压电执行器动态迟滞建模与LQG最优控制器设计

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%。

Dynamic hysteresis modeling and LQG optimal controller design of piezoelectric actuators

To improve the dynamic performance of a piezoelectric fast steering mirror in the space-telescope 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 the generalized 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 1 μm (at 1 Hz) to 0.512 3 μm (at 100 Hz), and relative errors from 0.003 1 (at 1 Hz) to 0.020 9 (at 100 Hz). The tracking accuracy of the LQG control algorithm increases by 48.6% and 27.02%, respectively, compared with the direct feedforward and PID controls, in the real-time tracking of the variable-frequency target displacement with an amplitude of 24.5 μm.