Design and Digital Implementation of Spatial Repetitive Control for Fast Tool Servo System
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摘要: 在非圆零件车削过程中, 快速刀具伺服(Fast tool servo, FTS)的运动精度直接影响零件的加工质量. 主轴变速加工使得FTS的参考目标信号周期时变而不确定, 这对实现其渐近跟踪提出了极大的挑战. 本文利用FTS的位置域周期特性, 提出一种基于位置域重复控制和时域速度反馈镇定的FTS系统复合控制设计方法, 并给出位置域改进型重复控制器(Spatial modified repetitive controller, SMRC)的数字实现算法, 实现对时变周期参考目标信号的高精度跟踪. 首先, 建立包含位置相关时变周期参考目标信号内模的SMRC, 并引入位置域相位超前装置对镇定补偿器引起的相位滞后进行补偿, 在此基础上构建复合控制律. 然后应用小增益定理和算子理论, 推导出闭环系统的稳定性条件, 在保持系统采样频率不变的条件下, 应用插值法建立SMRC的数字实现算法, 确保位置域重复控制和时域镇定控制器的同步执行. 最后, 通过仿真验证所设计的FTS控制系统具有满意的时变周期跟踪性能和鲁棒性, 并通过与其他位置域重复控制方法的比较, 说明所提方法同时具有更好的暂态和稳态性能.Abstract: The motion accuracy of the fast tool servo (FTS) directly affects the machining quality of the non-circular parts. In variable spindle speed machining, the period of the reference signal of the FTS is time-varying and uncertain, which poses a great challenge to achieve its asymptotic tracking. In this paper, we propose a composite control design method based on spatial repetitive control and temporal velocity feedback stabilization for an FTS system, and present a digital implementation algorithm of the spatial modified-repetitive-controller (SMRC) to achieve high-precision tracking of time-varying periodic reference signals. First, an SMRC with the internal mode of the position-dependent time-varying periodic reference input signal is constructed, in which a phase-lead alignment segment is used to compensate for the phase lag caused by the temporal stabilization controller. And based on this, a composite control law is established. Then, the stability criteria of the control system are derived using the small gain theorem and operator theory. A digital implementation algorithm of the SMRC is developed to ensure the synchronous execution of the SMRC and the temporal stabilization controller through a regular fixed time sampling. Finally, simulation results show that the designed FTS control system has satisfactory time-varying-period tracking performance and robustness. A comparison with other spatial repetitive control methods verifies that the proposed method achieves better performance in both transient and steady state aspects.
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图 3 位置域基本重复控制器和改进型重复控制器的零极点分布图和幅值特性曲线
Fig. 3 Zero-pole map and Bode plots of spatial basic repetitive controller and SMRC
图 5 时域纯时滞环节的数字实现
Fig. 5 Digital implementation of the pure time-delay link in the time domain
图 6 位置域时滞单元输入输出曲线
Fig. 6 Input and output curves of the delay element in position domain
图 7 位置相关周期信号等时采样示意图
Fig. 7 Diagram of isochronous sampling of a position-dependent periodic signal
图 8 位置域改进型重复控制器数字实现算法流程图
Fig. 8 Flowchart of the digital implementation algorithm for SMRC
图 12 不同相位补偿因子的跟踪误差对比
Fig. 12 Tracking errors with different phase compensation factors
图 14 无速度反馈时的$Q_{{{m}}}(s)$和$1+G^{\prime}(s)$伯德图
Fig. 14 Bode plots of $Q_{{{m}}}(s)$ and $1+G^{\prime}(s)$ without velocity feedback
图 15 本文方法与传统定周期时域重复控制方法的对比
Fig. 15 Comparison of our method with the conventional fixed-period time-domain repetitive control method
表 1 音圈式直线电机相关参数
Table 1 Parameters of the voice coil type linear motor
参数 符号 单位 数值 弹簧刚度 $ K $ ${\rm{N/m }} $ 4 980 阻尼系数 $ C $ ${\rm{N\cdot s\cdot m^{-1}}}$ 14.51 动子质量 $ M $ ${\rm{Kg }} $ 0.32 电机力常数 $ K_{m} $ ${\rm{N/A }} $ 12.325 放大器增益 $ K_a $ ${\rm{A/v}} $ 1.6 
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表 2 性能指标对比
Table 2 Comparison of performance indices
控制方法 $\max|e(t)|_{0<t\leq 20}$ $e_{pp}\;(0 < t\leq 20)$ $\max|e(t)|_{t>20} $ $e_{pp}\;(t > 20)$ CRC $8.744\times10^{-2} $ $17.393\times10^{-2} $ $9.707\times10^{-3} $ $1.580\times10^{-2} $ Liu等[32] $3.246\times10^{-2} $ $5.993\times10^{-2} $ $1.006\times10^{-3} $ $1.992\times10^{-3} $ Yao等[33] $2.315\times10^{-2} $ $3.684\times10^{-2} $ $6.737\times10^{-3} $ $1.347\times10^{-2} $ 本文方法 2.315 × 10−2 3.665 × 10−2 6.759 × 10−4 1.334 × 10−3
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