压电执行器动态迟滞建模与LQG最优控制器设计
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摘要
为提高空间天文望远镜稳像系统中压电快摆镜(Fast Steering Mirror,FSM)的动态性能,对压电执行器(Piezoelectric Actuator,PZT)动态迟滞补偿和控制进行研究。鉴于基于广义Play算子Prandtl-Ishlinskii(PI)模型的求逆复杂性和迟滞曲线的非对称性,构造一种基于广义Stop算子PI逆模型来补偿压电执行器迟滞非线性。采用Hammerstein模型对压电执行器动态迟滞特性进行建模,以广义PI模型和自回归遍历模型(Auto-regressive Exogenous Model,ARX)分别表征Hammerstein迟滞模型中的静态非线性和率相关性,并针对迟滞率相关模型不确定性问题,提出一种前馈补偿和线性二次型Gauss最优控制算法(Linear Quadratic Gaussian,LQG)相结合的复合控制策略。利用自适应差分进化算法(Adaptive Differential Evolution algorithm,ADE)辨识和整定模型及控制器参数。实验结果表明:该动态迟滞模型能够有效描述1～100Hz频率范围内压电执行器迟滞曲线,拟合均方根误差为0.077 1μm(@1 Hz)～0.512 3μm(@100Hz),相对误差为0.31%(@1Hz)～2.09%(@100Hz);实时跟踪幅值为24.5μm的变频目标位移,LQG控制算法的跟踪精度相比于直接前馈控制和PID控制分别提高48.6%和27.02%。
To improve the dynamic performance of a piezoelectric fast steering mirror in the spacetelescope image-stabilization system,the dynamic hysteresis compensation and control of a piezoelectric actuator are investigated.According to the inversion complexity of the PI model based on the generalized Play operator and the asymmetry of hysteresis curves,a PI inverse model based on thegeneralized Stop operator is constructed to compensate the hysteresis nonlinearity.The Hammerstein model is applied to model the dynamic hysteresis of the piezoelectric actuator and to describe the static nonlinearity and rate-dependent properties of the Hammerstein hysteresis model using the generalized PI and auto-regressive exogenous models,respectively.A compound counter strategy that combines the feedforward compensation and linear quadratic Gauss(LQG)optimal control algorithm is proposed to solve the hysteresis rate dependent model uncertainty.The adaptive differential evolution algorithm is used to identify the model parameters and tune the controller parameters.The test results show that the dynamic hysteresis model can effectively describe the hysteresis curve of the piezoelectric actuator in the frequency range of 1—100 Hz,fitting tracking root mean square errors from 0.077 1μm(at 1 Hz)to 0.512 3μm(at 100 Hz),and relative errors from 0.003 1(at 1 Hz)to0.020 9(at 100 Hz).The tracking accuracy of the LQG control algorithm increases by 48.6% and27.02%,respectively,compared with the direct feedforward and PID controls,in the real-time tracking of the variable-frequency target displacement with an amplitude of 24.5μm.
To improve the dynamic performance of a piezoelectric fast steering mirror in the spacetelescope image-stabilization system,the dynamic hysteresis compensation and control of a piezoelectric actuator are investigated.According to the inversion complexity of the PI model based on the generalized Play operator and the asymmetry of hysteresis curves,a PI inverse model based on thegeneralized Stop operator is constructed to compensate the hysteresis nonlinearity.The Hammerstein model is applied to model the dynamic hysteresis of the piezoelectric actuator and to describe the static nonlinearity and rate-dependent properties of the Hammerstein hysteresis model using the generalized PI and auto-regressive exogenous models,respectively.A compound counter strategy that combines the feedforward compensation and linear quadratic Gauss(LQG)optimal control algorithm is proposed to solve the hysteresis rate dependent model uncertainty.The adaptive differential evolution algorithm is used to identify the model parameters and tune the controller parameters.The test results show that the dynamic hysteresis model can effectively describe the hysteresis curve of the piezoelectric actuator in the frequency range of 1—100 Hz,fitting tracking root mean square errors from 0.077 1μm(at 1 Hz)to 0.512 3μm(at 100 Hz),and relative errors from 0.003 1(at 1 Hz)to0.020 9(at 100 Hz).The tracking accuracy of the LQG control algorithm increases by 48.6% and27.02%,respectively,compared with the direct feedforward and PID controls,in the real-time tracking of the variable-frequency target displacement with an amplitude of 24.5μm.
引文
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[15]张泉,尹达一,李清灵.高分辨率压电陶瓷微位移检测电路设计与实现[J].压电与声光,2017,39(6):903-906.ZHANG Q,YIN D Y,LI Q L.The design and implementation of high-resolution micrometric displacement signal of Piezoelectric Ceramic Drive[J].Piezoelectrics&Acoustooptics,2017,39(6):903-906.(in Chinese)
[16]张雅楠.动态系统的可逆性和逆建模方法的研究[D].沈阳:东北大学,2012.ZHANG Y N.Research on Invertibility and Inverse Modeling of Dynamical Systems[D].Shenyang:Northeastern University,2012.(in Chinese)
[2]CHEN W,CHEN S,WU X,et al..A new twodimensional fast steering mirror based on piezoelectric actuators[C].2014 4th IEEE International Conference on Information Science and Technology,2014:308-311.
[3]WANG G,BAI F.Robust tracking control of piezoelectric fast steering mirror with hysteresis and disturbances correction[C].2015 34th Chinese Control Conference,2015:389-394.
[4]王贞艳,张臻,等.压电作动器的动态迟滞建模与H∞鲁棒控制[J].控制理论与应用,2014,31(1):35-41.WANG ZH Y,ZHANG ZH,et al..Dynamic hysteresis modeling and H-infinity robust control of piezoelectric actuators[J].Control Theory and Applications,2014,31(1):35-41.(in Chinese)
[5]胡亮亮,米凤文,金伟其,等.基于PI逆模型的快速微摆反射镜的开环控制[J].红外与激光工程,2017,1(8):135-141.HU L L,MI F W,JIN W Q,et al..Open loop control of fast steering mirror based on PI inverse model[J].Infrared and Laser Engineering,2017,1(8):135-141.(in Chinese)
[6]方楚,郭劲,徐新行,等.压电陶瓷迟滞非线性前馈补偿器[J].光学精密工程,2016,24(9):2217-2223.FANG CH,GUO J,XU X X,et al..Compensating controller for hysteresis nonlinerity of piezoelectric ceramics[J].Opt.Precision Eng.,2016,24(9):2217-2223.(in Chinese)
[7]于志亮,王岩,曹开锐,等.压电陶瓷执行器迟滞补偿及复合控制[J].光学精密工程,2017,25(8):2113-2120.YU ZH L,WANG Y,CAO K R,et al..Hysteresis compensation and composite control for Piezoelectric actuator[J].Opt.Precision Eng.,2017,25(8):2113-2120.(in Chinese)
[8]田艳兵,王涛,王美玲.基于广义PI逆模型的超精密定位平台复合控制[J].机械工程学报,2015,51(2):198-206.TIAN Y B,WANG T,WANG M L,et al..Compounding control of ultra-precision positioning stage based on inverse generalized pi model[J].Journal of Mechanical Engineering,2015,51(2):198-206.(in Chinese)
[9]陈远晟.压电驱动器的迟滞非线性建模与控制[D].南京:南京航空航天大学,2013.CHEN Y SH.Hysteresis Modeling and Nonlinear Control of Piezoelectric Actuators[D].Nanjing:Nanjing University of Aeronautics and Astronautics,2013.(in Chinese)
[10]王钰锋,郭咏新,毛剑琴,等.压电作动器的率相关迟滞建模与跟踪控制[J].光学精密工程,2014,22(3):616-625.WANG Y F,GUO Y X,MAO J Q,et al..Ratedependent modeling and tracking control of piezoelectric actuators[J].Opt.Precision Eng.,2014,22(3):616-625.(in Chinese)
[11]ZHANG SH;WANG D q,ZHANG C H.A novel modelling method of hammerstein systems with a time-delay block[C].Chinese Automation Congress(CAC),2017:3921-3925.
[12]郭咏新,张臻,毛剑琴,等.超磁致伸缩作动器的率相关Hammerstein模型与H∞鲁棒跟踪控制[J].自动化学报,2014,40(2):197-207.GUO Y X,ZANG ZH,MAO J Q,et al..Ratedependent hammerstein model and H∞robust tracking control of giant magnetostrictive actuators[J].Acta Automatica Sinica,2014,40(2):197-207.(in Chinese)
[13]薛定宇.控制系统计算机辅助设计:MATLAB语言及应用(第3版)[M].北京:清华大学出版社,2012.XUE D Y.Computer Aided Control Systems Design:Using MATLAB Language(Third Edition)[M].Beijing:TsingHua University Press,2012.(in Chinese)
[14]戴小英.差分演化算法及其改进算法集成界面的实现[D].武汉:华中科技大学,2013.DAI X Y.The integrated interface of differential evolution algorithm and its modified algorithms[D].Wuhan:Huazhong University of Science&Technology,2013.(in Chinese)
[15]张泉,尹达一,李清灵.高分辨率压电陶瓷微位移检测电路设计与实现[J].压电与声光,2017,39(6):903-906.ZHANG Q,YIN D Y,LI Q L.The design and implementation of high-resolution micrometric displacement signal of Piezoelectric Ceramic Drive[J].Piezoelectrics&Acoustooptics,2017,39(6):903-906.(in Chinese)
[16]张雅楠.动态系统的可逆性和逆建模方法的研究[D].沈阳:东北大学,2012.ZHANG Y N.Research on Invertibility and Inverse Modeling of Dynamical Systems[D].Shenyang:Northeastern University,2012.(in Chinese)