基于子空间方法的闭环系统辨识
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摘要
闭环系统辨识对于工程实际具有重要的意义。一方面,出于安全性、质量控制或者系统不稳定等原因,系统辨识必须在闭环条件下进行；另一方面,随着面向控制的辨识等研究领域的不断发展,基于模型的控制系统设计同样也要求在闭环条件下对过程进行辨识。子空间辨识方法是一类新的状态空间模型辨识方法,与传统的辨识方法相比,具有对模型结构先验知识需求较少,数值计算具有一定鲁棒性,且尤其适用于多变量系统等优点。本文以子空间辨识方法这一新兴工具为理论基础,围绕闭环系统辨识的相关问题展开深入的研究。主要内容包括：
1.针对传统闭环子空间辨识方法在较大有色噪声干扰下产生估计偏差的问题,将子空间辨识框架从传统的输入输出数据扩展到相关函数数据,提出一种基于相关函数估计的闭环子空间辨识方法。该方法的基本思想是将整个辨识过程分成两步进行：第一步采用M序列作为闭环系统的外部输入信号计算相关函数估计序列,进而获得待辨识对象的相关函数状态空间方程;第二步基于相关函数的时移不变性,采用零空间投影估计广义能观性矩阵。仿真结果表明,该方法可以有效地抵抗不同方差水平噪声的干扰,尤其当系统中存在方差水平较大的有色噪声时,实现闭环系统待辨识模型的无偏估计。
2.针对大多数递推子空间辨识方法不适用于闭环系统的情形,提出一种基于正交分解的闭环递推子空间辨识方法,很好地实现了闭环条件下待辨识模型的递推估计。首先通过联合输入输出过程的正交分解获得模型的确定性状态空间实现,有效地避免了闭环条件下测量输入和噪声之间相关性的影响；在此基础上,对新获得的采样数据进行序列、递推处理,并结合子空间跟踪技术获取广义能观性矩阵的递推估计,进而实现辨识模型的递推更新。仿真结果表明,该方法在闭环系统控制器未知的情形下,不依赖任何待辨识对象的先验知识,较其他闭环递推子空间辨识方法得到了更为准确的模型估计。
3.针对一类观测输入和输出通道非均匀采样的多率闭环系统,提出一种基于新息估计的多率闭环子空间辨识方法,实现在反馈存在的条件下确定闭环系统中待辨识对象的离散状态空间模型。该方法在模型参数化的过程中,充分地考虑了由于提升导致的因果约束,将特定矩阵的每一行块转化为三角结构；然后逐行计算新息序列并将已获得的新息序列纳入回归变量,进而选取合适的权矩阵估计状态序列。仿真结果表明,该方法不仅可以有效地确定多率闭环系统中待辨识对象的离散状态空间模型参数,而且可以通过状态序列的奇异值分解确定模型的阶次。
Closed-loop system identification is greatly significant for the engineering practice. On the one hand, for the security, quality control, or the system instability, etc., the system identification must be carried out under the closed-loop condition; On the other hand, as the rise and the development of the researches related to control oriented identification, the design of the model-based control system also requires to identify the plant in the closed-loop. Subspace identification method (SIM) is a new class of state-space model identification methods. Compared to other traditional identification methods, SIM needs less a prior knowledge of the model structure, has the numerical robustness, and is especially suitable for the multivariable system. Based on this new theoretical tool, the dissertation focuses on the research of relevant problems in the closed-loop system identification. The main contents include:
1. For the traditional closed-loop subspace identification methods result in biased estimates under colored noises of large variances, the framework of SIM is extended from traditional input/output data to correlation function and a closed-loop subspace identification method based on correlation function estimates is proposed. The basic idea of the proposed method is to carry out the identification process into two steps. In the first step, the M-sequence is used to be the external input signal of the closed-loop system and to compute correlation function estimates sequences, and then the state-space equations of the identified object based on correlation function are obtained; In the second step, the null-space projection is developed to estimate the extended observability matrix based on the shifted-invariant of the correlation function. The simulation results show that the proposed method can effectively resist the noises of different variances in order to achieve unbiased estimates, especially colored noises of large variances.
2. For most of recursive subspace identification methods (RSIMs) are not suitable to the closed-loop system, a closed-loop RSIM based on the orthogonal decomposition is proposed which can realize the recursive update of the identified model under the closed-loop condition. Firstly, through the orthogonal decomposition of the input-output process, the deterministic state-space realization of the identified model is obtained which effectively avoids the influence of the correlation between the observed input and the noise in closed-loop. After that, the newly sampled data is sequentially and recursively processed and the extended observability matrix is recursively obtained based on subspace tracking technique to realize the update of the model. The simulation results show that the proposed method dose not require any information of the controller and a priori knowledge of the identified object in the closed-loop system and can obtain more accurate estimates of the identified model than other closed-loop recursive subspace identification methods.
3. Consider a class of the multirate system with the input-output non-uniformly sampled, a closed-loop subspace identification method based on the innovation estimates is proposed. The proposed method can determine the discrete state-space model of the identified object under the feedback. During the process of the identification parameterization, the causal constraint due to the lifting technique is fully considered. So the each row of the specific matrix is transformed into the triangular structure. Then the innovation sequences are computed row by row and the obtained innovations are included into the regression variables. Finally the state sequence is estimated by the appropriate weighting matrices. The simulation results show that the proposed method can effectively determine not only the paramters of the discrete state-space model of the identified object in the multirate closed-loop system, but also the order of the model by the singular values decomposition of the state sequences.
1.针对传统闭环子空间辨识方法在较大有色噪声干扰下产生估计偏差的问题,将子空间辨识框架从传统的输入输出数据扩展到相关函数数据,提出一种基于相关函数估计的闭环子空间辨识方法。该方法的基本思想是将整个辨识过程分成两步进行：第一步采用M序列作为闭环系统的外部输入信号计算相关函数估计序列,进而获得待辨识对象的相关函数状态空间方程;第二步基于相关函数的时移不变性,采用零空间投影估计广义能观性矩阵。仿真结果表明,该方法可以有效地抵抗不同方差水平噪声的干扰,尤其当系统中存在方差水平较大的有色噪声时,实现闭环系统待辨识模型的无偏估计。
2.针对大多数递推子空间辨识方法不适用于闭环系统的情形,提出一种基于正交分解的闭环递推子空间辨识方法,很好地实现了闭环条件下待辨识模型的递推估计。首先通过联合输入输出过程的正交分解获得模型的确定性状态空间实现,有效地避免了闭环条件下测量输入和噪声之间相关性的影响；在此基础上,对新获得的采样数据进行序列、递推处理,并结合子空间跟踪技术获取广义能观性矩阵的递推估计,进而实现辨识模型的递推更新。仿真结果表明,该方法在闭环系统控制器未知的情形下,不依赖任何待辨识对象的先验知识,较其他闭环递推子空间辨识方法得到了更为准确的模型估计。
3.针对一类观测输入和输出通道非均匀采样的多率闭环系统,提出一种基于新息估计的多率闭环子空间辨识方法,实现在反馈存在的条件下确定闭环系统中待辨识对象的离散状态空间模型。该方法在模型参数化的过程中,充分地考虑了由于提升导致的因果约束,将特定矩阵的每一行块转化为三角结构；然后逐行计算新息序列并将已获得的新息序列纳入回归变量,进而选取合适的权矩阵估计状态序列。仿真结果表明,该方法不仅可以有效地确定多率闭环系统中待辨识对象的离散状态空间模型参数,而且可以通过状态序列的奇异值分解确定模型的阶次。
Closed-loop system identification is greatly significant for the engineering practice. On the one hand, for the security, quality control, or the system instability, etc., the system identification must be carried out under the closed-loop condition; On the other hand, as the rise and the development of the researches related to control oriented identification, the design of the model-based control system also requires to identify the plant in the closed-loop. Subspace identification method (SIM) is a new class of state-space model identification methods. Compared to other traditional identification methods, SIM needs less a prior knowledge of the model structure, has the numerical robustness, and is especially suitable for the multivariable system. Based on this new theoretical tool, the dissertation focuses on the research of relevant problems in the closed-loop system identification. The main contents include:
1. For the traditional closed-loop subspace identification methods result in biased estimates under colored noises of large variances, the framework of SIM is extended from traditional input/output data to correlation function and a closed-loop subspace identification method based on correlation function estimates is proposed. The basic idea of the proposed method is to carry out the identification process into two steps. In the first step, the M-sequence is used to be the external input signal of the closed-loop system and to compute correlation function estimates sequences, and then the state-space equations of the identified object based on correlation function are obtained; In the second step, the null-space projection is developed to estimate the extended observability matrix based on the shifted-invariant of the correlation function. The simulation results show that the proposed method can effectively resist the noises of different variances in order to achieve unbiased estimates, especially colored noises of large variances.
2. For most of recursive subspace identification methods (RSIMs) are not suitable to the closed-loop system, a closed-loop RSIM based on the orthogonal decomposition is proposed which can realize the recursive update of the identified model under the closed-loop condition. Firstly, through the orthogonal decomposition of the input-output process, the deterministic state-space realization of the identified model is obtained which effectively avoids the influence of the correlation between the observed input and the noise in closed-loop. After that, the newly sampled data is sequentially and recursively processed and the extended observability matrix is recursively obtained based on subspace tracking technique to realize the update of the model. The simulation results show that the proposed method dose not require any information of the controller and a priori knowledge of the identified object in the closed-loop system and can obtain more accurate estimates of the identified model than other closed-loop recursive subspace identification methods.
3. Consider a class of the multirate system with the input-output non-uniformly sampled, a closed-loop subspace identification method based on the innovation estimates is proposed. The proposed method can determine the discrete state-space model of the identified object under the feedback. During the process of the identification parameterization, the causal constraint due to the lifting technique is fully considered. So the each row of the specific matrix is transformed into the triangular structure. Then the innovation sequences are computed row by row and the obtained innovations are included into the regression variables. Finally the state sequence is estimated by the appropriate weighting matrices. The simulation results show that the proposed method can effectively determine not only the paramters of the discrete state-space model of the identified object in the multirate closed-loop system, but also the order of the model by the singular values decomposition of the state sequences.
引文
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