制方法, 并通过引入拉格朗日乘子向量和松弛向量验证了该方法的最优性. 理论分析和仿真对比实验结果表明本文所提简化控制方法的有效性和优越性. 此外, 当考虑电熔镁砂熔炼过程中存在的不可测外部干扰时, 在上述简化的电极电流饱和约束算法的基础上设计了高阶干扰观测器, 理论分析和仿真结果验证了具有高阶干扰观测器的简化算法的优越性.Abstract: The fused magnesia smelting process melts the material through the electrode current, and uses the submerged arc method to feed the material while melting. The controlled object is a three-phase motor with the rotation direction and frequency as the inputs and the electrode current as the outputs. In this paper, by introducing an intermediate variable and transforming the control objective, the complex nonlinear control problem of the three-phase electrode current for the fused magnesia smelting process is simplified into a linear control problem, and a saturation constraint one-step optimal controller is proposed. The optimality of the method is verified by introducing Lagrangian multiplier vectors and relaxation vectors. Theoretical analysis and simulation results show the effectiveness and superiority of the simplified control method. Furthermore, when unmeasurable external disturbances are considered in the fused magnesia smelting process, a high-order disturbance observer is designed on the basis of the simplified electrode current saturation constraint algorithm. Theoretical analysis and simulation results verify the superiority of the simplified algorithm with the high-order disturbance observer.
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图 2 饱和约束一步最优控制结构图
Fig. 2 Structure diagram of one-step optimal control with saturation constraint
图 4 采用本文控制方法时A相电极电流
$y_1$ Fig. 4 A-phase electrode current
$y_1$ using the control method in this paper
图 8 采用本文控制方法时A相电极电流误差概率分布
Fig. 8 Error probability distribution of A-phase electrode current using the control method in this paper
图 9 加入不可测干扰时A相电极电流
$y_1$ Fig. 9 A-phase electrode current
$y_1$ when unmeasurable disturbance is introduced
图 10 加入不可测干扰时控制器输出
$u_1$ Fig. 10 Controller output
$u_1$ when unmeasurable disturbance is introduced
图 11 采用高阶干扰观测器控制时A相电极电流
$y_1$ Fig. 11 A-phase electrode current
$y_1$ using the proposed high-order disturbance observer based controller
图 12 采用高阶干扰观测器时控制器输出
$u_1$ Fig. 12 Controller output
$u_1$ using the proposed high-order disturbance observer based controller
表 1 电极电流动态模型中参数的符号及物理意义
Table 1 Symbols and meanings of parameters in dynamic model of electrode current
符号 物理意义 $f_{1}(\cdot)$ 随原料颗粒长度和杂质成分变化的埋弧电阻率 $f_{2}(\cdot)$ 随原料颗粒长度和杂质成分变化的熔池电阻率 $r_{\rm {iarc}}$ 埋弧等效弧柱半径 $h_{\rm {ipool }}(\cdot)$ 随原料颗粒长度、杂质和电极电流变化的熔池高度 $U$ 熔炼电压 $p$ 电极极对数 $r_{d}$ 升降机构等效齿轮半径 $s$ 转差率 
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表 2 采用文献[1]控制方法、文献[21]控制方法和本文控制方法时A相电极电流
$y_1$ 的性能评价Table 2 Performance evaluating of A-phase electrode current
$y_1$ using the control method proposed in this paper and described in [1] and [21]
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表 3 采用文献[24]控制方法和本文控制方法时A相电极电流
$y_1$ 的性能评价Table 3 Performance evaluating of A-phase electrode current
$y_1$ using the control method proposed in this paper and described in [24]${\rm {MSE}}$ ${\rm {IAE}}$ 采用本文第3节控制方法 $0.4970 \times 10^{6}$ $0.0854 \times 10^{6}$ 文献[24]的控制方法 $0.5906 \times 10^{6}$ $0.2879 \times 10^{6}$ 本文控制方法 $0.2951 \times 10^{6}$ $0.0784 \times 10^{6}$ 与第3节方法相比降低 $40.62 \;{\text{%} }$ $8.20\; {\text{%} }$ 与文献[24]方法相比降低 $50.03\; {\text{%} }$ $72.77\; {\text{%} }$
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