非光滑三明治系统的辨识和控制研究
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摘要
在实际工程系统中,通常存在着死区、间隙和迟滞等非光滑非线性特性,事实上,这些非光滑非线性特性存在于液压执行器驱动的飞机升降梯、带压电执行器的电子扫描隧道显微镜、超精密运动平台,齿轮传动的伺服定位平台等系统中。由此可见,这些非光滑非线性特性不是孤立存在的,而是与系统其他环节相联接,往往是夹在两个线性动态环节之间。为了研究问题的方便,我们称这样的系统为非光滑的三明治系统。在这类非光滑的三明治系统中,非光滑非线性环节的输入和输出往往不能直接测量,所以要对其进行辨识和控制存在着较大的困难。尤其当非线性环节为间隙或迟滞时,这些环节不仅具有非光滑性而且还具有多值映射的特性,使得传统的辨识与控制方法往往难以直接用于对其进行有效的辨识与控制。因此,本论文研究了带有死区、间隙和迟滞的非光滑的三明治系统的辨识与控制问题,主要研究成果有:
首先,根据关键项分离原则,提出了一种描述带有死区三明治系统的模型,这个模型具有新的表达形式。根据这个模型,我们提出了一种改进的一般性递推辨识算法,并且证明了算法的收敛性,将用于光滑线性动态系统辨识的方法推广应用到非光滑非线性动态系统的辨识中。有关仿真结果和对X-Y定位平台的辨识实验结果都验证了该方法的有效性。
其次,根据关键项分离原则,提出了一种间隙非线性特性的新形式的参数模型,将间隙的输入-输出之间的多值映射转换为一种单值映射。针对这类系统提出了改进的一般递推辨识算法,分析了算法的收敛性,拓宽了系统参数收敛的条件。随后分别研究了带有间隙的Hammerstein、Wiener和三明治系统的辨识问题。实现了这类带有间隙的动态系统模型的在线辨识。
由于带有死区或间隙的三明治系统是非光滑系统,所以我们引入了Clarke次梯度和Clarke次微分的概念,提出了一种基于梯度的递推辨识算法,根据bundle思想得到一种针对非凸和非光滑系统的次梯度方向的递推搜索方法,仿真结果表明了这种方法的有效性,为这类三明治系统的辨识和控制提供了一种新的选择。
由于迟滞的复杂特性,采用上述的递推辨识算法难以辨识其模型。因此我们将其视为黑箱系统,提出了一种改进的Prandtl-Ishlinksii (PI)模型。在这个改进的PI模型中,利用非对称广义间隙算子作为PI模型的基本算子,辨识过程中,提出了一种基于bundle思想的Levenberg-Marquardt算法对间隙算子的参数和模型的权值进行优化估计。与其他的PI模型相比,这种改进的PI模型不需费时费力地凭经验确定间隙算子的参数,而且还获得了简化的模型结构和较高的建模精度。对压电陶瓷驱动器和超声波电机中的迟滞建模实验获得了令人满意的结果。
通常迟滞的特性与其输入信号的频率有关,我们称这类迟滞特性为依赖输入频率的迟滞特性。在实际工程中,输入信号的频率往往是变化的。为了便于在线获得迟滞输入信号的频率的变化信息,我们证明了输入信号的导数与输入信号频率两者间存在一定的关系,从而可以用输入信号的导数对输入信号频率的变化进行估计。并用bundle思想估计迟滞的输出对其输入的Clarke次梯度,以提取迟滞运动方向的一些特征。通过构造包含输入导数、迟滞输出对其输入的次梯度等信息的扩展输入空间,把迟滞的多值映射转换为一个单值的映射。在此空间上,建立了一个基于神经网络的依赖输入频率的迟滞模型。这个方法同样也扩展到了带有迟滞的三明治系统的辨识中。这种方法用于具有迟滞特性的压电陶瓷执行器和X-Y微位移定位平台的带迟滞三明治系统的建模中,所获得的实验结果均表明:所提出的方法有潜在的工程应用价值。
最后,我们研究了一类带有间隙三明治系统的内模控制方法,这类三明治系统的第一个线性子系统的输出可测并且是最小相位系统。我们提出了一种改进的离散时间内模控制策略。在这个改进的控制策略中,首先利用第一个线性子系统的逆模型对第一个线性子系统进行补偿,使得对这类带间隙三明治系统的控制问题简化为一个带有间隙的Hammerstein系统的控制问题,然后提出了相应的模型的新的形式及其逆模型,使得模型误差中包含了间隙影响所产生的误差。考虑到带有间隙的系统在工作时会在不同的工作区切换,为了提高控制系统的鲁棒性与动态响应性能,我们根据模型不同的工作区,提出了控制滤波器的分段设计方法。仿真结果表明了我们所提出的改进内模控制方法的有效性。
Dead zone, backlash and hysteresis often exist in practical engineering systems. Actually, those non-smmoth nonlinearities exist in plane elevator droved by hydraulic actuators, electronic scanner microscope with piezoceramic actuators, ultraprecise moving stage and positioning servo control system with gear and so on. Therefore, those non-smooth nonlineairies do not exist isolatively but with the other sub-systems. In fact, they are more likely sandwiched between two linear subsystems in engineering applications. For the convenience of dealing with the problem of such systems, we define this kind of the systems as the sandwich systems. In the nonsmooth sandwich systems, the input and output of the nonlinear sub-system often cannot be measurable directly. Thus, the identification of this sort of systems becomes even more difficult especially when the nonlinear sub-system is backlash or hysteresis which is a nonsmooth nonlinearity with multi-valued mapping. So it is not easy to use the traditional identification and control methods to effectively handle the problems of modeling and control of such systems. In this dissertation, the identification and control on those nonsmooth systems, i.e., the sandwich system with dead zone, backlash or hysteresis are investigated. The main contributions are briefly described as follows:
Firstly, based on the key term separation principle, a novel form of the model to describe the sandwich system with dead zone is proposed. Then, a modified recursive general identification algorithm (MRGIA) is developed to estimate the model parameters. Also, the convergence of the algorithm is proved. Thus, the identification scheme usually used for smoothly linear dynamic systems is extended to the case of non-smoothly dynamic systems. Moreover, both the simulation and the modeling of the X-Y moving positioning stage demonstrate the effectiveness of the proposed method.
Secondly, based on the key term separation principle, we developed a parametrical model to describe the characteristic of the backlash. In the proposed scheme, the multi-valued mapping between the input and output of the backlash is transformed into a one-to-one mapping. Then, a MRGIA method is proposed for the identification of such kind of systems. The corresponding convergence of the algorithm is analyzed and the looser convergent conditions of the algorithm are obtained. Moreover, the extensions of the identification method to the Hammerstein, Wiener and sandwich systems with backlash are discussed respectively.
A gradient based recursive identification method is proposed for sandwich systems with dead zones or backlash. As the sandwich systems with dead zone or backlash are non-smooth systems, the gradients cannot exist at the nonsmooth points of the system. Thus, the Clarke subgradient and Clarke subdifferential are introduced to approximate the gradients at the nonsmooth points. The bundle method based recursive algorithm is proposed to search for the subgradient direction. The method to handle the case for nonsmooth and nonconvex systems is also investigated. The presented simulation results illustrate the effectiveness of the proposed method. This provides us with one of the new options for the identification and control of the nonsmooth sandwich systems.
Due to the complexity of hysteresis, it is hard for us to apply the above-mentioned recursive identification method to the modeling of the behavior of hysteresis. Hence, in this dissertation, we consider hysteresis as a black box. A modified Prandtl-Ishlinksii (PI) model is presented. In the modified PI model, an asymmetrically generalized backlash operator is used as the elementary operator of the PI model. In order to estimate the weights of the operators as well as the thresholds and slopes of the operators, the Levenberg-Marquardt algorithm based on the bundle method is developed. Compared with the other types of PI models which usually specify the parameters of the models by time-consuming empirical procedure, the proposed modified PI modeling method can determine the optimizing parameters of the model automatically. Moreover, the proposed modeling method can obtain a simplified model structure as well as the accurate model. The experimental results respectively on a piezoceramic actuator and an ultrasonic motor have shown the proposed method has achieved satisfactory modeling results.
As the performance of the hysteresis usually depends on the frequency of the input, especially the frequency of the input is increased to a certain value. This kind of hysteresis is called as rate-dependent hysteresis. In order to obtain the information of the frequency change of the input on-line, in his dissertation, we give the proof that the derivative of the input with respect to time has a certain relationship with the input frequency. Moreover, the movement direction of the hysteresis is estimated by using the derivative of the input based on the bundle method. Then, the Clarke subgradient of the hysteresis output with respect to its inputs is used to approximate the gradients at the nonsmooth points. By introducing the derivative of the input and the Clarke subgradient of the hysteresis output with respect to its input, an expanded input space is constructed to transform the multi-valued mapping of the hysteresis to a one-to-one mapping. Based on the expanded input space, the neural networks can be used to model the behavior of the rate-dependent hysteresis. Similarly, this method can also be utilized to identify and predict the sandwich systems with hysteresis. Furthermore, the experimental results on modeling of a piezoceramic actuator and an X-Y ultraprecise positioning stage are presented respectively to illustrate the performance of the proposed approach.
Finally, the modified discrete-time internl model control method for a class of sandwich system with backlash is discussed. In this sort of systems, the output of the first linear subsystem can be measured directliy. In the control scheme, an inverse model based compensator is introduced to compensate for the effect of the first linear subsystem. Therefore, the sandwich systems with backlash can be simplified as the Hammerstein systems with backlash. Then, a novel form of the system model and the corresponding inverse model are constructed respectively. Thus, the model errors caused by both linear part and the backlash are included into the consideration of the control design. As the backlash model is switched among the different operating zones, the piecewise robust filters are proposed to improve the robust stability and transient performance of the control system. Finally, the simulation results based on the proposed method are also presented.
首先,根据关键项分离原则,提出了一种描述带有死区三明治系统的模型,这个模型具有新的表达形式。根据这个模型,我们提出了一种改进的一般性递推辨识算法,并且证明了算法的收敛性,将用于光滑线性动态系统辨识的方法推广应用到非光滑非线性动态系统的辨识中。有关仿真结果和对X-Y定位平台的辨识实验结果都验证了该方法的有效性。
其次,根据关键项分离原则,提出了一种间隙非线性特性的新形式的参数模型,将间隙的输入-输出之间的多值映射转换为一种单值映射。针对这类系统提出了改进的一般递推辨识算法,分析了算法的收敛性,拓宽了系统参数收敛的条件。随后分别研究了带有间隙的Hammerstein、Wiener和三明治系统的辨识问题。实现了这类带有间隙的动态系统模型的在线辨识。
由于带有死区或间隙的三明治系统是非光滑系统,所以我们引入了Clarke次梯度和Clarke次微分的概念,提出了一种基于梯度的递推辨识算法,根据bundle思想得到一种针对非凸和非光滑系统的次梯度方向的递推搜索方法,仿真结果表明了这种方法的有效性,为这类三明治系统的辨识和控制提供了一种新的选择。
由于迟滞的复杂特性,采用上述的递推辨识算法难以辨识其模型。因此我们将其视为黑箱系统,提出了一种改进的Prandtl-Ishlinksii (PI)模型。在这个改进的PI模型中,利用非对称广义间隙算子作为PI模型的基本算子,辨识过程中,提出了一种基于bundle思想的Levenberg-Marquardt算法对间隙算子的参数和模型的权值进行优化估计。与其他的PI模型相比,这种改进的PI模型不需费时费力地凭经验确定间隙算子的参数,而且还获得了简化的模型结构和较高的建模精度。对压电陶瓷驱动器和超声波电机中的迟滞建模实验获得了令人满意的结果。
通常迟滞的特性与其输入信号的频率有关,我们称这类迟滞特性为依赖输入频率的迟滞特性。在实际工程中,输入信号的频率往往是变化的。为了便于在线获得迟滞输入信号的频率的变化信息,我们证明了输入信号的导数与输入信号频率两者间存在一定的关系,从而可以用输入信号的导数对输入信号频率的变化进行估计。并用bundle思想估计迟滞的输出对其输入的Clarke次梯度,以提取迟滞运动方向的一些特征。通过构造包含输入导数、迟滞输出对其输入的次梯度等信息的扩展输入空间,把迟滞的多值映射转换为一个单值的映射。在此空间上,建立了一个基于神经网络的依赖输入频率的迟滞模型。这个方法同样也扩展到了带有迟滞的三明治系统的辨识中。这种方法用于具有迟滞特性的压电陶瓷执行器和X-Y微位移定位平台的带迟滞三明治系统的建模中,所获得的实验结果均表明:所提出的方法有潜在的工程应用价值。
最后,我们研究了一类带有间隙三明治系统的内模控制方法,这类三明治系统的第一个线性子系统的输出可测并且是最小相位系统。我们提出了一种改进的离散时间内模控制策略。在这个改进的控制策略中,首先利用第一个线性子系统的逆模型对第一个线性子系统进行补偿,使得对这类带间隙三明治系统的控制问题简化为一个带有间隙的Hammerstein系统的控制问题,然后提出了相应的模型的新的形式及其逆模型,使得模型误差中包含了间隙影响所产生的误差。考虑到带有间隙的系统在工作时会在不同的工作区切换,为了提高控制系统的鲁棒性与动态响应性能,我们根据模型不同的工作区,提出了控制滤波器的分段设计方法。仿真结果表明了我们所提出的改进内模控制方法的有效性。
Dead zone, backlash and hysteresis often exist in practical engineering systems. Actually, those non-smmoth nonlinearities exist in plane elevator droved by hydraulic actuators, electronic scanner microscope with piezoceramic actuators, ultraprecise moving stage and positioning servo control system with gear and so on. Therefore, those non-smooth nonlineairies do not exist isolatively but with the other sub-systems. In fact, they are more likely sandwiched between two linear subsystems in engineering applications. For the convenience of dealing with the problem of such systems, we define this kind of the systems as the sandwich systems. In the nonsmooth sandwich systems, the input and output of the nonlinear sub-system often cannot be measurable directly. Thus, the identification of this sort of systems becomes even more difficult especially when the nonlinear sub-system is backlash or hysteresis which is a nonsmooth nonlinearity with multi-valued mapping. So it is not easy to use the traditional identification and control methods to effectively handle the problems of modeling and control of such systems. In this dissertation, the identification and control on those nonsmooth systems, i.e., the sandwich system with dead zone, backlash or hysteresis are investigated. The main contributions are briefly described as follows:
Firstly, based on the key term separation principle, a novel form of the model to describe the sandwich system with dead zone is proposed. Then, a modified recursive general identification algorithm (MRGIA) is developed to estimate the model parameters. Also, the convergence of the algorithm is proved. Thus, the identification scheme usually used for smoothly linear dynamic systems is extended to the case of non-smoothly dynamic systems. Moreover, both the simulation and the modeling of the X-Y moving positioning stage demonstrate the effectiveness of the proposed method.
Secondly, based on the key term separation principle, we developed a parametrical model to describe the characteristic of the backlash. In the proposed scheme, the multi-valued mapping between the input and output of the backlash is transformed into a one-to-one mapping. Then, a MRGIA method is proposed for the identification of such kind of systems. The corresponding convergence of the algorithm is analyzed and the looser convergent conditions of the algorithm are obtained. Moreover, the extensions of the identification method to the Hammerstein, Wiener and sandwich systems with backlash are discussed respectively.
A gradient based recursive identification method is proposed for sandwich systems with dead zones or backlash. As the sandwich systems with dead zone or backlash are non-smooth systems, the gradients cannot exist at the nonsmooth points of the system. Thus, the Clarke subgradient and Clarke subdifferential are introduced to approximate the gradients at the nonsmooth points. The bundle method based recursive algorithm is proposed to search for the subgradient direction. The method to handle the case for nonsmooth and nonconvex systems is also investigated. The presented simulation results illustrate the effectiveness of the proposed method. This provides us with one of the new options for the identification and control of the nonsmooth sandwich systems.
Due to the complexity of hysteresis, it is hard for us to apply the above-mentioned recursive identification method to the modeling of the behavior of hysteresis. Hence, in this dissertation, we consider hysteresis as a black box. A modified Prandtl-Ishlinksii (PI) model is presented. In the modified PI model, an asymmetrically generalized backlash operator is used as the elementary operator of the PI model. In order to estimate the weights of the operators as well as the thresholds and slopes of the operators, the Levenberg-Marquardt algorithm based on the bundle method is developed. Compared with the other types of PI models which usually specify the parameters of the models by time-consuming empirical procedure, the proposed modified PI modeling method can determine the optimizing parameters of the model automatically. Moreover, the proposed modeling method can obtain a simplified model structure as well as the accurate model. The experimental results respectively on a piezoceramic actuator and an ultrasonic motor have shown the proposed method has achieved satisfactory modeling results.
As the performance of the hysteresis usually depends on the frequency of the input, especially the frequency of the input is increased to a certain value. This kind of hysteresis is called as rate-dependent hysteresis. In order to obtain the information of the frequency change of the input on-line, in his dissertation, we give the proof that the derivative of the input with respect to time has a certain relationship with the input frequency. Moreover, the movement direction of the hysteresis is estimated by using the derivative of the input based on the bundle method. Then, the Clarke subgradient of the hysteresis output with respect to its inputs is used to approximate the gradients at the nonsmooth points. By introducing the derivative of the input and the Clarke subgradient of the hysteresis output with respect to its input, an expanded input space is constructed to transform the multi-valued mapping of the hysteresis to a one-to-one mapping. Based on the expanded input space, the neural networks can be used to model the behavior of the rate-dependent hysteresis. Similarly, this method can also be utilized to identify and predict the sandwich systems with hysteresis. Furthermore, the experimental results on modeling of a piezoceramic actuator and an X-Y ultraprecise positioning stage are presented respectively to illustrate the performance of the proposed approach.
Finally, the modified discrete-time internl model control method for a class of sandwich system with backlash is discussed. In this sort of systems, the output of the first linear subsystem can be measured directliy. In the control scheme, an inverse model based compensator is introduced to compensate for the effect of the first linear subsystem. Therefore, the sandwich systems with backlash can be simplified as the Hammerstein systems with backlash. Then, a novel form of the system model and the corresponding inverse model are constructed respectively. Thus, the model errors caused by both linear part and the backlash are included into the consideration of the control design. As the backlash model is switched among the different operating zones, the piecewise robust filters are proposed to improve the robust stability and transient performance of the control system. Finally, the simulation results based on the proposed method are also presented.
引文
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