一类非线性系统的参数辨识
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摘要
非线性系统在实际生活中广泛存在，例如通信系统，化工过程，生物医药等，因此非线性系统辨识研究具有重要的理论意义和实用价值。本文以国家自然科学基金项目为背景，提出了输入非线性系统辨识的课题。本文基于多新息辨识理论，辅助模型辨识思想，迭代辨识原理，梯度搜索原理和牛顿辨识方法研究输入非线性系统的辨识问题，并取得如下的研究成果。
1利用梯度搜索原理和多新息辨识理论，研究输入非线性有限脉冲系统(Hammerstein—FIR)的辨识问题，推导针对此模型的投影算法，随机梯度算法和多新息随机梯度算法来辨识参数，应用数字仿真对算法进行比较和分析。
2研究表明，根据牛顿方法推导基于输入非线性有限脉冲系统的牛顿递推和牛顿迭代算法。对于无噪声的系统牛顿递推算法有很好的辨识效果，但对于有噪声的系统辨识会产生波动。而牛顿迭代算法对于有噪声与无噪声的系统都有良好的辨识效果。
3考虑有色噪声的输入非线性系统(Hammerstein—cARMA)，通过梯度搜索原理推导增广投影算法，简化的增广投影算法和增广随机梯度算法，对系统参数进行辨识，用残差来替代不可测噪声项．投影算法对噪声的影响非常敏感而增广随机梯度算法的收敛速度慢且估计误差大，但在增广随机梯度算法中加入遗忘因子后算法会得到明显改善．进一步通过仿真来比较算法的估计误差以及收敛速度。为加以比较推导增广牛顿递推算法和增广牛顿迭代算法，仿真证明牛顿迭代算法可以得到高精度的参数估计。
4根据梯度搜索原理，辅助模型辨识思想，牛顿方法和迭代辨识原理，推导出输入非线性输出误差系统(Hammerstein一0E)的辅助模型投影算法，辅助模型随机梯度算法，辅助模型牛顿递推算法和辅助模型牛顿迭代算法，并给出相应的仿真例子。
论文最后给出了结论和展望，并对本课题的研究所面临的一些困难和有待深入研究的问题做了简单介绍，文中所给出的几种基于输入非线性系统的辨识算法的收敛性有待进一步证明。
Nonlinear systems widely exist in practical applications, like communication systems,chemical processes, biomedical systems and so on. Therefore, nonlinear systems identifica-tion is quite significant both in theory and application. This thesis presents the identificationalgorithms for a class of nonlinear systems based on the National Natural Science Foundationof China. By employing the multiinnovation principle, the auxiliary model identification, theiterative identification principle, the gradient search principle and the Newton method, theidentification of nonlinear systems are studied, the innovation research results are as follows:
1. The identification problem of FIR systems with input nonlinearity is investigated. Bymeans of the gradient search principle and the multiinnovation principle, the projectionalgorithm, the stochastic gradient algorithm and the multiinnovation stochastic gradientalgorithm are derived, where the combined parameters in the system model are identifiedseparately. Further, the simulations are carried out for comparison and analysis.
2. In order to reduce the sensitivity of the projection algorithm to noise, and to improve theconvergence rate of the stochastic gradient algorithm, a Newton recursive identificationalgorithm and a Newton iterative identification algorithm are derived by using the Newtonmethod. The simulation results show that the Newton iterative algorithm performs muchbetter for nonlinear systems with noise than the other algorithms.
3. Considering the identification of the input nonlinear systems with the colored noise, anextended projection algorithm, a simplified extended projection algorithm, and an extendedstochastic gradient algorithm by gradient search principle are derived to identify systemparameter with the residuals instead of the unpredictable noises. Since the projectionalgorithm is very sensitive to the noise and the extended stochastic gradient algorithmhas slow convergence rate and undesirable estimation accuracy, by introducing a forgettingfactor to the extended stochastic gradient algorithm, the convergence rate of the algorithmis faster. An extended Newton recursive algorithm and an extended Newton interactivealgorithm are derived for comparison. In the simulation, the results show that the Newtoniterative algorithm can get better accurate parameter estimates.
4. According to the gradient search principle, the auxiliary model identification, the Newtonmethod and iterative identification, the auxiliary model projection, the auxiliary modelstochastic gradient, the auxiliary model Newton recursive and the auxiliary model Newton iterative identification algorithms are derived for input nonlinear output error type systems.The simulation examples test the proposed algorithms and compare with each other. Theresults show that the auxiliary model Newton iterative identification algorithm works quitewell.
The main constructive algorithms and results of the dissertation are concluded, and theidentification di?culties for nonlinear systems and further research topics in this area are dis-cussed.
1利用梯度搜索原理和多新息辨识理论，研究输入非线性有限脉冲系统(Hammerstein—FIR)的辨识问题，推导针对此模型的投影算法，随机梯度算法和多新息随机梯度算法来辨识参数，应用数字仿真对算法进行比较和分析。
2研究表明，根据牛顿方法推导基于输入非线性有限脉冲系统的牛顿递推和牛顿迭代算法。对于无噪声的系统牛顿递推算法有很好的辨识效果，但对于有噪声的系统辨识会产生波动。而牛顿迭代算法对于有噪声与无噪声的系统都有良好的辨识效果。
3考虑有色噪声的输入非线性系统(Hammerstein—cARMA)，通过梯度搜索原理推导增广投影算法，简化的增广投影算法和增广随机梯度算法，对系统参数进行辨识，用残差来替代不可测噪声项．投影算法对噪声的影响非常敏感而增广随机梯度算法的收敛速度慢且估计误差大，但在增广随机梯度算法中加入遗忘因子后算法会得到明显改善．进一步通过仿真来比较算法的估计误差以及收敛速度。为加以比较推导增广牛顿递推算法和增广牛顿迭代算法，仿真证明牛顿迭代算法可以得到高精度的参数估计。
4根据梯度搜索原理，辅助模型辨识思想，牛顿方法和迭代辨识原理，推导出输入非线性输出误差系统(Hammerstein一0E)的辅助模型投影算法，辅助模型随机梯度算法，辅助模型牛顿递推算法和辅助模型牛顿迭代算法，并给出相应的仿真例子。
论文最后给出了结论和展望，并对本课题的研究所面临的一些困难和有待深入研究的问题做了简单介绍，文中所给出的几种基于输入非线性系统的辨识算法的收敛性有待进一步证明。
Nonlinear systems widely exist in practical applications, like communication systems,chemical processes, biomedical systems and so on. Therefore, nonlinear systems identifica-tion is quite significant both in theory and application. This thesis presents the identificationalgorithms for a class of nonlinear systems based on the National Natural Science Foundationof China. By employing the multiinnovation principle, the auxiliary model identification, theiterative identification principle, the gradient search principle and the Newton method, theidentification of nonlinear systems are studied, the innovation research results are as follows:
1. The identification problem of FIR systems with input nonlinearity is investigated. Bymeans of the gradient search principle and the multiinnovation principle, the projectionalgorithm, the stochastic gradient algorithm and the multiinnovation stochastic gradientalgorithm are derived, where the combined parameters in the system model are identifiedseparately. Further, the simulations are carried out for comparison and analysis.
2. In order to reduce the sensitivity of the projection algorithm to noise, and to improve theconvergence rate of the stochastic gradient algorithm, a Newton recursive identificationalgorithm and a Newton iterative identification algorithm are derived by using the Newtonmethod. The simulation results show that the Newton iterative algorithm performs muchbetter for nonlinear systems with noise than the other algorithms.
3. Considering the identification of the input nonlinear systems with the colored noise, anextended projection algorithm, a simplified extended projection algorithm, and an extendedstochastic gradient algorithm by gradient search principle are derived to identify systemparameter with the residuals instead of the unpredictable noises. Since the projectionalgorithm is very sensitive to the noise and the extended stochastic gradient algorithmhas slow convergence rate and undesirable estimation accuracy, by introducing a forgettingfactor to the extended stochastic gradient algorithm, the convergence rate of the algorithmis faster. An extended Newton recursive algorithm and an extended Newton interactivealgorithm are derived for comparison. In the simulation, the results show that the Newtoniterative algorithm can get better accurate parameter estimates.
4. According to the gradient search principle, the auxiliary model identification, the Newtonmethod and iterative identification, the auxiliary model projection, the auxiliary modelstochastic gradient, the auxiliary model Newton recursive and the auxiliary model Newton iterative identification algorithms are derived for input nonlinear output error type systems.The simulation examples test the proposed algorithms and compare with each other. Theresults show that the auxiliary model Newton iterative identification algorithm works quitewell.
The main constructive algorithms and results of the dissertation are concluded, and theidentification di?culties for nonlinear systems and further research topics in this area are dis-cussed.
引文
1胡寿松．自动控制原理[Ml_北京：科学出版社，2001
2胡跃明．非线性控制系统理论与应用(第二版)[M]北京：国防工业出版社，2001
3静大海，刘晓下．一种非线性模型的在线辨识方法[J]控制工程，2007，14(5)：482-484
4袁晓磊，白焰，董玲．基于遗传编程的非线性系统辨识[J]控制工程，2009，16(1)：52-55
5 Anorld C R，Narendra K S The Characterization anddentificationof Systems[R]Technical Report471，Harvard University,1965
6 Ding F，Chen T Identification of Hammerstein nonlinear ARMAX systems [J] Automatica，2005，41(9)：1479-1489
7 Ding F，Shi Y，Chen T Gradient-based identification methods for Hammerstein nonlinear AR-MAX models [J] Nonlinear Dynamics，2006，45(1-2)：31-43
8向微，陈宗海．基于Hammerstein模型描述的非线性系统辨识新方法[J]控制理论与应用，2007，24(1)：143-147
9 Narendra K S．Gallman P G An iterative method for the identification of nonlinear systems usinga Hammerstein model [J]_IEEE Transactions on Automatic Control，1996，11(3)：546-550
10 Stoica P On the convergence of an iterative algorithm used for Hammerstein system identification[J]_IEEE Transactions on Automatic Control，1981，26(4)：967-969
11 Rangan S，Wolodkin G，Poolla K Identification methods for Hammerstein systems[C]Proceed-ings of Control and Decision，New Orleans，1995，697-702
12 Chang F，Luus R A noniterative method for identification using Hammerstein model[J]_ IEEETransactions on Automatic Control，1971，16(5)：464-468
13 Cerone V，Regruto D Parameter bounds for discrete-time Hammerstein models with boundedoutput errors[J]_IEEE Transaction on Automatic Control，2003，48(10)：1855-1860
14 Bai E W，Li D Convergence of the iterative Hammerstein system identification algorithm [J]IEEE Transactions on Automatic Control，2004，49(11)：1929-1940
15 Vanbeylen L，Pintelon R，Schoukens J Blind maximum likelihood identification of Hammersteinsystems[J] Automatica，2008，44(12)：3139-3146
16 Giri F，Rochdi Y，Chaoui F Z，Brouri A Identification of Hammerstein systems in presence ofhysteresis-backlash and hysteresis-relay nonlinearities[J]_Automatica，2008，44(3)：767-775
17 Bai E W Identification of linear systems with hard inputn。nlinearities of known structure[J]Automatica，2002，38(5)：853-860
18 Golub G．Pereyra V Separable nonlinear least squares：the variable projection method and itsapplications [J] Inverse Problems，2003，19(2)：R1一R26
19 Bruls J，Chou C T，Haverkamp B，Verhaegen M Linear and nonlinear system identification usingseparable least squares [J] European Journal of Control，1999，5(11)：116-128
20 Westwick D，Kearney R Separable least squares identification of nonlinear Hammerstein models：Application to stretch reflex dynamics[J]_Annals ofBiomedical Engineering，2001，29(8)：707-718
21丁锋．系统辨识理论与方法+Matlab仿真[M]_北京：电力出皈社，2010
22 Ding F，Chen T Hierarchical gradient-based identification of multivariable discrete-time systems[J] Automatica，2005，41(2)：315-325
23丁锋．基于输出估计的多输入系统随机梯度估计算法[J]南京信息工程大学学报(自然科学版)，2010，2(6)：481-488
24方崇智，萧德云．过程辨识[Ml_北京：清华大学出皈社，1988
25谢新民，丁锋．自适应控制系统[M]_北京：清华大学出版社，2002
26 VOrOs J Parameter identification of discontinuousHammerstein systems[J]_Automatica，1997，33(61：1141-1146
27 Stoica P．S6derstr6m T Instrumental variable methods for identification of Hammerstein systems[J] International Journal of Control，1982，35(3)：459-476
28丁锋．系统辨识(1)：辨识导引[J]南京信息工程大学学报(自然科学皈)，2011，3(1)：1-22
29 Ding F，Chen T Combined parameter and output estimation of dual-rate systems using an aux-iliary model [J] Automatica，2004，40(10)：1739-1748
30丁锋，萧德云，丁韬．多新息随机梯度辨识方法[J]_控制理论与应用，2003，20(6)：870-874
31 Ding F，Chen T Parameter estimation of dual-rate stochastic systems by using an output errormethod [J] IEEE Transactions on Automatic Control，2005，50(9)：1436-1441
32丁锋，谢新民．传递函数阵子子模型参数递推估计：辅助模型方法[J]控制与决策，1991，6(6)：447-452
33丁锋，陈通文．多率系统模型与辨识[J]_自动化学报，2005，31(1)：105-122
34 Ding F，Chen T Identification of dual-rate systems based on finite impulse response models[J]_International Journal of Adaptive Control and Signal Processing，2004，18(7)：589-598
35 Ding F．Shi Y．Chen T Auxiliary model based least-squares identification methods for Hammer-stein output-errpr systems [J] Systems & Control Letters，2007，56(5)：373-380
36丁锋，杨家本．大系统的递阶辨识[J]_自动化学报，1999，25(5)：647-654
37 Ding F，Chen H B，Li M Multi-innovation least squares identification methods based on theauxiliary model for MISO systems [J] Applied Mathematics and Computation，2007，187(2)：658-668
38丁锋，丁韬．时变多变量系统多新息投影算法的均方收敛性[J]湖北工学院学报，2001，
16f41：16-20
39 Titteringto D M，Kitsos C P Recent Advances in Nonlinear Experimental Design [J] Technomet-rices，1989，(31)：49-60
40 Liu Y，Bai E W Iterative identification of Hammerstein systems [J] Automatica，2007，43(2)：346-354
41范伟，丁锋．Hammerstein非线性系统参数估计分离的三种方法[J]科学技术与工程，2008，8(6)：586-1589
42 Ding F，Liu P X，Liu G Identification methods for Hammerstein nonlinear systems[J] DigitalSignal Processing，2011，21(2)：215-238
43 Goodwin G C，Sin K S Adaptive Filtering，Prediction and Control[M]Englewood Cliffs，New"Jersey：Prentice-Hall，1984
2胡跃明．非线性控制系统理论与应用(第二版)[M]北京：国防工业出版社，2001
3静大海，刘晓下．一种非线性模型的在线辨识方法[J]控制工程，2007，14(5)：482-484
4袁晓磊，白焰，董玲．基于遗传编程的非线性系统辨识[J]控制工程，2009，16(1)：52-55
5 Anorld C R，Narendra K S The Characterization anddentificationof Systems[R]Technical Report471，Harvard University,1965
6 Ding F，Chen T Identification of Hammerstein nonlinear ARMAX systems [J] Automatica，2005，41(9)：1479-1489
7 Ding F，Shi Y，Chen T Gradient-based identification methods for Hammerstein nonlinear AR-MAX models [J] Nonlinear Dynamics，2006，45(1-2)：31-43
8向微，陈宗海．基于Hammerstein模型描述的非线性系统辨识新方法[J]控制理论与应用，2007，24(1)：143-147
9 Narendra K S．Gallman P G An iterative method for the identification of nonlinear systems usinga Hammerstein model [J]_IEEE Transactions on Automatic Control，1996，11(3)：546-550
10 Stoica P On the convergence of an iterative algorithm used for Hammerstein system identification[J]_IEEE Transactions on Automatic Control，1981，26(4)：967-969
11 Rangan S，Wolodkin G，Poolla K Identification methods for Hammerstein systems[C]Proceed-ings of Control and Decision，New Orleans，1995，697-702
12 Chang F，Luus R A noniterative method for identification using Hammerstein model[J]_ IEEETransactions on Automatic Control，1971，16(5)：464-468
13 Cerone V，Regruto D Parameter bounds for discrete-time Hammerstein models with boundedoutput errors[J]_IEEE Transaction on Automatic Control，2003，48(10)：1855-1860
14 Bai E W，Li D Convergence of the iterative Hammerstein system identification algorithm [J]IEEE Transactions on Automatic Control，2004，49(11)：1929-1940
15 Vanbeylen L，Pintelon R，Schoukens J Blind maximum likelihood identification of Hammersteinsystems[J] Automatica，2008，44(12)：3139-3146
16 Giri F，Rochdi Y，Chaoui F Z，Brouri A Identification of Hammerstein systems in presence ofhysteresis-backlash and hysteresis-relay nonlinearities[J]_Automatica，2008，44(3)：767-775
17 Bai E W Identification of linear systems with hard inputn。nlinearities of known structure[J]Automatica，2002，38(5)：853-860
18 Golub G．Pereyra V Separable nonlinear least squares：the variable projection method and itsapplications [J] Inverse Problems，2003，19(2)：R1一R26
19 Bruls J，Chou C T，Haverkamp B，Verhaegen M Linear and nonlinear system identification usingseparable least squares [J] European Journal of Control，1999，5(11)：116-128
20 Westwick D，Kearney R Separable least squares identification of nonlinear Hammerstein models：Application to stretch reflex dynamics[J]_Annals ofBiomedical Engineering，2001，29(8)：707-718
21丁锋．系统辨识理论与方法+Matlab仿真[M]_北京：电力出皈社，2010
22 Ding F，Chen T Hierarchical gradient-based identification of multivariable discrete-time systems[J] Automatica，2005，41(2)：315-325
23丁锋．基于输出估计的多输入系统随机梯度估计算法[J]南京信息工程大学学报(自然科学版)，2010，2(6)：481-488
24方崇智，萧德云．过程辨识[Ml_北京：清华大学出皈社，1988
25谢新民，丁锋．自适应控制系统[M]_北京：清华大学出版社，2002
26 VOrOs J Parameter identification of discontinuousHammerstein systems[J]_Automatica，1997，33(61：1141-1146
27 Stoica P．S6derstr6m T Instrumental variable methods for identification of Hammerstein systems[J] International Journal of Control，1982，35(3)：459-476
28丁锋．系统辨识(1)：辨识导引[J]南京信息工程大学学报(自然科学皈)，2011，3(1)：1-22
29 Ding F，Chen T Combined parameter and output estimation of dual-rate systems using an aux-iliary model [J] Automatica，2004，40(10)：1739-1748
30丁锋，萧德云，丁韬．多新息随机梯度辨识方法[J]_控制理论与应用，2003，20(6)：870-874
31 Ding F，Chen T Parameter estimation of dual-rate stochastic systems by using an output errormethod [J] IEEE Transactions on Automatic Control，2005，50(9)：1436-1441
32丁锋，谢新民．传递函数阵子子模型参数递推估计：辅助模型方法[J]控制与决策，1991，6(6)：447-452
33丁锋，陈通文．多率系统模型与辨识[J]_自动化学报，2005，31(1)：105-122
34 Ding F，Chen T Identification of dual-rate systems based on finite impulse response models[J]_International Journal of Adaptive Control and Signal Processing，2004，18(7)：589-598
35 Ding F．Shi Y．Chen T Auxiliary model based least-squares identification methods for Hammer-stein output-errpr systems [J] Systems & Control Letters，2007，56(5)：373-380
36丁锋，杨家本．大系统的递阶辨识[J]_自动化学报，1999，25(5)：647-654
37 Ding F，Chen H B，Li M Multi-innovation least squares identification methods based on theauxiliary model for MISO systems [J] Applied Mathematics and Computation，2007，187(2)：658-668
38丁锋，丁韬．时变多变量系统多新息投影算法的均方收敛性[J]湖北工学院学报，2001，
16f41：16-20
39 Titteringto D M，Kitsos C P Recent Advances in Nonlinear Experimental Design [J] Technomet-rices，1989，(31)：49-60
40 Liu Y，Bai E W Iterative identification of Hammerstein systems [J] Automatica，2007，43(2)：346-354
41范伟，丁锋．Hammerstein非线性系统参数估计分离的三种方法[J]科学技术与工程，2008，8(6)：586-1589
42 Ding F，Liu P X，Liu G Identification methods for Hammerstein nonlinear systems[J] DigitalSignal Processing，2011，21(2)：215-238
43 Goodwin G C，Sin K S Adaptive Filtering，Prediction and Control[M]Englewood Cliffs，New"Jersey：Prentice-Hall，1984