基于最小二乘支持向量机的在线建模与控制方法研究
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摘要
论文对最小二乘支持向量机(Least Squares Support Vector Machines, LSSVM)的有关算法及LSSVM在系统在线建模与控制中的若干应用方法展开研究。
主要研究内容包括：
在分析现有基于分块矩阵求逆公式的LSSVM增量式和减量式学习算法的基础上,为这些算法设计了一些加速实现策略,得到LSSVM快速在线式学习算法。基于这种快速LSSVM的多输入多输出(Multiple-Input Multiple-Output, MIMO)系统在线建模仿真显示这些加速策略能明显提高原有在线算法执行速度；加速策略需要的空间为少数几个缓存数组。
为了减少LSSVM基本剪枝算法的计算量,提出了一种快速剪枝算法。在分析剪枝前后两个LSSVM对应线性方程组系数矩阵之间关系的基础上,利用对调单位矩阵两行(或列)所得的初等方阵的逆等于其自身的性质和分块矩阵求逆公式,导出这两个系数矩阵的子阵的逆之间的递推关系,避免剪枝过程中多次进行高阶矩阵求逆,从而减少计算量。在不考虑递推计算误差时,该算法理论上得出与基本剪枝算法相同结果的稀疏LSSVM。仿真实验表明该算法比基本剪枝算法速度快,而且初始训练样本越多,加速比越大。
为了减少带时间窗型在线LSSVM的计算量和存储空间,提出了一种带时间窗型在线稀疏LSSVM。它利用滑动时间窗中部分时刻的样本作为训练样本集。新时刻的样本总是加入训练样本集；每次删除样本时,若滑动时间窗最前端时刻的样本在训练样本集中,则删除它,否则从训练样本集中选择留一法预测误差最小的样本删除。与现有的在线LSSVM相比,该在线稀疏LSSVM能用较少的样本学习系统较多的特性,能提高时空效率；系统建模仿真实验表明,本文在线稀疏LSSVM能节省时间和空间,具有较高的预测精度。
针对无偏置最小二乘支持向量机(non-bias LSSVM, NB-LSSVM),给出了删除贡献最小(或任意)样本的计算过程,并设计了稀疏在线无偏置最小二乘支持向量机(sparse online non-bias LSSVM, SONB-LSSVM).该样本删除技巧能提高学习样本集的多样性和代表性；与在线无偏置最小二乘支持向量机(online non-bias LSSVM, ONB-LSSVM)相比,SONB-LSSVM能学习系统更长时间范围内的特性；用于动态系统在线建模时,其泛化能力受输入信号频率影响较小。
针对直接利用LSSVM对动态过程在线建模时预测精度易受过程输出测量值上的粗大误差和噪声影响的问题,在分析样本序列结构特征和噪声作用特征基础上,提出一种基于NB-LSSVM的稳健在线过程建模方法。该方法在每一预测周期中根据预测误差与设定阈值之间的关系来识别和剔除异常测量值、识别和修正含噪声测量值,从而降低样本中的噪声,使得出的LSSVM较好地跟踪过程的动态特性。这种在线过程建模方法具有稳健性,能减少输出值上粗大误差和高斯白噪声对LSSVM预测精度的影响,提高预测精度。数字仿真显示了本文方法的有效性和优越性。
利用LSSVM在线辨识时变非线性过程时,设定其核参数较困难,设定的核参数不能适应过程变化而进行自动调节；针对此问题,提出了一种基于核参数分时段调节型LSSVM的在线过程辨识方法。该方法利用三个LSSVM,并将整个建模预测时期分为启动阶段和若干个工作周期,初始阶段末和每个工作周期末选定预测误差累加值最小的LSSVM作为后续工作周期的工作LSSVM,同时根据启发式规则为另两个LSSVM重新设定核参数值,它们作为后续工作周期的比较LSSVM.该方法设定核参数相对容易,而且核参数具有一定的自动调节能力。数字仿真显示,从统计角度而言,本文方法比传统方法有更好的适应性。
对于非线性系统预测控制问题,提出了一种基于SONB-LSSVM的有约束单步预测控制算法。该预测控制利用SONB-LSSVM在线建模预测系统输出,滚动优化环节采用粒子群优化(Particle Swarm Optimization,PSO)或Brent优化搜索非线性系统的控制量。由于SONB-LSSVM能及时学习过程新动态特性,该预测控制方法具有良好的自适应能力。在控制启动阶段利用比例-积分-微分(proportional-integral-derivative,PID)控制,同时采集过程输入输出数据构成样本逐步生成初始的NB-LSSVM,该控制方法使用方便。液位控制、连续搅拌反应釜浓度控制和水浴温度控制等仿真表明该预测控制方法是有效的。
探讨了SONB-LSSVM和ONB-LSSVM在非线性逆控制中的应用。提出基于SONB-LSSVM的自适应直接逆控制方法,该方法利用SONB-LSSVM对被控对象在线逆建模并计算控制量,该控制方法适合控制定常或参数(或特性)轻微变化的可逆对象。对于运行中参数(或特性)在较大范围中变化的可逆对象,提出一种ONB-LSSVM逆模型与PID相结合的自适应复合控制,在每个控制周期,控制信号由逆控制量与PID控制量合成,合成比例自动调节。逆控制在高频段发挥较大作用,PID控制在低频段发挥较大作用。因此,该复合控制方法有较宽的频率适应范围,并且对时变系统有较好控制效果。两种控制方法中初始NB-LSSVM在线生成,因此这两种控制方法使用方便。数字仿真表明了这两种控制方法的可行性和有效性。
Some algorithms about least squares support vector machines(LSSVM) and applying methods for system online modeling and control based on LSSVM are studied in this dissertation.
The main contents are outlined as follows:
The existing LSSVM incremental and decremental learning algorithms by the formula of solving inversion of partitioned matrix are analyzed, and some speeding up implementing tactics are designed for the algorithms; consequently, a fast online LSSVM learning algorithm is obtained. The numerical simulation of multiple-input multiple-output(MIMO) system online modeling employing the fast online LSSVM shows these speedy tactics can heighten time performance of the existing LSSVM online learning algorithm; the tactics only require space of several additional buffer arrays.
To reduce the computation amount of basic pruning algorithm(BPA) for LSSVM, a fast pruning algorithm(FPA) is proposed. The connection between two coefficient matrices of linear equations corresponding to LSSVM before pruning and to one after doing is analyzed, and the recursive relation between inversions of sub-matrices of the two coefficient matrices is derived by property of elementary matrix attained through swapping two columns or two rows that its inversion equalling to itself and by the calculation formula of solving inversion of partitioned matrix, therefore repeat calculation of inversion of higher order matrices is avoided in pruning process, and computation amount is decreased. FPA products the same resultant sparse LSSVM as BPA theoretically, when recursive calculating error is neglected. The numerical simulation results show the presented algorithm is quicker than BPA, and the more the training samples, the greater FPA's speedup rate to BPA is.
To reduce the computation time and storage space of online LSSVM with time window, an online sparse LSSVM with time window is proposed. It only takes samples ranking at partial moments among sliding time window as training samples set(TSS). The new sample is learned necessarily; when sample elimination is performed, if the sample ranking at the oldest moment among sliding time window exists in TSS, then it will be removed during decremental learning, otherwise, the sample with the smallest leave-one-out predicting error among TSS is selected and deleted. Compared with the existing online LSSVM, the presented online sparse LSSVM can learn more characteristic of system using less samples, and heighten time-space efficiency; compared with the existing online spare LSSVM, it can get rid of the obsolete sample, and adapt to time-variant properties of system better. The numerical simulation results for system modeling show the presented online sparse LSSVM can save time and space, provide accurate prediction.
For non-bias LSSVM(NB-LSSVM), computing process to delete the least important sample or any one is given, and sparse online non-bias LSSVM (SONB-LSSVM) is designed. The skill for deleting sample can improve diversity and representative capacity of the training sample set. Compared with online non-bias LSSVM(ONB-LSSVM), SONB-LSSVM can study system properties in longer time horizon; generalization of SONB-LSSVM is less affected by the input signal frequency when it is employed for dynamic system online modeling.
Aiming at the problem that predicting accuracy of LSSVM is influenced easily by gross errors and noises overriding on measure value of plant output when LSSVM applied to the dynamic process online modeling directly, a robust online process modeling method using NB-LSSVM is presented after characteristics of samples sequence structure and of noises action are analyzed. Abnormal measure data are recognized and eliminated, and measure data carrying noises are detected and rectified according to relation between predicting error and set threshold value during per predicting period, consequently less noises enter into samples, and obtained online LSSVM can track dynamics of process better. The modeling method is robust, can decrease effect of gross error and Gaussian white noise on LSSVM predicting accuracy and improve predicting accuracy. The numerical simulation shows the validity and advantage of the method.
To tackle the difficulty in setting the kernel parameter and in adjusting it to varying process employing LSSVM to identify time-varying nonlinear process online, an online process identification approach based on LSSVM using regulated kernel parameter during different term is proposed in this paper. Three LSSVMs are utilized and the whole modeling predicting times is divided into starting stage and working periods, at the end of which the LSSVM with smallest sum of predicting error is selected as working LSSVM for successive working period, and kernel parameters are reset for other two LSSVM according to heuristic rules and they become comparative LSSVMs during the following working period. The method is easy to set kernel parameters and has adjustability to a certain extent. The numerical simulation shows the adaptability of the method is better than that of traditional method statistically.
For the predictive control of nonlinear systems, a constrained single-step-ahead predictive control(PC) algorithm is proposed utilizing SONB-LSSVM. The presented control algorithm uses SONB-LSSVM to model online and to forecost the plant output value; the control values are obtained by the rolling optimization of particle swarm optimization(PSO) or Brent optimization. During starting stage of control, the control values are calculated through proportional-integral-derivative(PID) controller, and the plant input-output values are acquired to form samples and to train initial NB-LSSVM incrementally, which facilitates the usage of the presented PC. Because SONB-LSSVM can study new dynamic properties of process in time, the predictive control strategy possesses excellent adaptation. Simulation results of liquid-level process control, continuous stirred tank reactor(CSTR) concentration control and temperature control of water tank show the validity of the predictive control algorithm.
The applications of SONB-LSSVM and ONB-LSSVM in nonlinear inverse control are investigated. An adaptive direct inverse control is proposed utilizing SONB-LSSVM. SONB-LSSVM is used to build the nonlinear inverse model for the controlled object and to compute control value during per control period. The approach is suit for time-invariant ivertible system and one with parameters(or properties) time-varying slightly. For the invertible system with parameters(or properties) time-varying in wide range, a composite control strategy combining ONB-LSSVM inverse model with PID control is designed. During per controlling period, control signal is compounded from inverse control value and PID control value, and compound ratio of them changes automatically. Inverse control plays more important role in high frequency range, and PID control does so in low frequency range. Consequently, the control approach possesses wide frequency adaptation range and good control performances to time-varying systems. In both methods the initial NB-LSSVM is trained online, which facilitates the usage of the proposals. Simulation results indicate the feasibility and validity of the control methods.
主要研究内容包括：
在分析现有基于分块矩阵求逆公式的LSSVM增量式和减量式学习算法的基础上,为这些算法设计了一些加速实现策略,得到LSSVM快速在线式学习算法。基于这种快速LSSVM的多输入多输出(Multiple-Input Multiple-Output, MIMO)系统在线建模仿真显示这些加速策略能明显提高原有在线算法执行速度；加速策略需要的空间为少数几个缓存数组。
为了减少LSSVM基本剪枝算法的计算量,提出了一种快速剪枝算法。在分析剪枝前后两个LSSVM对应线性方程组系数矩阵之间关系的基础上,利用对调单位矩阵两行(或列)所得的初等方阵的逆等于其自身的性质和分块矩阵求逆公式,导出这两个系数矩阵的子阵的逆之间的递推关系,避免剪枝过程中多次进行高阶矩阵求逆,从而减少计算量。在不考虑递推计算误差时,该算法理论上得出与基本剪枝算法相同结果的稀疏LSSVM。仿真实验表明该算法比基本剪枝算法速度快,而且初始训练样本越多,加速比越大。
为了减少带时间窗型在线LSSVM的计算量和存储空间,提出了一种带时间窗型在线稀疏LSSVM。它利用滑动时间窗中部分时刻的样本作为训练样本集。新时刻的样本总是加入训练样本集；每次删除样本时,若滑动时间窗最前端时刻的样本在训练样本集中,则删除它,否则从训练样本集中选择留一法预测误差最小的样本删除。与现有的在线LSSVM相比,该在线稀疏LSSVM能用较少的样本学习系统较多的特性,能提高时空效率；系统建模仿真实验表明,本文在线稀疏LSSVM能节省时间和空间,具有较高的预测精度。
针对无偏置最小二乘支持向量机(non-bias LSSVM, NB-LSSVM),给出了删除贡献最小(或任意)样本的计算过程,并设计了稀疏在线无偏置最小二乘支持向量机(sparse online non-bias LSSVM, SONB-LSSVM).该样本删除技巧能提高学习样本集的多样性和代表性；与在线无偏置最小二乘支持向量机(online non-bias LSSVM, ONB-LSSVM)相比,SONB-LSSVM能学习系统更长时间范围内的特性；用于动态系统在线建模时,其泛化能力受输入信号频率影响较小。
针对直接利用LSSVM对动态过程在线建模时预测精度易受过程输出测量值上的粗大误差和噪声影响的问题,在分析样本序列结构特征和噪声作用特征基础上,提出一种基于NB-LSSVM的稳健在线过程建模方法。该方法在每一预测周期中根据预测误差与设定阈值之间的关系来识别和剔除异常测量值、识别和修正含噪声测量值,从而降低样本中的噪声,使得出的LSSVM较好地跟踪过程的动态特性。这种在线过程建模方法具有稳健性,能减少输出值上粗大误差和高斯白噪声对LSSVM预测精度的影响,提高预测精度。数字仿真显示了本文方法的有效性和优越性。
利用LSSVM在线辨识时变非线性过程时,设定其核参数较困难,设定的核参数不能适应过程变化而进行自动调节；针对此问题,提出了一种基于核参数分时段调节型LSSVM的在线过程辨识方法。该方法利用三个LSSVM,并将整个建模预测时期分为启动阶段和若干个工作周期,初始阶段末和每个工作周期末选定预测误差累加值最小的LSSVM作为后续工作周期的工作LSSVM,同时根据启发式规则为另两个LSSVM重新设定核参数值,它们作为后续工作周期的比较LSSVM.该方法设定核参数相对容易,而且核参数具有一定的自动调节能力。数字仿真显示,从统计角度而言,本文方法比传统方法有更好的适应性。
对于非线性系统预测控制问题,提出了一种基于SONB-LSSVM的有约束单步预测控制算法。该预测控制利用SONB-LSSVM在线建模预测系统输出,滚动优化环节采用粒子群优化(Particle Swarm Optimization,PSO)或Brent优化搜索非线性系统的控制量。由于SONB-LSSVM能及时学习过程新动态特性,该预测控制方法具有良好的自适应能力。在控制启动阶段利用比例-积分-微分(proportional-integral-derivative,PID)控制,同时采集过程输入输出数据构成样本逐步生成初始的NB-LSSVM,该控制方法使用方便。液位控制、连续搅拌反应釜浓度控制和水浴温度控制等仿真表明该预测控制方法是有效的。
探讨了SONB-LSSVM和ONB-LSSVM在非线性逆控制中的应用。提出基于SONB-LSSVM的自适应直接逆控制方法,该方法利用SONB-LSSVM对被控对象在线逆建模并计算控制量,该控制方法适合控制定常或参数(或特性)轻微变化的可逆对象。对于运行中参数(或特性)在较大范围中变化的可逆对象,提出一种ONB-LSSVM逆模型与PID相结合的自适应复合控制,在每个控制周期,控制信号由逆控制量与PID控制量合成,合成比例自动调节。逆控制在高频段发挥较大作用,PID控制在低频段发挥较大作用。因此,该复合控制方法有较宽的频率适应范围,并且对时变系统有较好控制效果。两种控制方法中初始NB-LSSVM在线生成,因此这两种控制方法使用方便。数字仿真表明了这两种控制方法的可行性和有效性。
Some algorithms about least squares support vector machines(LSSVM) and applying methods for system online modeling and control based on LSSVM are studied in this dissertation.
The main contents are outlined as follows:
The existing LSSVM incremental and decremental learning algorithms by the formula of solving inversion of partitioned matrix are analyzed, and some speeding up implementing tactics are designed for the algorithms; consequently, a fast online LSSVM learning algorithm is obtained. The numerical simulation of multiple-input multiple-output(MIMO) system online modeling employing the fast online LSSVM shows these speedy tactics can heighten time performance of the existing LSSVM online learning algorithm; the tactics only require space of several additional buffer arrays.
To reduce the computation amount of basic pruning algorithm(BPA) for LSSVM, a fast pruning algorithm(FPA) is proposed. The connection between two coefficient matrices of linear equations corresponding to LSSVM before pruning and to one after doing is analyzed, and the recursive relation between inversions of sub-matrices of the two coefficient matrices is derived by property of elementary matrix attained through swapping two columns or two rows that its inversion equalling to itself and by the calculation formula of solving inversion of partitioned matrix, therefore repeat calculation of inversion of higher order matrices is avoided in pruning process, and computation amount is decreased. FPA products the same resultant sparse LSSVM as BPA theoretically, when recursive calculating error is neglected. The numerical simulation results show the presented algorithm is quicker than BPA, and the more the training samples, the greater FPA's speedup rate to BPA is.
To reduce the computation time and storage space of online LSSVM with time window, an online sparse LSSVM with time window is proposed. It only takes samples ranking at partial moments among sliding time window as training samples set(TSS). The new sample is learned necessarily; when sample elimination is performed, if the sample ranking at the oldest moment among sliding time window exists in TSS, then it will be removed during decremental learning, otherwise, the sample with the smallest leave-one-out predicting error among TSS is selected and deleted. Compared with the existing online LSSVM, the presented online sparse LSSVM can learn more characteristic of system using less samples, and heighten time-space efficiency; compared with the existing online spare LSSVM, it can get rid of the obsolete sample, and adapt to time-variant properties of system better. The numerical simulation results for system modeling show the presented online sparse LSSVM can save time and space, provide accurate prediction.
For non-bias LSSVM(NB-LSSVM), computing process to delete the least important sample or any one is given, and sparse online non-bias LSSVM (SONB-LSSVM) is designed. The skill for deleting sample can improve diversity and representative capacity of the training sample set. Compared with online non-bias LSSVM(ONB-LSSVM), SONB-LSSVM can study system properties in longer time horizon; generalization of SONB-LSSVM is less affected by the input signal frequency when it is employed for dynamic system online modeling.
Aiming at the problem that predicting accuracy of LSSVM is influenced easily by gross errors and noises overriding on measure value of plant output when LSSVM applied to the dynamic process online modeling directly, a robust online process modeling method using NB-LSSVM is presented after characteristics of samples sequence structure and of noises action are analyzed. Abnormal measure data are recognized and eliminated, and measure data carrying noises are detected and rectified according to relation between predicting error and set threshold value during per predicting period, consequently less noises enter into samples, and obtained online LSSVM can track dynamics of process better. The modeling method is robust, can decrease effect of gross error and Gaussian white noise on LSSVM predicting accuracy and improve predicting accuracy. The numerical simulation shows the validity and advantage of the method.
To tackle the difficulty in setting the kernel parameter and in adjusting it to varying process employing LSSVM to identify time-varying nonlinear process online, an online process identification approach based on LSSVM using regulated kernel parameter during different term is proposed in this paper. Three LSSVMs are utilized and the whole modeling predicting times is divided into starting stage and working periods, at the end of which the LSSVM with smallest sum of predicting error is selected as working LSSVM for successive working period, and kernel parameters are reset for other two LSSVM according to heuristic rules and they become comparative LSSVMs during the following working period. The method is easy to set kernel parameters and has adjustability to a certain extent. The numerical simulation shows the adaptability of the method is better than that of traditional method statistically.
For the predictive control of nonlinear systems, a constrained single-step-ahead predictive control(PC) algorithm is proposed utilizing SONB-LSSVM. The presented control algorithm uses SONB-LSSVM to model online and to forecost the plant output value; the control values are obtained by the rolling optimization of particle swarm optimization(PSO) or Brent optimization. During starting stage of control, the control values are calculated through proportional-integral-derivative(PID) controller, and the plant input-output values are acquired to form samples and to train initial NB-LSSVM incrementally, which facilitates the usage of the presented PC. Because SONB-LSSVM can study new dynamic properties of process in time, the predictive control strategy possesses excellent adaptation. Simulation results of liquid-level process control, continuous stirred tank reactor(CSTR) concentration control and temperature control of water tank show the validity of the predictive control algorithm.
The applications of SONB-LSSVM and ONB-LSSVM in nonlinear inverse control are investigated. An adaptive direct inverse control is proposed utilizing SONB-LSSVM. SONB-LSSVM is used to build the nonlinear inverse model for the controlled object and to compute control value during per control period. The approach is suit for time-invariant ivertible system and one with parameters(or properties) time-varying slightly. For the invertible system with parameters(or properties) time-varying in wide range, a composite control strategy combining ONB-LSSVM inverse model with PID control is designed. During per controlling period, control signal is compounded from inverse control value and PID control value, and compound ratio of them changes automatically. Inverse control plays more important role in high frequency range, and PID control does so in low frequency range. Consequently, the control approach possesses wide frequency adaptation range and good control performances to time-varying systems. In both methods the initial NB-LSSVM is trained online, which facilitates the usage of the proposals. Simulation results indicate the feasibility and validity of the control methods.
引文
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