Markovian随机时滞系统的状态估计
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摘要
时滞现象广泛存在于各类实际的工程系统中,比如制造系统、电力系统、以及网络化控制系统等,时滞现象是导致系统不稳定、降低系统性能、或引起系统出现混沌现象的一个重要因素。因而时滞系统的研究具有重要的理论意义和应用价值。时滞系统状态估计的研究最早可追溯到20世纪60年代,然而已有的结论主要是围绕定常时滞系统展开的。随着无线传感器网络等新兴技术的发展和应用,人们正在面临着更加复杂的不确定性问题,即随机传输时滞等问题,随机时滞往往导致系统的结构发生随机变化。因此,随机时滞系统的估计问题要比定常时滞系统复杂很多。
本文系统地研究了几类不同时滞系统的状态估计问题。分别针对随机跳跃时滞系统、多步随机时滞系统、以及定常时滞Markov跳跃系统,运用观测重构、偏差分方程和偏微分方程等技术,并结合随机分析中的完全配方法、随机逼近原理和新息分析理论,设计有效的状态估计算法。具体的研究内容如下：
首先,研究了带有观测时滞的Markov跳跃系统的线性最小方差估计问题。对于离散时间Markov跳跃时滞系统,首先对观测信息重新组合,得出一组等价的无时滞变结构观测序列。基此定义新的状态变量,该变量既与状态有关,又与系统的跳参数有关。进一步采用新息分析理论设计线性最小方差估计器,得出新的状态变量估计值,其中估值器的解通过求解两个差分Riccati方程而得出。最后通过Markov跳跃特性,得出真实状态的估计值。采用同样的思想,我们研究了连续时间Ito型Markov跳跃时滞系统的状态估计问题。通过随机系统的新息分析理论和Ito微分法则设计线性最小方差估计器,估值器的解通过求解两个有限维的微分Riccati方程而得出。
其次,研究了随机跳跃时滞系统的最优估计问题。对于观测时滞服从Markov跳跃分布的离散时间线性系统,不仅考虑了时滞可测的情形,并且首次给出了时滞未知情形下滤波器的设计方法。针对这两种情形,我们首先引入一组二元函数,将单通道随机跳跃时滞系统转化为带有乘性噪声的双通道定常时滞系统,进一步引入等价的变结构观测序列,使其成为无时滞观测系统,通过定义新的Markov链,最终得出标准的Markov跳跃系统。其中,Markov链的统计特性由随机时滞决定。当跳参数可知时,采用完全配方法以及随机逼近的思想设计最优Markov跳跃滤波器,滤波器的表达式只与当前时刻跳参数有关,而与以前历史时刻无关。其解析解通过求解一组与原系统维数相同的耦合差分Riccati方程而得出；当跳参数未知时,采用新息分析方法设计最优估计器,其中估值器的增益通过求解一组扩展的差分Riccati方程而得出。对于观测时滞服从Markov跳跃分布的Ito型随机系统,我们也分别给出了时滞可知情形下的最优Markov滤波器与时滞未知情形下的线性最小方差估计器的设计方法。然而,不同于离散时间的情形,连续时间随机时滞系统的状态估计器须结合随机分析理论中的Ito微分法则来给出,并且其解析解是通过求解相应的微分Riccati方程而得出的。
最后,研究了多步随机时滞系统的最优估计问题。首先对于离散时间随机时滞系统,采用偏差分Riccati方程的方法设计线性最小方差估计器。其中随机时滞不仅存在于系统方程中,还包含于观测方程中,并且时滞在不同时刻的取值是相互独立的。首先引入随机时滞的二元示性函数,将随机时滞系统转化为带有乘性噪声的多步定常时滞系统,在此基础上应用投影定理设计最小方差平滑器和滤波器,其解析解通过求解一个基于Lyapunov方程的偏差分Riccati方程而得出,基于已得的滤波器进而求得最优预报器的解。对于连续时间随机时滞系统,其中只考虑相互独立的随机观测时滞的情形。同样的采用投影定理设计最小方差平滑器、滤波器和预报器,而所得的估计器的解需通过求解一个基于Lyapunov方程的偏微分Riccati方程而得出。需要指出的是,偏微方程的解析解一般很难得出,通常情况下需采用数值逼近的方法来求解。
It is well known that time-delays are frequently encountered in various prac-tical control systems, such as manufacturing systems, power systems, networked control systems, and etc. The aforementioned delays usually degrade the system performance, and are the source of potential instability, and even lead to the oc-currence of chaos phenomenon. So study on filtering problems with time delays is of great theoretical and practical significance. The state estimation for sys-tems with time delays can be traced back to the 1960's, while most of the existed works focused on the constant delayed systems. In fact, the time-delays occur in a random way, rather than a deterministic way, for a number of engineering applications, such as wireless sensor network systems, and etc. In such a random case, the overall system is no longer a deterministic model, and thus the filter design for the systems with random delays becomes more complicated.
This thesis is devoted to the study of state estimation problems for several kinds of random delayed systems. It is assumed that the random delay is either modeled by a finite state Markov chain, or characterized by a multiple Bernoulli distribution model. Also, we will give the filtering design method for the Markov jump linear systems with constant observation delays. The key techniques applied for treating the delay terms are the observation reconstructed method, partial difference or partial differential method, and then different kinds of estimators are obtained by using the complete square method, the stochastic approximation method and the innovation analysis method. The main contents of this paper are list as follows:
Firstly, we study the linear minimum mean square error estimation for Marko-vian jump linear systems with delayed measurements. For the discrete-time case, we first reconstruct a new delay free observation sequence which contains the same information as the original one. Then, we introduce a new state variable which is concerned on the original state and the Markov jumping parameters simulta-neously. The optimal filter on the new state variable is derived based on the innovation analysis method in the Hilbert space, and its analytical solution is ob-tained in terms of two Riccati difference equations. Finally, the estimator of the original state is obtained directly from the new state estimator via the Markov jumping properties. Also, the estimation problems for the continuous-time case are considered. Following the similar design procedure as in the discrete-time case, the optimal filter is obtained by using the innovation analysis method and the Ito differential rule, while its solution is given in terms of two differential Riccati equations.
Secondly, we consider the optimal estimation problems for the dynamic sys-tems with random observation delays, where the delay process is modeled as a finite state Markov chain. On the estimation of discrete-time systems, two situ-ations are considered:the first one deals with the class of optimal Markov jump filters where the jump delay is assumed accessible; while in the second situation the jump delay is not accessible, and we derive the minimum linear mean square error filter. For both of these situations, the single random delayed measure-ment is firstly rewritten as the two channel constant delayed measurements with multiplicative noises, and further transformed into the delay-free ones via the ob-servation reconstruction technique. Then a set of new Markov chains is defined according to the new observations, which representing the same jumping proper-ties as the random delays. As a result, a standard Markov jump linear system is obtained. Then for the case that the jumping parameter is known, an opti-mal Markov jump filter is obtained by using the complete square method and the stochastic approximation method. The filter depends just on the present value of the Markov parameter, rather on the entire past history of the modes, and the so-lution to the filter is given by solving a set of coupled Riccati difference equations, which have the same dimension as the original systems. In the situation that the Markov chain is not observed, a linear minimum mean square error estimator is derived by using the innovation analysis method and an analytical solution to this estimator is presented via two generalized Riccati difference equations. On the estimation of continuous-time systems, also two situations are considered:For the case that the random delay is known on-line, an optimal Markov jump filter is developed, and for the case that the random delay is unknown, a linear minimum mean square error estimator is developed. But distinguishing from the discrete-time cases, the Ito differential rule needs to be employed in the continuous-time domain, and at last the solutions are given in terms of two kinds of differential Riccati equations.
Finally, We investigate the estimation problems for systems with multiple random delays. For the discrete-time case, the optimal estimation for more general systems with random state and measurement delays is considered. The aim is to present a partial difference equation approach to the optimal estimation. By introducing a set of binary random variables, the system is firstly converted into the one with both multiplicative noises and constant delays. Then an estimator which includes the case of smoothing and filtering is derived via the projection formula, and the solution is given in terms of a partial difference Riccati equation with Lyapunov equations. Also, a predictor for such systems is presented based on the proposed filter and smoother. It can be found that the estimators have the same dimensions as the original system. For the continuous-time case, the optimal estimators including filter, predictor and smoother are developed in the linear minimum variance sense, and the solution is given in terms of partial differential Riccati equations. It should be pointed out that the partial differential equations are difficult to solve, and an analytical solution to these equations might not be possible. In general, the solution can be given in a numerical form by the approximation method.
本文系统地研究了几类不同时滞系统的状态估计问题。分别针对随机跳跃时滞系统、多步随机时滞系统、以及定常时滞Markov跳跃系统,运用观测重构、偏差分方程和偏微分方程等技术,并结合随机分析中的完全配方法、随机逼近原理和新息分析理论,设计有效的状态估计算法。具体的研究内容如下：
首先,研究了带有观测时滞的Markov跳跃系统的线性最小方差估计问题。对于离散时间Markov跳跃时滞系统,首先对观测信息重新组合,得出一组等价的无时滞变结构观测序列。基此定义新的状态变量,该变量既与状态有关,又与系统的跳参数有关。进一步采用新息分析理论设计线性最小方差估计器,得出新的状态变量估计值,其中估值器的解通过求解两个差分Riccati方程而得出。最后通过Markov跳跃特性,得出真实状态的估计值。采用同样的思想,我们研究了连续时间Ito型Markov跳跃时滞系统的状态估计问题。通过随机系统的新息分析理论和Ito微分法则设计线性最小方差估计器,估值器的解通过求解两个有限维的微分Riccati方程而得出。
其次,研究了随机跳跃时滞系统的最优估计问题。对于观测时滞服从Markov跳跃分布的离散时间线性系统,不仅考虑了时滞可测的情形,并且首次给出了时滞未知情形下滤波器的设计方法。针对这两种情形,我们首先引入一组二元函数,将单通道随机跳跃时滞系统转化为带有乘性噪声的双通道定常时滞系统,进一步引入等价的变结构观测序列,使其成为无时滞观测系统,通过定义新的Markov链,最终得出标准的Markov跳跃系统。其中,Markov链的统计特性由随机时滞决定。当跳参数可知时,采用完全配方法以及随机逼近的思想设计最优Markov跳跃滤波器,滤波器的表达式只与当前时刻跳参数有关,而与以前历史时刻无关。其解析解通过求解一组与原系统维数相同的耦合差分Riccati方程而得出；当跳参数未知时,采用新息分析方法设计最优估计器,其中估值器的增益通过求解一组扩展的差分Riccati方程而得出。对于观测时滞服从Markov跳跃分布的Ito型随机系统,我们也分别给出了时滞可知情形下的最优Markov滤波器与时滞未知情形下的线性最小方差估计器的设计方法。然而,不同于离散时间的情形,连续时间随机时滞系统的状态估计器须结合随机分析理论中的Ito微分法则来给出,并且其解析解是通过求解相应的微分Riccati方程而得出的。
最后,研究了多步随机时滞系统的最优估计问题。首先对于离散时间随机时滞系统,采用偏差分Riccati方程的方法设计线性最小方差估计器。其中随机时滞不仅存在于系统方程中,还包含于观测方程中,并且时滞在不同时刻的取值是相互独立的。首先引入随机时滞的二元示性函数,将随机时滞系统转化为带有乘性噪声的多步定常时滞系统,在此基础上应用投影定理设计最小方差平滑器和滤波器,其解析解通过求解一个基于Lyapunov方程的偏差分Riccati方程而得出,基于已得的滤波器进而求得最优预报器的解。对于连续时间随机时滞系统,其中只考虑相互独立的随机观测时滞的情形。同样的采用投影定理设计最小方差平滑器、滤波器和预报器,而所得的估计器的解需通过求解一个基于Lyapunov方程的偏微分Riccati方程而得出。需要指出的是,偏微方程的解析解一般很难得出,通常情况下需采用数值逼近的方法来求解。
It is well known that time-delays are frequently encountered in various prac-tical control systems, such as manufacturing systems, power systems, networked control systems, and etc. The aforementioned delays usually degrade the system performance, and are the source of potential instability, and even lead to the oc-currence of chaos phenomenon. So study on filtering problems with time delays is of great theoretical and practical significance. The state estimation for sys-tems with time delays can be traced back to the 1960's, while most of the existed works focused on the constant delayed systems. In fact, the time-delays occur in a random way, rather than a deterministic way, for a number of engineering applications, such as wireless sensor network systems, and etc. In such a random case, the overall system is no longer a deterministic model, and thus the filter design for the systems with random delays becomes more complicated.
This thesis is devoted to the study of state estimation problems for several kinds of random delayed systems. It is assumed that the random delay is either modeled by a finite state Markov chain, or characterized by a multiple Bernoulli distribution model. Also, we will give the filtering design method for the Markov jump linear systems with constant observation delays. The key techniques applied for treating the delay terms are the observation reconstructed method, partial difference or partial differential method, and then different kinds of estimators are obtained by using the complete square method, the stochastic approximation method and the innovation analysis method. The main contents of this paper are list as follows:
Firstly, we study the linear minimum mean square error estimation for Marko-vian jump linear systems with delayed measurements. For the discrete-time case, we first reconstruct a new delay free observation sequence which contains the same information as the original one. Then, we introduce a new state variable which is concerned on the original state and the Markov jumping parameters simulta-neously. The optimal filter on the new state variable is derived based on the innovation analysis method in the Hilbert space, and its analytical solution is ob-tained in terms of two Riccati difference equations. Finally, the estimator of the original state is obtained directly from the new state estimator via the Markov jumping properties. Also, the estimation problems for the continuous-time case are considered. Following the similar design procedure as in the discrete-time case, the optimal filter is obtained by using the innovation analysis method and the Ito differential rule, while its solution is given in terms of two differential Riccati equations.
Secondly, we consider the optimal estimation problems for the dynamic sys-tems with random observation delays, where the delay process is modeled as a finite state Markov chain. On the estimation of discrete-time systems, two situ-ations are considered:the first one deals with the class of optimal Markov jump filters where the jump delay is assumed accessible; while in the second situation the jump delay is not accessible, and we derive the minimum linear mean square error filter. For both of these situations, the single random delayed measure-ment is firstly rewritten as the two channel constant delayed measurements with multiplicative noises, and further transformed into the delay-free ones via the ob-servation reconstruction technique. Then a set of new Markov chains is defined according to the new observations, which representing the same jumping proper-ties as the random delays. As a result, a standard Markov jump linear system is obtained. Then for the case that the jumping parameter is known, an opti-mal Markov jump filter is obtained by using the complete square method and the stochastic approximation method. The filter depends just on the present value of the Markov parameter, rather on the entire past history of the modes, and the so-lution to the filter is given by solving a set of coupled Riccati difference equations, which have the same dimension as the original systems. In the situation that the Markov chain is not observed, a linear minimum mean square error estimator is derived by using the innovation analysis method and an analytical solution to this estimator is presented via two generalized Riccati difference equations. On the estimation of continuous-time systems, also two situations are considered:For the case that the random delay is known on-line, an optimal Markov jump filter is developed, and for the case that the random delay is unknown, a linear minimum mean square error estimator is developed. But distinguishing from the discrete-time cases, the Ito differential rule needs to be employed in the continuous-time domain, and at last the solutions are given in terms of two kinds of differential Riccati equations.
Finally, We investigate the estimation problems for systems with multiple random delays. For the discrete-time case, the optimal estimation for more general systems with random state and measurement delays is considered. The aim is to present a partial difference equation approach to the optimal estimation. By introducing a set of binary random variables, the system is firstly converted into the one with both multiplicative noises and constant delays. Then an estimator which includes the case of smoothing and filtering is derived via the projection formula, and the solution is given in terms of a partial difference Riccati equation with Lyapunov equations. Also, a predictor for such systems is presented based on the proposed filter and smoother. It can be found that the estimators have the same dimensions as the original system. For the continuous-time case, the optimal estimators including filter, predictor and smoother are developed in the linear minimum variance sense, and the solution is given in terms of partial differential Riccati equations. It should be pointed out that the partial differential equations are difficult to solve, and an analytical solution to these equations might not be possible. In general, the solution can be given in a numerical form by the approximation method.
引文
[1]N. N. Krasovskii, E. A. Lidskii, Analytical design of controllers in systems with random attributes [J], Automation and Remote Control, Parts Ⅰ-Ⅲ, 1961,22:1021-1025,1141-1146,1289-1294.
[2]W. P. J. Blair, D. D. Sworder, Continuous-time regulation of a class of econometric models [J], IEEE Transactions on Systems, Man and Cybernet-ics,1975,5(3):341-346.
[3]M. Mariton, P. Bertrand, Reliable flight control systems:components placement and feedback synthesis[C], Proceedings of the 10th IFAC World Congress, Munich, Germany,1987,150-154.
[4]R. Akella, P. R. Kumar, Optimal control of production rate in a failure prone manufacturing system[J], IEEE Transactions on Automatic Control, 1986,31(1):116-126.
[5]M. Mariton, Jump Linear Systems in Automatic Control[M], New York: Marcel Dekker Inc,1990.
[6]G. S. Ladde, D. D. Siljak, Multiplex control systems:stochastic stability and dynamic reliability[J], International Journal of Control,1983,38(4): 515-524.
[7]N. Wiener, Extrapolation Interpolation, and Smoothing of Stationary Time Series[M], New York:The Technology Press and Wiley,1950.
[8]V. Kuera, Polynomials and linear control systems[J], Automatica,1985, 21(1):111-112.
[9]M. J. Grimble, Polynomial approach to optimal linear filtering and predic-tion[J], International Journal of Control,1985,41(6):1545-1564.
[10]A. Ahlen, M. Sternad, Wiener filter design using polynomial equations[J], IEEE Transactions on Signal Processing,1991,39(11):2387-2399.
[11]R. E. Kalman, A new approach to linear filtering and prediction [J], Trans-actions of ASME-Journal of Basic Engineering,1960,82d:34-45.
[12]郭尚来,随机控制[M],北京：清华大学出版社,1999.
[13]S. Sangsuk-Iam, T. E. Bullock, Analysis of discrete-time Kalman filtering under incorrect noise covariance[J], IEEE Transactions on Automatic Con-trol,1990,35(12):1304-1309.
[14]C. de Souza, M. Gevers, G. Goodwin, Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices [J], IEEE Transactions on Automatic Control,1986,31(9):831-838.
[15]J. C. Spall, The kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions[J], Automatica,1995,31(10):1513-1517.
[16]邓自立,最优估计理论及其应用：建模、滤波、信息融合估计[M],哈尔滨：哈尔滨工业大学出版社,2005.
[17]中国科学院数学研究所概率组,离散时间系统滤波的数学方法[M],北京：国防工业出版社,1975.
[18]史忠科,最优估计的计算方法[M],北京：科学出版社,2001.
[19]S. H. Jim, J. B. Park, An LMI approach to guaranteed cost control of linear uncertain time-delay systems [J], IEE Proceedings Control Theory & Applications,2001,148(1):55-59.
[20]L. Yu, J. Chu, Robust H∞ filter for polytopic uncertain systems via convex optimization [J], Automatica,1999,35(6):1155-1159.
[21]关新平,陈彩莲,龙承念,刘奕昌,段广仁,不确定离散时滞系统的动态输出反馈鲁棒保性能控制[J],控制理论与应用,2003，20(4)：199-204.
[22]W. M. Hadded, D. S. Bernstein, Robust stabilization with positive real uncertainty:beyond small gain theorem [J], Systems & Control Letters,1991, 17(3):1914-1918.
[23]李志虎,邵惠鹤,王景成,线性时滞系统的耗散控制(英文)[J],控制理论与应用,2001,18(6)：838-842.
[24]B. D. O. Anderson, J. B. Moore, Optimal Filtering[M], Englewood Cliffs, N.J.:Prentice-Hall,1979.
[25]T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation[M], Englewood Cliffs, N.J.:Prentice-Hall,1999.
[26]G. C. Goodwin, K. S. Sin, Adaptive Filtering, Prediction and Control[M], Englewood Cliffs, N.J.:Prentice-Hall,1984.
[27]J. S. Meditch, Stochastic Optimal Linear Estimation and Control[M], New York:McGraw-Hill,1969.
[28]P. Bolzern, P. Colaneri, G. D. Nicolao, On discrete-time H∞ fixed-lag smoothing[J], IEEE Transactions on Signal Processing,2004,52(1):132-141.
[29]V. Kueera, Discrete Linear Control, the Polynomial Approach[M], Chich-ester:Wiley,1979.
[30]L. Chisci, E. Mosca,Polynomial equations for the linear MMSE state esti-mation[J], IEEE Transactions on Automatic Control,1992,37(5):623-626.
[31]L. Chisci, E. Mosca, MMSE deconvolution via polynomial method and its dual LQG regulation[J], Automatica,1991,30(7):1197-1202.
[32]V. Kucera, New results in state estimation and regulation[J], Automatica, 1981,17(9):745-748.
[33]R. Priemer, A. G. Vacroux, Estimation in linear discrete systems with mul-tiple time delays[J], IEEE Transactions on Automatic Control,1969,14(4): 384-387.
[34]J. Mishra, V. Rajamani, Least-squares state estimation in time-delay sys-tems with colored observation noise:an innovations approach [J], IEEE Transactions on Automatic Control,1975,20(1):140-142.
[35]H. Kwakernaak, Optimal filtering in linear systems with time delays[J], IEEE Transactions on Automatic Control,1967,12(2):169-173.
[36]D. Liang, On continuous-time estimation for linear delayed-systems with correlated state and observation noises [J], IEEE Transactions on Automatic Control,1977,22(3):472-474.
[37]R. Kwong, A. Willsky, Optimal filtering and filter stability of linear stochas-tic delay systems[J], IEEE Transactions on Automatic Control,1977,2(2): 196-201.
[38]M. Briggs, Linear filtering for time-delay systems[J], IMA Journal of Math-ematical Control and Information,1989,6(2):167-178.
[39]H. Zhang, L. Xie, Control and Estimation of Systems with Input/Output Delays[M], Berlin Heidelberg:Springer-Verlag,2007.
[40]M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems[M], New York:Marcel-Dekker,2000.
[41]H. Gao, C. Wang, Delay-dependent robust H∞ and L2-L∞ filtering for a class of uncertain nonlinear time-delay systems [J], IEEE Transactions on Automatic ontrol,2003,48(9):1661-1666.
[42]H. Gao, C. Wang, A delay-dependent approach to robust H∞ filtering for un-certain discrete-time state-delayed systems [J], IEEE Transactions on Signal Processing,2004,52(6):1631-1640.
[43]C. E. de Souza, R. M. Palhares, P. L. P. Peres, Robust H∞ filter design for uncertain linear systems with multiple time-varying state delays [J], IEEE Transactions on Signal Processing,2001,49(3):569-576.
[44]E. Friman, U. Shaked, A new H∞ filter design for linear time delay sys-tems[J], IEEE Transactions on Signal Processing,2001,49(11):2839-2843.
[45]S. Xu, J. Lam, T. Chen, Y. Zou, A delay-dependent approach to robust H∞ filtering for uncertain distributed delay systems [J], IEEE Transactions on Signal Processing,2005,53(10):3764-3772.
[46]S. Xu, T. Chen, An LMI approach to the H∞ filter design for uncertain sys-tems with distributed delays [J], IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,2004,51(4):195-201.
[47]M. S. Mahmoud, L. Xie, Y. C. Soh, Robust Kalman filtering for discrete state delay systems [J], IEE Proceedings Control Theory and Applications, 2000,147(6):613-618.
[48]Z. Wang, K. J. Burnham, Robust filtering for a class of stochastic uncer-tain nonlinear time-delay systems via exponential state estimation [J], IEEE Transactions on Signal Processing,2001,49(4):794-804.
[49]T. Vincent, J. Abedor, K. Nagpal, P. P. Khargonekar, Discrete-time estima-tors with guaranteed peak-to-peak performance [C], Proceedings of the 13th IFAC Triennial World Congress, San Francisco,1996,43-48.
[50]H. Gao, C. Wang, Robust L2-L∞ filtering for uncertain systems with multi-ple time-varying delays [J], IEEE Transactions on Circuit and Systems,2003, 50(4):594-599.
[51]H. Zhang, L. Xie, D. Zhang, Y. C. Soh, A reorganized innovation approach to linear estimation [J], IEEE Transactions on Automatic ontrol,2004,49(10): 1810-1814.
[52]H. Zhang, X. Lu, D. Cheng, Optimal estimation for continuous-time sys-tems with delayed measurements [J], IEEE Transactions on Automatic on-trol,2006,51(5):823-827.
[53]X. Lu, H. Zhang, W. Wang, K. L. Teo, Kalman filtering for multiple time-delay systems[J], Automatica,2005,41(8):1455-1461.
[54]H. Zhang, L. Xie, Optimal estimation for systems with time-varying de-lay [C], Proceedings of the 46th IEEE Conference on Decision and Control, 2007,4311-4316.
[55]W. Wang, H. Zhang, L. Xie, Optimal filtering for continuous-time linear systems with time-varying delay [C], Proceedings of the 6th International Conference on Information, Communications and Signal Processing,2007, 1-6.
[56]H. Zhang, G. Feng, G. Duan, X. Lu, H∞ filtering for multiple-time-delay measurements[J], IEEE Transactions on Signal Processing,2006,54(5): 1681-1688.
[57]H. Zhang, L. Xie, W. Wang, X. Lu, An innovation approach to H∞ fixed-lag smoothing for continuous time-varying systems [J], IEEE Transactions on Automatic ontrol,2004,49(12):2240-2244.
[58]H. Zhang, L. Xie, Y. C. Soh, D. Zhang, H∞ fixed-lag smoothing for discrete linear time-varying systems [J], Automatica,2005,41(5):839-846.
[59]R. Krtolica, U. Ozghner, H. Chan, H. Gokta, J. Winkelman, M. Liubakka, Stability of linear feedback systems with random communication delays [J], International Journal orf Control,1994,59(4):925-953.
[60]J. Nilsson, B. Bernhardsson, B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays [J], Automatica,1998,34(1): 57-64.
[61]L. Xiao, A. Hassibi, J. How, Control with random communication delays via a discrete-time jump system approach[C], Proceedings of the Amerecan Control Conference, Chicago, IL,2000,2199-2204.
[62]S. Hu, W. Zhu, Stochastic optimal control and analysis of stability of net-worked control systems with long delay [J], Automatica,2003,39(11):1877-1884.
[63]J.P. Hespanha, P. Naghshtabrizi, Y. Xu, A survey of recent results in net-worked control systems [J], IEEE Transactions on Automatic ontrol,2007, 95(1):138-162.
[64]L. Zhang, Y. Shi, T. Chen, A new method for stabilization of networked control systems with random delays [J], IEEE Transactions on Automatic ontrol,2005,50(8):1177-1181.
[65]W. Zhang, L. Yu, Modeling and control of networked control systems with both network-induced delay and packet-dropout [J], Automatica,2008, 44(12):3206-3210.
[66]M. Liu, D. W. C. Ho, Y. Niu, Stabilization of Markovian jump linear sys-tem over networks with random communication delay[J], Automatica,2009, 45(2):416-421.
[67]C. Lin, Z. Wang, F. Yang, Observer-based networked control for continuous-time systems with random sensor delays [J], Automatica,2009,45(2):578-584.
[68]E. Yaz, A. Ray, Linear unbiased state estimation under randomly varying bounded sensor delay [J], Applied Mathematics Letters,1998, 11(4):27-32.
[69]N. Tsai, A. Ray, Compensatability and optimal compensation under ran-domly varying distributed delays [J], International Journal of Control,1999, 72(9):826-832.
[70]N. Tsai, A. Ray, Stochastic optimal control under randomly varying dis-tributed delays[J], International Journal of Control,1997,68(5):1179-1202.
[71]C. Su, C. Lu, Interconnected network state estimation using randomly delayed measurements [J], IEEE Transactions on Power Systems,2001, 16(4):870-878.
[72]S. Nakamori, R. C. Aguila, A. H. Carazo, J. L. Perez, Recursive estimators of signals from measurements with stochastic delays using covariance infor-mation[J], Applied Mathematics and Computation,2005,162(1):65-79.
[73]A. H. Carazo, J. L. Perez, Extended and unscented filtering algorithms using one-step randomly delayed observations [J], Applied Mathematics and Computation,2007,190(2):1375-1393.
[74]Z. Wang, D. W. C. Ho, X. Liu, Robust filtering under randomly varying sensor delay measurements [J], IEEE Transactions on Circuits and Systems-II:Express Briefs,2004,51(6):320-326.
[75]S. Zhou, G. Feng, H∞ filtering for discrete-time systems with randomly varying sensor delays[J], Automatica,2008,44(7):1918-1922.
[76]B. Shen, Z. Wang, H. Shu, G. Wei, H∞ filtering for nonlinear discrete-time stochastic systems with randomly varying sensor delays [J], Automatica, 2009,45(4):1032-1037.
[77]M. Sahebsara, T. Chen, S. L. Shah, Optimal H2 filtering with random sensor delay, multiple packet dropout and uncertain observations [J], International Journal of Control,2007,80(2):292-301.
[78]M. Moayedi, Y. C. Soh, Y. K. Foo, Optimal Kalman filtering with random sensor delays, packet dropouts and missing measurements[C], Proceedings of the American Control Conference, USA, June 2009,3405-3410.
[79]K. Zhang, X. Li, Y. Zhu, Optimal update with out-of-sequence measure-ments[J], IEEE Transactions on Signal Processing,2005,53(6):1992-2004.
[80]X. Shen, Y. Zhu, E. Song, Y. Luo, Optimal centralized update with mul-tiple local out-of-sequence measurements [J], IEEE Transactions on Signal Processing,2009,57(4):1551-1562.
[81]X. Shen, E. Song, Y. Zhu, Y. Luo, Globally optimal distributed Kalman fu-sion with local out-of-sequence-measurement updates [J], IEEE Transactions on Automatic Control,2009,54(8):1928-1934.
[82]A. S. Matveev, A. V. Savkin, The problem of state estimation via asyn-chronous communication channels with irregular transmission times [J], IEEE Transactions on Automatic ontrol,2003,48(4):670-676.
[83]A. S. Matveev, A. V. Savkin, Optimal state estimation in networked systems with asynchronous communication channels and switched sensors [J], Journal of Optimization Theory and Applications,2006,128(1):139-165.
[84]N. E. Nahi, Optimal recursive estimation with uncertain observation [J], IEEE Transactions on Information Theory,1969,15(4):457-462.
[85]M. T. Hadidi, C. S. Schwartz, Linear recursive state estimators under uncer-tain observations [J], IEEE Transactions on Automatic Control,1979,24(6): 944-948.
[86]B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan, S. S. Sastry, Kalman filtering with intermittent observations [J], IEEE Transac-tions on Automatic Control,2004,49(9):1453-1463.
[87]K. Plarre, F. Bullo, On Kalman filtering for detectable systems with in-termittent observations [J], IEEE Transactions on Automatic Control,2009, 54(2):386-390.
[88]L. Schenato, Optimal estimation in networked control systems subject to random delay and packet drop[J], IEEE Transactions on Automatic Control, 2008,53(5):1311-1317.
[89]刘次华,随机过程M],武汉：华中科技大学出版社,2001.
[90]刘嘉焜,王公恕,应用随机过程[M],北京：科学出版社,2004.
[91]J. Evans, V. Krishnamurthy, Hidden Markov model state estimation with randomly delayed observations [J], IEEE Transactions on Automatic Control, 1999,47(8):2157-2166.
[92]H. Song, L. Yu, W. Zhang, H∞ filtering of network-based systems with random delay [J], Signal Processing,2009,89(4):615-622.
[93]J. K. Tugnait, Adaptive estimation and identification for discrete systems with Markov jump parameters [J], IEEE Transactions on Automatic Control, 1982,27(5):1054-1064.
[94]J. K. Tugnait, Detection and estimation for abruptly changing systems [J], Automatica,1982,18(5):607-615.
[95]R. J. Kenefic, A Bayesian approach to time delay estimation [J], IEEE Trans-actions on Acoustics, Speech, and Signal Processing,1981, ASSP-29(3):611-614.
[96]H. A. P. Blom, An efficient filter for abruptly changing systems[C], Proceed-ings of the 23rd IEEE Conference On Decision and Control, Las Vegas, NV, 1984,656-658.
[97]H. A. P. Blom, Continuous-discrete filtering for the systems with Markovian switching coefficients and simultaneous jump[C], Proceedings of the 21st Asilomar Conference on Signals, Systems and Computers, Pacific Grove, 1987,244-248.
[98]H. A. P. Blom, X. R. Li, The interacting multiple model algorithm for sys-tems with Markovian switching coefficients [J], IEEE Transactions on Auto-matic Control,1988,33(8):780-783.
[99]R. Elliot, F. Dufour, F. Sworder, Exact hybrid filters in discrete-time [J], IEEE Transactions on Automatic Control,1996,41(12):1807-1810.
[100]R. J. Elliot, W. P. Malcolm, Robust M-ary detection filters for continuous-time jump Markov systems[C], Proceedings of the 40th IEEE Conference on Decision and Control, Oriando, Florid USA,2001,1681-1686.
[101]F. Dufour, P. Bertrand, The filtering problem for continous-time linear sys-tems with Markovian switching coeficients[J], Systems & Control Letters, 1994,23(5):453-461.
[102]F. Dufour, P. Bertrand, An image based filter for discrete-time Markovian jump linear systems[J], Automatica,1996,32(2):241-247.
[103]F. Dufour, R. Elliot, Filtering with discrete state observations[J], Applied Mathematics and Optimization,1999,40(2):259-272.
[104]A. Doucet, C. Andrieu, Iterative algorithms for state estimation of jump Markov linear systems [J], IEEE Transactions on Automatic Control,2000, 49(6):1216-1227.
[105]Q. Zhang, Optimal filtering of discrete-time hybrid sy stems [J], Journal of Optimization Theory and Applications,1999,100(1):123-144.
[106]Q. Zhang, Hybrid filtering for linear systems with non-Gaussian distur-bances [J], IEEE Transactions on Automatic Control,2000,45(1):50-61.
[107]S. Allam, F. Dufour, P. Bertrand, Discrete-time estimation of a Markov chain with marked point process observations [J], IEEE Transactions on Au-tomatic Control,2001,46(6):903-908.
[108]Y. Bar-Shalom, X. R. Li, Estimation and Tracking. Principles, Techniques and Software[M], Boston:Artech House,1993.
[109]F. Dufour, D. Kannan, Discrete time nonlinear filtering with marked point process observations [J], Stochatic Analysis and Applications,1999,17(1): 99-115.
[110]T. Bjork, Finite dimensional optimal filters for a class of lto-processes with jumping parameters [J], Stochatics,1980,4(2):167-183.
[111]W. M. Wonham, Some applications of stochastic differential equations to optimal nonlinear filtering[J], SIAM Journal on Control and Optimization, 1964,2(3):347-369.
[112]O. L. V. Costa, Linear minimum mean square error estimation for discrete-time Markovian jump linear systems [J], IEEE Transactions on Automatic Control,1994,39(8):1685-1689.
[113]O. L. V. Costa, S. Guerra, Stationary filter for linear minimum mean square error estimator of discrete-time Markovian jump systems [J], IEEE Transac-tions on Automatic Control,2002,47(8):1351-1356.
[114]O. L. V. Costa, M. D. Fragoso, R. P. Marques, Discrete-Time Markov Jump Linear Systems[M], London:Springer-Verlag,2005.
[115]M. D. Fragoso, O. L. V. Costa, J. Baczynski, N. Rocha, Optimal linear mean square filter for continuous-time jump linear systems [J], IEEE Transactions on Automatic Control,2005,50(9):1364-1369.
[116]M. D. Fragoso, N. C. S. Rocha, Stationary filter for continuous-time Marko-vian jump linear systems [J], SI AM Journal on Control and Optimization, 2005,44(3):801-815.
[117]A. Doucet, A. Logothetis, V. Krishnamurthy, Stochastic sampling algo-rithms for state estimation of jump Markov linear systems [J], IEEE Trans-actions on Automatic Control,2000,45(2):188-202.
[118]K. Sun, L. Li, A. P. G. Panayoutopoulos, A. Packard, Extensions of jump Markov system estimation techniques [C], Proceedings of the American Con-trol Conference, Anchorage,2002,4209-4214.
[119]C. Carlemalm, A. Logothetis, V. Krishnamurthy, l1 state estimation of jump Markov linear systems[C], Proceedings of the 37th IEEE Conference on De-cision and Control, Tampa, Florida,1998,263-267.
[120]P. Shi, E. Boukas, R. K. Agarwal, Kalman filtering for continuous-time un-certain systems with Markovian jumping parameters [J], IEEE Transactions on Automatic Control,1999,44(8):1592-1597.
[121]C. E. de Souza, H∞ filtering for discrete-time linear systems with Marko-vian jumping parameters [J], International Journal of Robust and Nonlinear Control,2003; 13(14):1299-1361.
[122]C. E. de Souza, M. D. Fragoso, H∞ filtering for Markovian jump linear sys-tems[J], International Journal of Systems Science,2002,33(11):905-915.
[123]Z. Wang, J. Lam, X. Liu, Robust filtering for discrete-time Markovian jump delay systems[J], IEEE Signal Processing Letters,2004,11(8):659-662.
[124]M. S. Mahmoud, P. Shi, Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters [J], IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,2003,50(1):98-105.
[125]S. Xu, T. Chen, J. Lam, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters [J], IEEE Transactions on Au-tomatic Control,1999,44(8):1592-1597.
[126]S. Xu, T. Chen, J. Lam, Robust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays [J], IEEE Transactions on Auto-matic Control,2003,48(5):900-907.
[127]E. K. Boukas, Z. K. Liu, Robust H∞ filtering for polytopic uncertain time-delay systems with Markov jumps[J], Computers and Electrical Engineering, 2002,28(3):171-193.
[128]E. K. Boukas, Z. K. Liu, Robust H∞ filtering design for uncertain linear systems with multiple time-varying state delay [J], IEEE Transactions on Signal Processing,2001,49(3):569-576.
[129]R. M. Palhares, C. E. de Souza, P. L. D. Peres, Robust H∞, filtering for un-certain discrete-time state-delayed systems [J], IEEE Transactions on Signal Processing,2001,49(8):1696-1703.
[130]M. S. Mahmoud, Robust filtering for uncertain multi-state-delay jumping systems[C], Proceedings of the 7th Internstionnl Conference on Control, Am-tomation, Robotics and Virion, Singspore,2002,168-173.
[131]P. Shi, M. Mahmoud, S. K. Nguang, A. Ismail, Robust filtering for jumping systems with mode-dependent delays [J], Signal Processing,2006,86(1):140-152.
[132]H. Liu, F. Sun, K. He, Z. Sun, Design of reduced-order H∞ filter for Marko-vian jumping systems with time delay [J], IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,2004,51(11):607-612.
[133]Z. Wang, J. Lam, X. Liu, Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances [J], IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,2004,51(5):262-268.
[134]Z. Wang, J. Lam, X. Liu, Nonlinear filtering for state delayed systems with Markovian switching [J], IEEE Transactions on Signal Processing,2003, 51(9):2321-2328.
[135]Y. Fu, Y. Zhang, G. Duan, Q. Chai, Robust guaranteed cost filtering for un-certain timedelay systems with Markovian jumping parameters [J], Journal of Systems Engineering and Electronics,2005,16(4):852-857.
[136]L. Hu, P. Shi, Y. Cao, Delay-dependent Filtering Design for Time-delay Systems with Markovian Jumping Parameters [J], International Journal of Adaptive Control and Signal Processing,2006,21(5):434-448.
[137]M. H. A. Davis, R. B. Vinter, Stochastic Modeling and Control[M], New York:Chapman and Hall,1985.
[138]M. H. A. Davis, Linear Estimation and Stochastic Control[M], London: Chapman and Hall,1977.
[139]Y. Bar-Shalom, X. R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation:Theory Algorithms and Software[M], New York: Wiley,2001.
[140]V. Gupta, T. H. Chung, B. Hassibi, R. M. Murray, On a stochastic sen-sor selection algorithm with applications in sensor scheduling and sensor coverage[J], Automatica,2006,42(2):251-260.
[2]W. P. J. Blair, D. D. Sworder, Continuous-time regulation of a class of econometric models [J], IEEE Transactions on Systems, Man and Cybernet-ics,1975,5(3):341-346.
[3]M. Mariton, P. Bertrand, Reliable flight control systems:components placement and feedback synthesis[C], Proceedings of the 10th IFAC World Congress, Munich, Germany,1987,150-154.
[4]R. Akella, P. R. Kumar, Optimal control of production rate in a failure prone manufacturing system[J], IEEE Transactions on Automatic Control, 1986,31(1):116-126.
[5]M. Mariton, Jump Linear Systems in Automatic Control[M], New York: Marcel Dekker Inc,1990.
[6]G. S. Ladde, D. D. Siljak, Multiplex control systems:stochastic stability and dynamic reliability[J], International Journal of Control,1983,38(4): 515-524.
[7]N. Wiener, Extrapolation Interpolation, and Smoothing of Stationary Time Series[M], New York:The Technology Press and Wiley,1950.
[8]V. Kuera, Polynomials and linear control systems[J], Automatica,1985, 21(1):111-112.
[9]M. J. Grimble, Polynomial approach to optimal linear filtering and predic-tion[J], International Journal of Control,1985,41(6):1545-1564.
[10]A. Ahlen, M. Sternad, Wiener filter design using polynomial equations[J], IEEE Transactions on Signal Processing,1991,39(11):2387-2399.
[11]R. E. Kalman, A new approach to linear filtering and prediction [J], Trans-actions of ASME-Journal of Basic Engineering,1960,82d:34-45.
[12]郭尚来,随机控制[M],北京：清华大学出版社,1999.
[13]S. Sangsuk-Iam, T. E. Bullock, Analysis of discrete-time Kalman filtering under incorrect noise covariance[J], IEEE Transactions on Automatic Con-trol,1990,35(12):1304-1309.
[14]C. de Souza, M. Gevers, G. Goodwin, Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices [J], IEEE Transactions on Automatic Control,1986,31(9):831-838.
[15]J. C. Spall, The kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions[J], Automatica,1995,31(10):1513-1517.
[16]邓自立,最优估计理论及其应用：建模、滤波、信息融合估计[M],哈尔滨：哈尔滨工业大学出版社,2005.
[17]中国科学院数学研究所概率组,离散时间系统滤波的数学方法[M],北京：国防工业出版社,1975.
[18]史忠科,最优估计的计算方法[M],北京：科学出版社,2001.
[19]S. H. Jim, J. B. Park, An LMI approach to guaranteed cost control of linear uncertain time-delay systems [J], IEE Proceedings Control Theory & Applications,2001,148(1):55-59.
[20]L. Yu, J. Chu, Robust H∞ filter for polytopic uncertain systems via convex optimization [J], Automatica,1999,35(6):1155-1159.
[21]关新平,陈彩莲,龙承念,刘奕昌,段广仁,不确定离散时滞系统的动态输出反馈鲁棒保性能控制[J],控制理论与应用,2003，20(4)：199-204.
[22]W. M. Hadded, D. S. Bernstein, Robust stabilization with positive real uncertainty:beyond small gain theorem [J], Systems & Control Letters,1991, 17(3):1914-1918.
[23]李志虎,邵惠鹤,王景成,线性时滞系统的耗散控制(英文)[J],控制理论与应用,2001,18(6)：838-842.
[24]B. D. O. Anderson, J. B. Moore, Optimal Filtering[M], Englewood Cliffs, N.J.:Prentice-Hall,1979.
[25]T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation[M], Englewood Cliffs, N.J.:Prentice-Hall,1999.
[26]G. C. Goodwin, K. S. Sin, Adaptive Filtering, Prediction and Control[M], Englewood Cliffs, N.J.:Prentice-Hall,1984.
[27]J. S. Meditch, Stochastic Optimal Linear Estimation and Control[M], New York:McGraw-Hill,1969.
[28]P. Bolzern, P. Colaneri, G. D. Nicolao, On discrete-time H∞ fixed-lag smoothing[J], IEEE Transactions on Signal Processing,2004,52(1):132-141.
[29]V. Kueera, Discrete Linear Control, the Polynomial Approach[M], Chich-ester:Wiley,1979.
[30]L. Chisci, E. Mosca,Polynomial equations for the linear MMSE state esti-mation[J], IEEE Transactions on Automatic Control,1992,37(5):623-626.
[31]L. Chisci, E. Mosca, MMSE deconvolution via polynomial method and its dual LQG regulation[J], Automatica,1991,30(7):1197-1202.
[32]V. Kucera, New results in state estimation and regulation[J], Automatica, 1981,17(9):745-748.
[33]R. Priemer, A. G. Vacroux, Estimation in linear discrete systems with mul-tiple time delays[J], IEEE Transactions on Automatic Control,1969,14(4): 384-387.
[34]J. Mishra, V. Rajamani, Least-squares state estimation in time-delay sys-tems with colored observation noise:an innovations approach [J], IEEE Transactions on Automatic Control,1975,20(1):140-142.
[35]H. Kwakernaak, Optimal filtering in linear systems with time delays[J], IEEE Transactions on Automatic Control,1967,12(2):169-173.
[36]D. Liang, On continuous-time estimation for linear delayed-systems with correlated state and observation noises [J], IEEE Transactions on Automatic Control,1977,22(3):472-474.
[37]R. Kwong, A. Willsky, Optimal filtering and filter stability of linear stochas-tic delay systems[J], IEEE Transactions on Automatic Control,1977,2(2): 196-201.
[38]M. Briggs, Linear filtering for time-delay systems[J], IMA Journal of Math-ematical Control and Information,1989,6(2):167-178.
[39]H. Zhang, L. Xie, Control and Estimation of Systems with Input/Output Delays[M], Berlin Heidelberg:Springer-Verlag,2007.
[40]M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems[M], New York:Marcel-Dekker,2000.
[41]H. Gao, C. Wang, Delay-dependent robust H∞ and L2-L∞ filtering for a class of uncertain nonlinear time-delay systems [J], IEEE Transactions on Automatic ontrol,2003,48(9):1661-1666.
[42]H. Gao, C. Wang, A delay-dependent approach to robust H∞ filtering for un-certain discrete-time state-delayed systems [J], IEEE Transactions on Signal Processing,2004,52(6):1631-1640.
[43]C. E. de Souza, R. M. Palhares, P. L. P. Peres, Robust H∞ filter design for uncertain linear systems with multiple time-varying state delays [J], IEEE Transactions on Signal Processing,2001,49(3):569-576.
[44]E. Friman, U. Shaked, A new H∞ filter design for linear time delay sys-tems[J], IEEE Transactions on Signal Processing,2001,49(11):2839-2843.
[45]S. Xu, J. Lam, T. Chen, Y. Zou, A delay-dependent approach to robust H∞ filtering for uncertain distributed delay systems [J], IEEE Transactions on Signal Processing,2005,53(10):3764-3772.
[46]S. Xu, T. Chen, An LMI approach to the H∞ filter design for uncertain sys-tems with distributed delays [J], IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,2004,51(4):195-201.
[47]M. S. Mahmoud, L. Xie, Y. C. Soh, Robust Kalman filtering for discrete state delay systems [J], IEE Proceedings Control Theory and Applications, 2000,147(6):613-618.
[48]Z. Wang, K. J. Burnham, Robust filtering for a class of stochastic uncer-tain nonlinear time-delay systems via exponential state estimation [J], IEEE Transactions on Signal Processing,2001,49(4):794-804.
[49]T. Vincent, J. Abedor, K. Nagpal, P. P. Khargonekar, Discrete-time estima-tors with guaranteed peak-to-peak performance [C], Proceedings of the 13th IFAC Triennial World Congress, San Francisco,1996,43-48.
[50]H. Gao, C. Wang, Robust L2-L∞ filtering for uncertain systems with multi-ple time-varying delays [J], IEEE Transactions on Circuit and Systems,2003, 50(4):594-599.
[51]H. Zhang, L. Xie, D. Zhang, Y. C. Soh, A reorganized innovation approach to linear estimation [J], IEEE Transactions on Automatic ontrol,2004,49(10): 1810-1814.
[52]H. Zhang, X. Lu, D. Cheng, Optimal estimation for continuous-time sys-tems with delayed measurements [J], IEEE Transactions on Automatic on-trol,2006,51(5):823-827.
[53]X. Lu, H. Zhang, W. Wang, K. L. Teo, Kalman filtering for multiple time-delay systems[J], Automatica,2005,41(8):1455-1461.
[54]H. Zhang, L. Xie, Optimal estimation for systems with time-varying de-lay [C], Proceedings of the 46th IEEE Conference on Decision and Control, 2007,4311-4316.
[55]W. Wang, H. Zhang, L. Xie, Optimal filtering for continuous-time linear systems with time-varying delay [C], Proceedings of the 6th International Conference on Information, Communications and Signal Processing,2007, 1-6.
[56]H. Zhang, G. Feng, G. Duan, X. Lu, H∞ filtering for multiple-time-delay measurements[J], IEEE Transactions on Signal Processing,2006,54(5): 1681-1688.
[57]H. Zhang, L. Xie, W. Wang, X. Lu, An innovation approach to H∞ fixed-lag smoothing for continuous time-varying systems [J], IEEE Transactions on Automatic ontrol,2004,49(12):2240-2244.
[58]H. Zhang, L. Xie, Y. C. Soh, D. Zhang, H∞ fixed-lag smoothing for discrete linear time-varying systems [J], Automatica,2005,41(5):839-846.
[59]R. Krtolica, U. Ozghner, H. Chan, H. Gokta, J. Winkelman, M. Liubakka, Stability of linear feedback systems with random communication delays [J], International Journal orf Control,1994,59(4):925-953.
[60]J. Nilsson, B. Bernhardsson, B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays [J], Automatica,1998,34(1): 57-64.
[61]L. Xiao, A. Hassibi, J. How, Control with random communication delays via a discrete-time jump system approach[C], Proceedings of the Amerecan Control Conference, Chicago, IL,2000,2199-2204.
[62]S. Hu, W. Zhu, Stochastic optimal control and analysis of stability of net-worked control systems with long delay [J], Automatica,2003,39(11):1877-1884.
[63]J.P. Hespanha, P. Naghshtabrizi, Y. Xu, A survey of recent results in net-worked control systems [J], IEEE Transactions on Automatic ontrol,2007, 95(1):138-162.
[64]L. Zhang, Y. Shi, T. Chen, A new method for stabilization of networked control systems with random delays [J], IEEE Transactions on Automatic ontrol,2005,50(8):1177-1181.
[65]W. Zhang, L. Yu, Modeling and control of networked control systems with both network-induced delay and packet-dropout [J], Automatica,2008, 44(12):3206-3210.
[66]M. Liu, D. W. C. Ho, Y. Niu, Stabilization of Markovian jump linear sys-tem over networks with random communication delay[J], Automatica,2009, 45(2):416-421.
[67]C. Lin, Z. Wang, F. Yang, Observer-based networked control for continuous-time systems with random sensor delays [J], Automatica,2009,45(2):578-584.
[68]E. Yaz, A. Ray, Linear unbiased state estimation under randomly varying bounded sensor delay [J], Applied Mathematics Letters,1998, 11(4):27-32.
[69]N. Tsai, A. Ray, Compensatability and optimal compensation under ran-domly varying distributed delays [J], International Journal of Control,1999, 72(9):826-832.
[70]N. Tsai, A. Ray, Stochastic optimal control under randomly varying dis-tributed delays[J], International Journal of Control,1997,68(5):1179-1202.
[71]C. Su, C. Lu, Interconnected network state estimation using randomly delayed measurements [J], IEEE Transactions on Power Systems,2001, 16(4):870-878.
[72]S. Nakamori, R. C. Aguila, A. H. Carazo, J. L. Perez, Recursive estimators of signals from measurements with stochastic delays using covariance infor-mation[J], Applied Mathematics and Computation,2005,162(1):65-79.
[73]A. H. Carazo, J. L. Perez, Extended and unscented filtering algorithms using one-step randomly delayed observations [J], Applied Mathematics and Computation,2007,190(2):1375-1393.
[74]Z. Wang, D. W. C. Ho, X. Liu, Robust filtering under randomly varying sensor delay measurements [J], IEEE Transactions on Circuits and Systems-II:Express Briefs,2004,51(6):320-326.
[75]S. Zhou, G. Feng, H∞ filtering for discrete-time systems with randomly varying sensor delays[J], Automatica,2008,44(7):1918-1922.
[76]B. Shen, Z. Wang, H. Shu, G. Wei, H∞ filtering for nonlinear discrete-time stochastic systems with randomly varying sensor delays [J], Automatica, 2009,45(4):1032-1037.
[77]M. Sahebsara, T. Chen, S. L. Shah, Optimal H2 filtering with random sensor delay, multiple packet dropout and uncertain observations [J], International Journal of Control,2007,80(2):292-301.
[78]M. Moayedi, Y. C. Soh, Y. K. Foo, Optimal Kalman filtering with random sensor delays, packet dropouts and missing measurements[C], Proceedings of the American Control Conference, USA, June 2009,3405-3410.
[79]K. Zhang, X. Li, Y. Zhu, Optimal update with out-of-sequence measure-ments[J], IEEE Transactions on Signal Processing,2005,53(6):1992-2004.
[80]X. Shen, Y. Zhu, E. Song, Y. Luo, Optimal centralized update with mul-tiple local out-of-sequence measurements [J], IEEE Transactions on Signal Processing,2009,57(4):1551-1562.
[81]X. Shen, E. Song, Y. Zhu, Y. Luo, Globally optimal distributed Kalman fu-sion with local out-of-sequence-measurement updates [J], IEEE Transactions on Automatic Control,2009,54(8):1928-1934.
[82]A. S. Matveev, A. V. Savkin, The problem of state estimation via asyn-chronous communication channels with irregular transmission times [J], IEEE Transactions on Automatic ontrol,2003,48(4):670-676.
[83]A. S. Matveev, A. V. Savkin, Optimal state estimation in networked systems with asynchronous communication channels and switched sensors [J], Journal of Optimization Theory and Applications,2006,128(1):139-165.
[84]N. E. Nahi, Optimal recursive estimation with uncertain observation [J], IEEE Transactions on Information Theory,1969,15(4):457-462.
[85]M. T. Hadidi, C. S. Schwartz, Linear recursive state estimators under uncer-tain observations [J], IEEE Transactions on Automatic Control,1979,24(6): 944-948.
[86]B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan, S. S. Sastry, Kalman filtering with intermittent observations [J], IEEE Transac-tions on Automatic Control,2004,49(9):1453-1463.
[87]K. Plarre, F. Bullo, On Kalman filtering for detectable systems with in-termittent observations [J], IEEE Transactions on Automatic Control,2009, 54(2):386-390.
[88]L. Schenato, Optimal estimation in networked control systems subject to random delay and packet drop[J], IEEE Transactions on Automatic Control, 2008,53(5):1311-1317.
[89]刘次华,随机过程M],武汉：华中科技大学出版社,2001.
[90]刘嘉焜,王公恕,应用随机过程[M],北京：科学出版社,2004.
[91]J. Evans, V. Krishnamurthy, Hidden Markov model state estimation with randomly delayed observations [J], IEEE Transactions on Automatic Control, 1999,47(8):2157-2166.
[92]H. Song, L. Yu, W. Zhang, H∞ filtering of network-based systems with random delay [J], Signal Processing,2009,89(4):615-622.
[93]J. K. Tugnait, Adaptive estimation and identification for discrete systems with Markov jump parameters [J], IEEE Transactions on Automatic Control, 1982,27(5):1054-1064.
[94]J. K. Tugnait, Detection and estimation for abruptly changing systems [J], Automatica,1982,18(5):607-615.
[95]R. J. Kenefic, A Bayesian approach to time delay estimation [J], IEEE Trans-actions on Acoustics, Speech, and Signal Processing,1981, ASSP-29(3):611-614.
[96]H. A. P. Blom, An efficient filter for abruptly changing systems[C], Proceed-ings of the 23rd IEEE Conference On Decision and Control, Las Vegas, NV, 1984,656-658.
[97]H. A. P. Blom, Continuous-discrete filtering for the systems with Markovian switching coefficients and simultaneous jump[C], Proceedings of the 21st Asilomar Conference on Signals, Systems and Computers, Pacific Grove, 1987,244-248.
[98]H. A. P. Blom, X. R. Li, The interacting multiple model algorithm for sys-tems with Markovian switching coefficients [J], IEEE Transactions on Auto-matic Control,1988,33(8):780-783.
[99]R. Elliot, F. Dufour, F. Sworder, Exact hybrid filters in discrete-time [J], IEEE Transactions on Automatic Control,1996,41(12):1807-1810.
[100]R. J. Elliot, W. P. Malcolm, Robust M-ary detection filters for continuous-time jump Markov systems[C], Proceedings of the 40th IEEE Conference on Decision and Control, Oriando, Florid USA,2001,1681-1686.
[101]F. Dufour, P. Bertrand, The filtering problem for continous-time linear sys-tems with Markovian switching coeficients[J], Systems & Control Letters, 1994,23(5):453-461.
[102]F. Dufour, P. Bertrand, An image based filter for discrete-time Markovian jump linear systems[J], Automatica,1996,32(2):241-247.
[103]F. Dufour, R. Elliot, Filtering with discrete state observations[J], Applied Mathematics and Optimization,1999,40(2):259-272.
[104]A. Doucet, C. Andrieu, Iterative algorithms for state estimation of jump Markov linear systems [J], IEEE Transactions on Automatic Control,2000, 49(6):1216-1227.
[105]Q. Zhang, Optimal filtering of discrete-time hybrid sy stems [J], Journal of Optimization Theory and Applications,1999,100(1):123-144.
[106]Q. Zhang, Hybrid filtering for linear systems with non-Gaussian distur-bances [J], IEEE Transactions on Automatic Control,2000,45(1):50-61.
[107]S. Allam, F. Dufour, P. Bertrand, Discrete-time estimation of a Markov chain with marked point process observations [J], IEEE Transactions on Au-tomatic Control,2001,46(6):903-908.
[108]Y. Bar-Shalom, X. R. Li, Estimation and Tracking. Principles, Techniques and Software[M], Boston:Artech House,1993.
[109]F. Dufour, D. Kannan, Discrete time nonlinear filtering with marked point process observations [J], Stochatic Analysis and Applications,1999,17(1): 99-115.
[110]T. Bjork, Finite dimensional optimal filters for a class of lto-processes with jumping parameters [J], Stochatics,1980,4(2):167-183.
[111]W. M. Wonham, Some applications of stochastic differential equations to optimal nonlinear filtering[J], SIAM Journal on Control and Optimization, 1964,2(3):347-369.
[112]O. L. V. Costa, Linear minimum mean square error estimation for discrete-time Markovian jump linear systems [J], IEEE Transactions on Automatic Control,1994,39(8):1685-1689.
[113]O. L. V. Costa, S. Guerra, Stationary filter for linear minimum mean square error estimator of discrete-time Markovian jump systems [J], IEEE Transac-tions on Automatic Control,2002,47(8):1351-1356.
[114]O. L. V. Costa, M. D. Fragoso, R. P. Marques, Discrete-Time Markov Jump Linear Systems[M], London:Springer-Verlag,2005.
[115]M. D. Fragoso, O. L. V. Costa, J. Baczynski, N. Rocha, Optimal linear mean square filter for continuous-time jump linear systems [J], IEEE Transactions on Automatic Control,2005,50(9):1364-1369.
[116]M. D. Fragoso, N. C. S. Rocha, Stationary filter for continuous-time Marko-vian jump linear systems [J], SI AM Journal on Control and Optimization, 2005,44(3):801-815.
[117]A. Doucet, A. Logothetis, V. Krishnamurthy, Stochastic sampling algo-rithms for state estimation of jump Markov linear systems [J], IEEE Trans-actions on Automatic Control,2000,45(2):188-202.
[118]K. Sun, L. Li, A. P. G. Panayoutopoulos, A. Packard, Extensions of jump Markov system estimation techniques [C], Proceedings of the American Con-trol Conference, Anchorage,2002,4209-4214.
[119]C. Carlemalm, A. Logothetis, V. Krishnamurthy, l1 state estimation of jump Markov linear systems[C], Proceedings of the 37th IEEE Conference on De-cision and Control, Tampa, Florida,1998,263-267.
[120]P. Shi, E. Boukas, R. K. Agarwal, Kalman filtering for continuous-time un-certain systems with Markovian jumping parameters [J], IEEE Transactions on Automatic Control,1999,44(8):1592-1597.
[121]C. E. de Souza, H∞ filtering for discrete-time linear systems with Marko-vian jumping parameters [J], International Journal of Robust and Nonlinear Control,2003; 13(14):1299-1361.
[122]C. E. de Souza, M. D. Fragoso, H∞ filtering for Markovian jump linear sys-tems[J], International Journal of Systems Science,2002,33(11):905-915.
[123]Z. Wang, J. Lam, X. Liu, Robust filtering for discrete-time Markovian jump delay systems[J], IEEE Signal Processing Letters,2004,11(8):659-662.
[124]M. S. Mahmoud, P. Shi, Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters [J], IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,2003,50(1):98-105.
[125]S. Xu, T. Chen, J. Lam, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters [J], IEEE Transactions on Au-tomatic Control,1999,44(8):1592-1597.
[126]S. Xu, T. Chen, J. Lam, Robust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays [J], IEEE Transactions on Auto-matic Control,2003,48(5):900-907.
[127]E. K. Boukas, Z. K. Liu, Robust H∞ filtering for polytopic uncertain time-delay systems with Markov jumps[J], Computers and Electrical Engineering, 2002,28(3):171-193.
[128]E. K. Boukas, Z. K. Liu, Robust H∞ filtering design for uncertain linear systems with multiple time-varying state delay [J], IEEE Transactions on Signal Processing,2001,49(3):569-576.
[129]R. M. Palhares, C. E. de Souza, P. L. D. Peres, Robust H∞, filtering for un-certain discrete-time state-delayed systems [J], IEEE Transactions on Signal Processing,2001,49(8):1696-1703.
[130]M. S. Mahmoud, Robust filtering for uncertain multi-state-delay jumping systems[C], Proceedings of the 7th Internstionnl Conference on Control, Am-tomation, Robotics and Virion, Singspore,2002,168-173.
[131]P. Shi, M. Mahmoud, S. K. Nguang, A. Ismail, Robust filtering for jumping systems with mode-dependent delays [J], Signal Processing,2006,86(1):140-152.
[132]H. Liu, F. Sun, K. He, Z. Sun, Design of reduced-order H∞ filter for Marko-vian jumping systems with time delay [J], IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,2004,51(11):607-612.
[133]Z. Wang, J. Lam, X. Liu, Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances [J], IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,2004,51(5):262-268.
[134]Z. Wang, J. Lam, X. Liu, Nonlinear filtering for state delayed systems with Markovian switching [J], IEEE Transactions on Signal Processing,2003, 51(9):2321-2328.
[135]Y. Fu, Y. Zhang, G. Duan, Q. Chai, Robust guaranteed cost filtering for un-certain timedelay systems with Markovian jumping parameters [J], Journal of Systems Engineering and Electronics,2005,16(4):852-857.
[136]L. Hu, P. Shi, Y. Cao, Delay-dependent Filtering Design for Time-delay Systems with Markovian Jumping Parameters [J], International Journal of Adaptive Control and Signal Processing,2006,21(5):434-448.
[137]M. H. A. Davis, R. B. Vinter, Stochastic Modeling and Control[M], New York:Chapman and Hall,1985.
[138]M. H. A. Davis, Linear Estimation and Stochastic Control[M], London: Chapman and Hall,1977.
[139]Y. Bar-Shalom, X. R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation:Theory Algorithms and Software[M], New York: Wiley,2001.
[140]V. Gupta, T. H. Chung, B. Hassibi, R. M. Murray, On a stochastic sen-sor selection algorithm with applications in sensor scheduling and sensor coverage[J], Automatica,2006,42(2):251-260.