基于LMI的集员估计算法在不确定系统中的研究及应用
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摘要
在确定模型和噪声边界的情况下,Fogel和Huang提出了基于最优外定界椭球的集员估计方法,成为了集员估计理论的一个重要分支。通常情况下,运用各种方法所得到的系统模型仅能尽量精准地反映系统所具有的特性,并且系统模型的精确描述和复杂性存在着不可调和的矛盾。针对这种情况,过去一段时间的大量研究工作以系统模型精确可知为前提,集中探索了如何将不确定性隐藏在状态扰动和测量噪声中的问题。在此种假设下取得了大量成果,但假定本身存在着无法弥补的缺陷。
本论文以半正定规划问题为主要框架,结合线性矩阵不等式形式的约束条件,对具有可参数化不确定离散系统的状态估计问题进行了深入研究。首先,将具有范数边界的可参数化不确定系统转化为线性分块表达式,从而将鲁棒技术引入状态估计问题。其次,根据外定界椭球集员估计算法的思想,以时间更新和测量更新为主要结构,在每一步更新过程中结合上一步的信息通过求解半正定规划问题从而得到状态估计值。
本论文围绕线性系统,改进了用于可参数化不确定系统的外定界椭球算法。围绕非线性系统,同样采用时间更新和测量更新结构的半正定规划方法进行求解。在线性化过程中围绕状态估计值对非线性模型采用二阶泰勒展开公式,保留二次项的有效信息,从而提高算法的精确性。从实际需求考虑,本论文进一步将此算法应用于里程计传感器和视觉传感器组合而成的联合定位系统中,替代了传统的扩展卡尔曼滤波算法,实现了对移动机器人位姿的状态估计。针对以上三部分进行的仿真实验结果证明了算法的有效性,不但提高了状态估计的精度,更具意义的是增强了算法的鲁棒性。
In the case of the accurate system model and noise boundary, Fogel and Huang proposed the optimal bounded ellipsoid set-membership estimation. This algorithm is set to become a member of an important branch of state estimation theory. Typically, the use of various methods can only try to get the system model accurately reflects characteristic of the system, The error always exists between them. There are irreconcilable contradictions bewteen precision and complexity of the system. For this situation, large number of studies over a period of time as a prerequisite for the accurate system model, studied the problem that how to expression the uncertainty of the system through the state disturbance and measurement noise. In this hypothesis, the researchers made a lot of results. But suppose there can not make up their own shortcomings.
In this thesis, semidefinite programming as the main framework, the new set-membership state estimation algorithm was proposed for the robust problem of uncertain system using Linear Matrix Inequality (LMI) constraints. First, the uncertainties structured are norm bounded and represented by linear fractional resprestation (LFR), so that the estimation of the relative situation is computed by robust filtering techniques. Second, the proposed technique is based on a classical prediction/measurement update recursion which requires at each step the solution of a convex semidefinite optimization problem under LMI.
For linear system, the main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using set-membership state estimation for convex optimization. For nonlinear system, the efficient scheme takes into explicit account the effects of nonlinearities via second-order information of Taylor formula, thus to improve the accuracy of estimation. From the actual needs, And then, replacing the traditional extended Kalman filter algorithm, LMI-based set-membership state estimation algorithm is applied to mobile robot location using odometer sensor and visual sensor. Simulation results demonstrate the effectiveness of the algorithm. Not only the system state estimation accuracy improved, but more significant is the greatly enhanced the robustness of the algorithm.
本论文以半正定规划问题为主要框架,结合线性矩阵不等式形式的约束条件,对具有可参数化不确定离散系统的状态估计问题进行了深入研究。首先,将具有范数边界的可参数化不确定系统转化为线性分块表达式,从而将鲁棒技术引入状态估计问题。其次,根据外定界椭球集员估计算法的思想,以时间更新和测量更新为主要结构,在每一步更新过程中结合上一步的信息通过求解半正定规划问题从而得到状态估计值。
本论文围绕线性系统,改进了用于可参数化不确定系统的外定界椭球算法。围绕非线性系统,同样采用时间更新和测量更新结构的半正定规划方法进行求解。在线性化过程中围绕状态估计值对非线性模型采用二阶泰勒展开公式,保留二次项的有效信息,从而提高算法的精确性。从实际需求考虑,本论文进一步将此算法应用于里程计传感器和视觉传感器组合而成的联合定位系统中,替代了传统的扩展卡尔曼滤波算法,实现了对移动机器人位姿的状态估计。针对以上三部分进行的仿真实验结果证明了算法的有效性,不但提高了状态估计的精度,更具意义的是增强了算法的鲁棒性。
In the case of the accurate system model and noise boundary, Fogel and Huang proposed the optimal bounded ellipsoid set-membership estimation. This algorithm is set to become a member of an important branch of state estimation theory. Typically, the use of various methods can only try to get the system model accurately reflects characteristic of the system, The error always exists between them. There are irreconcilable contradictions bewteen precision and complexity of the system. For this situation, large number of studies over a period of time as a prerequisite for the accurate system model, studied the problem that how to expression the uncertainty of the system through the state disturbance and measurement noise. In this hypothesis, the researchers made a lot of results. But suppose there can not make up their own shortcomings.
In this thesis, semidefinite programming as the main framework, the new set-membership state estimation algorithm was proposed for the robust problem of uncertain system using Linear Matrix Inequality (LMI) constraints. First, the uncertainties structured are norm bounded and represented by linear fractional resprestation (LFR), so that the estimation of the relative situation is computed by robust filtering techniques. Second, the proposed technique is based on a classical prediction/measurement update recursion which requires at each step the solution of a convex semidefinite optimization problem under LMI.
For linear system, the main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using set-membership state estimation for convex optimization. For nonlinear system, the efficient scheme takes into explicit account the effects of nonlinearities via second-order information of Taylor formula, thus to improve the accuracy of estimation. From the actual needs, And then, replacing the traditional extended Kalman filter algorithm, LMI-based set-membership state estimation algorithm is applied to mobile robot location using odometer sensor and visual sensor. Simulation results demonstrate the effectiveness of the algorithm. Not only the system state estimation accuracy improved, but more significant is the greatly enhanced the robustness of the algorithm.
引文
[1] Kalman. R.E. A New Approach to Linear Filtering and Prediction Problems. Transactions of the ASME-Journal of Basic Engineering,1960,82:35-45
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[6] Fogel E,Huang Y F. On the value of information in system identification bounded noise case. Automatica,1982,18(20):229-238
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[9] Liung L. Convergence Analysis of Parametric Identification Methods. IEEE Trans. On Automatic Control,1978,23:770-783
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[13] Jazwinski A H. Stochastic Processes and Filtering Theory. New York: Academic Press, 1970
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[2]宋文尧,张牙.卡尔曼滤波.北京:科学出版社,1991,2-153
[3] Fred E. Daum. Beyond Kalman filters: practical design of nonlinear filters. SPIE, 1995,2561:252-262
[4] Schweppe F.C. Recursive state estimation: unknown but bounded errors and system inputs. IEEE Trans, Autom, Control,1968,13:22-28
[5] Fogel E. System identification via membership set constraints with energy constrained noise. IEEE Trans. Aut. Control,1979,24(5):752-1009
[6] Fogel E,Huang Y F. On the value of information in system identification bounded noise case. Automatica,1982,18(20):229-238
[7] Bertsekas D.P., Rhodes, I.B. Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Autom. Control,1970, 16:117-128
[8] Zadeh L A. Form Circuit Theory to System Theory. Proceedings of the Institute of Radio Engineers,1962,50(5): 856-865
[9] Liung L. Convergence Analysis of Parametric Identification Methods. IEEE Trans. On Automatic Control,1978,23:770-783
[10]王志贤.最优状态估计与系统辨识.西安:西北工业大学出版社,2004,3-152
[11] Kalman R E, Bucy R S. New results in linear filtering and Prediction theory. J. Basic Engineering, Trans. ASME,1961,83(3):95-108
[12] Luenberger D G. An Introduction to Observers. IEEE Trans. Autom. Control, 1971,16(6):596-602
[13] Jazwinski A H. Stochastic Processes and Filtering Theory. New York: Academic Press, 1970
[14] Misawa E A, Hedrick J K. Nonlinear observers: A state of the survey. ASME J. Dyn. System Meas. Control,1989,111(3):34-352
[15] Liang D F .Comparisons of nonlinear recursive filters with nonneglible nonlinearities. Control and Dynamic Systems. NewYork: Academic Press,1983
[16] Gordon N J, Salmond D J, Smith A F M. Novel approach to nonlinear /non-Gaussian Bayesian state estimation. IEEE Proceedings Radar and Signal Processing, 1993, 140:107-113
[17] Zirnger G. State Observation by online Minimization. Int. J. Control,1994, 60(4): 595- 606
[18]韩正之,潘丹杰,张钟俊.非线性系统的能观性和状态观测器.控制理论与应用,1990,7(4):l-9
[19] Vargas J A R, Hemerly E M. Adaptive observers for unknown general nonlinear systems. IEEE Trans. System, Man and Cybernetics-Part B,2001,31(5):683-690
[20] Johansson A. Nonlinear observers with applications in the steel industry. Doctoral thesis.Sweden,2001
[21]周东华,叶银忠.现代故障诊断与容错控制.北京:清华大学出版社,2000
[22] Xia X H, Gao W B. Nonlinear observer design by observer error linearization. SIAM J. Control and Optimization,1989,27(l):199-216
[23] Xia X H, Gao W B. On exponential observers for nonlinear systems. Sys. And Contr. Lett.,1988,11(4):319-325
[24]韩京清.一类不确定对象的扩张状态观测器.控制与决策,1995,l0(1):85-88
[25]张洪才,张友民,贺志斌.一种鲁棒自适应推广Kalman滤波及其在飞行状态估计中的应用.信息与控制,1992,21(6):343-348
[26] Chu J, Ohshima M, Hashimoto I, Takamatsu T, Wang J C. A design method for a class of robust nonlinear observers. J. Chemical Engineering of Japan, 1989, 22(3):225-235
[27]朱瑞军,柴天佑.一类不确定非线性系统鲁棒自适应观测器设计的新方法明.自动化学报,1998,24(6):780-783
[28]梅生伟,申铁龙,刘康志.现代鲁棒控制理论与应用(第2版).北京:清华大学出版社,2008,2-90
[29] Kunpeng Sun, Andy Packard. Robust H2 and H∞filters for uncertain LFT systems. IEEE Transactions on Automatic Control,2005,5 (50):715-720
[30]何青.集员估计理论方法及其应用研究:[湖南大学博士学位论文].长沙:湖南大学,2002,1-78
[31]柴伟.集员估计理论、方法及其应用:[北京航空航天大学博士学位论文].北京:北京航空航天大学,2008,1-58
[32] Chernousko F L. Optimal guaranteed estimates of the indeterminacy with the aid of ellipsoid. Engineering Cybernetics,1980,18(4):1-9
[33] Boris T. Polyak, A.Nazin, Cécile Durieu, Eric Walter. Ellipsoidal parameter or state estimation under model uncertainly. Automatic,2004,40:1171-1179
[34] S. Dussy, L. EI Ghaoui. Measurement-scheduled control for the RTAC problem: An LMI approach. Int. J. Robust Nonlinear Control,1998,8:337-400
[35] S. Boyd, V. Balakrishnan, E. Feron, L. El Ghaoui. Control system analysis and synthesis via linear matrix inequalities. In: Eckman lecture, American Control Conference, San Francisco,1993,2:2147-2154
[36] L. EI Ghaoui and G.Scorletti. Performance Control of rational systems using linear fractional representation and linear matrix inequalities. In: the 33rd IEEE Conference on Deision and Control,1996,3:2792-2797
[37] L. EI Ghaoui and G.Calafiore. Worst-case simulation of uncertain systems.Lecture Notes in Control and Information Sciences,1999,245:134-146
[38] L. EI Ghaoui and G.Calafiore. Robust filtering for discrete-time systems with bounds noise and parametric uncertainty. IEEE Trans. Autom. Control, 2001, 46(7):1084-1089
[39] G. Calafiore and L. EI Ghaoui. Ellipsoidal bounds for uncertain linear equations and dynamical systems. IEEE, Trans. Autom,2004,40:773-787
[40]孙先仿.集员辨识-算法、收敛性和鲁棒性:[中国科学院自动化研究所博士学位论文].北京:中国科学院自动化研究所,1994,5-46
[41] S. Boyd, L. EI Ghaoui, E. Ferron, V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.Part A
[42]俞立著.鲁棒控制:线性矩阵不等式处理方法.北京:清华大学出版社
[43] Lieven Vandenberghe, Venkataramanan Balakrishnan. Algorithms and software for LMI problem in control. IEEE Control Systems Magazine,1997,89-95
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