线性乘性噪声系统估计与二次最优控制
|
摘要
随机系统具有广泛的应用背景，存在于无线传感器网络信号处理，经济管理，通讯系统,金融数学等许多领域中.因而,随机系统的控制与估计问题得到了众多学者的关注.但是,随机因素的存在使得这类问题的研究变得非常复杂,许多基础性理论问题尚未得到彻底的解决.因此,随机系统的估计与二次最优控制问题有待于进一步的研究与完善.
本文研究了具有输入时滞乘性噪声系统的线性二次最优控制问题,乘性噪声系统估计与最优控制问题之间的对偶关系,观测丢包系统的新型估计器设计问题，具有输入时滞It(?)随机系统的二次最优控制问题和具有输出时滞It(?)随机系统的Kalman滤波问题.主要研究成果包括以下几点：
●针对两通道单时滞输入随机系统,引入倒向随机系统,应用动态规划方法和完全平方和技术,解决了具有输入时滞随机系统的二次最优控制问题.通过计算一个与原系统维数相同的偏差分方程,设计出控制器.该方法与扩维方法相比计算量较小,尤其当输入时滞很大时.
●针对含有乘性噪声的随机系统，提出了一种新型估计器,建立了乘性噪声系统估计与控制之间的对偶关系,并在可镇与精确能观的条件下证明了一正向广义黎卡提方程的收敛性.作为一个应用,所设计的新型估计器可以应用到观测丢包系统上，利用时间戳技术,给出了滤波器、平滑器计算公式,将估计器增益归结为求解一个广义黎卡提方程,并在标准条件下给出了估计器收敛性分析,最后证明了所设计的估计器在性能上是线性均方差估计与间歇卡尔曼滤波的折衷.
●针对单通道单时滞It(?)随机系统,研究了线性二次最优控制问题.通过构造个新的李亚普诺夫泛函,利用It(?)微分法则及完全平方和技术,将最优控制器的设计归结为求解偏微分方程.另外,针对一类特殊的单通道单时滞系统,研究了二次最优控制问题,将控制器的设计归结到求解一个黎卡提方程.
●针对带有观测时滞的It(?)随机系统,研究了线性最优估计问题.利用It(?)微分法则及新息重组理论,将最优估计器的设计归结为求解一个李亚普诺夫方程和两个黎卡提方程.
·●针对It(?)随机系统,提出了新型估计器,将估计器增益归结到求解正向广义黎卡提方程,进而建立了估计与二次最优控制之间的对偶原理.
总之,本文围绕着随机系统的估计与二次最优控制与问题展开了研究.所得结果进一步丰富了随机系统的估计与最优控制理论.
Stochastic systems are of wide application background, and are available in many fields such as signal processing in wireless sensor networks, economic man-agement, communication systems, and financial mathematics. Hence, the control and estimation problems for the stochastic systems have been intriguing many researchers for decades. However, they are known to be complicated due to the presence of stochastic uncertainties, some fundamental and theoretical problems stay unsettled as a great challenge. The control and estimation problems for stochastic systems remain to be perfected further.
The dissertation focuses on the quadratic optimal control for multiplicative noise systems with input delay, duality between estimation and optimal control for multiplicative noise systems, the novel estimator problem for measurement missing systems, the quadratic optimal control for ItO stochastic systems with input delay and the Kalman filtering problem for ltO stochastic systems with measurement delay. Our main results are as follows.
●It resolves the quadratic optimal control problem for multiplicative noise systems with two channels single input delay via dynamic programming approach and completing square technique. The controller is designed by solving partial difference equation with the same dimension as the original systems. It is superior to the augmentation method in computation espe-cially when the time delay is large.
●It proposes a novel estimator for stochastic system with multiplicative noise, establishes the duality between the optimal control and estimation, and testi-fies the convergence of a forward generalized Riccati equation as the system is stabilizable and exactly observable. Associating time stamp technique with the idea above, we present the filter and smoother for measurement missing systems, which can be designed by solving a generalized Riccati equation, analyze the convergence of the estimator and show that the pro-posed estimator is a trade-off over performance between the MMSE filter and the intermittent Kalman filter.
●It considers the linear quadratic optimal control for ItO stochastic systems with single channel single input delay. By constructing a novel Lyapunov functional, making use of ItO formula and complete square technique, we provide the optimal controller via solving partial differential equation. In addition, we also apply discretization approach to study the problem in a special case and give the optimal controller based on a differential Riccati equation.
●It considers the optimal estimation problem for ItO stochastic systems with measurement delay. By taking advantage of the reorganized innovation anal-ysis and making use of Ito formula, the optimal estimator is designed via solving two Riccati equations and a Lyapunov equation.
●It proposes a new estimator for ItO stochastic systems where the estimator gain is calculated by solving a forward generalized Riccati equation, and establishes the duality between estimation and optimal control problem.
In a word, this dissertation focuses on the estimation and quadratic opti-mal control problem for stochastic systems, the obtained results make stochastic estimation and optimal control theory better.
本文研究了具有输入时滞乘性噪声系统的线性二次最优控制问题,乘性噪声系统估计与最优控制问题之间的对偶关系,观测丢包系统的新型估计器设计问题，具有输入时滞It(?)随机系统的二次最优控制问题和具有输出时滞It(?)随机系统的Kalman滤波问题.主要研究成果包括以下几点：
●针对两通道单时滞输入随机系统,引入倒向随机系统,应用动态规划方法和完全平方和技术,解决了具有输入时滞随机系统的二次最优控制问题.通过计算一个与原系统维数相同的偏差分方程,设计出控制器.该方法与扩维方法相比计算量较小,尤其当输入时滞很大时.
●针对含有乘性噪声的随机系统，提出了一种新型估计器,建立了乘性噪声系统估计与控制之间的对偶关系,并在可镇与精确能观的条件下证明了一正向广义黎卡提方程的收敛性.作为一个应用,所设计的新型估计器可以应用到观测丢包系统上，利用时间戳技术,给出了滤波器、平滑器计算公式,将估计器增益归结为求解一个广义黎卡提方程,并在标准条件下给出了估计器收敛性分析,最后证明了所设计的估计器在性能上是线性均方差估计与间歇卡尔曼滤波的折衷.
●针对单通道单时滞It(?)随机系统,研究了线性二次最优控制问题.通过构造个新的李亚普诺夫泛函,利用It(?)微分法则及完全平方和技术,将最优控制器的设计归结为求解偏微分方程.另外,针对一类特殊的单通道单时滞系统,研究了二次最优控制问题,将控制器的设计归结到求解一个黎卡提方程.
●针对带有观测时滞的It(?)随机系统,研究了线性最优估计问题.利用It(?)微分法则及新息重组理论,将最优估计器的设计归结为求解一个李亚普诺夫方程和两个黎卡提方程.
·●针对It(?)随机系统,提出了新型估计器,将估计器增益归结到求解正向广义黎卡提方程,进而建立了估计与二次最优控制之间的对偶原理.
总之,本文围绕着随机系统的估计与二次最优控制与问题展开了研究.所得结果进一步丰富了随机系统的估计与最优控制理论.
Stochastic systems are of wide application background, and are available in many fields such as signal processing in wireless sensor networks, economic man-agement, communication systems, and financial mathematics. Hence, the control and estimation problems for the stochastic systems have been intriguing many researchers for decades. However, they are known to be complicated due to the presence of stochastic uncertainties, some fundamental and theoretical problems stay unsettled as a great challenge. The control and estimation problems for stochastic systems remain to be perfected further.
The dissertation focuses on the quadratic optimal control for multiplicative noise systems with input delay, duality between estimation and optimal control for multiplicative noise systems, the novel estimator problem for measurement missing systems, the quadratic optimal control for ItO stochastic systems with input delay and the Kalman filtering problem for ltO stochastic systems with measurement delay. Our main results are as follows.
●It resolves the quadratic optimal control problem for multiplicative noise systems with two channels single input delay via dynamic programming approach and completing square technique. The controller is designed by solving partial difference equation with the same dimension as the original systems. It is superior to the augmentation method in computation espe-cially when the time delay is large.
●It proposes a novel estimator for stochastic system with multiplicative noise, establishes the duality between the optimal control and estimation, and testi-fies the convergence of a forward generalized Riccati equation as the system is stabilizable and exactly observable. Associating time stamp technique with the idea above, we present the filter and smoother for measurement missing systems, which can be designed by solving a generalized Riccati equation, analyze the convergence of the estimator and show that the pro-posed estimator is a trade-off over performance between the MMSE filter and the intermittent Kalman filter.
●It considers the linear quadratic optimal control for ItO stochastic systems with single channel single input delay. By constructing a novel Lyapunov functional, making use of ItO formula and complete square technique, we provide the optimal controller via solving partial differential equation. In addition, we also apply discretization approach to study the problem in a special case and give the optimal controller based on a differential Riccati equation.
●It considers the optimal estimation problem for ItO stochastic systems with measurement delay. By taking advantage of the reorganized innovation anal-ysis and making use of Ito formula, the optimal estimator is designed via solving two Riccati equations and a Lyapunov equation.
●It proposes a new estimator for ItO stochastic systems where the estimator gain is calculated by solving a forward generalized Riccati equation, and establishes the duality between estimation and optimal control problem.
In a word, this dissertation focuses on the estimation and quadratic opti-mal control problem for stochastic systems, the obtained results make stochastic estimation and optimal control theory better.
引文
[1]Y. A. Phillis, A smoothing algorithm for systems with multiplicative noise[J], IEEE Trans. Autom. Control,1988,33(4):401-403.
[2]Y. A. Phillis, Estimation and control of systems with unknown covariance and multiplicative noise[J], IEEE Trans. Autom. Control,1989,34(10):1075-1078.
[3]Y. A. Phillis, Controller design of systems with multiplicative noise[J], IEEE Trans. Autom. Control,1988,30(10):1017-1019.
[4]L. Xiao, A. Hassibi, and J. P. How, Control with random communication delays via a discrete-time jump system approach[C], Proceedings of the 2000 American Control Conference, Chicago,2000,2199-2204.
[5]M. L. Sichitiu, P. H. Bauer, and K. Premaratne. The effect of uncertain time-variant delays in ATM networks with explicit rate feedback:a control theoretic approach[J], IEEE/ACM Trans. Networking,2003,11(4):628-637.
[6]E. Altman, T. Basar, and R. Srikant, Congestion control as a stochastic control problem with action delays [J], Automatica,1999,35(12):1937-1950.
[7]G. Tadmor and L. Mirkin, H_∞ control and estimation with preview-part I:matrix ARE solutions in continuous-time [J], IEEE Trans. Autom. Control, 2005,50(1):19-28.
[8]G. Tadmor and L. Mirkin, H_∞ control and estimation with preview-part II:matrix ARE solutions in discrete-time [J], IEEE Trans. Autom. Control, 2005,50(1):29-39.
[9]A. Kojima and S. Ishijima, H_∞, performance of preview control sy stems [J], Automatica,2003,39(4):693-701.
[10]N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications[M], New York:Wiley,1949.
[11]R. E. Kalman, A new approach to linear filtering and prediction problems[J], Transactions of the ASME-Journal of Basic Engineering,1960,82(Series D):35-45.
[12]邓自立,现代时问序列分析-建模、滤波、去卷、预报和控制[M],北京：知识出版社,1989.
[13]B. D.O. Anderson and J. B. Moore. Optimal Filter ing [M], Prentice-Hall, Englewood Cliffs, NJ,1979.
[14]Y. S. Hung and H. T. Ho, A Kalman filter approach to direct depth estima-tion incorporating surface struucture[J], IEEE Trans. Pattern Anal. Machine Intell.,1999,21(6):570-575.
[15]T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation[M], Prentice-Hall, Englewood. Cliffs, N.J,1999.
[16]A. M. Tekalp and G. Pavlovic, Image restoration with multiplicative noise: Incorporating the sensor nonlinearity[J], IEEE Trans. Signal Process.,1991, 39(9):2132-2136.
[17]W. L. De Koning, Optimal estimation of linear discrete-time systems with stochastic parameters [J], Automatica,1984,20(1):113-116.
[18]Y. L. Chen and B. S. Chen, Minimax robust deconvolution filters under stochastic parameter and noise uncertainties [J], IEEE Trans. Signal Process., 1994,42(1):32-45.
[19]H. Zhang, D. Zhang, L. Xie, and J. Lin, Robust filtering under stochastic parametric uncertainties [J], Automatica,2004,40(9):1583-1589.
[20]F. Carravetta, A. Germani, and M. Raimondi, Polynoomial filtering of discrete-time stochastic linear systems with multiplicative state noise[J], IEEE Trans. Autom. Control,1997,42(8):1107-1126.
[21]E. N. Nahi, Optimal recursive estimation with uncetain observation [J], IEEE Trans. Inform. Theory,1969,15(4):457-462.
P. K. Rajasekaran, N. Satyanarayana, and M. D. Srinath, Optimum linear estimation of stochastic signals in presence of multiplicative noise[J], IEEE Trans. Aerosp. Electron. Syst.,1971,7(3):462-468.
[23]A. G. Jaffer and S. C. Gupta, Recursive Bayesian estimation with uncer-tain observation[J], IEEE Trans. Inform. Theory,1971, IT-17:614-616.
[24]M. T. Hadidi and S. C. Schwarts, Linear recursive state estimators under uncertain observations [J], IEEE Trans. Autom. Control 1979,24(6):944-947.
[25]J. K. Tugnait, Stability of optimal linear estimations of stochastic signals in white multiplicative noise [J], IEEE Trans. Autom. Control,1981,26(3): 757-761.
[26]B. S. Chow and W. P. Birkemeier, A new recursive filter for systems with multiplicative noise[J], IEEE Trans. Inform. Theory,1990,36(6):1430-1435.
[27]褚东升,带乘性噪声系统的估计理论与应用,哈尔滨工业大学博士论文[D],1994.
[28]F. Yang, Z. Wang, and Y. S. Hung, Trace bounds on the covariances of continuous-time systems with multiplicative noise [J], IEEE Trans. Autom. Control,1993,38(7):138-142.
[29]H. Zhang, X. Lu, W. Zhang, and W. Wang, Kalman filtering for linear time-delayed continuous-time systems with stochastic multiplicative noises [J], In-ternational Journal of Control, Automation, and Systems,2007,5(4):355-363.
[30]J. C. Jimeneza and T. Ozakib, Linear estimation of continuous-discrete linear state space models with multiplicative noise [J], Systems Control Lett.,2002, 47(1):91-1-01.
[31]E. Gershon, U. Shaked, and I. Yaesh, H_(∞) control and filtering of discrete-time stochastic systems with multplicative noise[J], Automatica,2001,37(3): 409-417.
[32]E. Gershon, D. J. N. Limebeer, U. Shaked, and I. Yaesh, Stochastic H_2/H_∞ tracking with preview for state-multiplicative systems [J], IEEE Trans. Au-tom. Control,2004,49(11):2061-2068.
E. Gershon, U. Shaked, and N. Berman, H_∞ control and estimation of re-tarded state-multiplicative stochastic systems[J], IEEE Trans. Autom. Con-trol,2007,52(9):1773-1779.
[34]E. Gershon, D. J. N-. Limebeer, U. Shaked, and I. Yaesh, Robust H_∞ filtering of stationary continuous-time linear systems with stochastic uncertainties [J], IEEE Trans. Autom. Control,2001,46(11):1788-1793.
[35]F. Yang, Z. Wang, and Y. S. Hung, Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noise[J], IEEE Trans. Autom. Control,2002,47(7):1179-1183.
[36]F. Wang and V. Balakrishnan, Robust Kalman filters for linear time-varying systems with stochatsic parameter uncertainties [J], IEEE Trans. Signal Pro-cess.,2002,50(4):803-813.
[37]F. Wang and V. Balakrishnan, Robust steady-state filtering for systems with deterministic and stochastic uncertainties [J], IEEE Trans. Signal Process., 2003,51(10):2550-2558.
[38]F. Carravetta, A. Germani, and M. K. Shuakayev, A new suboptimal ap-proach to the filtering problem for bilinear stochastic differential systems [J], SIAM J. Control Optim.,2000,38(4):1171-1203.
[39]D. L. Kleinman, Optimal control of linear systems with time-delay and ob-servation noise [J],IEEE Trans. Autom. Control,1969,14(5):524-527.
[40]H. Kwakernaak, Optimal filtering in linear systems with time delays [J], IEEE Trans. Autom. Control,1967,12(2):169-173.
[41]H. Zhang, X. Lu, and D. Cheng, Optimal estimation for continuous-time systems with delayed measurements [J], IEEE Trans. Autom. Control,2006, 51(5):823-827.
[42]H. Zhang, D. Zhang, and L. Xie, An innovation approach to H_∞ prediction with application to systems with time delayed measurements[J], Automatica, 2004,40(7):1253-1261.
[43]H. Zhao, H. Zhang, and C. Zhang, Optimal robust estimation for linear un-certain systems with single delayed measurement [J], Acta Automatica Sinica, 2008,34(2):202-207.
[44]H. Zhao.H. Zhang, and C. Zhang, Optimal robust filtering for a linear un-certain systems with multiple delayed measurements [J], Journal of Shandong University (Engineering Science),2007,37(4):6-12.
[45]罗成新,不定随机线性二次最优控制问题理论研究[D],大连理工大学博士论文,2004.
[46]郑大钟,线性系统理论(第二版)[M],北京：清华大学出版社,2002.
[47]解学书,最优控制理论与应用[M],北京：清华大学出版社,1986.
[48]陈翰馥,离散时间系统的递推估计与随机控制[M],北京：科学出版社,1980.
[49]郭尚来,随机控制[M],北京：清华大学出版社,1999.
[50]方洋旺,随机系统最优控制[M],北京：清华大学出版社,2005.
[51]M. M. Mohler and W. J. Kolodziej, An overview of stochastic bilinear.control processes[J], IEEE Trans, on Systems, Man and Cybernetics,1980,10(12): 913-919.
[52]O. L. V. Costa and C. S. Kubrusly, State-feedback H_∞ control for discrete-time infinite-dimensional stochastic bilinear systems[J], J. Math. Syst., Es-tim., Control,1996,6:1-32.
[53]L. E. Ghaoui, State-feedback control of systems with multiplicative noise via linear matrix inequalities [J], Systems Control Lett.,1995,24(3):223-228.
[54]A. E. Bouhtouri, D. Hinrichsen, and A. J. Pritchard,H_∞-Type control for discrete-time stochastic systems [J], Int. J. Robust Nonlinear Control,1999, 13(9):923-948.
[55]B. S. Chen and W. Zhang, Stochastic H_2/H_∞ control with state-dependent noise[J], Automatica,2004,45(1):45-57.
[56]W. Zhang, H. Zhang, and B. S. Chen, Stochastic H_2/H_∞ control with (x, u,v)-dependent noise:Finite horizon case[J], Automatica,2006,42(11): 1891-1898.
[57]W. Zhang, Y. Huang, and H. Zhang, Stochastic H_2/H_∞ control for discrete-time systems with state and disturbance dependent noise[J], Automatica, 2007,43(3):513-521.
[58]B. D. O. Anderson and J. B. Moore, Linear Optimal Control[M], Prentice-Hall, Englewood Cliffs, NJ,1971.
[59]B. Hassibi, A. H. Sayed, and T. Kailath. Indefinite Quadratic Estimation and Control:A Unified Approach to H_2 and H_∞ Theories[M], SIAM Studies in Applied Mathematics series,1998.
[60]W. M. Wonham, On a matrix Riccati equation of stochastic control[J], SIAM J. Control Optim.,1968,6(4):681-697.
[61]G. Freiling and G. Jank, Generalized Riccati difference and differential equa-tions[J], Linear Algebra Appl.,1996,241-243:291-303.
[62]A. C. M. Ran and R. Vreugdenhil, Existence and comparison theorems for al-gebraic Riccati equations for continuous-and discrete-time systems [J], Linear Algebra Appl.,1988,99:63-83.
[63]P. J. Mclane, Optimal stochastic control of linear systems with state-and control-dependent disturbance [J], IEEE Trans. Autom. Control,1971,16(6): 793-798.
[64]J. L. Willems and J. C. Willems, Feedback stabilizability for stochastic sys-tems with state and control dependent noise [J], Automatica,1976,12(3): 277-283.
[65]M. A. Rami, X. Chen, and X. Y. Zhou, Discrete-time indefinite LQ control with state and control dependent noises [J], J. Global Optim.,2002,23(3): 245-265.
[66]M. A. Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls [J], IEEE Trans. Autom. Control,2000,45(6):245-265.
[67]A. E. B. Lim and X. Y. Zhou, Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights[J], IEEE Trans. Autom. Control,1999,44(7):1359-1369.
[68]M. A. Rami, X. Chen, J. B. Moorse, and X. Y. Zhou, Solvability and asymp-totic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls[J], IEEE Trans. Autom. Control,2001,46(3):428-440.
[69]C. Luo and E. Feng, Generalized differential Riccati equation and indefinite stochastic LQ control with cross term[J], Appl. Math. Comput.,2004,155(1): 121-135.
[70]W. Zhang, H. Zhang, and B. Chen, Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion[J], IEEE Trans. Autom. Control,2008,53(7):1630-1642.
[71]S. Peng, A general stochastic maximum principle for optimal control prob-lems[J], SIAM J. Control Optim.,1990,28(4):966-979.
[72]G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information[J], IEEE Trans. Autom. Control, 2009,54(6):1230-1242.
[73]S. Chen, X. Li, and X. Y. Zhou, Stochastic linear quadratic regulars with indefinite control weight costs[J], SIAM J. Control Optim.,1998,36(5): 1685-1702.
[74]W. Zhang and B. S. Chen, On stabilizability and exact observability of stochastic systems with their applications [J], Automatica,2004,40(1):87-94.
[75]J. B. Moore, X. Y. Zhou, and A. E. B. Lim, Discrete time LQG controls with control dependent noise[J], Systems Control Lett.,1999,36(3):199-206.
[76]A. Beghi and D. D'alessandro, Discrete-time optimal control with control-dependent noise and generalized Riccati difference equations[J], Automatica, 1998,34(8):1031-1034.
[77]Y. Huang, W. Zhang, and H. Zhang, Infinite horizon LQ optimal control for discrete-time stochastic systems[C], The Sixth World Congress on Intelligent Control and Automation,2006,1:252-256.
[78]J. S. Meditch, Stochastic Optimal Linear Estimation and Control[M], New York:McGraw-Hill,1969.
[79]S. M. Joshi, On optimal control of linear systems in the presence of multi-plicative noise[J], IEEE Trans. Aerosp. Electron. Syst.,1976,12(1):80-85.
[80]F. Carravetta and G. Mavelli, Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise:The incomplete information case[J], Automatica,2007,43(5):751-757.
[81]J. Nilsson, B. Bernhardsson, and B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays [J], Automatica,1998, 34(1):57-64.
[82]L. Mariani and B. Nicoletti, Optimal discrete-time systems with pure de-lays[J], IEEE Trans. Autom. Control,1973,18(3):311-313.
[83]L. Mirkin and G. Tadmour,H_∞ control of system with I/O delay:a review of some problem-oriented methods [J], IMA J. Math. Control Inform.,2002, 19(1-2):185-199.
[84]G. Meinsma and L. Mirkin, H_∞ control of systems with multiple I/O delays via decomposition to adobe problems [J], IEEE Trans. Autom. Control,2005, 50(2):199-211.
[85]M. N. Oguztoreli, Time-Lag Control Systems[M], Academic Press, NY,1966.
[86]D. Ross, Controller design for time lag systems via a quadratic criterion[J], IEEE Trans. Autom. Control,1971,16(6):664-672.
[87]M. A. Soliman and W. H. Ray, Optimal feedback control for linear-quadratic systems having time delays[J], Int. J. Control,1972,15(4):609-627.
[88]Y. A. Fiagbedzi and A. E. Pearson, Feedback stabilization of linear au-tonomous time lag systems[J], IEEE Trans. Autom. Control,1986,31(9): 847-855.
[89]A. Kojima, K. Uchida, E. Shimemura, and S. Ishijima, Robust stabilization of a system with delays in control[J], IEEE Trans. Autom. Control,1994, 39(8):1694-1698.
[90]D. H. Chyung, Discrete systems with delays in control[J], IEEE Trans. Au-tomat. Control,1969,14(2):196-197.
[91]R. S. Pindyck, The discrete-time tracking problem with a time delay in the control[J], IEEE Trans. Autom. Control,1972,17(3):397-398.
[92]H. Zhang, G. Duan, and L. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays[J], Automatica,2006,42(9):1465-1476.
[93]A. Kojima and S. Ishijima, Formulas on preview and delayed H_∞ control[C], Proceedings of the 42th IEEE Conference on Decision and Control, Hawaii, 2003,6532-6538.
[94]S. Liu, L. Xie, and H. Zhang, Linear quadratic tracking problem for discrete-time systems [J], Int. J. Robust Nonlinear Control,2010, on-line.
[95]R. E. Kalman, Contributions to the theory of optimal control [J], Bol. Soc. Math. Mexicana.1960,5(2):102-119.
[96]S. Liu, L. Xie, and H. Zhang, Linear quadratic regulation for discrete-time systems with multiple delays in single input channel[C], Proceedings of the 17th IFAC World Congress, Seoul, Korea,2008:13336-13341.
[97]M. C. Delfour, Linear optimal control of systems with state control variable delays[J], Automatica,1984,20(1):69-77.
[98]M. C. Delfour, The linear quadratic optimal control problem with delays in the state and control variables:a state space approach [J], SI AM J. Control Optim.,1986,24(5):835-883.
[99]G. Tadmor, The standard H_∞ problem in systems with a single input delay [J], IEEE Trans. Autom. Control,2000,45(3):382-396.
[100]A. Moelja and G. Meinsma, H_2-optimal control of systems with multiple i/o delays:Time domain approach[J], Automatica,2005,41(7):1229-1238.
[101]S. Liu, L. Xie, and H. Zhang, Linear quadratic regulation for continuous-time systems with time-varying input delay [J], Int. J. Robust Nonlinear Control, 2010, accepted for publication.
[102]R. Bellman, Some problems in the theory of dynamic programming[J], Econometrica,1954,22(1):37-48.
[103]P. Blackmore and R. Bitmead, Duality between the discrete-time Kalman filter and LQ control law[J], IEEE Trans. Autom. Control,1995,40(8):1442-1444.
[104]H. Zhang, L. Xie, and G. Duan, H_∞ control of discrete-time systems with multiple input delays[J], IEEE Trans. Autom. Control,2007,52(2):271-283.
[105]B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, and S. Sastry, Time varying optimal control with packet losses [C], Proceedings of the 43rd Conference on Decision and Control, Atlantis, Paradises Island, Bahamas,2004,1938-1943.
[106]K. Plarre and F. Bullo, On Kalman filtering for detective systems with intermittent observations [J], IEEE Trans. Autom. Control,2009,54(2):386-390.
[107]V. Gupta, A. F. Dana, J. P. Hespanha, R. M. Murray, and B. Hassibi, Data transmission over networks for estimation and control[J], IEEE Trans. Automat. Control,2009,54(8):1807-1819.
[108]A. S. Leong. S. Dey, and J. S. Evans, On Kalman smoothing with random packet loss[J], IEEE Trans. Autom. Control,2008,56(7):3346-3351.
[109]M. Sahebsara, T. Chen, and S. L. Shah, Optimal H2 filtering with random sensor delay, multiple packet dropout and uncertain observations[J], Int. J. Control,2007,80(2):292-301.
[110]L. Schenato. Optimal estimation in networkd control systems subject to random delay and packet drop[J], IEEE Trans. Autom. Control,2008,53(5): 1311-1317.
[111]B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, and S. Sas-try, Kalman filtering with intermittent observations[J], IEEE Trans. Autom. Control,2004,49(9):1453-1464.
[112]S. Kar, B. Sinopoli, and J. M. F. Moura, Kalman filtering with intermittent observations:weak convergence to a stationary distribution[J], IEEE Trans. Automat. Control, to be appeared.
[113]X. Liu and A. J. Goldsmith, Kalman filtering with partial observation losses[C], Proceedings of the 43rd IEEE Conference on Decision Control, Par-adise Island, Bahamas,2004,1413-1418.
[114]Z. Jin, V. Gupta, and R. M. Murray, State estimation over packet dropping networks using multiple description coding[J], Automatica,2006,42(9):1413-1418.
[115]M. Huang and S. Dey, Stabilty of Kalman filtering with Markovian packet losses [J], Automatica,2007,43(4):598-607.
[116]S. Sun, L. Xie, W. Xiao, and Y. C. Soh, Optimal linear estimation for systems with multiple packet dropouts[J], Automatica,2008,44(5):1333-1342.
[117]S. Sun, Optimal estimators for systems with finite consecutive packet dropouts[J], IEEE Trans. Signal Processing Letters,2009,16(7):557-560.
[118]S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix In-equality in Systems and Control Theory[M], Philadelphia, PA:SIAM,1994.
[119]X. Song, H. Zhang, and L. Xie, Linear quadratic regulation for discrete-time stochastic systems with input delays[C], Proceedings of the 27th Chinese Control Conference,2008,432-436.
[120]H. Zhang and X. Song, New estimator and duality of control for multiplica-tive noise systems with applications to packet dropping and multiple input delay systems[C], Proceedings of the 7th IEEE International Conference on Control and Automation, Christchurch, New Zealand,2009,75-80.
[2]Y. A. Phillis, Estimation and control of systems with unknown covariance and multiplicative noise[J], IEEE Trans. Autom. Control,1989,34(10):1075-1078.
[3]Y. A. Phillis, Controller design of systems with multiplicative noise[J], IEEE Trans. Autom. Control,1988,30(10):1017-1019.
[4]L. Xiao, A. Hassibi, and J. P. How, Control with random communication delays via a discrete-time jump system approach[C], Proceedings of the 2000 American Control Conference, Chicago,2000,2199-2204.
[5]M. L. Sichitiu, P. H. Bauer, and K. Premaratne. The effect of uncertain time-variant delays in ATM networks with explicit rate feedback:a control theoretic approach[J], IEEE/ACM Trans. Networking,2003,11(4):628-637.
[6]E. Altman, T. Basar, and R. Srikant, Congestion control as a stochastic control problem with action delays [J], Automatica,1999,35(12):1937-1950.
[7]G. Tadmor and L. Mirkin, H_∞ control and estimation with preview-part I:matrix ARE solutions in continuous-time [J], IEEE Trans. Autom. Control, 2005,50(1):19-28.
[8]G. Tadmor and L. Mirkin, H_∞ control and estimation with preview-part II:matrix ARE solutions in discrete-time [J], IEEE Trans. Autom. Control, 2005,50(1):29-39.
[9]A. Kojima and S. Ishijima, H_∞, performance of preview control sy stems [J], Automatica,2003,39(4):693-701.
[10]N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications[M], New York:Wiley,1949.
[11]R. E. Kalman, A new approach to linear filtering and prediction problems[J], Transactions of the ASME-Journal of Basic Engineering,1960,82(Series D):35-45.
[12]邓自立,现代时问序列分析-建模、滤波、去卷、预报和控制[M],北京：知识出版社,1989.
[13]B. D.O. Anderson and J. B. Moore. Optimal Filter ing [M], Prentice-Hall, Englewood Cliffs, NJ,1979.
[14]Y. S. Hung and H. T. Ho, A Kalman filter approach to direct depth estima-tion incorporating surface struucture[J], IEEE Trans. Pattern Anal. Machine Intell.,1999,21(6):570-575.
[15]T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation[M], Prentice-Hall, Englewood. Cliffs, N.J,1999.
[16]A. M. Tekalp and G. Pavlovic, Image restoration with multiplicative noise: Incorporating the sensor nonlinearity[J], IEEE Trans. Signal Process.,1991, 39(9):2132-2136.
[17]W. L. De Koning, Optimal estimation of linear discrete-time systems with stochastic parameters [J], Automatica,1984,20(1):113-116.
[18]Y. L. Chen and B. S. Chen, Minimax robust deconvolution filters under stochastic parameter and noise uncertainties [J], IEEE Trans. Signal Process., 1994,42(1):32-45.
[19]H. Zhang, D. Zhang, L. Xie, and J. Lin, Robust filtering under stochastic parametric uncertainties [J], Automatica,2004,40(9):1583-1589.
[20]F. Carravetta, A. Germani, and M. Raimondi, Polynoomial filtering of discrete-time stochastic linear systems with multiplicative state noise[J], IEEE Trans. Autom. Control,1997,42(8):1107-1126.
[21]E. N. Nahi, Optimal recursive estimation with uncetain observation [J], IEEE Trans. Inform. Theory,1969,15(4):457-462.
P. K. Rajasekaran, N. Satyanarayana, and M. D. Srinath, Optimum linear estimation of stochastic signals in presence of multiplicative noise[J], IEEE Trans. Aerosp. Electron. Syst.,1971,7(3):462-468.
[23]A. G. Jaffer and S. C. Gupta, Recursive Bayesian estimation with uncer-tain observation[J], IEEE Trans. Inform. Theory,1971, IT-17:614-616.
[24]M. T. Hadidi and S. C. Schwarts, Linear recursive state estimators under uncertain observations [J], IEEE Trans. Autom. Control 1979,24(6):944-947.
[25]J. K. Tugnait, Stability of optimal linear estimations of stochastic signals in white multiplicative noise [J], IEEE Trans. Autom. Control,1981,26(3): 757-761.
[26]B. S. Chow and W. P. Birkemeier, A new recursive filter for systems with multiplicative noise[J], IEEE Trans. Inform. Theory,1990,36(6):1430-1435.
[27]褚东升,带乘性噪声系统的估计理论与应用,哈尔滨工业大学博士论文[D],1994.
[28]F. Yang, Z. Wang, and Y. S. Hung, Trace bounds on the covariances of continuous-time systems with multiplicative noise [J], IEEE Trans. Autom. Control,1993,38(7):138-142.
[29]H. Zhang, X. Lu, W. Zhang, and W. Wang, Kalman filtering for linear time-delayed continuous-time systems with stochastic multiplicative noises [J], In-ternational Journal of Control, Automation, and Systems,2007,5(4):355-363.
[30]J. C. Jimeneza and T. Ozakib, Linear estimation of continuous-discrete linear state space models with multiplicative noise [J], Systems Control Lett.,2002, 47(1):91-1-01.
[31]E. Gershon, U. Shaked, and I. Yaesh, H_(∞) control and filtering of discrete-time stochastic systems with multplicative noise[J], Automatica,2001,37(3): 409-417.
[32]E. Gershon, D. J. N. Limebeer, U. Shaked, and I. Yaesh, Stochastic H_2/H_∞ tracking with preview for state-multiplicative systems [J], IEEE Trans. Au-tom. Control,2004,49(11):2061-2068.
E. Gershon, U. Shaked, and N. Berman, H_∞ control and estimation of re-tarded state-multiplicative stochastic systems[J], IEEE Trans. Autom. Con-trol,2007,52(9):1773-1779.
[34]E. Gershon, D. J. N-. Limebeer, U. Shaked, and I. Yaesh, Robust H_∞ filtering of stationary continuous-time linear systems with stochastic uncertainties [J], IEEE Trans. Autom. Control,2001,46(11):1788-1793.
[35]F. Yang, Z. Wang, and Y. S. Hung, Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noise[J], IEEE Trans. Autom. Control,2002,47(7):1179-1183.
[36]F. Wang and V. Balakrishnan, Robust Kalman filters for linear time-varying systems with stochatsic parameter uncertainties [J], IEEE Trans. Signal Pro-cess.,2002,50(4):803-813.
[37]F. Wang and V. Balakrishnan, Robust steady-state filtering for systems with deterministic and stochastic uncertainties [J], IEEE Trans. Signal Process., 2003,51(10):2550-2558.
[38]F. Carravetta, A. Germani, and M. K. Shuakayev, A new suboptimal ap-proach to the filtering problem for bilinear stochastic differential systems [J], SIAM J. Control Optim.,2000,38(4):1171-1203.
[39]D. L. Kleinman, Optimal control of linear systems with time-delay and ob-servation noise [J],IEEE Trans. Autom. Control,1969,14(5):524-527.
[40]H. Kwakernaak, Optimal filtering in linear systems with time delays [J], IEEE Trans. Autom. Control,1967,12(2):169-173.
[41]H. Zhang, X. Lu, and D. Cheng, Optimal estimation for continuous-time systems with delayed measurements [J], IEEE Trans. Autom. Control,2006, 51(5):823-827.
[42]H. Zhang, D. Zhang, and L. Xie, An innovation approach to H_∞ prediction with application to systems with time delayed measurements[J], Automatica, 2004,40(7):1253-1261.
[43]H. Zhao, H. Zhang, and C. Zhang, Optimal robust estimation for linear un-certain systems with single delayed measurement [J], Acta Automatica Sinica, 2008,34(2):202-207.
[44]H. Zhao.H. Zhang, and C. Zhang, Optimal robust filtering for a linear un-certain systems with multiple delayed measurements [J], Journal of Shandong University (Engineering Science),2007,37(4):6-12.
[45]罗成新,不定随机线性二次最优控制问题理论研究[D],大连理工大学博士论文,2004.
[46]郑大钟,线性系统理论(第二版)[M],北京：清华大学出版社,2002.
[47]解学书,最优控制理论与应用[M],北京：清华大学出版社,1986.
[48]陈翰馥,离散时间系统的递推估计与随机控制[M],北京：科学出版社,1980.
[49]郭尚来,随机控制[M],北京：清华大学出版社,1999.
[50]方洋旺,随机系统最优控制[M],北京：清华大学出版社,2005.
[51]M. M. Mohler and W. J. Kolodziej, An overview of stochastic bilinear.control processes[J], IEEE Trans, on Systems, Man and Cybernetics,1980,10(12): 913-919.
[52]O. L. V. Costa and C. S. Kubrusly, State-feedback H_∞ control for discrete-time infinite-dimensional stochastic bilinear systems[J], J. Math. Syst., Es-tim., Control,1996,6:1-32.
[53]L. E. Ghaoui, State-feedback control of systems with multiplicative noise via linear matrix inequalities [J], Systems Control Lett.,1995,24(3):223-228.
[54]A. E. Bouhtouri, D. Hinrichsen, and A. J. Pritchard,H_∞-Type control for discrete-time stochastic systems [J], Int. J. Robust Nonlinear Control,1999, 13(9):923-948.
[55]B. S. Chen and W. Zhang, Stochastic H_2/H_∞ control with state-dependent noise[J], Automatica,2004,45(1):45-57.
[56]W. Zhang, H. Zhang, and B. S. Chen, Stochastic H_2/H_∞ control with (x, u,v)-dependent noise:Finite horizon case[J], Automatica,2006,42(11): 1891-1898.
[57]W. Zhang, Y. Huang, and H. Zhang, Stochastic H_2/H_∞ control for discrete-time systems with state and disturbance dependent noise[J], Automatica, 2007,43(3):513-521.
[58]B. D. O. Anderson and J. B. Moore, Linear Optimal Control[M], Prentice-Hall, Englewood Cliffs, NJ,1971.
[59]B. Hassibi, A. H. Sayed, and T. Kailath. Indefinite Quadratic Estimation and Control:A Unified Approach to H_2 and H_∞ Theories[M], SIAM Studies in Applied Mathematics series,1998.
[60]W. M. Wonham, On a matrix Riccati equation of stochastic control[J], SIAM J. Control Optim.,1968,6(4):681-697.
[61]G. Freiling and G. Jank, Generalized Riccati difference and differential equa-tions[J], Linear Algebra Appl.,1996,241-243:291-303.
[62]A. C. M. Ran and R. Vreugdenhil, Existence and comparison theorems for al-gebraic Riccati equations for continuous-and discrete-time systems [J], Linear Algebra Appl.,1988,99:63-83.
[63]P. J. Mclane, Optimal stochastic control of linear systems with state-and control-dependent disturbance [J], IEEE Trans. Autom. Control,1971,16(6): 793-798.
[64]J. L. Willems and J. C. Willems, Feedback stabilizability for stochastic sys-tems with state and control dependent noise [J], Automatica,1976,12(3): 277-283.
[65]M. A. Rami, X. Chen, and X. Y. Zhou, Discrete-time indefinite LQ control with state and control dependent noises [J], J. Global Optim.,2002,23(3): 245-265.
[66]M. A. Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls [J], IEEE Trans. Autom. Control,2000,45(6):245-265.
[67]A. E. B. Lim and X. Y. Zhou, Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights[J], IEEE Trans. Autom. Control,1999,44(7):1359-1369.
[68]M. A. Rami, X. Chen, J. B. Moorse, and X. Y. Zhou, Solvability and asymp-totic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls[J], IEEE Trans. Autom. Control,2001,46(3):428-440.
[69]C. Luo and E. Feng, Generalized differential Riccati equation and indefinite stochastic LQ control with cross term[J], Appl. Math. Comput.,2004,155(1): 121-135.
[70]W. Zhang, H. Zhang, and B. Chen, Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion[J], IEEE Trans. Autom. Control,2008,53(7):1630-1642.
[71]S. Peng, A general stochastic maximum principle for optimal control prob-lems[J], SIAM J. Control Optim.,1990,28(4):966-979.
[72]G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information[J], IEEE Trans. Autom. Control, 2009,54(6):1230-1242.
[73]S. Chen, X. Li, and X. Y. Zhou, Stochastic linear quadratic regulars with indefinite control weight costs[J], SIAM J. Control Optim.,1998,36(5): 1685-1702.
[74]W. Zhang and B. S. Chen, On stabilizability and exact observability of stochastic systems with their applications [J], Automatica,2004,40(1):87-94.
[75]J. B. Moore, X. Y. Zhou, and A. E. B. Lim, Discrete time LQG controls with control dependent noise[J], Systems Control Lett.,1999,36(3):199-206.
[76]A. Beghi and D. D'alessandro, Discrete-time optimal control with control-dependent noise and generalized Riccati difference equations[J], Automatica, 1998,34(8):1031-1034.
[77]Y. Huang, W. Zhang, and H. Zhang, Infinite horizon LQ optimal control for discrete-time stochastic systems[C], The Sixth World Congress on Intelligent Control and Automation,2006,1:252-256.
[78]J. S. Meditch, Stochastic Optimal Linear Estimation and Control[M], New York:McGraw-Hill,1969.
[79]S. M. Joshi, On optimal control of linear systems in the presence of multi-plicative noise[J], IEEE Trans. Aerosp. Electron. Syst.,1976,12(1):80-85.
[80]F. Carravetta and G. Mavelli, Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise:The incomplete information case[J], Automatica,2007,43(5):751-757.
[81]J. Nilsson, B. Bernhardsson, and B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays [J], Automatica,1998, 34(1):57-64.
[82]L. Mariani and B. Nicoletti, Optimal discrete-time systems with pure de-lays[J], IEEE Trans. Autom. Control,1973,18(3):311-313.
[83]L. Mirkin and G. Tadmour,H_∞ control of system with I/O delay:a review of some problem-oriented methods [J], IMA J. Math. Control Inform.,2002, 19(1-2):185-199.
[84]G. Meinsma and L. Mirkin, H_∞ control of systems with multiple I/O delays via decomposition to adobe problems [J], IEEE Trans. Autom. Control,2005, 50(2):199-211.
[85]M. N. Oguztoreli, Time-Lag Control Systems[M], Academic Press, NY,1966.
[86]D. Ross, Controller design for time lag systems via a quadratic criterion[J], IEEE Trans. Autom. Control,1971,16(6):664-672.
[87]M. A. Soliman and W. H. Ray, Optimal feedback control for linear-quadratic systems having time delays[J], Int. J. Control,1972,15(4):609-627.
[88]Y. A. Fiagbedzi and A. E. Pearson, Feedback stabilization of linear au-tonomous time lag systems[J], IEEE Trans. Autom. Control,1986,31(9): 847-855.
[89]A. Kojima, K. Uchida, E. Shimemura, and S. Ishijima, Robust stabilization of a system with delays in control[J], IEEE Trans. Autom. Control,1994, 39(8):1694-1698.
[90]D. H. Chyung, Discrete systems with delays in control[J], IEEE Trans. Au-tomat. Control,1969,14(2):196-197.
[91]R. S. Pindyck, The discrete-time tracking problem with a time delay in the control[J], IEEE Trans. Autom. Control,1972,17(3):397-398.
[92]H. Zhang, G. Duan, and L. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays[J], Automatica,2006,42(9):1465-1476.
[93]A. Kojima and S. Ishijima, Formulas on preview and delayed H_∞ control[C], Proceedings of the 42th IEEE Conference on Decision and Control, Hawaii, 2003,6532-6538.
[94]S. Liu, L. Xie, and H. Zhang, Linear quadratic tracking problem for discrete-time systems [J], Int. J. Robust Nonlinear Control,2010, on-line.
[95]R. E. Kalman, Contributions to the theory of optimal control [J], Bol. Soc. Math. Mexicana.1960,5(2):102-119.
[96]S. Liu, L. Xie, and H. Zhang, Linear quadratic regulation for discrete-time systems with multiple delays in single input channel[C], Proceedings of the 17th IFAC World Congress, Seoul, Korea,2008:13336-13341.
[97]M. C. Delfour, Linear optimal control of systems with state control variable delays[J], Automatica,1984,20(1):69-77.
[98]M. C. Delfour, The linear quadratic optimal control problem with delays in the state and control variables:a state space approach [J], SI AM J. Control Optim.,1986,24(5):835-883.
[99]G. Tadmor, The standard H_∞ problem in systems with a single input delay [J], IEEE Trans. Autom. Control,2000,45(3):382-396.
[100]A. Moelja and G. Meinsma, H_2-optimal control of systems with multiple i/o delays:Time domain approach[J], Automatica,2005,41(7):1229-1238.
[101]S. Liu, L. Xie, and H. Zhang, Linear quadratic regulation for continuous-time systems with time-varying input delay [J], Int. J. Robust Nonlinear Control, 2010, accepted for publication.
[102]R. Bellman, Some problems in the theory of dynamic programming[J], Econometrica,1954,22(1):37-48.
[103]P. Blackmore and R. Bitmead, Duality between the discrete-time Kalman filter and LQ control law[J], IEEE Trans. Autom. Control,1995,40(8):1442-1444.
[104]H. Zhang, L. Xie, and G. Duan, H_∞ control of discrete-time systems with multiple input delays[J], IEEE Trans. Autom. Control,2007,52(2):271-283.
[105]B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, and S. Sastry, Time varying optimal control with packet losses [C], Proceedings of the 43rd Conference on Decision and Control, Atlantis, Paradises Island, Bahamas,2004,1938-1943.
[106]K. Plarre and F. Bullo, On Kalman filtering for detective systems with intermittent observations [J], IEEE Trans. Autom. Control,2009,54(2):386-390.
[107]V. Gupta, A. F. Dana, J. P. Hespanha, R. M. Murray, and B. Hassibi, Data transmission over networks for estimation and control[J], IEEE Trans. Automat. Control,2009,54(8):1807-1819.
[108]A. S. Leong. S. Dey, and J. S. Evans, On Kalman smoothing with random packet loss[J], IEEE Trans. Autom. Control,2008,56(7):3346-3351.
[109]M. Sahebsara, T. Chen, and S. L. Shah, Optimal H2 filtering with random sensor delay, multiple packet dropout and uncertain observations[J], Int. J. Control,2007,80(2):292-301.
[110]L. Schenato. Optimal estimation in networkd control systems subject to random delay and packet drop[J], IEEE Trans. Autom. Control,2008,53(5): 1311-1317.
[111]B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, and S. Sas-try, Kalman filtering with intermittent observations[J], IEEE Trans. Autom. Control,2004,49(9):1453-1464.
[112]S. Kar, B. Sinopoli, and J. M. F. Moura, Kalman filtering with intermittent observations:weak convergence to a stationary distribution[J], IEEE Trans. Automat. Control, to be appeared.
[113]X. Liu and A. J. Goldsmith, Kalman filtering with partial observation losses[C], Proceedings of the 43rd IEEE Conference on Decision Control, Par-adise Island, Bahamas,2004,1413-1418.
[114]Z. Jin, V. Gupta, and R. M. Murray, State estimation over packet dropping networks using multiple description coding[J], Automatica,2006,42(9):1413-1418.
[115]M. Huang and S. Dey, Stabilty of Kalman filtering with Markovian packet losses [J], Automatica,2007,43(4):598-607.
[116]S. Sun, L. Xie, W. Xiao, and Y. C. Soh, Optimal linear estimation for systems with multiple packet dropouts[J], Automatica,2008,44(5):1333-1342.
[117]S. Sun, Optimal estimators for systems with finite consecutive packet dropouts[J], IEEE Trans. Signal Processing Letters,2009,16(7):557-560.
[118]S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix In-equality in Systems and Control Theory[M], Philadelphia, PA:SIAM,1994.
[119]X. Song, H. Zhang, and L. Xie, Linear quadratic regulation for discrete-time stochastic systems with input delays[C], Proceedings of the 27th Chinese Control Conference,2008,432-436.
[120]H. Zhang and X. Song, New estimator and duality of control for multiplica-tive noise systems with applications to packet dropping and multiple input delay systems[C], Proceedings of the 7th IEEE International Conference on Control and Automation, Christchurch, New Zealand,2009,75-80.