多率系统多尺度融合及不确定系统稳定性分析
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摘要
在多速率信号处理当中,往往要求有多个传感器同时在不同尺度上对研究的现象或过程进行观测。怎样将不同类型、不同尺度上的传感器获得的信息进行有效的综合是目前普遍关注的工作,其中多尺度分析和多尺度建模是其中的一个重要研究方向。
随着多尺度系统理论的不断发展,将小波分析理论与Kalman滤波技术结合起来,并将它有效地用到对多速率动态系统的研究和应用当中也已受到相关领域科研技术工作者的关注。利用Kalman滤波的实时性以及小波变换的多分辨率分析能力可以在不同尺度上得到所研究对象的统计特性,由此可以推导出该对象的多尺度表示方法,进而获得高效、并行的迭代算法。实现上述目标的关键是要建立正确的多尺度动态模型,并且找出有效的数据融合算法,它是获取具有多尺度特征的数据分析和信号处理问题的重要方式。
本文利用新的分块技术与多尺度变换方法,建立一个动态系统基于时域与频域相结合的多尺度联合滤波器。本文主要是针对单传感器单模型以及多传感器(包括同步、采样率成任意整数倍关系)单模型的情形分别给出了递归的多尺度融合算法。在单传感器情况下,首先将时域中描述的状态方程和观测方程分别改写为块状态方程和块观测方程的;然后,利用多尺度变换技术对系统进行多尺度建模;最后,类似与经典Kalman滤波的估计步骤,利用新得到的状态方程与测量方程对状态进行估计。在多传感器情况下的多尺度估计与单传感器情形下的不同之处在于更新时的不同,其利用了序贯滤波的思想,建立了应用于动态系统的多尺度估计联合滤波器。这些混合滤波器均是实时的、递归的。
本文研究的另外一部分内容是利用估计误差协方差的限制问题来处理一类带有测量丢失以及时间延迟的离散随机系统。所研究系统的状态矩阵中同样存在不确定性,而带有丢失的测量用满足贝努利分布的二值序列来刻画。解决这类系统的关键是线性滤波器的设计问题,它需要能够使得在所有可允许参数不确定情况下,随机系统的误差状态是均方有界的,并且每个状态的估计误差方差分别小于给定的值。在对系统的求解过程中,系统的稳定性问题部分可以转化为代数矩阵不等式以及二次矩阵不等式的求解问题,通过解方程得到所研究系统稳定的一个充分条件。
In multi-rate signal processing, several sensors are used to observe the system, how to combine the different type sensors on different scales and how to utilize the observation to obtain the effective fusion is a research aspect, while multi-scale analysis and multi-scale modeling is an important research aspect.
With the development of the multiscale theory, it has been intensively suggested to combine the wavelet analysis with Kalman filter and use it to effectively estimate the multi-rate system research and application attract more and more attention. With the real time estimate property of Kalman filter and the multi-scale ability, new algorithm can be used to analyze the statistic property of object on different scales, based on which the multiscale form of the object is expressed to obtain the effective, parallel recursive algorithm. The key to realize the object is accurate multi-scale modeling and finding the effective fusion algorithm, which is an important manner to obtain the multi-scale data analysis and signal processing.
This paper use a new parting technology and multi-scale transform method to found a new dynamic system based on time-frequency domain. This paper gives the recursive multi-scale data fusion algorithm of the system with single sensor single model and multi-sensor single model, respectively. With single sensor, first, reformulate the original state and measurement dynamic systems in a new blocked data form, then the multi-scale model is founded based on the multi-transform, last, adopt the main idea of Kalman filter to develop the new algorithm. The difference of multi-sensor is that it combines the idea of sequential filtering to obtain optimal hybrid state estimator in the multiscale domain. These algorithms are real-time, recursive.
In the other part of this paper, we are concerned with a new control problem for uncertain discrete-time stochastic systems with time delay and missing measurements using the constrained method of estimate error covariance. The parameter uncertainties are allowed to be norm-bounded and enter into the state matrix. The probability of the occurrence of missing data is assumed to be known and assumed to be a Bernoulli distributed white sequence. The purpose of this problem is to design an output feedback controller such that, for all admissible parameter uncertainties and all possible incomplete observations, the system state of the closed-loop system is mean square bounded, and the steady-state variance of each state is not more than the individual prescribed upper bound. We show that the addressed problem can be solved by means of algebraic matrix inequalities and quadratic matrix inequalities and obtain an effective condition of the system stability.
随着多尺度系统理论的不断发展,将小波分析理论与Kalman滤波技术结合起来,并将它有效地用到对多速率动态系统的研究和应用当中也已受到相关领域科研技术工作者的关注。利用Kalman滤波的实时性以及小波变换的多分辨率分析能力可以在不同尺度上得到所研究对象的统计特性,由此可以推导出该对象的多尺度表示方法,进而获得高效、并行的迭代算法。实现上述目标的关键是要建立正确的多尺度动态模型,并且找出有效的数据融合算法,它是获取具有多尺度特征的数据分析和信号处理问题的重要方式。
本文利用新的分块技术与多尺度变换方法,建立一个动态系统基于时域与频域相结合的多尺度联合滤波器。本文主要是针对单传感器单模型以及多传感器(包括同步、采样率成任意整数倍关系)单模型的情形分别给出了递归的多尺度融合算法。在单传感器情况下,首先将时域中描述的状态方程和观测方程分别改写为块状态方程和块观测方程的;然后,利用多尺度变换技术对系统进行多尺度建模;最后,类似与经典Kalman滤波的估计步骤,利用新得到的状态方程与测量方程对状态进行估计。在多传感器情况下的多尺度估计与单传感器情形下的不同之处在于更新时的不同,其利用了序贯滤波的思想,建立了应用于动态系统的多尺度估计联合滤波器。这些混合滤波器均是实时的、递归的。
本文研究的另外一部分内容是利用估计误差协方差的限制问题来处理一类带有测量丢失以及时间延迟的离散随机系统。所研究系统的状态矩阵中同样存在不确定性,而带有丢失的测量用满足贝努利分布的二值序列来刻画。解决这类系统的关键是线性滤波器的设计问题,它需要能够使得在所有可允许参数不确定情况下,随机系统的误差状态是均方有界的,并且每个状态的估计误差方差分别小于给定的值。在对系统的求解过程中,系统的稳定性问题部分可以转化为代数矩阵不等式以及二次矩阵不等式的求解问题,通过解方程得到所研究系统稳定的一个充分条件。
In multi-rate signal processing, several sensors are used to observe the system, how to combine the different type sensors on different scales and how to utilize the observation to obtain the effective fusion is a research aspect, while multi-scale analysis and multi-scale modeling is an important research aspect.
With the development of the multiscale theory, it has been intensively suggested to combine the wavelet analysis with Kalman filter and use it to effectively estimate the multi-rate system research and application attract more and more attention. With the real time estimate property of Kalman filter and the multi-scale ability, new algorithm can be used to analyze the statistic property of object on different scales, based on which the multiscale form of the object is expressed to obtain the effective, parallel recursive algorithm. The key to realize the object is accurate multi-scale modeling and finding the effective fusion algorithm, which is an important manner to obtain the multi-scale data analysis and signal processing.
This paper use a new parting technology and multi-scale transform method to found a new dynamic system based on time-frequency domain. This paper gives the recursive multi-scale data fusion algorithm of the system with single sensor single model and multi-sensor single model, respectively. With single sensor, first, reformulate the original state and measurement dynamic systems in a new blocked data form, then the multi-scale model is founded based on the multi-transform, last, adopt the main idea of Kalman filter to develop the new algorithm. The difference of multi-sensor is that it combines the idea of sequential filtering to obtain optimal hybrid state estimator in the multiscale domain. These algorithms are real-time, recursive.
In the other part of this paper, we are concerned with a new control problem for uncertain discrete-time stochastic systems with time delay and missing measurements using the constrained method of estimate error covariance. The parameter uncertainties are allowed to be norm-bounded and enter into the state matrix. The probability of the occurrence of missing data is assumed to be known and assumed to be a Bernoulli distributed white sequence. The purpose of this problem is to design an output feedback controller such that, for all admissible parameter uncertainties and all possible incomplete observations, the system state of the closed-loop system is mean square bounded, and the steady-state variance of each state is not more than the individual prescribed upper bound. We show that the addressed problem can be solved by means of algebraic matrix inequalities and quadratic matrix inequalities and obtain an effective condition of the system stability.
引文
[1] 文成林,周东华. 多尺度估计理论及其应用. 北京: 清华大学出版社, 2002
[2] 何友,王国宏,陆大鑫等.多传感器信息融合及应用.北京:电子工业出版社, 2000
[3] X.R.Li. Comparison of two measurement fusion methods for Kalman-Filter-Based multisensor data fusion. IEEE Transactions on Aerospace and Electronic Systems, 2001, 37(1):273~280
[4] 文成林, 吕冰, 葛泉波. 一种基于分步式滤波的数据融合算法. 电子学报, 2004, 32(8):1264~1267
[5] B. S. Rao, H. F. Durrant-Whyte. Fully decentralized algorithm for multisensor Kalman filtering. IEEE Proceedings-D, 1991, 138(5):413~420
[6] Kessler etc., Functional Description of data fusion Process, Tech Rep., Office of Naval Technol., Naval Air development Ctr., Waminster, PA, 1992
[7] 史忠科.最优估计的计算方法.北京:科学出版社, 2001
[8] Stephen B.HBruder. Heterogeneous multi-sensor integration for unstructured non-static robotic environments. PHD. Queen’s University, Kingston, Ontario, Canada. 1994
[9] S. Blackman and S. Multiple-target tracking with radar application. London: Artech House, Dedham, MA, 350-380, 1986
[10] Blackman S, Popoli R. Design and analysis of modem tracking systems. London: Artech House, 1999
[11] Nayebi, K., Barnwell, T. and M. J. T.Smith. A general time domain analysis and design framework for exactly reconstructing FIR analysis/synthesis filter banks. In Proceeding of IEEE ISCAS: 1990, 2022-2025
[12] 刘炯明.数据融合及应用.北京:国防工业出版社, 1999
[13] 何友,彭应宁.多级式多传感器信息融合中的状态估计.电子学报, 1998, 27(8):60-63
[14] 徐毓.雷达网数据融合问题研究.清华大学博士论文. 北京:清华大学, 2003
[15] A. Benveniste, R. Nikoukhah, A. S. willsky. Multiscale system theory. In: Proc. 29th IEEE Conference on Decision and Control, Honolulu, HI, 1990, 2484-2487
[16] M. Basseville, A. Benveniste, K.Chou, S. Golden, R. Nikoukhah, and A. S. Willsky. Modeling and Estimation of Multiresolution Stochastic Processes. IEEE Transactions on Information Theory. 1992, 38(2):766-784
[17] K. Chou, A. S. Willsky and R. Nikoukhah. Multiscale systems, Kalman filters, and Riccatic equations. IEEE Trans. Automatic Control. 1994, 39(3):479-492
[18] M. Luettgen, W. Karl, A. S. Willsky. Efficient Multiscale Regularization with Applications to the Computation of Optimal Flow. IEEE Transactions on Image Processing, 1993,3(1):41-64
[19] L. Hong. Multiresolution Filtering Using Wavelet Transform. IEEE Transactions on Aerospace and Electronic Systems. 29(4): 1993, 1244-1251
[20] L. Hong. Multi-resolutional distributed filtering. IEEE Tranactions on Automatic Control, 1994, 39(4): 853-856
[21] L. Hong. Multiresolutional Multiple-model Target Tracking. IEEE Transactions on Areospace Electrical Systems, 1994, 30(2): 1025-1035
[22] L. Hong, G. Chen and C. K. Chui. A Filter-Bank-Based Kalman Filtering Technique for Wavelet Estimation and Decomposition of Random Signals. IEEE Transactions on Circuits and Systems—II: Analog and Digital Signal Proecssing, 1998, 45(2):237-241
[23] 文成林, 潘泉, 张洪才, 戴冠中. 多传感器多模型动态系统多尺度分布式融合系统估计算法. 自动化学报. 2000, 6(增刊 B): 66-70
[24] 文成林, 周东华, 潘泉, 张洪才. 多尺度动态模型单传感器的动态系统分布式信息融合, 自动化学报, 2001, 27(2): 158-165
[25] L. Zhang, Q. Pan, P. Bao and H. Zhang. The Discrete Kalman Filtering of a Class of Dynamic Multi-scale Systems, IEEE Transactions on Circults and systems- II:Analogand Digital Signal Processing, 2002, 49(10):668-676
[26] L. Zhang, X. Wu, Q. Pan and H. Zhang. Multiresolution modeling and estimation of multisensor data. IEEE Transactions on Signal Processing 2004, 52 (11):3170-3182
[27] 崔培玲, 潘泉, 张磊, 张洪才. 一类动态多尺度系统的模型和估计算法. 系统仿真学报. 2004, 16(3): 405-410
[28] 崔培玲, 潘泉, 张磊, 张洪才. 基于 Haar 小波的一类动态多尺度系统最优估计算法. 西北工业大学学报, 2004, 22(2):149-152
[29] 崔培玲, 潘泉, 张磊, 张洪才. 非线性状态方程约束下的一类动态多尺度系统建模和估计算法. 计算机工程与应用. 2004, 40(3): 23-25
[30] L. Yan, B. Liu and D. Zhou. The modeling and estimation of asynchronous multirate multisensor dynamic systems, Aerospace Science and Technology. 2006, 10(1):63-71
[31] R.E. Kalman, A new approach to linear filtering and prediction problems, ASME J. Basic Eng., 1960, 82 :34-45
[32] R.E. Kalman and R.S. Bucy, New Results in Linear Filtering and Prediction Theory, ASME J. Basic Eng., 1961, 80:193-196
[33] I. Daubechies. Ten Lectures on Wavelets, Philadelphia: SIAM, 1992
[34] G. Strang, T. Nguyen. Wavelets and Filter Banks. USA: Wellesley-Cambridge, 1997
[35] Donald B. Percival and Andrew T. Walden. Wavelet methods for time series analysis. London: Cambridge, 2000
[36] Albert Boggess and Francis J. Narcowich. 小波分析与傅立叶分析基础. 北京: 电子工业出版社, 2002
[37] R. G. Agniel and E. I. Jury, “Almost sure boundedness of randomly sampled systems,” SIAM J. Contr., 1971, 9:372–384
[38] J.M. Chen and B.S. Chen, “System parameter estimation with input/output noisy data and missing measurements,” IEEE Transactions on Signal Processing, 2000, 48(6):1548-1558
[39] Z. Wang and Q. Hong, “Robust Filtering for Bilinear Uncertain Stochastic Discrete-Time Systems” IEEE Trans. Signal Processing, 2002, 50:560-56
[40] W. NaNacara and E.Yaz, “Recursive estimator for linear and nonlinear systems with uncertain observations,” Signal Processing, 1997, 62:215-228
[41] W. L. DeKoning, “Infinite horizon optimal control of linear discrete time systems with stochastic parameters,” Automatica, 1982, 18:443–453
[42] A.V. Savkin and I.R. Petersen, “Robust filtering with missing data and a deterministic description of noise and uncertainty,” Int. J. Syst. Sci., 1997, 28(4):373-390
[43] E. Yaz, “Full and reduced-order observer design for discrete stochastic bilinearsystems,” IEEE Transactions on Automatic Control, 1992, 37:503-505
[44] R. E. Agniel and E. I. Jury, “Almost sure boundedness of randomly sampled systems,” SIAMJ. Contr., 1971, 9:372-384
[45] Z. Wang, W.C. Daniel and X. Liu, “Varianc-Contrained Control for Uncertain Stochastic Systems With Missing Measurement,” IEEE Transactions on Systems, Man, and Cybernetics-oart A:Systems and Humans, 2005, 35(5):746-753
[46] N.E. Nahi, “Optimal recursive estimation with uncertain observations,”IEEE Transactions on Information Theory, 1969, 15:457-462
[47] W.L.dekoning, “Optimal estimation of linear discrete time systems with stochastic parameters,” Automatica. 1984, 20:113-115
[48] Z.Wang and K. J. Burnham, “Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation,” IEEE Transactions on Signal Processing, 2001, 49: 794–804
[49] Y. Wang, L. Xie and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Syst. Contr. Lett., 1992, 19:139-149
[50] Z. Wang and B. Huang, “Robust H ∞ filtering for linear systems with error variance constraints,” IEEE Transactions on Signal Processing, 2000, 48: 2463–2467
[51] Y. Bar-Shalom, X. Li, T. Kirubarajan, Estimation with applications to tracking and navigation. New York: Wiley, 2001
[2] 何友,王国宏,陆大鑫等.多传感器信息融合及应用.北京:电子工业出版社, 2000
[3] X.R.Li. Comparison of two measurement fusion methods for Kalman-Filter-Based multisensor data fusion. IEEE Transactions on Aerospace and Electronic Systems, 2001, 37(1):273~280
[4] 文成林, 吕冰, 葛泉波. 一种基于分步式滤波的数据融合算法. 电子学报, 2004, 32(8):1264~1267
[5] B. S. Rao, H. F. Durrant-Whyte. Fully decentralized algorithm for multisensor Kalman filtering. IEEE Proceedings-D, 1991, 138(5):413~420
[6] Kessler etc., Functional Description of data fusion Process, Tech Rep., Office of Naval Technol., Naval Air development Ctr., Waminster, PA, 1992
[7] 史忠科.最优估计的计算方法.北京:科学出版社, 2001
[8] Stephen B.HBruder. Heterogeneous multi-sensor integration for unstructured non-static robotic environments. PHD. Queen’s University, Kingston, Ontario, Canada. 1994
[9] S. Blackman and S. Multiple-target tracking with radar application. London: Artech House, Dedham, MA, 350-380, 1986
[10] Blackman S, Popoli R. Design and analysis of modem tracking systems. London: Artech House, 1999
[11] Nayebi, K., Barnwell, T. and M. J. T.Smith. A general time domain analysis and design framework for exactly reconstructing FIR analysis/synthesis filter banks. In Proceeding of IEEE ISCAS: 1990, 2022-2025
[12] 刘炯明.数据融合及应用.北京:国防工业出版社, 1999
[13] 何友,彭应宁.多级式多传感器信息融合中的状态估计.电子学报, 1998, 27(8):60-63
[14] 徐毓.雷达网数据融合问题研究.清华大学博士论文. 北京:清华大学, 2003
[15] A. Benveniste, R. Nikoukhah, A. S. willsky. Multiscale system theory. In: Proc. 29th IEEE Conference on Decision and Control, Honolulu, HI, 1990, 2484-2487
[16] M. Basseville, A. Benveniste, K.Chou, S. Golden, R. Nikoukhah, and A. S. Willsky. Modeling and Estimation of Multiresolution Stochastic Processes. IEEE Transactions on Information Theory. 1992, 38(2):766-784
[17] K. Chou, A. S. Willsky and R. Nikoukhah. Multiscale systems, Kalman filters, and Riccatic equations. IEEE Trans. Automatic Control. 1994, 39(3):479-492
[18] M. Luettgen, W. Karl, A. S. Willsky. Efficient Multiscale Regularization with Applications to the Computation of Optimal Flow. IEEE Transactions on Image Processing, 1993,3(1):41-64
[19] L. Hong. Multiresolution Filtering Using Wavelet Transform. IEEE Transactions on Aerospace and Electronic Systems. 29(4): 1993, 1244-1251
[20] L. Hong. Multi-resolutional distributed filtering. IEEE Tranactions on Automatic Control, 1994, 39(4): 853-856
[21] L. Hong. Multiresolutional Multiple-model Target Tracking. IEEE Transactions on Areospace Electrical Systems, 1994, 30(2): 1025-1035
[22] L. Hong, G. Chen and C. K. Chui. A Filter-Bank-Based Kalman Filtering Technique for Wavelet Estimation and Decomposition of Random Signals. IEEE Transactions on Circuits and Systems—II: Analog and Digital Signal Proecssing, 1998, 45(2):237-241
[23] 文成林, 潘泉, 张洪才, 戴冠中. 多传感器多模型动态系统多尺度分布式融合系统估计算法. 自动化学报. 2000, 6(增刊 B): 66-70
[24] 文成林, 周东华, 潘泉, 张洪才. 多尺度动态模型单传感器的动态系统分布式信息融合, 自动化学报, 2001, 27(2): 158-165
[25] L. Zhang, Q. Pan, P. Bao and H. Zhang. The Discrete Kalman Filtering of a Class of Dynamic Multi-scale Systems, IEEE Transactions on Circults and systems- II:Analogand Digital Signal Processing, 2002, 49(10):668-676
[26] L. Zhang, X. Wu, Q. Pan and H. Zhang. Multiresolution modeling and estimation of multisensor data. IEEE Transactions on Signal Processing 2004, 52 (11):3170-3182
[27] 崔培玲, 潘泉, 张磊, 张洪才. 一类动态多尺度系统的模型和估计算法. 系统仿真学报. 2004, 16(3): 405-410
[28] 崔培玲, 潘泉, 张磊, 张洪才. 基于 Haar 小波的一类动态多尺度系统最优估计算法. 西北工业大学学报, 2004, 22(2):149-152
[29] 崔培玲, 潘泉, 张磊, 张洪才. 非线性状态方程约束下的一类动态多尺度系统建模和估计算法. 计算机工程与应用. 2004, 40(3): 23-25
[30] L. Yan, B. Liu and D. Zhou. The modeling and estimation of asynchronous multirate multisensor dynamic systems, Aerospace Science and Technology. 2006, 10(1):63-71
[31] R.E. Kalman, A new approach to linear filtering and prediction problems, ASME J. Basic Eng., 1960, 82 :34-45
[32] R.E. Kalman and R.S. Bucy, New Results in Linear Filtering and Prediction Theory, ASME J. Basic Eng., 1961, 80:193-196
[33] I. Daubechies. Ten Lectures on Wavelets, Philadelphia: SIAM, 1992
[34] G. Strang, T. Nguyen. Wavelets and Filter Banks. USA: Wellesley-Cambridge, 1997
[35] Donald B. Percival and Andrew T. Walden. Wavelet methods for time series analysis. London: Cambridge, 2000
[36] Albert Boggess and Francis J. Narcowich. 小波分析与傅立叶分析基础. 北京: 电子工业出版社, 2002
[37] R. G. Agniel and E. I. Jury, “Almost sure boundedness of randomly sampled systems,” SIAM J. Contr., 1971, 9:372–384
[38] J.M. Chen and B.S. Chen, “System parameter estimation with input/output noisy data and missing measurements,” IEEE Transactions on Signal Processing, 2000, 48(6):1548-1558
[39] Z. Wang and Q. Hong, “Robust Filtering for Bilinear Uncertain Stochastic Discrete-Time Systems” IEEE Trans. Signal Processing, 2002, 50:560-56
[40] W. NaNacara and E.Yaz, “Recursive estimator for linear and nonlinear systems with uncertain observations,” Signal Processing, 1997, 62:215-228
[41] W. L. DeKoning, “Infinite horizon optimal control of linear discrete time systems with stochastic parameters,” Automatica, 1982, 18:443–453
[42] A.V. Savkin and I.R. Petersen, “Robust filtering with missing data and a deterministic description of noise and uncertainty,” Int. J. Syst. Sci., 1997, 28(4):373-390
[43] E. Yaz, “Full and reduced-order observer design for discrete stochastic bilinearsystems,” IEEE Transactions on Automatic Control, 1992, 37:503-505
[44] R. E. Agniel and E. I. Jury, “Almost sure boundedness of randomly sampled systems,” SIAMJ. Contr., 1971, 9:372-384
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