静电悬浮系统离散滑模控制的研究
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摘要
静电悬浮是依靠可控的静电力将悬浮体无接触、无摩擦地支承起来，其显著特点是既适用于导体和非导体的悬浮，又适用于磁性体和非磁性体的悬浮，克服了磁悬浮技术仅适用于磁性体的局限性。静电陀螺是静电悬浮技术的重要应用，它是当今国际上公认的高精度自主式惯性导航仪表。由于静电陀螺悬浮系统存在系统参数变化和外部扰动的影响，因此悬浮控制系统的设计就成为静电陀螺的关键技术之一。通常的线性控制方法鲁棒性较差，难以满足高性能悬浮系统的指标要求，而离散滑模控制具有很强的鲁棒性，对系统参数变化和外部扰动具有很强的抑制能力，因而很适合用于设计静电陀螺悬浮控制系统。
本文将离散滑模控制理论应用于静电陀螺悬浮系统，根据陀螺转子静电力方程以及各环节特性(包括高压放大器、测量环节和数字控制器的滞后环节)，建立考虑系统参数变化的广义被控对象离散状态空间表达式。在离散域设计线性扰动观测器，进行前馈补偿，抵消一部分或全部扰动力，然后设计离散滑模观测器，使得状态误差在存在外部扰动和系统参数变化的情况下，可以在有限拍内收敛到零，提高观测器鲁棒性。在离散域设计积分型离散滑模控制器，首先采用极点配置法设计带有积分项的线性控制项，然后设计不连续控制项和带有边界层的扰动控制项。为了便于控制算法之间的比较，设计了静电悬浮系统的滞后—超前校正和线性状态反馈控制。
采用MATLAB SIMULINK实现以上控制算法，然后从时域、频域和鲁棒性等方面比较各种控制算法的性能。与滞后—超前校正相比，采用离散滑模控制时，阶跃响应的超调量降低40％，调整时间缩短60％；对于外部的振动和过载冲击，陀螺转子偏移量分别减小80％和87％：闭环带宽仍位于600Hz～800Hz之间，闭环谐振峰降低6.32dB，而且在高频段具有更大衰减率；刚度曲线在低频段提高了一个数量级，在中频段刚度曲线最低点的刚度值提高2.85倍；抑制系统参数变化和外部扰动的能力显著增强，充分体现了滑模控制的强鲁棒性。
采用DSP仿真系统进行电模拟实验，将控制算法编制成汇编语言，经调试通过后，在DSP仿真系统上运行，比较各种算法性能，进一步验证离散滑模控制的鲁棒性。
将滑模控制理论应用于静电陀螺悬浮控制系统可以极大地提高悬浮系统的鲁棒性，对于静电陀螺长时间稳定工作具有重要意义。
Electrostatic levitation in which electrostatic forces are applied to suspend levitators without contact and friction is superior to traditional electromagnetic levitation in that electrostatic levitation suspends not only electromagnetic levitators but also non-electromagnetic levitators such as conductors and nonconductors. Electrostatic Suspension Gyroscope(ESG) which is acknowledged as a highly precise active inertial navigation instrument is an important application of electrostatic levitation. Due to plant parameter variation and exterior disturbance, controller design is of vital importance to realize stable suspension. Commonly adopted linear control algorithm is inadequate to suppress these uncertainties while sliding mode control algorithm is robust enough to overcome this shortcoming, so discrete sliding mode control of electrostatic levitation system is presented in this dissertation in order to improve the robustness of the levitation system.
Discrete sliding mode theory has been applied to electrostatic levitation system in which general discrete state space expressions with parameter variation have been set up based on gyro rotor electrostatic force equation and other annuluses including high-voltage amplifier, position sensor and time delay. Discrete linear disturbance observer designed in discrete domain as a feed-forward compensation is applied to cancel part or all of exterior disturbance forces. Meanwhile, state variable errors in discrete sliding mode observer designed in discrete field tend to zeros in finite number of sampling periods with the existence of parameter variation and exterior disturbance. Discrete sliding mode controller designed in discrete domain consists of integral linear control component which is designed by pole placement method, discontinuous control component and disturbance control component with boundary layer. Meanwhile, continuous lag-lead control algorithm and discrete linear state feedback control algorithm a
re applied to electrostatic levitation system respectively in comparison with sliding mode control algorithm.
All the three control systems mentioned have been simulated respectively by using MATLAB SIMULINK in which dynamical simulation models of three control systems have been set up. In order to testify the strong robustness of sliding mode controller, the properties of three control systems in time domain, frequency domain are compared respectively in consideration of plant parameter variation and exterior disturbance force. Compared with lag-lead controller, sliding mode controller has more satisfactory performance in that its overshoot and settling time in step response have been reduced 40 percent and shortened 60 percent respectively, and the deviations of the rotor from the sphere center in the stimulation of exterior vibration and impulse have been reduced 80 percent and 87 percent respectively, and its resonant peak has been reduced 6.32dB and its damping factor has been increased at high frequency band while keeping the bandwidth still at the range between 600Hz and 800Hz, and its stiffness at lower freq
uency band(<10Hz) and the lowest stiffness at medium frequency band have been raised about 10 times and 2.85 times respectively. The simulation results show that sliding mode controller has strong robustness in suppressing parameter variation and exterior disturbance.
In order to further testify this result, electronic circuit experiment with DSP simulation environment has been carried on, in which three control algorithms have been transferred into instructions of assemble language, and then is simulated.
All in all, the robustness of electrostatic gyro levitation system has greatly been improved by applying sliding mode control theory to the system. It is of vital significance to secure the long-time and stable operation of ESG.
本文将离散滑模控制理论应用于静电陀螺悬浮系统，根据陀螺转子静电力方程以及各环节特性(包括高压放大器、测量环节和数字控制器的滞后环节)，建立考虑系统参数变化的广义被控对象离散状态空间表达式。在离散域设计线性扰动观测器，进行前馈补偿，抵消一部分或全部扰动力，然后设计离散滑模观测器，使得状态误差在存在外部扰动和系统参数变化的情况下，可以在有限拍内收敛到零，提高观测器鲁棒性。在离散域设计积分型离散滑模控制器，首先采用极点配置法设计带有积分项的线性控制项，然后设计不连续控制项和带有边界层的扰动控制项。为了便于控制算法之间的比较，设计了静电悬浮系统的滞后—超前校正和线性状态反馈控制。
采用MATLAB SIMULINK实现以上控制算法，然后从时域、频域和鲁棒性等方面比较各种控制算法的性能。与滞后—超前校正相比，采用离散滑模控制时，阶跃响应的超调量降低40％，调整时间缩短60％；对于外部的振动和过载冲击，陀螺转子偏移量分别减小80％和87％：闭环带宽仍位于600Hz～800Hz之间，闭环谐振峰降低6.32dB，而且在高频段具有更大衰减率；刚度曲线在低频段提高了一个数量级，在中频段刚度曲线最低点的刚度值提高2.85倍；抑制系统参数变化和外部扰动的能力显著增强，充分体现了滑模控制的强鲁棒性。
采用DSP仿真系统进行电模拟实验，将控制算法编制成汇编语言，经调试通过后，在DSP仿真系统上运行，比较各种算法性能，进一步验证离散滑模控制的鲁棒性。
将滑模控制理论应用于静电陀螺悬浮控制系统可以极大地提高悬浮系统的鲁棒性，对于静电陀螺长时间稳定工作具有重要意义。
Electrostatic levitation in which electrostatic forces are applied to suspend levitators without contact and friction is superior to traditional electromagnetic levitation in that electrostatic levitation suspends not only electromagnetic levitators but also non-electromagnetic levitators such as conductors and nonconductors. Electrostatic Suspension Gyroscope(ESG) which is acknowledged as a highly precise active inertial navigation instrument is an important application of electrostatic levitation. Due to plant parameter variation and exterior disturbance, controller design is of vital importance to realize stable suspension. Commonly adopted linear control algorithm is inadequate to suppress these uncertainties while sliding mode control algorithm is robust enough to overcome this shortcoming, so discrete sliding mode control of electrostatic levitation system is presented in this dissertation in order to improve the robustness of the levitation system.
Discrete sliding mode theory has been applied to electrostatic levitation system in which general discrete state space expressions with parameter variation have been set up based on gyro rotor electrostatic force equation and other annuluses including high-voltage amplifier, position sensor and time delay. Discrete linear disturbance observer designed in discrete domain as a feed-forward compensation is applied to cancel part or all of exterior disturbance forces. Meanwhile, state variable errors in discrete sliding mode observer designed in discrete field tend to zeros in finite number of sampling periods with the existence of parameter variation and exterior disturbance. Discrete sliding mode controller designed in discrete domain consists of integral linear control component which is designed by pole placement method, discontinuous control component and disturbance control component with boundary layer. Meanwhile, continuous lag-lead control algorithm and discrete linear state feedback control algorithm a
re applied to electrostatic levitation system respectively in comparison with sliding mode control algorithm.
All the three control systems mentioned have been simulated respectively by using MATLAB SIMULINK in which dynamical simulation models of three control systems have been set up. In order to testify the strong robustness of sliding mode controller, the properties of three control systems in time domain, frequency domain are compared respectively in consideration of plant parameter variation and exterior disturbance force. Compared with lag-lead controller, sliding mode controller has more satisfactory performance in that its overshoot and settling time in step response have been reduced 40 percent and shortened 60 percent respectively, and the deviations of the rotor from the sphere center in the stimulation of exterior vibration and impulse have been reduced 80 percent and 87 percent respectively, and its resonant peak has been reduced 6.32dB and its damping factor has been increased at high frequency band while keeping the bandwidth still at the range between 600Hz and 800Hz, and its stiffness at lower freq
uency band(<10Hz) and the lowest stiffness at medium frequency band have been raised about 10 times and 2.85 times respectively. The simulation results show that sliding mode controller has strong robustness in suppressing parameter variation and exterior disturbance.
In order to further testify this result, electronic circuit experiment with DSP simulation environment has been carried on, in which three control algorithms have been transferred into instructions of assemble language, and then is simulated.
All in all, the robustness of electrostatic gyro levitation system has greatly been improved by applying sliding mode control theory to the system. It is of vital significance to secure the long-time and stable operation of ESG.
引文
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[2] 章海军，黄峰．光电反馈式静电悬浮的机制研究[J]．光子学报，2000，29(1)：72-77．
[3] HUI S，ZAK，STANISLAW H．On discrete-time variable structure sliding mode control[J]. Systems&Control Letters,1999，38(4-5)：283-288．
[4] 姚琼荟，黄继起，吴汉松．变结构控制系统[M]．第一版．重庆：重庆大学出版社1997．14-28．168-244．
[5] 田宏奇．滑模控制理论及其应用[M]．第一版．武汉：武汉出版社，1995，5-32，73-77，201-218，234-239．
[6] JEON J U, HIGUICHI T. Electrostatic levitator for hard disk media[J]. IEEE Trans. Ind. Electron., 1995，42(6): 938-946．
[7] 吴兰．磁悬浮与光电反馈式静电悬浮[J]．光学仪器．1999，21(3)：20-24．
[8] WU C H. DC electrostatic gyro suspension system for the gravity probe B experiment[D]. U S A: Dept. of Aeronautic and Astronautics, Standford University, 1993.
[9] WANG J, RAD A B, CHAN P T, Indirect adaptive fuzzy sliding mode control: part Ⅰ: fuzzy switching[J]. Fuzzy sets and systems, 2001, 122(1): 21～30.
[10] CHANG W, PARK J B, JOO Y H, et al. Design of robust fuzzy-model-based controller with sliding mode control for SISO nonlinear systems[J]. Fuzzy sets and systems, 2002, 125(1): 1～22.
[11] 高为炳．变结构控制基础[M]．北京：科学出版社，1989．
[12] 刘迎澍．高速磁悬浮机床主轴的离散变结构控制理论及应用[J]．控制理论与应用，2002，19(2)：304-307，310．
[13] 李永华，王荣瑞．离散滑模控制系统参数优化设计[J]．武汉汽车工业大学学报．1996，18[6]：1～6．
[14] HAN F T, GAO Z Y, WANG Y L. A differential capacitance to voltage converter for electrostatic levitation application[J]. Sensors and actuators: physical, 2002, 99(3):249～255.
[15] FRANKLIN G F, POWELL J D, WORKMAN M L. Digital control of dynamic systems[M]. 2nd ED. USA: Addison Wesley publishing company Inc., 1990, 280～299.
[16] 高金源．计算机控制系统—理论、设计与实现[M]．第一版．北京：北京航空航天大学出版社，2001．160～171．
[17] 胡寿松．自动控制原理[M]．第四版．北京：科学出版社，2002．501～510．
[18] SYRMOS, VASSILIS L. On the discrete generalized Lyapunov equation[J]. Automatica, 1995, 31(2): 297～301.
[19] CHANG K F. Strictly positive definite function[J]. Journal of Approximation Theory, 1996, 87(2): 149～158.
[20] 毕卫红，申石虎．Z域正实函数及其定理[J]．东北重型机械学院学报．1994，18(3)：239～242．
[21] GUTIERREZ H M. A mechatronic framework for high precision machining: modeling and control of magnetic servo levitation[D]. U S A: The graduate faculty of North Carolina State University, 1997.
[22] LEE J H, ALLAIRE P E, TAO G, et al. Integral sliding-mode control of magnetically suspended balance beam: analysis, simulation, and experiment[J]. IEEE/ASME Transactions
on mechatronics, 2001, 6(3):338～346.
[23] CHAN C Y. Discrete adaptive sliding mode control of a state-space system with a bounded disturbance[J]. Automatica. 1998, 34( 12): 1631～1635.
[24] EDMONDS S L, PIEPER J K. Discrete sliding mode control of magnetic bearings[J]. IEEE conference on control Applications-proceedings, 2000,: 658～663.
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