多变量系统辨识方法及性能分析
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摘要
系统辨识与模型参数估计是基于模型的控制问题的基础。各个领域的控制对象大部分是结构复杂，干扰不确定的多变量系统。因此研究多变量系统的辨识具有普遍意义及广泛的应用前景。多变量系统结构复杂、变量多、参数化后维数增大，导致辨识算法的计算量加大。论文利用耦合辨识概念，数据滤波辨识技术、递阶辨识原理等技术研究计算量小又能保证参数估计精度的辨识方法，并通过理论分析和数值仿真分析算法的收敛性能。这对丰富和发展辨识理论具有理论意义。本文的具体工作如下。
1.针对一类多变量线性回归辨识模型，利用随机过程与鞅理论分别分析了这类模型的最小二乘算法和随机梯度算法的收敛性能。分析表明当噪声均值为零向量、方差向量有界时，最小二乘算法在强持续激励条件下参数估计一致收敛于真参数；随机梯度算法在满足信号持续激励，噪声无需平稳性假设时，系统参数估计收敛到真参数。理论分析的结果得到了仿真例子的验证。
2.针对滑动平均噪声干扰的多变量线性回归模型，给出相应的递推增广最小二乘算法与增广随机梯度算法，利用随机过程鞅收敛定理分析了不同持续激励条件下算法的收敛性能，并通过多个仿真例子验证算法的性能。
3.单变量系统的输入输出变量经过非均匀周期采样得到的离散模型可等价为多变量模型。基于耦合辨识概念，以一类非均匀周期采样系统为研究对象，推导了多变量系统当子系统参数向量完全相同时的耦合辨识方法，以及子系统参数向量部分相同时的部分耦合辨识方法。这种耦合辨识方法既可以有效减小算法的计算量。仿真例子验证了算法的有效性。
4.针对有色噪声干扰的多变量线性回归系统，基于数据滤波辨识技术，根据噪声模型结构选择合适的数据滤波器对输入输出数据进行滤波，然后交互估计滤波后的系统模型和噪声模型的参数，从而提出多变量系统基于滤波的递推最小二乘算法。这种辨识方法由于将辨识问题转化为两个维数较小的模型(即滤波后的系统模型和噪声模型)进行辨识，因而具有较少的计算量。数值仿真说明基于滤波的辨识算法和递推广义最小二乘算法有类似的收敛性能，而且当噪声方差较小时，基于滤波的辨识方法拥有更高的参数估计精度。此外还推导了多变量系统模型基于滤波的迭代最小二乘辨识方法。
5.单变量系统的输入输出变量经过非均匀周期采样和一般双率采样得到的离散模型为多变量模型。以一类非均匀采样系统和一般双率系统为研究对象，基于递阶辨识原理，对多输入单输出系统和多输入多输出系统的递阶最小二乘辨识方法分别进行了研究，并对这两种算法的收敛性能进行了理论分析。分析表明递阶辨识算法可以大大减少计算量，在一定条件下参数估计一致收敛。基于这种递阶辨识原理，进一步探讨有色噪声干扰多变量回归模型的递阶辨识方法，分析该模型不同算法的计算量，并通过仿真例子说明算法的有效性。
System identifcation and parameter estimation are important for model-based control problems. Mostof the practical control plants are multivariable systems, with complex structures and uncertain distur-bances. Therefore, the study of identifcation methods and their convergence properties for multivariablesystems is of universal signifcance and has wide application prospects. In recent years, research of theidentifcation methods for multivariable systems has received much attention and a number of achievementshave been made. However, it is not mature enough compared with the identifcation methods for scalarsystems. Especially when the number of variables is large, the identifcation models are of high dimensions,resulting in signifcant computational costs in the identifcation algorithms. By using the coupled identi-fcation concept, the data fltering technique and the hierarchical identifcation principle, this dissertationaims to develop more efcient identifcation methods for multivariable systems and to study their conver-gence properties, which is important to enrich the system identifcation theory. The main contributionsare summarized as follows.
1. The convergence properties of the recursive least squares (RLS) method and the stochastic gradient(SG) algorithm for multivariable systems that can be parameterized into a class of multivariate linearregression models are studied, by using the stochastic process theory and the martingale theorem.The theoretical analysis indicates that the parameter estimation error given by the RLS algorithmapproaches to zero when the input signal is persistently exciting and the noise has zero mean vectorand fnite variance, and the parameter estimates given by the SG algorithm can converge to their truevalues even for non-stationary noises.
2. For the multivariable linear systems disturbed by moving average noises, the recursive extended leastsquares algorithm and the extended stochastic gradient algorithm are presented. The performanceanalyses of the proposed algorithms under diferent conditions are studied by using the stochastic pro-cess theory and the martingale theorem. Simulation examples are given to illustrate the efectivenessof the algorithms.
3. For the non-uniformly sampled multirate systems, the discrete-time input-output representations ofwhich are multivariable models, a coupled least squares (C-LS) algorithm and a partially coupledrecursive least squares (PC-RLS) algorithms are derived to estimate the model parameters with theadvantage of not involving the computation of a matrix inverse in each recursion step. The proposed(C-LS) algorithm is equivalent to the standard RLS algorithm and thus has the same convergenceproperties. Even the signals for the subsystems are not persistently exciting, the coupled identifcationalgorithms can still converge. Simulation examples verify the efectiveness of the algorithms.
4. For multivariable linear regression models with autoregressive noises, a fltering-based recursive leastsquares (F-RLS) algorithm is presented. The idea is to transfer the system with a colored noise intoa system with a white noise by fltering the input-output data with a specifc flter, and then toidentify the fltered model and the noise model interactively. The flter is selected according to thenoise model structure. Since the algorithm transfers the identifcation model into two systems withmuch lower dimensions (one is the fltered system model and the other is the noise model), it hasless computational burden. The simulation results show that the F-RLS algorithm provides a higher accuracy of the parameter estimates under lower noise levels than the recursive generalized recursiveleast squares algorithm. When a batch of input-output data are obtained, a fltering based iterativeleast squares method is further derived.
5. Based on the hierarchical identifcation principle, for a class of non-uniformly sampled systems and thegeneral dural-rate systems, whose discrete-time models are equivalent to multiple input single outputsystems and multi-input multi-output systems, respectively, the hierarchical least squares algorithmtogether with their convergence analyses are studied. The proposed algorithms can greatly save thecomputational cost. The performance analyses indicate that parameter estimates can converge totheir true values under certain conditions. The simulation tests confrm the convergence results. Thehierarchical identifcation principle is also employed to identify multivariable linear regression modelswith colored noises, compared with other methods, the hierarchical identifcation methods has theleast computational cost. A simulation confrms that the proposed algorithm is efective.
1.针对一类多变量线性回归辨识模型，利用随机过程与鞅理论分别分析了这类模型的最小二乘算法和随机梯度算法的收敛性能。分析表明当噪声均值为零向量、方差向量有界时，最小二乘算法在强持续激励条件下参数估计一致收敛于真参数；随机梯度算法在满足信号持续激励，噪声无需平稳性假设时，系统参数估计收敛到真参数。理论分析的结果得到了仿真例子的验证。
2.针对滑动平均噪声干扰的多变量线性回归模型，给出相应的递推增广最小二乘算法与增广随机梯度算法，利用随机过程鞅收敛定理分析了不同持续激励条件下算法的收敛性能，并通过多个仿真例子验证算法的性能。
3.单变量系统的输入输出变量经过非均匀周期采样得到的离散模型可等价为多变量模型。基于耦合辨识概念，以一类非均匀周期采样系统为研究对象，推导了多变量系统当子系统参数向量完全相同时的耦合辨识方法，以及子系统参数向量部分相同时的部分耦合辨识方法。这种耦合辨识方法既可以有效减小算法的计算量。仿真例子验证了算法的有效性。
4.针对有色噪声干扰的多变量线性回归系统，基于数据滤波辨识技术，根据噪声模型结构选择合适的数据滤波器对输入输出数据进行滤波，然后交互估计滤波后的系统模型和噪声模型的参数，从而提出多变量系统基于滤波的递推最小二乘算法。这种辨识方法由于将辨识问题转化为两个维数较小的模型(即滤波后的系统模型和噪声模型)进行辨识，因而具有较少的计算量。数值仿真说明基于滤波的辨识算法和递推广义最小二乘算法有类似的收敛性能，而且当噪声方差较小时，基于滤波的辨识方法拥有更高的参数估计精度。此外还推导了多变量系统模型基于滤波的迭代最小二乘辨识方法。
5.单变量系统的输入输出变量经过非均匀周期采样和一般双率采样得到的离散模型为多变量模型。以一类非均匀采样系统和一般双率系统为研究对象，基于递阶辨识原理，对多输入单输出系统和多输入多输出系统的递阶最小二乘辨识方法分别进行了研究，并对这两种算法的收敛性能进行了理论分析。分析表明递阶辨识算法可以大大减少计算量，在一定条件下参数估计一致收敛。基于这种递阶辨识原理，进一步探讨有色噪声干扰多变量回归模型的递阶辨识方法，分析该模型不同算法的计算量，并通过仿真例子说明算法的有效性。
System identifcation and parameter estimation are important for model-based control problems. Mostof the practical control plants are multivariable systems, with complex structures and uncertain distur-bances. Therefore, the study of identifcation methods and their convergence properties for multivariablesystems is of universal signifcance and has wide application prospects. In recent years, research of theidentifcation methods for multivariable systems has received much attention and a number of achievementshave been made. However, it is not mature enough compared with the identifcation methods for scalarsystems. Especially when the number of variables is large, the identifcation models are of high dimensions,resulting in signifcant computational costs in the identifcation algorithms. By using the coupled identi-fcation concept, the data fltering technique and the hierarchical identifcation principle, this dissertationaims to develop more efcient identifcation methods for multivariable systems and to study their conver-gence properties, which is important to enrich the system identifcation theory. The main contributionsare summarized as follows.
1. The convergence properties of the recursive least squares (RLS) method and the stochastic gradient(SG) algorithm for multivariable systems that can be parameterized into a class of multivariate linearregression models are studied, by using the stochastic process theory and the martingale theorem.The theoretical analysis indicates that the parameter estimation error given by the RLS algorithmapproaches to zero when the input signal is persistently exciting and the noise has zero mean vectorand fnite variance, and the parameter estimates given by the SG algorithm can converge to their truevalues even for non-stationary noises.
2. For the multivariable linear systems disturbed by moving average noises, the recursive extended leastsquares algorithm and the extended stochastic gradient algorithm are presented. The performanceanalyses of the proposed algorithms under diferent conditions are studied by using the stochastic pro-cess theory and the martingale theorem. Simulation examples are given to illustrate the efectivenessof the algorithms.
3. For the non-uniformly sampled multirate systems, the discrete-time input-output representations ofwhich are multivariable models, a coupled least squares (C-LS) algorithm and a partially coupledrecursive least squares (PC-RLS) algorithms are derived to estimate the model parameters with theadvantage of not involving the computation of a matrix inverse in each recursion step. The proposed(C-LS) algorithm is equivalent to the standard RLS algorithm and thus has the same convergenceproperties. Even the signals for the subsystems are not persistently exciting, the coupled identifcationalgorithms can still converge. Simulation examples verify the efectiveness of the algorithms.
4. For multivariable linear regression models with autoregressive noises, a fltering-based recursive leastsquares (F-RLS) algorithm is presented. The idea is to transfer the system with a colored noise intoa system with a white noise by fltering the input-output data with a specifc flter, and then toidentify the fltered model and the noise model interactively. The flter is selected according to thenoise model structure. Since the algorithm transfers the identifcation model into two systems withmuch lower dimensions (one is the fltered system model and the other is the noise model), it hasless computational burden. The simulation results show that the F-RLS algorithm provides a higher accuracy of the parameter estimates under lower noise levels than the recursive generalized recursiveleast squares algorithm. When a batch of input-output data are obtained, a fltering based iterativeleast squares method is further derived.
5. Based on the hierarchical identifcation principle, for a class of non-uniformly sampled systems and thegeneral dural-rate systems, whose discrete-time models are equivalent to multiple input single outputsystems and multi-input multi-output systems, respectively, the hierarchical least squares algorithmtogether with their convergence analyses are studied. The proposed algorithms can greatly save thecomputational cost. The performance analyses indicate that parameter estimates can converge totheir true values under certain conditions. The simulation tests confrm the convergence results. Thehierarchical identifcation principle is also employed to identify multivariable linear regression modelswith colored noises, compared with other methods, the hierarchical identifcation methods has theleast computational cost. A simulation confrms that the proposed algorithm is efective.
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