机械系统的随机建模、控制和应用
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摘要
众所周知,机械系统经常受到随机干扰,这些干扰会以某种不确定的方式影响系统的性能.随着随机稳定性的发展,随机力学及控制的研究已经成为一个活跃的领域.本文针对机械系统的随机建模,跟踪控制及其应用的若干重要问题,发展了一些基本分析工具,并基于这些工具,深入研究和解决了这些问题.主要成果包括:
1.通过合理的引入随机噪声,我们构造一类随机拉格朗日控制系统和一类随机哈密顿控制系统来描述遭受随机干扰的机械系统的运动.
2.对一类带有未知参数的随机拉格朗日控制系统,在一些合理的假设下,设计了一种自适应跟踪控制器使得跟踪误差的均方可通过调节设计参数收敛到一个零的任意小邻域.假设的合理性和控制策略的有效性可由一个在随机振动环境中的机械模型说明.
3.对于一类速度不可测的随机拉格朗日控制系统,在一些合理的假设下,应用拉格朗日系统的结构性质,巧妙的构造了一个降阶的观测器来估计速度.基于这个观测器,设计了一个输出反馈跟踪控制器使得跟踪误差的均方通过调节参数可以收敛到一个零的任意小邻域内.控制器的有效性通过一个随机力学模型来说明.
4.对一类带有未知扩散和漂移函数的随机哈密顿控制系统,设计了一个向量形式的自适应backstepping控制器使得闭环随机哈密顿系统有唯一解且是依概率全局有界,并且跟踪误差的收敛到零的任意小的邻域.作为一个应用,研究了随机环境中的弹簧摆的建模和控制.
5.对一个二级刚性机械臂,构造了一个随机拉格朗日模型来描述其在随机振动环境中的运动.基于这个模型,针对所有状态可测的情形,设计了一个状态反馈控制器使得误差系统是4阶矩指数实用稳定.当速度不可测时,设计了一个输出反馈控制器使得闭环系统的位移尽可能的跟踪一个给定的参考信号.
6.对于在随机振动环境中带有多个旋转结点的机械臂,通过分析环境对质点的影响和引入等价的随机噪声过程,建立了一个随机哈密顿动态模型来描述其运动.基于这个模型,设计一个控制器使得闭环系统的位移近似的跟踪一个给定的参考信号.
7.对于状态依赖切换的随机非线性系统,当给定的活动区域集可以由它的内部来代替时,通过定义一列停时作为切换时间,构造了切换系统的局部解并通过Lyapunov方法提出了解的全局存在和稳定性的判定.对于活动区域集不能由它的内部来代替这种情形,切换系统不一定有解.对这一基本问题构造了准解并提出了有界性判定.
It is well known that mechanical systems are often subjected to random disturbances,which significantly affects the performance in a uncertain manner. With the development ofstochastic stability, the research on stochastic mechanics and control has been an active field.In this paper, for several important problems on stochastic modeling, tracking control and theapplication of mechanical systems, some basic analysis tools are developed, and based on sometools, these problems are deeply investigated and solved. The main results include:
1. By reasonably introducing random noise, a class of stochastic Lagrangian control sys-tems and a class of stochastic Hamiltonian control systems are constructed to describe the mo-tion of the mechanical systems subjected to random disturbance.
2. For a class of stochastic Lagrangian control systems with unknown parameters, undersome milder assumptions, an adaptive tracking controller is designed such that the mean squareof the tracking error converges to an arbitrarily small neighborhood of zero by tuning design pa-rameters. The reasonability of assumptions and the efficiency of the controller are demonstratedby a mechanics model in random vibration environment.
3. For a class of stochastic Lagrangian systems with the unmeasurable velocity, undersome milder assumptions, using the structural properties of Lagrangian systems, a reduced-order observer is skillfully constructed to estimate the velocity. Based on the observer, anoutput feedback tracking controller is designed such that the mean square of the tracking er-ror converges to an arbitrarily small neighborhood of zero by tuning design parameters. Theefficiency of the controller is demonstrated by a stochastic mechanical model.
4. For a class of stochastic Hamiltonian control systems with unknown drift and diffusionfunctions, a vector form of adaptive backstepping controller is designed such that the closed-loop stochastic Hamiltonian system has a unique solution that is globally bounded in probabilityand the tracking error converges to an arbitrarily small neighborhood of zero. As application,the modeling and the control for spring pendulum in stochastic surroundings are researched.
5. For a two-link planar rigid robot manipulator, a stochastic Lagrangian model is con-structed to describe the motion of the manipulator in random vibration environment. Based on the constructed model, for the case that all states are measurable, a state feedback controller isdesigned such that the error system is4-th moment exponentially practically stable. When thevelocity is unmeasurable, a output feedback controller is designed such that the configurationof the closed-loop system can track a given smooth reference signal as close as possible.
6. For the manipulator with multi-revolute joints in random vibration environment, byanalyzing the effect of environment to the mass points and introducing an equivalent stochasticnoise process, a stochastic Hamiltonian dynamic model is constructed to describe the motionof the manipulator. Based on the constructed model, a state feedback controller is designedsuch that the configuration of the closed-loop system can approximatively track a given smoothreference signal.
7. For stochastic nonlinear systems with state-dependent switching, when the given active-region set can be replaced by its interior, the local solution of the switched system is constructedby defining a series of stopping times as switching instants, and the criteria on global existenceand stability of solution are presented by Lyapunov approach. For the case where the active-region set can not be replaced by its interior, the switched systems do not necessarily havesolutions, thereby quasi-solution to the underlying problem is constructed and the boundednesscriterion is proposed.
1.通过合理的引入随机噪声,我们构造一类随机拉格朗日控制系统和一类随机哈密顿控制系统来描述遭受随机干扰的机械系统的运动.
2.对一类带有未知参数的随机拉格朗日控制系统,在一些合理的假设下,设计了一种自适应跟踪控制器使得跟踪误差的均方可通过调节设计参数收敛到一个零的任意小邻域.假设的合理性和控制策略的有效性可由一个在随机振动环境中的机械模型说明.
3.对于一类速度不可测的随机拉格朗日控制系统,在一些合理的假设下,应用拉格朗日系统的结构性质,巧妙的构造了一个降阶的观测器来估计速度.基于这个观测器,设计了一个输出反馈跟踪控制器使得跟踪误差的均方通过调节参数可以收敛到一个零的任意小邻域内.控制器的有效性通过一个随机力学模型来说明.
4.对一类带有未知扩散和漂移函数的随机哈密顿控制系统,设计了一个向量形式的自适应backstepping控制器使得闭环随机哈密顿系统有唯一解且是依概率全局有界,并且跟踪误差的收敛到零的任意小的邻域.作为一个应用,研究了随机环境中的弹簧摆的建模和控制.
5.对一个二级刚性机械臂,构造了一个随机拉格朗日模型来描述其在随机振动环境中的运动.基于这个模型,针对所有状态可测的情形,设计了一个状态反馈控制器使得误差系统是4阶矩指数实用稳定.当速度不可测时,设计了一个输出反馈控制器使得闭环系统的位移尽可能的跟踪一个给定的参考信号.
6.对于在随机振动环境中带有多个旋转结点的机械臂,通过分析环境对质点的影响和引入等价的随机噪声过程,建立了一个随机哈密顿动态模型来描述其运动.基于这个模型,设计一个控制器使得闭环系统的位移近似的跟踪一个给定的参考信号.
7.对于状态依赖切换的随机非线性系统,当给定的活动区域集可以由它的内部来代替时,通过定义一列停时作为切换时间,构造了切换系统的局部解并通过Lyapunov方法提出了解的全局存在和稳定性的判定.对于活动区域集不能由它的内部来代替这种情形,切换系统不一定有解.对这一基本问题构造了准解并提出了有界性判定.
It is well known that mechanical systems are often subjected to random disturbances,which significantly affects the performance in a uncertain manner. With the development ofstochastic stability, the research on stochastic mechanics and control has been an active field.In this paper, for several important problems on stochastic modeling, tracking control and theapplication of mechanical systems, some basic analysis tools are developed, and based on sometools, these problems are deeply investigated and solved. The main results include:
1. By reasonably introducing random noise, a class of stochastic Lagrangian control sys-tems and a class of stochastic Hamiltonian control systems are constructed to describe the mo-tion of the mechanical systems subjected to random disturbance.
2. For a class of stochastic Lagrangian control systems with unknown parameters, undersome milder assumptions, an adaptive tracking controller is designed such that the mean squareof the tracking error converges to an arbitrarily small neighborhood of zero by tuning design pa-rameters. The reasonability of assumptions and the efficiency of the controller are demonstratedby a mechanics model in random vibration environment.
3. For a class of stochastic Lagrangian systems with the unmeasurable velocity, undersome milder assumptions, using the structural properties of Lagrangian systems, a reduced-order observer is skillfully constructed to estimate the velocity. Based on the observer, anoutput feedback tracking controller is designed such that the mean square of the tracking er-ror converges to an arbitrarily small neighborhood of zero by tuning design parameters. Theefficiency of the controller is demonstrated by a stochastic mechanical model.
4. For a class of stochastic Hamiltonian control systems with unknown drift and diffusionfunctions, a vector form of adaptive backstepping controller is designed such that the closed-loop stochastic Hamiltonian system has a unique solution that is globally bounded in probabilityand the tracking error converges to an arbitrarily small neighborhood of zero. As application,the modeling and the control for spring pendulum in stochastic surroundings are researched.
5. For a two-link planar rigid robot manipulator, a stochastic Lagrangian model is con-structed to describe the motion of the manipulator in random vibration environment. Based on the constructed model, for the case that all states are measurable, a state feedback controller isdesigned such that the error system is4-th moment exponentially practically stable. When thevelocity is unmeasurable, a output feedback controller is designed such that the configurationof the closed-loop system can track a given smooth reference signal as close as possible.
6. For the manipulator with multi-revolute joints in random vibration environment, byanalyzing the effect of environment to the mass points and introducing an equivalent stochasticnoise process, a stochastic Hamiltonian dynamic model is constructed to describe the motionof the manipulator. Based on the constructed model, a state feedback controller is designedsuch that the configuration of the closed-loop system can approximatively track a given smoothreference signal.
7. For stochastic nonlinear systems with state-dependent switching, when the given active-region set can be replaced by its interior, the local solution of the switched system is constructedby defining a series of stopping times as switching instants, and the criteria on global existenceand stability of solution are presented by Lyapunov approach. For the case where the active-region set can not be replaced by its interior, the switched systems do not necessarily havesolutions, thereby quasi-solution to the underlying problem is constructed and the boundednesscriterion is proposed.
引文
[1] M. G. Calkin. Lagrangian and Hamiltonian Mechanics [M]. Singapore: World ScientificPublishing Co. Pte. Ltd.,1996.
[2] L. N. Hand and J. D. Finch. Analytical Mechanics [M]. Cambridge: Cambridge Univer-sity Press,1998.
[3] A. D. Lewis. Lagrangian Mechanics, Dynamics and Control [M]. Kingston: Queen Uni-versity,2004.
[4] O. D. Johns. Analytical Mechanics for Relativity and Quantum Mechanics [M]. NewYork: Oxford University Press,2005.
[5] R. Z. Khas’minskii. Stochastic Stability of Differential Equations (2nd ed)[M]. Berlin,Heidelberg: Springer-Verlag,2012.
[6] B. ksendal. Stochastic Differential Equations-An Introduction with Applications (6thEdition)[M]. New York: Springer-Verlag,2003.
[7] X. Mao. Stochastic Differential Equations and Applications [M]. New York: Horwood,1997.
[8] Z. J. Wu, M. Y. Cui, X. J. Xie, and P. Shi. Theory of stochastic dissipative systems [J].IEEE Transactions on Automatic Control,2011,56(7):1650–1655.
[9] M. Krstic′and H. Deng. Stabilization of Nonlinear Uncertain Systems [M]. New York:Springer-Verlag,1998.
[10] J. M. Yong and X. Y. Zhou. Stochastic Controls, Hamiltonian Systems and HJB Equa-tions [M]. New York: Springer-Verlag,1999.
[11] Z. J. Wu, J. Yang, and P. Shi. Adaptive tracking for stochastic nonlinear systems withmarkovian switching [J]. IEEE Transactions on Automatic Control,2010,55(9):2135–2141.
[12] J. A. La′zaro-Cam′and J. P. Ortega. Stochastic hamiltonian dynamical systems [R]. Re-ports on Mathematical Physics,2008,61:65–122.
[13]朱位秋.非线性随机动力学与控制: Hamilton理论体系框架[M].北京:科学出版社,2003.
[14] A. Loria and H. Nijmeijer. Bounded output feedback tracking control of fully actuatedeuler-lagrange systems [J]. Systems&Control Letters,1998,33:151–161.
[15] G. Besanc on. Global output feedback tracking control for a class of lagrangian systems[J]. Automatica,2000,36(12):1915–1921.
[16] B. Jayawardhana and G. Weiss. Tracking and disturbance rejection for fully actuatedmechanical systems [J]. Automatica,2008,44(11):2863–2868.
[17] P. M. Patre, W. MacKunis, K. Dupree, and W. E. Dixon. Modular adaptive control ofuncertain Euler-Lagrange systems with additive disturbances [J]. IEEE Transactions onAutomatic Control,2011,56(1):155–160.
[18] R. Ortega, A. Lor′a, P. J. Nicklasson, and H. Sira-Ram′rez. Passivity-based Control ofEuler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications
[M]. London: Springer-Verlag,1998.
[19] A. J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control [M]. Lon-don: Springer-Verlag, Berlin,2000.
[20] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland. Dissipative Systems Analysis andControl: Theory and Applications (2nd edition)[M]. London: Springer-Verlag,2007.
[21] S. Nicosia and P. Tomei. Robot control by using only joint position measurements [J].IEEE Transactions on Automatic Control,1990,35(9):1058–1061.
[22] F. Zhang, D. M. Dawson, M. S. de Queiroz, and W. E. Dixon. Global adaptive outputfeedback tracking control of robot manipulators [J]. IEEE Transactions on AutomaticControl,2000,45(6):1203–1208.
[23] S. Islam and P. X. Liu. Output feedback sliding mode control for robot manipulators [J].Robotica,2010,28(7):975–987.
[24]. N. Stamnes, O. M. Aamo, and G. O. Kaasa. A constructive speed observer design forgeneral Euler-Lagrange systems [J]. Automatica,2011,47(10):2233–2238.
[25]. N. Stamnes, O. M. Aamo, and G. O. Kaasa. Global output feedback tracking controlof euler-lagrange systems [C]. In The18th IFAC World Congress Milano (Italy) August28-September2,2011:215–220.
[26] A. Kovaleva and L. Akulenko. Approximation of escape time for Lagrangian systemswith fast noise [J]. IEEE Transactions on Automatic Control,2007,52(12):2338–2341.
[27] W. R. Hamilton. On the application to dynamics of a general mathematical method pre-viously applied to optics [C]. In British Association Report, Edinburgh, UK,1834:513–518.
[28] R. Ortega, A. van der Schaft, B. Maschke, and G. Escobar. Interconnection and dampingassignment passivity-based control of port-controlled Hamiltonian systems [J]. Automat-ica,2002,38(4):585–596.
[29] B. M. Maschke, R. Ortega, and A. J. van der Schaft. Energy-based Lyapunov functionsfor forced Hamiltonian systems with dissipation [J]. IEEE Transactions on AutomaticControl,2000,45(8):1498–1502.
[30] A. Astolfi and L. Menini. Noninteracting control with stability for Hamiltonian systems[J]. IEEE Transactions on Automatic Control,2000,45(8):1470–1482.
[31] K. Fujimoto and T. Sugie. Canonical transformations and stabilization of generalizedHamiltonian systems [J]. Systems&Control Letters,2001,42(3):217–227.
[32] Y. Z. Wang, C. W. Li, and D. Z. Cheng. Generalized Hamiltonian realization of time-invariant nonlinear systems [J]. Automatica,2003,39(8):1437–1443.
[33] S. G. Peng. Monotonic limit theorem of BSDE and nonlinear decomposition theorem ofdoob-meyer’s type [J]. Probability Theory and Related Fields,1999,113(4):473–499.
[34] S. G. Peng and Z. Wu. Fully coupled forward-backward stochastic differential equationsand applications to optimal control [J]. SIAM Journal on Control and Optimization,1999,37(3):825–843.
[35] J. M. Yong. A stochastic linear quadratic optimal control problem with generalized ex-pectation [J]. Stochastic Analysis and Applications,2008,26(6):1136–1160.
[36] W. Q. Zhu, Z. L. Huang, and Y. Q. Yang. Stochastic averaging of quasi-integrable Hamil-tonian systems [J]. ASME Journal of Applied Mechanics,1997,64(4):975–984.
[37] Z. H. Liu and W. Q. Zhu. Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control [J]. Automatica,2008,44(7):1923–1928.
[38] M. W. Spong, S. Hutchinson, and M. Vidyasagar. Robot Modeling and control [M]. NewYork: John Wiley&Sons Inc,2006.
[39] C. C. De Wit, B. Siciliano, and B. Bastin. Theory of Robot Control [M]. London:Springer-Verlag,1996.
[40] L. Sciavicco and B. Siciliano. Modelling and Control of Robot Manipulator [M]. London:Springer-Verlag,2000.
[41] F. L. Lewis, D. M. Dawson, and C. T. Abdallah. Robot Manipulator Control-Theoryand Practice [M]. New York: Marcel Dekke,2004.
[42] S. S. Ge, T. H. Lee, and C. J. Harris. Adaptive Neural Network Control of Robotic Ma-nipulators [M]. Singapore: World Scientific Publishing Co. Pte. Ltd.,1998.
[43] S. H. Crandall and W. D. Mark. Random Vibration in Mechanical Systems [M]. NewYork: Academic Press,1963.
[44] H. J. Kushner. Stochastic Stability and Control [M]. New York: Academic Press,1967.
[45] A. Friedman. Stochastic Differential Equations and Their Applications [M]. New York:Academic Press,1976.
[46] Z. G. Pan and T. Bas ar. Backstepping controller design for nonlinear stochastic systemsunder a risk-sensitive cost criterion [J]. SIAM Journal on Control and Optimization,1999,37(3):957–995.
[47] Z. J. Wu, X. J. Xie, P. Shi, and Y. Q. Xia. Backstepping controller design for a classof stochastic nonlinear systems with markovian switching [J]. Automatica,2009,45(4):997–1004.
[48] Y. G. Liu and J. F. Zhang. Practical output-feedback risk-sensitive control for stochasticnonlinear systems with stable zero-dynamics [J]. SIAM Journal on Control and Opti-mization,2006,45(3):885–926.
[49] S. J. Liu, J. F. Zhang, and Z. P. Jiang. Decentralized adaptive output-feedback stabiliza-tion for large-scale stochastic nonlinear systems [J]. Automatica,2007,43(2):238–251.
[50] Z. J. Wu, X. J. Xie, and S. Y. Zhang. Adaptive backstepping controller design usingstochastic small-gain theorem [J]. Automatica,2007,43(4):608–620.
[51] H. Deng and M. Krstic′. Output-feedback stochastic nonlinear stabilization [J]. IEEETransactions on Automatic Control,1999,44(2):328–333.
[52] H. Deng and M. Krstic′. Output-feedback stabilization of stochastic nonlinear systemsdriven by noise of unknown covariance [J]. Systems&Control Letters,2000,39(3):173–182.
[53] H. B. Ji and H. S. Xi. Adaptive output-feedback tracking of stochastic nonlinear systems[J]. IEEE Transactions on Automatic Control,2006,51(2):355–360.
[54] X. W. Mu and H. J. Liu. Stabilization for a class of stochastic nonlinear systems viaoutput feedback [J]. IEEE Transactions on Automatic Control,2008,53(1):360–367.
[55] S. J. Liu and J. F. Zhang. Output-feedback control of a class of stochastic nonlinearsystems with linearly bounded unmeasurable states [J]. International Journal of Robustand Nonlinear Control,2008,18(6):665–687.
[56] D. Liberzon. Switching in Systems and Control [M]. Boston: MA: Birkhauser,2003.
[57] Y. Ji and H. J. Chizeck. Controllability, stabilizability and continuous-time markovianjump linear quadratic control [J]. IEEE Transactions on Automatic Control,1990,35(7):777–788.
[58] El-K. Boukas. Stochastic Switching Systems: Analysis and Design [M]. Boston:Birkhauser,2006.
[59] P. Shi, E. K. Boukas, and R. K. Agarwal. Kalman filtering for continuous-time uncer-tain systems with markovian jumping parameters [J]. IEEE Transactions on AutomaticControl,1999,44(8):1592–1597.
[60] P. Shi, Y. Q. Xia, G. P. Liu, and D. Rees. On designing of sliding-mode control forstochastic jump systems [J]. IEEE Transactions on Automatic Control,2006,51(1):97–103.
[61] P. Shi and E. K. Boukas. H-infinity control for markovian jumping linear systems withparametric uncertainties [J]. Journal of Optimization Theory and Applications,1997,95:75–99.
[62] X. Mao. Stability of stochastic differential equations with Markovian switching [J].Stochastic Processes and Their Applications,1999,79(1):45–67.
[63] C. Yuan and X. Mao. Robust stability and controllability of stochastic differential delayequations with markovian switching [J]. Automatica,2004,40(3):343–354.
[64] X. Mao and C. Yuan. Stochastic Differential Equations with Markovian Switching [M].London: Imperial College Press,2006.
[65] W. Feng and J. F. Zhang. Stability analysis and stabilization control of multivariableswitched stochastic systems [J]. Automatica,2006,42(1):169–176.
[66] W. Feng, J. Tian, and P. Zhao. Stability analysis of switched stochastic systems [J]. Au-tomatica,2011,47(1):148–157.
[67] D. Chatterjee and D. Liberzon. Stability analysis of deterministic and stochastic switchedsystems via a comparison principle and multiple Lyapunov functions [J]. SIAM Journalon Control and Optimization,2006,45(1):174–206.
[68] H. Deng, M. Krstic′, and R. J. Williams. Stabilization of stochastic nonlinear systemsdriven by noise of unknown covariance [J]. IEEE Transactions on Automatic Control,2001,46(8):1237–1253.
[69] J. P. LaSalle and S. Lefschetz. Stability by Lyapunov’s Direct Method with Applications
[M]. New York: Academic Press,1961.
[70] V. Lakshmikantham, S. Leela, and A. A. Martynyuk. Practical Stability of NonlinearSystems [M]. Singapore: World Science Publisher,1990.
[71] E. Wong and M. Zakai. Riemann-stieltjes approximations of stochastic integrals [J]. Z.Wahrscheinlichkeitstheorie verw. Geb,1969,12:87–97.
[72] Y. G. Liu and J. F. Zhang. Reduced-order observer-based control design for nonlinearstochastic systems [J]. Systems&Control Letters,2004,52(2):123–135.
[73] Z. G. Pan. Differential geometric condition for feedback complete linearization ofstochastic nonlinear system [J]. Automatica,2001,37(1):145–149.
[74] Z. G. Pan. Canonical forms for stochastic nonlinear systems [J]. Automatica,2002,38(7):1163–1170.
[75] J. M. Bismut. Me′canique Ale′atoire [C]. Lecture Notes in Mathematics,866, Springer,1981.
[76] Z. G. Ying and W. Q. Zhu. Astochastically averaged optimal control strategy for quasi-Hamiltonian systemswith actuator saturation [J]. Automatica,2006,42:1577–1582.
[77] W. Lin and C. J. Qian. Adaptive control of nonlinearly parameterized systems: A nons-mooth feedback framework [J]. IEEE Transactions on Automatic Control,2002,47(5):757–774.
[78] M. M. Bridges, D. M. Dawson, Z. H. Qu, and S. C. Martindale. Robust control of rigid-link flexible-joint robots with redundant joint actuators [J]. IEEE Transactions on Sys-tems, Man and Cybernetics,1994,24(7):961–970.
[79] J. M. Knudsen and P. G. Hjorth. Elements of Newtonian Mechanics: Including NonlinearDynamics [M]. New York: Springer-Verlag,2000.
[80] D. L. Goodstein, R. P. Olenick, and T. M. Apostol. The Mechanical Universe: Introduc-tion to Mechanics and Heat [M]. New York: Cambridge University Press,2008.
[81] Z. J. Wu, X. J. Xie, and P. Shi. Robust adaptive output-feedback control for nonlinearsystems with output unmodeled dynamics [J]. International Journal of Robust and Non-linear Control,2008,18(11):1162–1187.
[82] A. Loria. Global tracking control of one degree of freedom euler-lagrange systems with-out velocity measurements [J]. European Journal of Control,1996,2:144–151.
[83] W. E. Dixon, E. Zergeroglu, and D. M. Dawson. Global robust output feedback trackingcontrol of robot manipulators [J]. Robotica,2004,22(4):351–357.
[84] J. Moreno-Valenzuela, V. Santiba′nez, and R. Campa. On output feedback tracking ccon-trol of robot manipulators with bounded torque input [J]. International Journal of control,Automation, and Systems,2008,6(1):76–85.
[85] S. Purwar, I. N. Kar, and A. N. Jha. Adaptive output feedback tracking control of robotmanipulators using position measurements only [J]. Expert Systems with Applications,2008,34(4):2789–2798.
[86] E. Kim. Output feedback tracking control of robot manipulators with model uncertaintyvia adaptive fuzzy logic [J]. IEEE Transactions on Fuzzy Systems,2004,12(3):368–378.
[87] O. M. Aamo, M. Arcak, T. I. Fossen, and P. V. Kokotovic. Global output tracking controlof a class of Euler-Lagrange systems. International Journal of Control,2000,74:649–658.
[88] K. D. Do, Z. P. Jiang, and J. Pan. Global partial-state feedback and output-feedbacktracking controllers for underactuated ships [J]. Systems&Control Letters,2005,54(10):1015–1036.
[89] D. Carnevale, D. Karagiannis, and A. Astolfi. A condition for certainty equivalence out-put feedback stabilization of nonlinear systems [J]. IEEE Transactions on AutomaticControl,2010,55(5):1180–1185.
[90] A. Astolfi, R. Ortega, and A. Venkatraman. A globally exponentially convergent immer-sion and invariance speed observer for mechanical systems with non-holonomic con-straints [J]. Automatica,2010,46(1):182–189.
[91] F. Ghorbel, B. Srinivasan, and M. W. Spong. On the uniform boundedness of the inertiamatrix of serial robot manipulators [J]. Journal of Robotic Systems,1998,15(1):17–28.
[92] L. Praly. Asymptotic stabilization via output feedback for lower triangular systems withoutput dependent incremental rate [J]. IEEE Transactions on Automatic Control,2003,48(6):1103–1108.
[93] P. Krishnamurthy, F. Khorrami, and Z. P. Jiang. Global output feedback tracking fornonlinear systems in generalized output-feedback canonical form [J]. IEEE Transactionson Automatic Control,2002,47(5):814–819.
[94] D. Karagiannis, M. Sassano, and A. Astolfi. Dynamic scaling and observer design withapplication to adaptive control [J]. Automatica,2009,45(12):2883–2889.
[95] A. Fasano and S. Marmi. Analytical Mechanics: An Introduction(Translated by B. Pel-loni)[M]. London: Oxford University Press,2006.
[96] Z. P. Jiang and L. Praly. Design of robust adaptive controllers for nonlinear systems withdynamic uncertainties [J]. Automatica,1998,34(7):825–840.
[97] Z. P. Jiang. A combined backstepping and small-gain approach to adaptive output feed-back control [J]. Automatica,1999,35(6):1131–1139.
[98] M. W. Spong and M. Vidyasagar. Robot Dynamics and Control [M]. New York: Wiley,1989.
[99] R. Kelly, V. Santiba′n ez and A. Lor′a Control of robot manipulators in joint space [M].London: Springer-Verlag,2005.
[100] F. L. Lewis, S. Jagannathan, and A. Yes ildirek. Neural network control of robot manipu-lators and nonlinear systems [M]. Taylor&Francis, Inc, Bristol, PA, USA,1998.
[101] A. C. Huang and M. C. Chien. Adaptive Control of Robot Manipulators: A UnifiedRegressor-free Approach [M]. Singapore: World Scientific Publishing Co. Pte. Ltd.,2010.
[102] H. D. Patino, R. Carelli, and B. R. Kuchen. Neural networks for advanced control ofrobot manipulators [J]. IEEE Transactions on Neural Networks,2002,13(2):343–354.
[103] J. Kasac, B. Novakovic, D. Majetic, and D. Brezak. Passive finite-dimensional repetitivecontrol of robot manipulators [J]. IEEE Transactions on Control Systems Technology,2008,16(3):570–576.
[104] K. Najim, E. Ikonen, and Eduardo G. Ram′rez. Trajectory tracking control based on agenealogical decision tree controller for robot manipulators [J]. International Journal ofInnovative Computing, Information and Control,2008,4(1):53–62.
[105] X. J. Xie and N. Duan. Output tracking of high-order stochastic nonlinear systems withapplication to benchmark mechanical system [J]. IEEE Transactions on Automatic Con-trol,2010,55(5):1197–1202.
[106] Y. Yin, P. Shi, and F. Liu. Gain scheduled pi tracking control on stochastic nonlinearsystems with partially known transition probabilities [J]. Journal of Franklin Institute,2011,348(4):685–702.
[107] B. Jiang, Z. Gao, P. Shi, and Y. Xu. Adaptive fault-tolerant tracking control of nearspace vehicle using takagi-sugeno fuzzy models [J]. IEEE Transactions on Fuzzy Sys-tems,2010,18(5):1000–1007.
[108] Q. Zhou, P. Shi, J. Lu, and S. Xu. Adaptive output feedback fuzzy tracking control for aclass of nonlinear systems [J]. IEEE Transactions on Fuzzy Systems,2011,19(5):972–982.
[109] J. J. Craig. Introduction to Robotics: Mechanics and Control (3rd Edition)[M]. UpperSaddle River, NJ: Prentice Hall,2004.
[110] F. L. Lewis, C. T. Abdallah, and D. M. Dawson. Control of Robot Manipulators [M].New York: Macmillan Pub. Co.,1993.
[111]E′. Dombre and W. Khalil. Robot Manipulators: Modeling, Performance Analysis andControl [M]. London: ISTE Publishing,2007.
[112] L. Villani, C. C. De Wit, and B. Brogliato. An exponentially stable adaptive control forforce and position tracking of robot manipulators [J]. IEEE Transactions on AutomaticControl,1999,44(4):798–802.
[113] C. C. Cheah, M. Hirano, S. Kawamura, and S. Arimoto. Approximate jacobian controlwith task-space damping for robot manipulators [J]. IEEE Transactions on AutomaticControl,2004,49(5):752–757.
[114] E. M. Jafarov, M. N. A. Parlakci, and Y. Istefanopulos. A new variable structure PID-controller design for robot manipulators [J]. IEEE Transactions on Control Systems Tech-nology,2005,13(1):122–130.
[115] H. Hu and P. Y. Woo. Fuzzy supervisory sliding-mode and neural-network control forrobotic manipulators [J]. IEEE Transactions on Industrial Electronics2006,53(3):929–940.
[116] H. B. Ji and H. S. Xi. Adaptive output-feedback tracking of stochastic nonlinear systems[J]. IEEE Transactions on Automatic Control,2006,51(2):355–360.
[117] A. J. van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems
[M]. Berlin: Springer-Verlag,1999.
[118] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King. Stability criteria for switched andhybrid systems [J]. SIAM Review,2005,49(4):545–592.
[119] A. P. Molchanov and Ye. S. Pyatnitskiy. Criteria of asymptotic stability of differentialand difference inclusions encountered in control theory [J]. Systems&Control Letters,1989,13(1):59–64.
[120] D. Liberzon and A. S. Morse. Basic problems in stability and design of switched systems[J]. IEEE Control Systems,1999,19(5):59–70.
[121] W. P. Dayawansa and C. F. Martin. A converse Lyapunov theorem for a class of dy-namical systems which undergo switching [J]. IEEE Transactions on Automatic Control,1999,44(4):751–760.
[122] M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched andhybrid systems [J]. IEEE Transactions on Automatic Control,1998,43(4):475–482.
[123] M. D. Lemmon, K. X. He, and I. Markovsky. Supervisory hybrid systems [J]. IEEEControl Systems,1999,19(4):42–55.
[124] A. N. Michel. Recent trends in the stability analysis of hybrid dynamical systems [J].IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,1999,46(1):120–134.
[125] Hui Ye, A. N. Michel, and Ling Hou. Stability theory for hybrid dynamical systems [J].IEEE Transactions on Automatic Control,1998,43(4):461–474.
[126] X. M. Sun, J. Zhao, and D. J. Hill. Stability and L2-gain analysis for switched delaysystems: A delay-dependent method [J]. Automatica,2006,42(10):1769–1774.
[127] J. P. Hespanha. Stochastic hybrid systems: applications to communication networks [C].in HSCC, ser. Lecture Notes in Computer Science, R. Alur and G. J. Pappas, Eds. NewYork: Springer Verlag,2004,2993:387–401.
[128] D. Liberzon, J. P. Hespanha, and A. S. Morse. Stability of switched systems: a lie-algebraic condition [J]. Systems&Control Letters,1999,37(3):117–122.
[129] J. P. Hespanha. Uniform stability of switched linear systems: extensions of LaSalle’sinvariance principle [J]. IEEE Transactions on Automatic Control,2004,49(4):470–82.
[130] K. S. Narenda and J. Balakrishnan. A common Lyapunov function for stable LTI systemswith commuting A-matrices [J]. IEEE Transactions on Automatic Control,1994,39(12):2469–2471.
[131] J. Krystul. Modelling of stochastic hybrid systems with applications to accident riskassessment [D]. PhD thesis, Ph.D. dissertation, University of Twente, Enschede, TheNetherlands,2001.
[132] G. Pola, M. L. Bujorianu, J. Lygeros, and M. D. D. Benedetto. Stochastic hybrid models:An overview [C]. In IFAC Conference on Analysis and Design of Hybrid Systems,2003.
[133] B. Fralix. Foster-type criteria for markov chains on general spaces [J]. Journal of AppliedProbability,2006,43(4):1194–1200.
[134] J. Leth, J. G. Rasmussen, H. Schioler, and R. Wisniewski. A class of stochastic hybridsystems with state-dependent switching noise [C]. In The51st IEEE Conference on De-cision and Control (Hawaii, America),2012.
[135] M. L. Bujorianu. Stochastic hybrid system: modelling and verification [D]. PhD thesis,Ph.D. dissertation, University of Stirling,2005.
[136] M. L. Bujorianu. Stochastic reachability: from markov chains to stochastic hybrid sys-tems [C]. In18th IFAC World Congress, Milano (Italy), August28,2011:4534–4539.
[137] S. P. Meyn and R. L. Tweedie. State-dependent criteria for convergence of markov chains[J]. Annals of Applied Probability,1994,4(1):149–168.
[138] J. Dai and S. Meyn. Stability and convergence of moments for multiclass queueing net-works via fluid limit models [J]. IEEE Transactions on Automatic Control,1995,40(11):1889–1904.
[139] S. Yuksel and S. P. Meyn. Random-time, state-dependent stochastic drift for markovchains and application to stochastic stabilization over erasure channels [J]. IEEE Trans-actions on Automatic Control,2013,58(1):47–59.
[140] R. W. Brockett. Asymptotic stabilitty and feedback stabilization, in Differential Geomet-ric Control Theory (R. W. Brockett, R. S. Millman and H. J. Sussmann eds.)[M], Boston:Birkhauser,1983:181-191.
[141] E. Sontag. Mathematical Control Theory. Deterministic Finite-Dimensional Systems [C],volume6.2nd ed., ser. Texts in Applied Mathematics. New York: Springer-Verlag,1998.
[142] H. K. Khalil. Nonlinear Systems,3nd edition [M]. Englewood Cliffs, NJ:Prentice Hall,2002.
[143] I. I. Gihman and A. V. Skorohod. The Theory of Stochastic Processes I (translated fromthe Russian by S. Kotz)[M]. New York: Springer-Verlag,1974.
[144] P. Florchinger. A universal formula for the stabilization of control stochastic differentialequations [J]. Stochastic Analysis and Applications,1993,11(2):155–162.
[145] J. Tsinias. Stochastic input-to-state stability and applications to global feedback stabiliza-tion (special issue on breakthrough in the control of nonlinear systems)[J]. InternationalJournal of Control,1998,71(5):907–930.
[146] X. Yu and X. J. Xie. Output feedback regulation of stochastic nonlinear systems withstochastic iISS inverse dynamics [J]. IEEE Transactions on Automatic Control,2010,55(2):304–320.
[147] X. Yu, X. J. Xie, and N. Duan. Small-gain control method for stochastic nonlinear sys-tems with stochastic iISS inverse dynamics [J]. Automatica,2010,46(12):1790–1798.
[148] Z. J. Wu, Y. Q. Xia, and X. J. Xie. Stochastic barbalat lemma and its applications [J].IEEE Transactions on Automatic Control,2012,57(6):1537–1543.
[149] X. Yu and Z. J. Wu. Corrections on ‘stochastic barbalat’s lemma and its applications’[J].IEEE Transactions on Automatic Control (submmited),2013.
[150] E. D. Sontag. On the input-to-state stability property [J]. European Journal of Control,1995,1:24–36.
[151] J. P. Hespanha. Logic-Based Switching Algorithms in Control [D]. PhD thesis, Dept. ofElectrical Eng., Yale University, New Haven, CT,1998.
[152] D. Applebaum, Le′vy Processes and Stochastic Calculus [M]. New York: CambridgeUniversity Press,2004.
[2] L. N. Hand and J. D. Finch. Analytical Mechanics [M]. Cambridge: Cambridge Univer-sity Press,1998.
[3] A. D. Lewis. Lagrangian Mechanics, Dynamics and Control [M]. Kingston: Queen Uni-versity,2004.
[4] O. D. Johns. Analytical Mechanics for Relativity and Quantum Mechanics [M]. NewYork: Oxford University Press,2005.
[5] R. Z. Khas’minskii. Stochastic Stability of Differential Equations (2nd ed)[M]. Berlin,Heidelberg: Springer-Verlag,2012.
[6] B. ksendal. Stochastic Differential Equations-An Introduction with Applications (6thEdition)[M]. New York: Springer-Verlag,2003.
[7] X. Mao. Stochastic Differential Equations and Applications [M]. New York: Horwood,1997.
[8] Z. J. Wu, M. Y. Cui, X. J. Xie, and P. Shi. Theory of stochastic dissipative systems [J].IEEE Transactions on Automatic Control,2011,56(7):1650–1655.
[9] M. Krstic′and H. Deng. Stabilization of Nonlinear Uncertain Systems [M]. New York:Springer-Verlag,1998.
[10] J. M. Yong and X. Y. Zhou. Stochastic Controls, Hamiltonian Systems and HJB Equa-tions [M]. New York: Springer-Verlag,1999.
[11] Z. J. Wu, J. Yang, and P. Shi. Adaptive tracking for stochastic nonlinear systems withmarkovian switching [J]. IEEE Transactions on Automatic Control,2010,55(9):2135–2141.
[12] J. A. La′zaro-Cam′and J. P. Ortega. Stochastic hamiltonian dynamical systems [R]. Re-ports on Mathematical Physics,2008,61:65–122.
[13]朱位秋.非线性随机动力学与控制: Hamilton理论体系框架[M].北京:科学出版社,2003.
[14] A. Loria and H. Nijmeijer. Bounded output feedback tracking control of fully actuatedeuler-lagrange systems [J]. Systems&Control Letters,1998,33:151–161.
[15] G. Besanc on. Global output feedback tracking control for a class of lagrangian systems[J]. Automatica,2000,36(12):1915–1921.
[16] B. Jayawardhana and G. Weiss. Tracking and disturbance rejection for fully actuatedmechanical systems [J]. Automatica,2008,44(11):2863–2868.
[17] P. M. Patre, W. MacKunis, K. Dupree, and W. E. Dixon. Modular adaptive control ofuncertain Euler-Lagrange systems with additive disturbances [J]. IEEE Transactions onAutomatic Control,2011,56(1):155–160.
[18] R. Ortega, A. Lor′a, P. J. Nicklasson, and H. Sira-Ram′rez. Passivity-based Control ofEuler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications
[M]. London: Springer-Verlag,1998.
[19] A. J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control [M]. Lon-don: Springer-Verlag, Berlin,2000.
[20] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland. Dissipative Systems Analysis andControl: Theory and Applications (2nd edition)[M]. London: Springer-Verlag,2007.
[21] S. Nicosia and P. Tomei. Robot control by using only joint position measurements [J].IEEE Transactions on Automatic Control,1990,35(9):1058–1061.
[22] F. Zhang, D. M. Dawson, M. S. de Queiroz, and W. E. Dixon. Global adaptive outputfeedback tracking control of robot manipulators [J]. IEEE Transactions on AutomaticControl,2000,45(6):1203–1208.
[23] S. Islam and P. X. Liu. Output feedback sliding mode control for robot manipulators [J].Robotica,2010,28(7):975–987.
[24]. N. Stamnes, O. M. Aamo, and G. O. Kaasa. A constructive speed observer design forgeneral Euler-Lagrange systems [J]. Automatica,2011,47(10):2233–2238.
[25]. N. Stamnes, O. M. Aamo, and G. O. Kaasa. Global output feedback tracking controlof euler-lagrange systems [C]. In The18th IFAC World Congress Milano (Italy) August28-September2,2011:215–220.
[26] A. Kovaleva and L. Akulenko. Approximation of escape time for Lagrangian systemswith fast noise [J]. IEEE Transactions on Automatic Control,2007,52(12):2338–2341.
[27] W. R. Hamilton. On the application to dynamics of a general mathematical method pre-viously applied to optics [C]. In British Association Report, Edinburgh, UK,1834:513–518.
[28] R. Ortega, A. van der Schaft, B. Maschke, and G. Escobar. Interconnection and dampingassignment passivity-based control of port-controlled Hamiltonian systems [J]. Automat-ica,2002,38(4):585–596.
[29] B. M. Maschke, R. Ortega, and A. J. van der Schaft. Energy-based Lyapunov functionsfor forced Hamiltonian systems with dissipation [J]. IEEE Transactions on AutomaticControl,2000,45(8):1498–1502.
[30] A. Astolfi and L. Menini. Noninteracting control with stability for Hamiltonian systems[J]. IEEE Transactions on Automatic Control,2000,45(8):1470–1482.
[31] K. Fujimoto and T. Sugie. Canonical transformations and stabilization of generalizedHamiltonian systems [J]. Systems&Control Letters,2001,42(3):217–227.
[32] Y. Z. Wang, C. W. Li, and D. Z. Cheng. Generalized Hamiltonian realization of time-invariant nonlinear systems [J]. Automatica,2003,39(8):1437–1443.
[33] S. G. Peng. Monotonic limit theorem of BSDE and nonlinear decomposition theorem ofdoob-meyer’s type [J]. Probability Theory and Related Fields,1999,113(4):473–499.
[34] S. G. Peng and Z. Wu. Fully coupled forward-backward stochastic differential equationsand applications to optimal control [J]. SIAM Journal on Control and Optimization,1999,37(3):825–843.
[35] J. M. Yong. A stochastic linear quadratic optimal control problem with generalized ex-pectation [J]. Stochastic Analysis and Applications,2008,26(6):1136–1160.
[36] W. Q. Zhu, Z. L. Huang, and Y. Q. Yang. Stochastic averaging of quasi-integrable Hamil-tonian systems [J]. ASME Journal of Applied Mechanics,1997,64(4):975–984.
[37] Z. H. Liu and W. Q. Zhu. Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control [J]. Automatica,2008,44(7):1923–1928.
[38] M. W. Spong, S. Hutchinson, and M. Vidyasagar. Robot Modeling and control [M]. NewYork: John Wiley&Sons Inc,2006.
[39] C. C. De Wit, B. Siciliano, and B. Bastin. Theory of Robot Control [M]. London:Springer-Verlag,1996.
[40] L. Sciavicco and B. Siciliano. Modelling and Control of Robot Manipulator [M]. London:Springer-Verlag,2000.
[41] F. L. Lewis, D. M. Dawson, and C. T. Abdallah. Robot Manipulator Control-Theoryand Practice [M]. New York: Marcel Dekke,2004.
[42] S. S. Ge, T. H. Lee, and C. J. Harris. Adaptive Neural Network Control of Robotic Ma-nipulators [M]. Singapore: World Scientific Publishing Co. Pte. Ltd.,1998.
[43] S. H. Crandall and W. D. Mark. Random Vibration in Mechanical Systems [M]. NewYork: Academic Press,1963.
[44] H. J. Kushner. Stochastic Stability and Control [M]. New York: Academic Press,1967.
[45] A. Friedman. Stochastic Differential Equations and Their Applications [M]. New York:Academic Press,1976.
[46] Z. G. Pan and T. Bas ar. Backstepping controller design for nonlinear stochastic systemsunder a risk-sensitive cost criterion [J]. SIAM Journal on Control and Optimization,1999,37(3):957–995.
[47] Z. J. Wu, X. J. Xie, P. Shi, and Y. Q. Xia. Backstepping controller design for a classof stochastic nonlinear systems with markovian switching [J]. Automatica,2009,45(4):997–1004.
[48] Y. G. Liu and J. F. Zhang. Practical output-feedback risk-sensitive control for stochasticnonlinear systems with stable zero-dynamics [J]. SIAM Journal on Control and Opti-mization,2006,45(3):885–926.
[49] S. J. Liu, J. F. Zhang, and Z. P. Jiang. Decentralized adaptive output-feedback stabiliza-tion for large-scale stochastic nonlinear systems [J]. Automatica,2007,43(2):238–251.
[50] Z. J. Wu, X. J. Xie, and S. Y. Zhang. Adaptive backstepping controller design usingstochastic small-gain theorem [J]. Automatica,2007,43(4):608–620.
[51] H. Deng and M. Krstic′. Output-feedback stochastic nonlinear stabilization [J]. IEEETransactions on Automatic Control,1999,44(2):328–333.
[52] H. Deng and M. Krstic′. Output-feedback stabilization of stochastic nonlinear systemsdriven by noise of unknown covariance [J]. Systems&Control Letters,2000,39(3):173–182.
[53] H. B. Ji and H. S. Xi. Adaptive output-feedback tracking of stochastic nonlinear systems[J]. IEEE Transactions on Automatic Control,2006,51(2):355–360.
[54] X. W. Mu and H. J. Liu. Stabilization for a class of stochastic nonlinear systems viaoutput feedback [J]. IEEE Transactions on Automatic Control,2008,53(1):360–367.
[55] S. J. Liu and J. F. Zhang. Output-feedback control of a class of stochastic nonlinearsystems with linearly bounded unmeasurable states [J]. International Journal of Robustand Nonlinear Control,2008,18(6):665–687.
[56] D. Liberzon. Switching in Systems and Control [M]. Boston: MA: Birkhauser,2003.
[57] Y. Ji and H. J. Chizeck. Controllability, stabilizability and continuous-time markovianjump linear quadratic control [J]. IEEE Transactions on Automatic Control,1990,35(7):777–788.
[58] El-K. Boukas. Stochastic Switching Systems: Analysis and Design [M]. Boston:Birkhauser,2006.
[59] P. Shi, E. K. Boukas, and R. K. Agarwal. Kalman filtering for continuous-time uncer-tain systems with markovian jumping parameters [J]. IEEE Transactions on AutomaticControl,1999,44(8):1592–1597.
[60] P. Shi, Y. Q. Xia, G. P. Liu, and D. Rees. On designing of sliding-mode control forstochastic jump systems [J]. IEEE Transactions on Automatic Control,2006,51(1):97–103.
[61] P. Shi and E. K. Boukas. H-infinity control for markovian jumping linear systems withparametric uncertainties [J]. Journal of Optimization Theory and Applications,1997,95:75–99.
[62] X. Mao. Stability of stochastic differential equations with Markovian switching [J].Stochastic Processes and Their Applications,1999,79(1):45–67.
[63] C. Yuan and X. Mao. Robust stability and controllability of stochastic differential delayequations with markovian switching [J]. Automatica,2004,40(3):343–354.
[64] X. Mao and C. Yuan. Stochastic Differential Equations with Markovian Switching [M].London: Imperial College Press,2006.
[65] W. Feng and J. F. Zhang. Stability analysis and stabilization control of multivariableswitched stochastic systems [J]. Automatica,2006,42(1):169–176.
[66] W. Feng, J. Tian, and P. Zhao. Stability analysis of switched stochastic systems [J]. Au-tomatica,2011,47(1):148–157.
[67] D. Chatterjee and D. Liberzon. Stability analysis of deterministic and stochastic switchedsystems via a comparison principle and multiple Lyapunov functions [J]. SIAM Journalon Control and Optimization,2006,45(1):174–206.
[68] H. Deng, M. Krstic′, and R. J. Williams. Stabilization of stochastic nonlinear systemsdriven by noise of unknown covariance [J]. IEEE Transactions on Automatic Control,2001,46(8):1237–1253.
[69] J. P. LaSalle and S. Lefschetz. Stability by Lyapunov’s Direct Method with Applications
[M]. New York: Academic Press,1961.
[70] V. Lakshmikantham, S. Leela, and A. A. Martynyuk. Practical Stability of NonlinearSystems [M]. Singapore: World Science Publisher,1990.
[71] E. Wong and M. Zakai. Riemann-stieltjes approximations of stochastic integrals [J]. Z.Wahrscheinlichkeitstheorie verw. Geb,1969,12:87–97.
[72] Y. G. Liu and J. F. Zhang. Reduced-order observer-based control design for nonlinearstochastic systems [J]. Systems&Control Letters,2004,52(2):123–135.
[73] Z. G. Pan. Differential geometric condition for feedback complete linearization ofstochastic nonlinear system [J]. Automatica,2001,37(1):145–149.
[74] Z. G. Pan. Canonical forms for stochastic nonlinear systems [J]. Automatica,2002,38(7):1163–1170.
[75] J. M. Bismut. Me′canique Ale′atoire [C]. Lecture Notes in Mathematics,866, Springer,1981.
[76] Z. G. Ying and W. Q. Zhu. Astochastically averaged optimal control strategy for quasi-Hamiltonian systemswith actuator saturation [J]. Automatica,2006,42:1577–1582.
[77] W. Lin and C. J. Qian. Adaptive control of nonlinearly parameterized systems: A nons-mooth feedback framework [J]. IEEE Transactions on Automatic Control,2002,47(5):757–774.
[78] M. M. Bridges, D. M. Dawson, Z. H. Qu, and S. C. Martindale. Robust control of rigid-link flexible-joint robots with redundant joint actuators [J]. IEEE Transactions on Sys-tems, Man and Cybernetics,1994,24(7):961–970.
[79] J. M. Knudsen and P. G. Hjorth. Elements of Newtonian Mechanics: Including NonlinearDynamics [M]. New York: Springer-Verlag,2000.
[80] D. L. Goodstein, R. P. Olenick, and T. M. Apostol. The Mechanical Universe: Introduc-tion to Mechanics and Heat [M]. New York: Cambridge University Press,2008.
[81] Z. J. Wu, X. J. Xie, and P. Shi. Robust adaptive output-feedback control for nonlinearsystems with output unmodeled dynamics [J]. International Journal of Robust and Non-linear Control,2008,18(11):1162–1187.
[82] A. Loria. Global tracking control of one degree of freedom euler-lagrange systems with-out velocity measurements [J]. European Journal of Control,1996,2:144–151.
[83] W. E. Dixon, E. Zergeroglu, and D. M. Dawson. Global robust output feedback trackingcontrol of robot manipulators [J]. Robotica,2004,22(4):351–357.
[84] J. Moreno-Valenzuela, V. Santiba′nez, and R. Campa. On output feedback tracking ccon-trol of robot manipulators with bounded torque input [J]. International Journal of control,Automation, and Systems,2008,6(1):76–85.
[85] S. Purwar, I. N. Kar, and A. N. Jha. Adaptive output feedback tracking control of robotmanipulators using position measurements only [J]. Expert Systems with Applications,2008,34(4):2789–2798.
[86] E. Kim. Output feedback tracking control of robot manipulators with model uncertaintyvia adaptive fuzzy logic [J]. IEEE Transactions on Fuzzy Systems,2004,12(3):368–378.
[87] O. M. Aamo, M. Arcak, T. I. Fossen, and P. V. Kokotovic. Global output tracking controlof a class of Euler-Lagrange systems. International Journal of Control,2000,74:649–658.
[88] K. D. Do, Z. P. Jiang, and J. Pan. Global partial-state feedback and output-feedbacktracking controllers for underactuated ships [J]. Systems&Control Letters,2005,54(10):1015–1036.
[89] D. Carnevale, D. Karagiannis, and A. Astolfi. A condition for certainty equivalence out-put feedback stabilization of nonlinear systems [J]. IEEE Transactions on AutomaticControl,2010,55(5):1180–1185.
[90] A. Astolfi, R. Ortega, and A. Venkatraman. A globally exponentially convergent immer-sion and invariance speed observer for mechanical systems with non-holonomic con-straints [J]. Automatica,2010,46(1):182–189.
[91] F. Ghorbel, B. Srinivasan, and M. W. Spong. On the uniform boundedness of the inertiamatrix of serial robot manipulators [J]. Journal of Robotic Systems,1998,15(1):17–28.
[92] L. Praly. Asymptotic stabilization via output feedback for lower triangular systems withoutput dependent incremental rate [J]. IEEE Transactions on Automatic Control,2003,48(6):1103–1108.
[93] P. Krishnamurthy, F. Khorrami, and Z. P. Jiang. Global output feedback tracking fornonlinear systems in generalized output-feedback canonical form [J]. IEEE Transactionson Automatic Control,2002,47(5):814–819.
[94] D. Karagiannis, M. Sassano, and A. Astolfi. Dynamic scaling and observer design withapplication to adaptive control [J]. Automatica,2009,45(12):2883–2889.
[95] A. Fasano and S. Marmi. Analytical Mechanics: An Introduction(Translated by B. Pel-loni)[M]. London: Oxford University Press,2006.
[96] Z. P. Jiang and L. Praly. Design of robust adaptive controllers for nonlinear systems withdynamic uncertainties [J]. Automatica,1998,34(7):825–840.
[97] Z. P. Jiang. A combined backstepping and small-gain approach to adaptive output feed-back control [J]. Automatica,1999,35(6):1131–1139.
[98] M. W. Spong and M. Vidyasagar. Robot Dynamics and Control [M]. New York: Wiley,1989.
[99] R. Kelly, V. Santiba′n ez and A. Lor′a Control of robot manipulators in joint space [M].London: Springer-Verlag,2005.
[100] F. L. Lewis, S. Jagannathan, and A. Yes ildirek. Neural network control of robot manipu-lators and nonlinear systems [M]. Taylor&Francis, Inc, Bristol, PA, USA,1998.
[101] A. C. Huang and M. C. Chien. Adaptive Control of Robot Manipulators: A UnifiedRegressor-free Approach [M]. Singapore: World Scientific Publishing Co. Pte. Ltd.,2010.
[102] H. D. Patino, R. Carelli, and B. R. Kuchen. Neural networks for advanced control ofrobot manipulators [J]. IEEE Transactions on Neural Networks,2002,13(2):343–354.
[103] J. Kasac, B. Novakovic, D. Majetic, and D. Brezak. Passive finite-dimensional repetitivecontrol of robot manipulators [J]. IEEE Transactions on Control Systems Technology,2008,16(3):570–576.
[104] K. Najim, E. Ikonen, and Eduardo G. Ram′rez. Trajectory tracking control based on agenealogical decision tree controller for robot manipulators [J]. International Journal ofInnovative Computing, Information and Control,2008,4(1):53–62.
[105] X. J. Xie and N. Duan. Output tracking of high-order stochastic nonlinear systems withapplication to benchmark mechanical system [J]. IEEE Transactions on Automatic Con-trol,2010,55(5):1197–1202.
[106] Y. Yin, P. Shi, and F. Liu. Gain scheduled pi tracking control on stochastic nonlinearsystems with partially known transition probabilities [J]. Journal of Franklin Institute,2011,348(4):685–702.
[107] B. Jiang, Z. Gao, P. Shi, and Y. Xu. Adaptive fault-tolerant tracking control of nearspace vehicle using takagi-sugeno fuzzy models [J]. IEEE Transactions on Fuzzy Sys-tems,2010,18(5):1000–1007.
[108] Q. Zhou, P. Shi, J. Lu, and S. Xu. Adaptive output feedback fuzzy tracking control for aclass of nonlinear systems [J]. IEEE Transactions on Fuzzy Systems,2011,19(5):972–982.
[109] J. J. Craig. Introduction to Robotics: Mechanics and Control (3rd Edition)[M]. UpperSaddle River, NJ: Prentice Hall,2004.
[110] F. L. Lewis, C. T. Abdallah, and D. M. Dawson. Control of Robot Manipulators [M].New York: Macmillan Pub. Co.,1993.
[111]E′. Dombre and W. Khalil. Robot Manipulators: Modeling, Performance Analysis andControl [M]. London: ISTE Publishing,2007.
[112] L. Villani, C. C. De Wit, and B. Brogliato. An exponentially stable adaptive control forforce and position tracking of robot manipulators [J]. IEEE Transactions on AutomaticControl,1999,44(4):798–802.
[113] C. C. Cheah, M. Hirano, S. Kawamura, and S. Arimoto. Approximate jacobian controlwith task-space damping for robot manipulators [J]. IEEE Transactions on AutomaticControl,2004,49(5):752–757.
[114] E. M. Jafarov, M. N. A. Parlakci, and Y. Istefanopulos. A new variable structure PID-controller design for robot manipulators [J]. IEEE Transactions on Control Systems Tech-nology,2005,13(1):122–130.
[115] H. Hu and P. Y. Woo. Fuzzy supervisory sliding-mode and neural-network control forrobotic manipulators [J]. IEEE Transactions on Industrial Electronics2006,53(3):929–940.
[116] H. B. Ji and H. S. Xi. Adaptive output-feedback tracking of stochastic nonlinear systems[J]. IEEE Transactions on Automatic Control,2006,51(2):355–360.
[117] A. J. van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems
[M]. Berlin: Springer-Verlag,1999.
[118] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King. Stability criteria for switched andhybrid systems [J]. SIAM Review,2005,49(4):545–592.
[119] A. P. Molchanov and Ye. S. Pyatnitskiy. Criteria of asymptotic stability of differentialand difference inclusions encountered in control theory [J]. Systems&Control Letters,1989,13(1):59–64.
[120] D. Liberzon and A. S. Morse. Basic problems in stability and design of switched systems[J]. IEEE Control Systems,1999,19(5):59–70.
[121] W. P. Dayawansa and C. F. Martin. A converse Lyapunov theorem for a class of dy-namical systems which undergo switching [J]. IEEE Transactions on Automatic Control,1999,44(4):751–760.
[122] M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched andhybrid systems [J]. IEEE Transactions on Automatic Control,1998,43(4):475–482.
[123] M. D. Lemmon, K. X. He, and I. Markovsky. Supervisory hybrid systems [J]. IEEEControl Systems,1999,19(4):42–55.
[124] A. N. Michel. Recent trends in the stability analysis of hybrid dynamical systems [J].IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,1999,46(1):120–134.
[125] Hui Ye, A. N. Michel, and Ling Hou. Stability theory for hybrid dynamical systems [J].IEEE Transactions on Automatic Control,1998,43(4):461–474.
[126] X. M. Sun, J. Zhao, and D. J. Hill. Stability and L2-gain analysis for switched delaysystems: A delay-dependent method [J]. Automatica,2006,42(10):1769–1774.
[127] J. P. Hespanha. Stochastic hybrid systems: applications to communication networks [C].in HSCC, ser. Lecture Notes in Computer Science, R. Alur and G. J. Pappas, Eds. NewYork: Springer Verlag,2004,2993:387–401.
[128] D. Liberzon, J. P. Hespanha, and A. S. Morse. Stability of switched systems: a lie-algebraic condition [J]. Systems&Control Letters,1999,37(3):117–122.
[129] J. P. Hespanha. Uniform stability of switched linear systems: extensions of LaSalle’sinvariance principle [J]. IEEE Transactions on Automatic Control,2004,49(4):470–82.
[130] K. S. Narenda and J. Balakrishnan. A common Lyapunov function for stable LTI systemswith commuting A-matrices [J]. IEEE Transactions on Automatic Control,1994,39(12):2469–2471.
[131] J. Krystul. Modelling of stochastic hybrid systems with applications to accident riskassessment [D]. PhD thesis, Ph.D. dissertation, University of Twente, Enschede, TheNetherlands,2001.
[132] G. Pola, M. L. Bujorianu, J. Lygeros, and M. D. D. Benedetto. Stochastic hybrid models:An overview [C]. In IFAC Conference on Analysis and Design of Hybrid Systems,2003.
[133] B. Fralix. Foster-type criteria for markov chains on general spaces [J]. Journal of AppliedProbability,2006,43(4):1194–1200.
[134] J. Leth, J. G. Rasmussen, H. Schioler, and R. Wisniewski. A class of stochastic hybridsystems with state-dependent switching noise [C]. In The51st IEEE Conference on De-cision and Control (Hawaii, America),2012.
[135] M. L. Bujorianu. Stochastic hybrid system: modelling and verification [D]. PhD thesis,Ph.D. dissertation, University of Stirling,2005.
[136] M. L. Bujorianu. Stochastic reachability: from markov chains to stochastic hybrid sys-tems [C]. In18th IFAC World Congress, Milano (Italy), August28,2011:4534–4539.
[137] S. P. Meyn and R. L. Tweedie. State-dependent criteria for convergence of markov chains[J]. Annals of Applied Probability,1994,4(1):149–168.
[138] J. Dai and S. Meyn. Stability and convergence of moments for multiclass queueing net-works via fluid limit models [J]. IEEE Transactions on Automatic Control,1995,40(11):1889–1904.
[139] S. Yuksel and S. P. Meyn. Random-time, state-dependent stochastic drift for markovchains and application to stochastic stabilization over erasure channels [J]. IEEE Trans-actions on Automatic Control,2013,58(1):47–59.
[140] R. W. Brockett. Asymptotic stabilitty and feedback stabilization, in Differential Geomet-ric Control Theory (R. W. Brockett, R. S. Millman and H. J. Sussmann eds.)[M], Boston:Birkhauser,1983:181-191.
[141] E. Sontag. Mathematical Control Theory. Deterministic Finite-Dimensional Systems [C],volume6.2nd ed., ser. Texts in Applied Mathematics. New York: Springer-Verlag,1998.
[142] H. K. Khalil. Nonlinear Systems,3nd edition [M]. Englewood Cliffs, NJ:Prentice Hall,2002.
[143] I. I. Gihman and A. V. Skorohod. The Theory of Stochastic Processes I (translated fromthe Russian by S. Kotz)[M]. New York: Springer-Verlag,1974.
[144] P. Florchinger. A universal formula for the stabilization of control stochastic differentialequations [J]. Stochastic Analysis and Applications,1993,11(2):155–162.
[145] J. Tsinias. Stochastic input-to-state stability and applications to global feedback stabiliza-tion (special issue on breakthrough in the control of nonlinear systems)[J]. InternationalJournal of Control,1998,71(5):907–930.
[146] X. Yu and X. J. Xie. Output feedback regulation of stochastic nonlinear systems withstochastic iISS inverse dynamics [J]. IEEE Transactions on Automatic Control,2010,55(2):304–320.
[147] X. Yu, X. J. Xie, and N. Duan. Small-gain control method for stochastic nonlinear sys-tems with stochastic iISS inverse dynamics [J]. Automatica,2010,46(12):1790–1798.
[148] Z. J. Wu, Y. Q. Xia, and X. J. Xie. Stochastic barbalat lemma and its applications [J].IEEE Transactions on Automatic Control,2012,57(6):1537–1543.
[149] X. Yu and Z. J. Wu. Corrections on ‘stochastic barbalat’s lemma and its applications’[J].IEEE Transactions on Automatic Control (submmited),2013.
[150] E. D. Sontag. On the input-to-state stability property [J]. European Journal of Control,1995,1:24–36.
[151] J. P. Hespanha. Logic-Based Switching Algorithms in Control [D]. PhD thesis, Dept. ofElectrical Eng., Yale University, New Haven, CT,1998.
[152] D. Applebaum, Le′vy Processes and Stochastic Calculus [M]. New York: CambridgeUniversity Press,2004.