基于非光滑Lipschitz曲面的控制设计方法研究
|
摘要
在控制科学领域的工程实践和理论研究中,会遇到大量具有非光滑动态的系统。无论系统本身是非光滑的,还是由于施加了非光滑的控制,都属于非光滑控制系统的研究范畴,这方面的研究已经逐渐成为控制科学与系统科学的研究热点之一。在非光滑控制系统理论框架下对非光滑控制设计方法进行深入研究,对改进和完善非光滑控制系统理论具有重要意义。本文从研究系统轨迹同状态空间中单一或若干非光滑Lipschitz曲面的相对关系出发,讨论了由Filippov微分包含解描述的非光滑控制系统的分析与设计问题。分别针对切换控制、滑模变结构控制以及开关控制,进行了以下几方面的研究工作:
首先,针对一类系统参数存在摄动的线性系统,提出了一种基于非光滑Lipschitz曲面的切换控制设计方法。依据相依锥判据的穿越条件和不相交条件,借鉴递推的思想构造了一系列的Lipschitz域,并分析了Lipschitz域内部系统轨迹关于平衡点的收敛性。在此基础上,以Lipschitz域的边界曲面作为控制设计的切换面,基于相依锥判据进一步给出了切换控制律,使得从状态空间中Lipschitz域外部和边界出发的系统轨迹能够在有限时间进入并保持在Lipschitz域的内部。
其次,针对一类含有不确定性或外部扰动的非线性系统,提出了一种基于非光滑Lipschitz曲面的滑模变结构控制设计方法。利用相依锥判据作为控制设计时的滑模到达条件和滑模存在条件,分别给出了基于全局相依锥判据和非全局相依锥判据的滑模变结构控制设计方法。为进一步解决非光滑Lipschitz滑模面的构造问题,针对一类不确定线性系统,结合自稳定域的概念给出了一种非光滑Lipschitz滑模面的构造方法。
然后,针对一类控制输入为开关量的线性系统,提出了一种基于非光滑Lipschitz曲面的开关控制设计方法。利用允许光滑切换面的不唯一性,通过多个光滑切换面的组合来构造新的非光滑Lipschitz切换面,给出了系统的平衡点集,并利用非光滑系统的LaSalle不变集原理证明了系统轨迹关于平衡点集的收敛性。本文提出的开关控制设计方法可以依据轨迹所在区域选取切换面,从而提高了控制设计的灵活性。
最后,对某型大气层外动能拦截器末制导段的开关式导引律设计问题进行了研究。根据弹-目相对运动的特点,设计了一种分段线性Lipschitz开关曲线,并利用相依锥判据的穿越条件,分析了制导系统的稳定性。通过仿真,验证了本文方法的有效性,与现有基于光滑开关曲线的导引律设计方法进行比较,显示了本文提出的方法在减少发动机开关次数方面的优势。
Nonsmooth systems arise in a large number of nature and physical science. In a control science point of view, there exist large numbers of dynamical systems with nonsmooth dynamics in the fields of engineering applications and theoretical studies. No matter the system is nonsmooth itself or is drived by a nonsmooth control input, the resulting system is called nonsmooth control system. The issues have received more and more attentions recently in control science and system science. It is important and theoretically meaningful to improve and develop nonsmooth control design methods in the framework of nonsmooth systems theory. Based on the relation between the trajectories of the closed-loop system and Lipschitz surfaces in the state space, the dissertation attempts to make a discussion on the analysis and synthesis of nonsmooth dynamical systems which can be described in the sense of Filippov differential inclusion, utilizing nonsmooth analysis and nonsmooth stability analysis as the mathematical tools. Several tops, such as switched control design, variable structure control design with sliding mode and on-off control design, are discussed as follows.
Firstly, the switched control design problem with nonsmooth Lipschitz surfaces is investigated for a class of linear systems with parameter uncertainties. A series of Lipschitz domains with Lipschitz surfaces as their boundaries are constructed in a recursive way using the notion of self-stable region. The equilibrium is discussed, and the convergence of the trajectories to the equilibrium in the interior of the Lipschitz domain is studied utilizing contingent cone criteria. In addition, taking the boundaries of the Lipschitz domain as switching surfaces, a switched control design method is proposed to drive the trajectories entering the interior of the Lipschitz domain in finite time and maintaining in it, such that the global stability is achieved.
Secondly, the sliding mode control design problem with nonsmooth Lipschitz surfaces is investigated for a class of nonlinear systems with uncertainties and perturbations. Contingent cone criteria are used as new approaching condition and existence condition of sliding mode. According to the new conditions, a sliding mode control design method based on the global contingent cone criteria and a sliding mode control design method based on the non-global contingent cone criteria, which is considered to reduce the conservative property, are proposed to drive the trajectories approaching the Lipschitz surface in finite time and sliding on it. In order to obtain the Lipschitz switching surface of the sliding mode control, a class of linear uncertain systems is considered, and a sliding mode control design method based on self-stable region is proposed. The stability of the new switching surface is analyzed and its properties are illustrated by comparing with some existing methods.
Thirdly, the on-off control design problem with nonsmooth Lipschitz surfaces is investigated for a class of linear controllable systems with on-off control input. The new nonsmooth Lipschitz on-off switching surfaces are constructed by means of combination of finite alternative smooth subsurfaces. The equilibria set of the closed-loop system is discussed. The on-off control design method is proposed and the glaoblly asymptotic stability is analyzed utilizing LaSalle’s invariant principle of nonsmooth system. The switching surfaces can be designed with respect to the domains in the state space and the flexibility of the on-off control design can be improved.
Finally, the on-off guidance law design problem with nonsmooth Lipschitz thresholds of divert thrusters for a certain type of exo-atmospheric interceptror is investigated. The Lipschitz threshold is constructed by means of combination of finite linear surfaces, considering the characteristics of the relative motion between the interceptror and the target. The stability of the guidance system is analyzed by utilizing contingent cone criteria. By comparing with some existing methods with smooth thresholds, the effectiveness and advantage of the proposed method are illustrated by some numerical simulations.
首先,针对一类系统参数存在摄动的线性系统,提出了一种基于非光滑Lipschitz曲面的切换控制设计方法。依据相依锥判据的穿越条件和不相交条件,借鉴递推的思想构造了一系列的Lipschitz域,并分析了Lipschitz域内部系统轨迹关于平衡点的收敛性。在此基础上,以Lipschitz域的边界曲面作为控制设计的切换面,基于相依锥判据进一步给出了切换控制律,使得从状态空间中Lipschitz域外部和边界出发的系统轨迹能够在有限时间进入并保持在Lipschitz域的内部。
其次,针对一类含有不确定性或外部扰动的非线性系统,提出了一种基于非光滑Lipschitz曲面的滑模变结构控制设计方法。利用相依锥判据作为控制设计时的滑模到达条件和滑模存在条件,分别给出了基于全局相依锥判据和非全局相依锥判据的滑模变结构控制设计方法。为进一步解决非光滑Lipschitz滑模面的构造问题,针对一类不确定线性系统,结合自稳定域的概念给出了一种非光滑Lipschitz滑模面的构造方法。
然后,针对一类控制输入为开关量的线性系统,提出了一种基于非光滑Lipschitz曲面的开关控制设计方法。利用允许光滑切换面的不唯一性,通过多个光滑切换面的组合来构造新的非光滑Lipschitz切换面,给出了系统的平衡点集,并利用非光滑系统的LaSalle不变集原理证明了系统轨迹关于平衡点集的收敛性。本文提出的开关控制设计方法可以依据轨迹所在区域选取切换面,从而提高了控制设计的灵活性。
最后,对某型大气层外动能拦截器末制导段的开关式导引律设计问题进行了研究。根据弹-目相对运动的特点,设计了一种分段线性Lipschitz开关曲线,并利用相依锥判据的穿越条件,分析了制导系统的稳定性。通过仿真,验证了本文方法的有效性,与现有基于光滑开关曲线的导引律设计方法进行比较,显示了本文提出的方法在减少发动机开关次数方面的优势。
Nonsmooth systems arise in a large number of nature and physical science. In a control science point of view, there exist large numbers of dynamical systems with nonsmooth dynamics in the fields of engineering applications and theoretical studies. No matter the system is nonsmooth itself or is drived by a nonsmooth control input, the resulting system is called nonsmooth control system. The issues have received more and more attentions recently in control science and system science. It is important and theoretically meaningful to improve and develop nonsmooth control design methods in the framework of nonsmooth systems theory. Based on the relation between the trajectories of the closed-loop system and Lipschitz surfaces in the state space, the dissertation attempts to make a discussion on the analysis and synthesis of nonsmooth dynamical systems which can be described in the sense of Filippov differential inclusion, utilizing nonsmooth analysis and nonsmooth stability analysis as the mathematical tools. Several tops, such as switched control design, variable structure control design with sliding mode and on-off control design, are discussed as follows.
Firstly, the switched control design problem with nonsmooth Lipschitz surfaces is investigated for a class of linear systems with parameter uncertainties. A series of Lipschitz domains with Lipschitz surfaces as their boundaries are constructed in a recursive way using the notion of self-stable region. The equilibrium is discussed, and the convergence of the trajectories to the equilibrium in the interior of the Lipschitz domain is studied utilizing contingent cone criteria. In addition, taking the boundaries of the Lipschitz domain as switching surfaces, a switched control design method is proposed to drive the trajectories entering the interior of the Lipschitz domain in finite time and maintaining in it, such that the global stability is achieved.
Secondly, the sliding mode control design problem with nonsmooth Lipschitz surfaces is investigated for a class of nonlinear systems with uncertainties and perturbations. Contingent cone criteria are used as new approaching condition and existence condition of sliding mode. According to the new conditions, a sliding mode control design method based on the global contingent cone criteria and a sliding mode control design method based on the non-global contingent cone criteria, which is considered to reduce the conservative property, are proposed to drive the trajectories approaching the Lipschitz surface in finite time and sliding on it. In order to obtain the Lipschitz switching surface of the sliding mode control, a class of linear uncertain systems is considered, and a sliding mode control design method based on self-stable region is proposed. The stability of the new switching surface is analyzed and its properties are illustrated by comparing with some existing methods.
Thirdly, the on-off control design problem with nonsmooth Lipschitz surfaces is investigated for a class of linear controllable systems with on-off control input. The new nonsmooth Lipschitz on-off switching surfaces are constructed by means of combination of finite alternative smooth subsurfaces. The equilibria set of the closed-loop system is discussed. The on-off control design method is proposed and the glaoblly asymptotic stability is analyzed utilizing LaSalle’s invariant principle of nonsmooth system. The switching surfaces can be designed with respect to the domains in the state space and the flexibility of the on-off control design can be improved.
Finally, the on-off guidance law design problem with nonsmooth Lipschitz thresholds of divert thrusters for a certain type of exo-atmospheric interceptror is investigated. The Lipschitz threshold is constructed by means of combination of finite linear surfaces, considering the characteristics of the relative motion between the interceptror and the target. The stability of the guidance system is analyzed by utilizing contingent cone criteria. By comparing with some existing methods with smooth thresholds, the effectiveness and advantage of the proposed method are illustrated by some numerical simulations.
引文
[1] Sontag E D. Mathematical Control Theory: Deterministic Finite Dimensional Systems[M]. Springer, 1998.
[2] Cortés J. Discontinuous Dynamical Systems[J]. IEEE Control Systems Magazine, 2008, 28(3): 36-73.
[3] Shen T. Analysis and Control of Discontinuous Dynamical Systems[C]// Proceedings of the 26th Chinese Control Conference. Zhang Jiajie, China, 2007: 37-45.
[4] Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction[J]. Automatica, 1994, 30(7): 1083-1138.
[5] Hori Y, Toyoda Y, Tsuruoka Y. Traction Control of Electric Vehicle: Basic Experimental Results Using the Test EV 'UOT Electric March'[J]. IEEE Transactions on Industry Applications, 1998, 34(5): 1131–1138.
[6] Ray L R. Nonlinear Tire Force Estimation and Road Friction Identification: Simulation and Experiments[J]. Automatica, 1997, 33(10): 1819-1833.
[7] Shen T. Robust Nonlinear Traction Control with Direct Torque Controlled Induction Motor[C]//Proceedings of Asian Electric Vehicle Conference. Seoul, Korea, 2004.
[8] Guzzella L, Onder C H. Introduction to Modeling and Control of Internal Combustion Engine Systems[M]. Berlin: Springer, 2004.
[9] Balluchi A, Benvenuti L, Benedetto M D D, et al. Automotive Engine Control and Hybrid Systems: Challenges and Opportunities[J]. Proceedings of the IEEE, 2000, 88(7): 888-912.
[10] Brockett R W, Liberzon D. Quantized Feedback Stabilization of Linear Systems [J]. IEEE Transaction on Automatic Control, 2000, 45(7): 1279-1289.
[11] Claasen T A C M, Mecklenbrauker W F G, Peek J B H. Effects of Quantization and Overflow in Recursive Digital Filters[J]. IEEE Transactions on Acoustics, Speech and Signal Processing, 1976, 24(6): 517-530.
[12] Fagnani F, Zampieri S. Stability Analysis and Synthesis for Scalar Linear Systems with a Quantized Feedback[J]. IEEE Transactions on Automatic Control, 2003, 48(9): 1569-1584.
[13] Goncalves J M, Megretski A, Dahleh M A. Global Stability of Relay Feedback Systems[J]. IEEE Transactions on Automatic Control, 2001, 46(4): 550-562.
[14] Bowes S R. Advanced Regular-sampled PWM Control Techniques for Drivesand Static Power Converters[J]. IEEE Transactions on Industrial Electronics, 1995, 42(4): 367-373.
[15] Brogliato B. Nonsmooth Mechanics: Models, Dynamics and Control(2nd)[M]. New York: Springer-Verlag, 1999.
[16] Agrachev A A, Sachkov Y. Control Theory from the Geometric Viewpoint[M]. New York: Springer-Verlag, 2004.
[17] Edwards C, Spurgeon S K. Sliding Mode Control: Theory and Applications[M]. New York: Taylor & Francis, 1998.
[18] Brockett R W. Asymptotic Stability and Feedback Stabilization[C]//Differential Geometric Control Theory. Boston, MA: Birkhauser, 1983: 181-191.
[19] Monforte J C. Geometric, Control and Numerical Aspects of Nonholonomic Systems[M]. Springer, 2002.
[20] Martin P, Rouchon P. Feedback Linearization and Driftless Systems[J]. Mathematics of Control, Signals and Systems, 1994, 7(3): 235-254.
[21] Sontag E D, Sussmann H. Remarks on Continuous Feedback Control[C]// Proceedings of IEEE Conferecse on Decision and Control. 1980: 916-921.
[22] Bloch A, Drakunov S. Stabilization and Tracking in the Nonholnomic Inter- grator via Sliding Modes[J]. Systems & Control Letters, 1996, 29(2): 91-99.
[23] Liberzon D. Switchings in Systems and Control[M]. Boston: Birkhauser, 2003.
[24] Liberzon D, Morse A S. Basic Problems in Stability and Design of Switched Systems[J]. IEEE Control Systems Magazine, 1999, 19(5): 59-70.
[25] Casagrande D, Astolfi A, Parisini T. A Stabilizing Time-switching Control Strategy for the Rolling Sapere[C]//Proceedings of CDC-ECC Conference. 2005: 3297-3302.
[26] Utkin V I. VSS with Sliding Mode[J]. IEEE Transaction on Automatic Control, 1977, 22(2): 212-222.
[27] Unkin V I.Sliding Modes in Control and Optimization[M].New York: Springer- Verlag, 1992.
[28] Xu J X. Variable Structure Systems: Towards the 21st Century[M]. Springer, 2002.
[29] Akasaka D, Liu K Z. Stabilization of a Class of On-off Feedback Systems by Nonlinear State Feedback[C]//Proceedings of European Control Conference. Kos, Greece, 2007: 667-672.
[30] Akasaka D, Liu K Z. Nonlinear Feedback Control of a Class of On-off Control Systems: An Approach Based on Analytical Solution of PDE's[C]//Proceedings of the IEEE Conference on Decision and Control. New Orleans, USA, 2007: 86-91.
[31] Polycarpou M M, Ioannou P A. On the Existence and Uniqueness of Solutions in Adaptive Control Systems[J]. IEEE Transactions on Automatic Control, 1993, 38(3): 474-479.
[32] Clarke F H. Optimization and Nonsmooth Analysis[M]. New York: John Wiley & Sons, 1983.
[33] Bhaya A, Kaszkurewicz E. Control Perspectives on Numerical Algorithms and Matrix Problems[M]. Philadelphia, PA: SIAM, 2006.
[34] Filppov A F. Differential Equations with Discontinuous Righthand Side[J]. American Mathematical Sociaty Translations, 1964, 42(2): 199-231.
[35] Filippov A F. Differential Equations with Discontinuous Right-hand Sides[M]. Dordrecht, Netherlands: Kluwer Academic Publishers, 1988.
[36] Zheng K, Shen T, Yao Y. New Approaching Condition for Sliding Mode Control Design with a Lipschitz Switching Surface[J]. Science in China Series F: Information Sciences, 2009, 52(11): 2032-2044.
[37] Zheng K, Shen T, Yao Y. Filippov Solutions on a Lipschitz Continuous Surface [C]//Proceedings of the 27th Chinese Control Conference. Kunming, China, 2008: 577-581.
[38] Yao Y, Zheng K, Ma K M, et al. Sliding Mode Control Design with a Class of Piecewise Smooth Switching Surfaces[J]. International Journal of Modelling, Identification and Control, 2009, 7(3): 294-304.
[39] Clarke F H, Ledyaev Y S, Stern R J, et al. Nonsmooth Analysis and Control Theory[M]. New York: Springer-Verlag, 1998.
[40] Marques M D P M. Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[M]. Cambridge, MA: Birkhauser, 1993.
[41] Brogliato B. Some Perspectives on the Analysis and Control of Complemen- tarity Systems[J]. IEEE Transactions on Automatic Control, 2003, 48(6): 918-935.
[42] Goebel R, Teel A R. Solutions for Hybrid Inclusions via Set and Graphical Convergence with Stability Theory Applications[J]. Automatica, 2006, 42(4): 573-587.
[43] Krasovskii N N, Subbotin A I. Game-Theoretical Control Problems[M]. New York: Springer-Verlag, 1988.
[44] Krasovskii N N. Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay[M]. Stanford, CA: Stanford University Press, 1963.
[45] Sentis R.Equations Differentielles a Second Membre Mesurable[J].Boll. Unione Matematica Italiana, 1978, 5(15-B): 724-742.
[46] Bacciotti A. Some Remarks on Generalized Solutions of Discontinuous Differential Equations[J]. International Journal of Pure and Applied Mathematics, 2004, 10(3): 257-266.
[47] Hermes H. Discontinuous Vector Fields and Feedback Control[M]. Differential Equations and Dynamical Systems. New York: Academic Press, 1967: 155-165.
[48] Hajek O. Discontinuous Differential Equations I[J]. Journal of Differential Equations, 1979, 32(2): 149-170.
[49] Hajek O. Discontinuous Differential Equations II[J]. Journal of Differential Equations, 1979, 32(2): 171-185.
[50] Ambrosio L. A Lower Closure Theorem for Autonomous Orientor Fields[J]. Proc.R.Soc.Edinb.A, Math, 1988, A110(3/4): 249-254.
[51] Yakubovich V A, Leonov G A, Gelig A K. Stabilituy of Stationary Sets in Control Systems with Discontinuous Nonlinearities[M]. Stability, Vibration and Control of Systes, Series A. Singapore: World Scientific Publishing, 2004, 14.
[52] Spraker J S, Biles D C. A Comparison of the Caratheodory and Filippov Solution Sets[J]. Journal of Mathematical Analysis and Applications, 1996, 198(2): 571-580.
[53] Hu S. Differential Equations with Discontinuous Right-hand Sides[J]. Journal of Mathematical Analysis and Applications, 1991, 154(2): 377–390.
[54] Ceragioli F. Discontinuous Ordinary Differential Equations and Stabilization [D]. University of Firenze, Italy, 1999.
[55] Rockafellar R T. Nonsmooth Optimization in Mathematical Programming: State of the Art[J]. University of Michigan Press, 1994: 248-258.
[56] Sussmann H J, Willems J C. 300 Years of Optimal Control: From the Brachi- stochrone to the Maximum Principle[J]. IEEE Control Systems Magazine, 1997, 17(3): 32-44.
[57] Schirotzek W. Nonsmooth Analysis[M]. Berlin Heidelberg: Springer-Verlag, 2007.
[58] Clarke F H. Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations[D]. University of Washington, USA, 1973.
[59] Clarke F H. Generalized Gradients and Applications[J]. Transactions of the American Mathematical Society, 1975, 205: 247-262.
[60] Clarke F H, Vinter R B. Applications of Optimal Multiprocesses[J]. SIAM Journal on Control and Optimization, 1989, 27(5): 1048-1071.
[61] Caines P E, Clarke F H, Liu X, et al. A Maximum Principle for Hybrid Optimal Control Problems with Pathwise State Constraints[C]//Proceedings of the 45th Conference on Decision and Control. San Diego, CA, USA, 2006: 4821-4825.
[62] Clarke F H. Maximum Principles without Differentiability[J]. Bulletin American Mathematical Society, 1975, 81(1): 219-222.
[63] Clarke F H. The Maximum Principle in Optimal Control, Then and Now[J]. Journal of Control & Cybernetics, 2005, 34(3): 709-722.
[64] Clarke F H, Rifford L, Stern R J. Feedback in State Constrained Optimal Control [J]. ESAIM: COCV, 2002, 7: 97–133.
[65] Clarke F H, Stern R J. State Constrained Feedback Stabilization[J]. SIAM Journal on Control and Optimization, 2003, 42(1): 422–441.
[66] Clarke F H, Stern R J. Hamilton-Jacobi Characterization of the State- constrained Value[J]. Nonlinear Analysis, 2005, 61(5): 725–734.
[67] Clarke F H, Stern R J. Lyapunov and Feedback Characterizations of State Constrained Controllability and Stabilization[J]. Systems and Control Letters, 2005, 54(8): 747-752.
[68] Clarke F H, Ledyaev Y S, Sontag E D, et al. Asymptotic Controllability Implies Feedback Stabilization[J]. IEEE Transactions on Automatic Control, 1997, 42(10): 1394-1407.
[69] Rockafellar R T. Convex Analysis(2nd)[M]. Princeton, New Jersey: Princeton University Press, 1972.
[70] Bouligand G. Introduction a La Geometrie Infinitesimale Directe[M]. Gauthier- Villard, Paris, 1932.
[71] Nagumo M. Uber Die Lage Der Integralkurven Gewohnlicher Differentialgle- ichungen[J]. Proceedings of the Physico-Mathematical Society of Japan, 1942, 24(3): 551-559.
[72] Ladde G S, Lakshmikantham V. On Flow-invariant Sets[J]. Pasific Journal of Mathematics, 1974, 51(1): 215-220.
[73] Terp C. Flow-invariant Sets[J]. Note di Matematica, 1989, 9(2): 221-227.
[74] Aubin J P, Cellina A. Differential Inclusions: Set-valued Maps and Viability Theory[M]. Spriger-Verlag, 1984.
[75] Bacciotti A, Rosier L. Liapunov Functions and Stability in Control Theory(2nd) [M]. NewYork, Springer-Verlag, 2005.
[76] Clarke F H. Nonsmooth Analysis in Control Theory: A Survey[J]. European Journal of Control, 2001, 7(3): 63-78.
[77] Clarke F H. Lyapunov Functions and Feedback in Nonlinear Control[C]// Lecture Notes in Control and Information Science. Springer-Verlag, 2004.
[78] Clarke F H, Ledyaev Y S, Stern R J. Asymptotic Stability and Smooth Lyapunov Functions[J]. Journal of Differential Equations, 1998, 149(11): 69-114.
[79] Rosier L. Smooth Lyapunov Functions for Discontinuous Stable Systems[J]. Set-Valued Analysis, 1999, 7(4): 375-405.
[80] Wu Q, Onyshko S, Sepehri N, et al. On Construction of Smooth Lyapunov Functions for Nonsmooth Systems[J]. International Journal of Control, 1998, 69(3): 442-457.
[81] Artstein Z. Stabilization with Relax Controls[J]. Nonlinear Analysis, Theory, Methods & Applications, 1983, 7(11): 1163-1173.
[82] Paden B, Sastry S. A Calculus for Computing Filippov's Differential Inclusion with Application to the Variable Structure Control of Robot Manipulators[J]. IEEE Transactionsm on Circuits Systems, 1987, 34(1): 73-82.
[83] Shevitz D, Paden B. Lyapunov Stability Theory of Nonsmooth Systems[J]. IEEE Transactions on Automatic Control, 1994, 39(9): 1910-1914.
[84] Bacciotti A, Ceragioli F. Stability and Stabilization of Discontnuous Systems and Nonsmooth Lyapunov Functions[J]. ESAIM: COCV, 1999, 4: 361-376.
[85] Ceragioli F. Some Remarks on Stabilization by Means of Discontinuous Feedbacks[J]. Systems & Control Letters, 2002, 45(4): 271-281.
[86] Guo Z Y, Huang L H. Generalized Lyapunov Method for Discontinuous Systems[J]. Nonlinear Analysis, 2009, 71(7-8): 3083-3092.
[87] Wu Q, Sepehri N. On Lyapunov's Stability Analysis of Non-smooth Systems with Applications to Control Engineering[J]. International Journal of Non- Linear Mechanics, 2001, 36(7): 1153-1161.
[88] Alvarez J, Orlov I, Acho L. An Invariance Principle for Discontinuous Dynamic Systems with Application to a Coulomb Friction Oscillator[J]. Journal of Dynamic Systems, Measurement, and Control, 2000, 122(4): 687-690.
[89] Cheng G, Mu X. LaSalle's Invariant Principle via Vector Lyapunov Functions of a Class of Discontinuous Systems[C]//Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai, China, 2009: 6317-6320.
[90] Ledyaev Y S, Sontag E D. A Lyapunov Characterization of Robust Stabilization [J]. Nonlinear Analysis, 1999, 37(7): 813-840.
[91] Clarke F H, Ledyaev Y S, Rifford L, et al. Feedback Stabilization and Lyapunov Functions[J]. SIAM Journal on Control and Optimization, 2000, 39(1): 25-48.
[92] Rifford L. Existence of Lipschitz and Semiconcave Control-Lyapunov Functions[J]. SIAM Journal on Control and Optimization, 2000, 39(4): 1043-1064.
[93] Rifford L. On the Existence of Nonsmooth Control-Lyapunov Functions in the Sense of Generalized Gradients[J]. ESAIM: COCV, 2001, 6: 539-611.
[94] Rifford L. Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks [J]. SIAM Journal on Control and Optimization, 2002, 41(3): 659-681.
[95] Rifford L. Singularities of Viscosity Solutions and the Stabilization Problem in the Plane[J]. Indiana University Mathematics Journal, 2003, 52(5): 1373-1396.
[96] Ancona F, Bressan A. Patchy Vector Fields and Asymptotic Stabilization[J]. ESAIM: COCV, 1999, 4: 445-472.
[97] Ancona F, Bressan A. Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization[J]. SIAM Journal on Control and Optimization, 2002, 41(5): 1455-1476.
[98] Ancona F, Bressan A. Stability Rates for Patchy Vector Fields[J]. ESAIM: COCV, 2004, 10: 168-200.
[99] Pogromsky A Y, Heemels W P M H, Nijmeijer H. On Solution Concepts and Well-posedness of Linear Relay Systems[J]. Automatica, 2003, 39(12): 2139-2147.
[100] Kim S J, Ha I J. On the Existence of Carathéodory Solutions in Nonlinear Systems with Discontinuous Switching Control Laws[J]. IEEE Transactions on Automatic Control, 2004, 49(7): 1167-1171.
[101] Bacciotti A, Ceragioli F. Nonpathological Lyapunov Functions and Discon- tinuous Carathéodory Systems[J]. Automatica, 2006, 42(3): 453-458.
[102] Bacciotti A. Genaralized Solustions of Differential Inclusions and Stability[J]. Italian Journal of Pure and Applied Mathematics, 2005, 17: 183-192.
[103] Shima H, Teel A R. Asymptotic Controllability and Observability Imply Semi- global Practical Asymptotic Stabilizability by Sample-data Output Feedback[J]. Automatica, 2003, 39(3): 441-454.
[104] Prieur C. Asymptotic Controllability and Robust Asymptotic Stabilizability[J]. SIAM Journal on Control and Optimization, 2005, 43(5): 1888-1912.
[105] Clarke F H, Vinter R B. Stability Analysis of Sliding-mode Feedback Control [J]. Journal of Control & Cybernetics, 2009, 38(4A): 1169-1192.
[106] Clarke F H. Control of Systems to Sets and Their Interior[J]. Journal of Mathematical Analysis and Applications, 1996, 88(1): 3-23.
[107] Nakakuki T, Tamura K. Control Design of Nonlinear Mechanical Systems with Friction[C]//Proceedings of 43rd IEEE Conference on Decision and Control. Atlantis, USA, 2004: 2966-2971.
[108] Akasaka D, Liu K Z. Nonlinear Output Feedback Control for Linear Systems with Input Saturation[C]//Proceedings of the IFAC World Congress. Seoul, Korea, 2008: 15154-15159.
[109] Wouw N V D, Leine R I. Attructivity of Equilibrium Sets of Systems with DryFriction[J]. Nonlinear Dynamics, 2004, 35(1):19-39.
[110] Adly S. Atractivity Theory for Second Order Nonsmooth Dynamical Systems with Application to Dry Friction[J]. Journal of Mathematical Analysis and Applications, 2006, 322(2): 1055-1070.
[111] Bacciotti A, Ceragioli F. L2-gain Stabilizability with Respect to Filippov Solutions[C]//Proceedings of the 41st IEEE Conference on Decision and Control. Las Vegas, USA, 2002: 713-716.
[112] Bacciotti A, Ceragioli F. Finite L2-gain with Nondifferentiable Storage Functions[J]. Nonlinear Differential Equations and Applications, 2006, 12(4): 401-418.
[113] Moulay E, Perruquetti W. Finite Time Stability of Differential Inclusions[J]. IMA Journal of Mathematical Control and Information, 2005, 22(4): 465-475.
[114] Cortés J. Finite-time Convergent Gradient Flows with Applications to Network Consensus[J]. Automatica, 2006, 42(11): 1993-2000.
[115] Ryan E P. On Brockett’s Condition for Smooth Stabilizability and Its Necessity in a Context of Nonsmooth Feedback[J]. SIAM Journal on Control and Optimization, 1994, 32(6): 1597-1604.
[116] Coron J M, Rosier L. A Relation between Continuous Time-varying and Discontinuous Feedback Stabilization[J]. Journal of Mathematical Systems Estimation, and Control, 1994, 4(1): 67-84.
[117] Frederick D K, Franklin G F. Design of Piecewise-linear Switching Functions for Relay Control Systems[J]. IEEE Transactions Automatic Control, 1967, 12(4): 380-387.
[118] Bacciotti A, Mazzi L. An Invariance Principle for Nonlinear Switched Systems [J]. Systems & Control Letters, 2005, 54(11): 1109-1119.
[119] Bacciotti A. Stabilization by Means of State Space Depending Switching Rules[J]. Systems & Control Letters, 2004, 53(3-4): 195-201.
[120] Bacciotti A, Ceragioli F. Closed Loop Stabilization of Planar Bilinear Switched Systems[J]. International Journal of Control, 2006, 79(1): 14-23.
[121] Ceragioli F. Finite Valued Feedback Laws and Piecewise Classical Solutions[J]. Nonlinear Analysis, 2006, 65(5): 984-998.
[122] Ceragioli F, Persis C D. Discontinuous Stabilization of Nonlinear Systems: Quantized and Switching Controls[J]. Systems & Control Letter, 2007, 56(7-8): 461-473.
[123] Orlov Y. Extended Invariance Principle for Nonautonomous Switched Systems [J]. IEEE Transactions on Automatic Control, 2003, 48(8): 1448-1452.
[124] Skafidas E, Evans R J, Savkin A V, et al. Stability Results for SwitchedController Systems[J]. Automatica, 1999, 35(4): 553-564.
[125]高为炳.变结构控制理论基础[M].中国科学技术出版社, 1990.
[126]胡跃明.变结构控制理论与应用[M].科学出版社, 2003.
[127]刘金琨,孙富春.滑模变结构控制理论及其算法研究与进展[J].控制理论与应用, 2007, 24(3): 407-418.
[128]冯勇,安澄全,李涛.采用双滑模平面减小一类非线性系统稳态误差[J].控制与决策, 2000, 15(3): 361-364.
[129] Yu X, Wu Y. Adaptive Terminal Sliding Mode Control of Uncertain Nonlinear Systems Backstepping Approach[J]. Control Theory and Applications, 1998, 15(6): 900-907.
[130] Wang W J, Lin H R. Fuzzy Control Design for the Trajectory Tracking on Uncertain Nonlinear Systems[J]. IEEE Transactions on Fuzzy Systems, 1999, 7(1): 53-62.
[131] Dotoli M. Fuzzy Sliding Mode Control with Piecewise Linear Switching Manifold[J]. Asian Journal of Control, 2003, 5(4): 528-542.
[132] Nakakuki T. Controller Design for a Class of Nonsmooth Systems and its Application to Mechanical Systems[D]. Sophia University, Japan, 2006.
[133] Nakakuki T, Shen T, Tamura K. Adaptive Control Design for a Class of Nonsmooth Nonlinear Systems with Matched and Linearly Parameterized Uncertainty[J]. International Journal of Robust and Nonlinear Control, 2008, 19(2): 243-255.
[134] Zhang J, Shen T. Functianal Differential Inclusion Based Approach to Control of Discontinuous Nonlinear System with Time Delay[C]//Proceedings of the 47th IEEE Conference on Decision and Control. Cancun, Mexico, 2008: 5300-5305.
[135] Zhang J, Shen T. Feedback Stabilization of a Class of Time-delay Systems with Discontinuity: A Functional Differential Inclusion-based Approach[C]//Inter- national Conference on Control, Automation and Systems. Seoul, Korea, 2008: 1240-1244.
[136] Tanner H G, Kyriakopoulos K J. Backstepping for Nonsmooth Systems[J]. Automatica, 2003, 39(7): 1259-1265.
[137] Zheng K, Shen T, He F, et al. Nonsmooth Backstepping Design for a Class of Parametric Strict-feedback Nonlinear Systems Based on Lipschitz Feedback[C] //American Control Conference. Baltimore, USA, 2010: 6351-6356.
[138] Zheng K, Zhang G, Du J, et al. Nonsmooth Backstepping Design for Motion Control of a Class of Euler-Lagrange Systems[C]//Proceedings of the 29th Chinese Control Conference. Beijing, China, 2010: 570-574.
[139] Zhou J, Wen C. Adaptive Backstepping Control of Uncertain Systems: Non- smooth Nonlinearities, Interactions or Time-Variations[M]. Springer, 2008.
[140] Zhou J, Wen C, Zhang Y. Adaptive Backstepping Control of a Class of Uncertain Nonlinear Systems with Unknown Backlash-like Hysteresis[J]. IEEE Transactions on Automatic Control, 2004, 49(10): 1751-1759.
[141] Lin W, Qian C. Adaptive Control of Nonlinearly Parameterized Systems: A Nonsmooth Feedback Framework[J]. IEEE Transactions on Automatic Control, 2002, 47(5): 757-774.
[142] Zhang J, Shen T, Jiao X. L2-gain Analysis and Feedback Design for Discon- tinuous Time-delay Systems Based on Functional Differential Inclusion[C]// Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai, China, 2009: 5114-5119.
[143] Soravia P. Feedback Stabilization and H∞Control of Nonlinear Systems Affected by Disturbance: the Differential Games Approach. Dynamics, Bifurcation, and Control[M]. London: Springer-Verlag, 2002.
[144] Kellett C M, Shim H, Teel A R. Robustness of Discontinuous Feedback via Sample and Hold Control[C]//Proceedings of American Control Conference. Anchorage, USA, 2002: 3512-3517.
[145] Apkarian P, Noll D. Nonsmooth H∞Synthesis[J]. IEEE Transactions on Automatic Control, 2006, 51(1): 71-86.
[146] Bompart V, Apkarian P, Noll D. Control Design in the Time and Frequency Domain Using Nonsmooth Techniques[J]. Systems & Control Letters, 2008, 57(3): 271-282.
[147] Apkarian P, Noll D. Nonsmooth Optimization for Multiband Frequency Domain Control Design[J]. Automatica, 2007, 43(4): 724-731.
[148] Fontes F A C C. A General Framework to Design Stabilizing Nonlinear Model Predictive Controller[J]. Systems & Control Letters, 2001, 42(2): 127-143.
[149] Fontes F A C C, Magni L. Min-max Model Predictive Control of Nonlinear Systems Using Discontinuous Feedbacks[J]. IEEE Transactions on Automatic Control, 2003, 48(10): 1750-1755.
[150] Messina M J, Tuna S E, Teel A R . Discrete-time Certainty Equivalence Output Feedback: Allowing Discontinuous Control Laws Including those from Model Predictive Control[J]. Automatica, 2005, 41(4): 617-628.
[151] Zhou X Y. Nonsmooth Analysis on Stochastic Controls: A Survey[C]// Proceedings of the 31th IEEE Conference on Decision and Control. Melbourne, USA, 1992: 2406-2411.
[152] Han J Q. From PID to Active Disturbance Rejection Control[J]. IEEE Trans-actions on Industrial Electronics, 2009, 56(3): 900-906.
[153] Burenkov V I. Sobolev Spaces on Domains[M]. B.G.Teubner Verlag, 1998.
[154] Dacorogna B. Introduction to the Calculus of Variations[M]. London: Imperial College Press, 2004.
[155] Simon L, Hungerbuhler N. Theorems on Regularity and Singularity of Energy Minimizing Maps[M]. Birkhauser, 1996.
[156] Neittaanmaki P, Rudnicki M, Savini A. Inverse Problems and Optimal Design in Electricity and Magnetism[M]. Oxford University Press, 1996.
[157] Delfour M C, Zolesio J P. Shapes and Geometries: Analysis, Differential Calculus and Optimization[M]. SIAM, 2001.
[158] Clarke F H, Ledyaev Y S. Qualitivie Properties of Trajectories of Control Systems: A Survey[J]. Journal of Dynamical and Control Systems, 1995, 1(1): 1-48.
[159] Aubin J P. A Survey of Viability Theory[J]. SIAM Journal on Control and Optimization, 1990, 28(4): 749-788.
[160] Khalil H K. Nonlinear Systems[M]. Prentice-Hall, 2002.
[161] Han J Q, Wang W, Huang Y. A Synthesis Method for Two-order Uncertain Systems: Auto-stable Region Approach[J]. Systems Science and Mathematical Sciences, 1997, 10(2), 152-159.
[162] Huang Y, Han J Q. The Self-stable Region Approach for Second Order Nonlinear Uncertain Systems[C]//Proceedings of 1999 IFAC World Congress, 1999, E: 135-140.
[163] Huang Y, Han J Q. A New Synthesis Method for Uncertain Systems: the Self-stable Region Approach[J]. International Journal of Systems Science, 1999, 30(1): 33-38.
[164]郑凯.基于Filippov微分包含解的非平滑控制系统设计方法研究[D].哈尔滨:哈尔滨工业大学控制科学与工程学科博士学位论文, 2009.
[165] Isidori A. Nonlinear Control Systems(3rd)[M]. New York: Springer, 1995.
[166] Curtis H D. Orbital Mechanics: for Engineering Students[M]. Elsevier, 2005.
[167] Gurfil P. Zero-Miss-Distance Guidance Law Based on Line-of-Sight Rate Measurement Only[J]. AIAA 2001-4277: 1-11.
[168] Hablani H B, Pearson D W. Determination of Guidance Parameters of Exo- atmospheric Interceptors via Miss Distance Error Analysis[J]. AIAA 2001-4279: 1-17.
[169] Hablani H B, Pearson D W. Miss Distance Error Analysis of Exoatmospheric Interceptors[J]. Journal of Guidance, Control, and Dynamics, 2004, 27(2): 283-289.
[170] Hablani H B. Endgame Guidance and Relative Navigation of Strategic Inter- ceptors with Delays[J]. Journal of Guidance, Control, and Dynamics. 2006, 29(1): 82-94.
[171] Hablani H B. Pulsed Guidance of Exoatmospheric Interceptors with Image Processing Delays in Angle Measurements[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Denver, USA, 2000: 1-15.
[172]程国采.战术导弹导引方法[M].北京:国防工业出版社, 1996.
[173]刘世勇,吴瑞林,周伯昭.大气层外拦截器末段轨控发动机开关规律[J].宇航学报, 2005, 26(sup): 106-109.
[174]张雅声,程国采,陈克俊.高空动能拦截器末制导导引方法设计与实现[J].现代防御技术, 2001, 29(2): 31-34.
[175]杨宝庆.基于预测控制的大气层外KKV制导控制规律研究[D].哈尔滨:哈尔滨工业大学控制科学与工程学科博士学位论文, 2009.
[176] Lawrence R V. Interceptor Line-of-Sight Rate Steering: Necessary Conditions for a Direct Hit[J]. Journal of Guidance, Control, and Dynamics, 1998, 21(3): 471-476.
[2] Cortés J. Discontinuous Dynamical Systems[J]. IEEE Control Systems Magazine, 2008, 28(3): 36-73.
[3] Shen T. Analysis and Control of Discontinuous Dynamical Systems[C]// Proceedings of the 26th Chinese Control Conference. Zhang Jiajie, China, 2007: 37-45.
[4] Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction[J]. Automatica, 1994, 30(7): 1083-1138.
[5] Hori Y, Toyoda Y, Tsuruoka Y. Traction Control of Electric Vehicle: Basic Experimental Results Using the Test EV 'UOT Electric March'[J]. IEEE Transactions on Industry Applications, 1998, 34(5): 1131–1138.
[6] Ray L R. Nonlinear Tire Force Estimation and Road Friction Identification: Simulation and Experiments[J]. Automatica, 1997, 33(10): 1819-1833.
[7] Shen T. Robust Nonlinear Traction Control with Direct Torque Controlled Induction Motor[C]//Proceedings of Asian Electric Vehicle Conference. Seoul, Korea, 2004.
[8] Guzzella L, Onder C H. Introduction to Modeling and Control of Internal Combustion Engine Systems[M]. Berlin: Springer, 2004.
[9] Balluchi A, Benvenuti L, Benedetto M D D, et al. Automotive Engine Control and Hybrid Systems: Challenges and Opportunities[J]. Proceedings of the IEEE, 2000, 88(7): 888-912.
[10] Brockett R W, Liberzon D. Quantized Feedback Stabilization of Linear Systems [J]. IEEE Transaction on Automatic Control, 2000, 45(7): 1279-1289.
[11] Claasen T A C M, Mecklenbrauker W F G, Peek J B H. Effects of Quantization and Overflow in Recursive Digital Filters[J]. IEEE Transactions on Acoustics, Speech and Signal Processing, 1976, 24(6): 517-530.
[12] Fagnani F, Zampieri S. Stability Analysis and Synthesis for Scalar Linear Systems with a Quantized Feedback[J]. IEEE Transactions on Automatic Control, 2003, 48(9): 1569-1584.
[13] Goncalves J M, Megretski A, Dahleh M A. Global Stability of Relay Feedback Systems[J]. IEEE Transactions on Automatic Control, 2001, 46(4): 550-562.
[14] Bowes S R. Advanced Regular-sampled PWM Control Techniques for Drivesand Static Power Converters[J]. IEEE Transactions on Industrial Electronics, 1995, 42(4): 367-373.
[15] Brogliato B. Nonsmooth Mechanics: Models, Dynamics and Control(2nd)[M]. New York: Springer-Verlag, 1999.
[16] Agrachev A A, Sachkov Y. Control Theory from the Geometric Viewpoint[M]. New York: Springer-Verlag, 2004.
[17] Edwards C, Spurgeon S K. Sliding Mode Control: Theory and Applications[M]. New York: Taylor & Francis, 1998.
[18] Brockett R W. Asymptotic Stability and Feedback Stabilization[C]//Differential Geometric Control Theory. Boston, MA: Birkhauser, 1983: 181-191.
[19] Monforte J C. Geometric, Control and Numerical Aspects of Nonholonomic Systems[M]. Springer, 2002.
[20] Martin P, Rouchon P. Feedback Linearization and Driftless Systems[J]. Mathematics of Control, Signals and Systems, 1994, 7(3): 235-254.
[21] Sontag E D, Sussmann H. Remarks on Continuous Feedback Control[C]// Proceedings of IEEE Conferecse on Decision and Control. 1980: 916-921.
[22] Bloch A, Drakunov S. Stabilization and Tracking in the Nonholnomic Inter- grator via Sliding Modes[J]. Systems & Control Letters, 1996, 29(2): 91-99.
[23] Liberzon D. Switchings in Systems and Control[M]. Boston: Birkhauser, 2003.
[24] Liberzon D, Morse A S. Basic Problems in Stability and Design of Switched Systems[J]. IEEE Control Systems Magazine, 1999, 19(5): 59-70.
[25] Casagrande D, Astolfi A, Parisini T. A Stabilizing Time-switching Control Strategy for the Rolling Sapere[C]//Proceedings of CDC-ECC Conference. 2005: 3297-3302.
[26] Utkin V I. VSS with Sliding Mode[J]. IEEE Transaction on Automatic Control, 1977, 22(2): 212-222.
[27] Unkin V I.Sliding Modes in Control and Optimization[M].New York: Springer- Verlag, 1992.
[28] Xu J X. Variable Structure Systems: Towards the 21st Century[M]. Springer, 2002.
[29] Akasaka D, Liu K Z. Stabilization of a Class of On-off Feedback Systems by Nonlinear State Feedback[C]//Proceedings of European Control Conference. Kos, Greece, 2007: 667-672.
[30] Akasaka D, Liu K Z. Nonlinear Feedback Control of a Class of On-off Control Systems: An Approach Based on Analytical Solution of PDE's[C]//Proceedings of the IEEE Conference on Decision and Control. New Orleans, USA, 2007: 86-91.
[31] Polycarpou M M, Ioannou P A. On the Existence and Uniqueness of Solutions in Adaptive Control Systems[J]. IEEE Transactions on Automatic Control, 1993, 38(3): 474-479.
[32] Clarke F H. Optimization and Nonsmooth Analysis[M]. New York: John Wiley & Sons, 1983.
[33] Bhaya A, Kaszkurewicz E. Control Perspectives on Numerical Algorithms and Matrix Problems[M]. Philadelphia, PA: SIAM, 2006.
[34] Filppov A F. Differential Equations with Discontinuous Righthand Side[J]. American Mathematical Sociaty Translations, 1964, 42(2): 199-231.
[35] Filippov A F. Differential Equations with Discontinuous Right-hand Sides[M]. Dordrecht, Netherlands: Kluwer Academic Publishers, 1988.
[36] Zheng K, Shen T, Yao Y. New Approaching Condition for Sliding Mode Control Design with a Lipschitz Switching Surface[J]. Science in China Series F: Information Sciences, 2009, 52(11): 2032-2044.
[37] Zheng K, Shen T, Yao Y. Filippov Solutions on a Lipschitz Continuous Surface [C]//Proceedings of the 27th Chinese Control Conference. Kunming, China, 2008: 577-581.
[38] Yao Y, Zheng K, Ma K M, et al. Sliding Mode Control Design with a Class of Piecewise Smooth Switching Surfaces[J]. International Journal of Modelling, Identification and Control, 2009, 7(3): 294-304.
[39] Clarke F H, Ledyaev Y S, Stern R J, et al. Nonsmooth Analysis and Control Theory[M]. New York: Springer-Verlag, 1998.
[40] Marques M D P M. Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[M]. Cambridge, MA: Birkhauser, 1993.
[41] Brogliato B. Some Perspectives on the Analysis and Control of Complemen- tarity Systems[J]. IEEE Transactions on Automatic Control, 2003, 48(6): 918-935.
[42] Goebel R, Teel A R. Solutions for Hybrid Inclusions via Set and Graphical Convergence with Stability Theory Applications[J]. Automatica, 2006, 42(4): 573-587.
[43] Krasovskii N N, Subbotin A I. Game-Theoretical Control Problems[M]. New York: Springer-Verlag, 1988.
[44] Krasovskii N N. Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay[M]. Stanford, CA: Stanford University Press, 1963.
[45] Sentis R.Equations Differentielles a Second Membre Mesurable[J].Boll. Unione Matematica Italiana, 1978, 5(15-B): 724-742.
[46] Bacciotti A. Some Remarks on Generalized Solutions of Discontinuous Differential Equations[J]. International Journal of Pure and Applied Mathematics, 2004, 10(3): 257-266.
[47] Hermes H. Discontinuous Vector Fields and Feedback Control[M]. Differential Equations and Dynamical Systems. New York: Academic Press, 1967: 155-165.
[48] Hajek O. Discontinuous Differential Equations I[J]. Journal of Differential Equations, 1979, 32(2): 149-170.
[49] Hajek O. Discontinuous Differential Equations II[J]. Journal of Differential Equations, 1979, 32(2): 171-185.
[50] Ambrosio L. A Lower Closure Theorem for Autonomous Orientor Fields[J]. Proc.R.Soc.Edinb.A, Math, 1988, A110(3/4): 249-254.
[51] Yakubovich V A, Leonov G A, Gelig A K. Stabilituy of Stationary Sets in Control Systems with Discontinuous Nonlinearities[M]. Stability, Vibration and Control of Systes, Series A. Singapore: World Scientific Publishing, 2004, 14.
[52] Spraker J S, Biles D C. A Comparison of the Caratheodory and Filippov Solution Sets[J]. Journal of Mathematical Analysis and Applications, 1996, 198(2): 571-580.
[53] Hu S. Differential Equations with Discontinuous Right-hand Sides[J]. Journal of Mathematical Analysis and Applications, 1991, 154(2): 377–390.
[54] Ceragioli F. Discontinuous Ordinary Differential Equations and Stabilization [D]. University of Firenze, Italy, 1999.
[55] Rockafellar R T. Nonsmooth Optimization in Mathematical Programming: State of the Art[J]. University of Michigan Press, 1994: 248-258.
[56] Sussmann H J, Willems J C. 300 Years of Optimal Control: From the Brachi- stochrone to the Maximum Principle[J]. IEEE Control Systems Magazine, 1997, 17(3): 32-44.
[57] Schirotzek W. Nonsmooth Analysis[M]. Berlin Heidelberg: Springer-Verlag, 2007.
[58] Clarke F H. Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations[D]. University of Washington, USA, 1973.
[59] Clarke F H. Generalized Gradients and Applications[J]. Transactions of the American Mathematical Society, 1975, 205: 247-262.
[60] Clarke F H, Vinter R B. Applications of Optimal Multiprocesses[J]. SIAM Journal on Control and Optimization, 1989, 27(5): 1048-1071.
[61] Caines P E, Clarke F H, Liu X, et al. A Maximum Principle for Hybrid Optimal Control Problems with Pathwise State Constraints[C]//Proceedings of the 45th Conference on Decision and Control. San Diego, CA, USA, 2006: 4821-4825.
[62] Clarke F H. Maximum Principles without Differentiability[J]. Bulletin American Mathematical Society, 1975, 81(1): 219-222.
[63] Clarke F H. The Maximum Principle in Optimal Control, Then and Now[J]. Journal of Control & Cybernetics, 2005, 34(3): 709-722.
[64] Clarke F H, Rifford L, Stern R J. Feedback in State Constrained Optimal Control [J]. ESAIM: COCV, 2002, 7: 97–133.
[65] Clarke F H, Stern R J. State Constrained Feedback Stabilization[J]. SIAM Journal on Control and Optimization, 2003, 42(1): 422–441.
[66] Clarke F H, Stern R J. Hamilton-Jacobi Characterization of the State- constrained Value[J]. Nonlinear Analysis, 2005, 61(5): 725–734.
[67] Clarke F H, Stern R J. Lyapunov and Feedback Characterizations of State Constrained Controllability and Stabilization[J]. Systems and Control Letters, 2005, 54(8): 747-752.
[68] Clarke F H, Ledyaev Y S, Sontag E D, et al. Asymptotic Controllability Implies Feedback Stabilization[J]. IEEE Transactions on Automatic Control, 1997, 42(10): 1394-1407.
[69] Rockafellar R T. Convex Analysis(2nd)[M]. Princeton, New Jersey: Princeton University Press, 1972.
[70] Bouligand G. Introduction a La Geometrie Infinitesimale Directe[M]. Gauthier- Villard, Paris, 1932.
[71] Nagumo M. Uber Die Lage Der Integralkurven Gewohnlicher Differentialgle- ichungen[J]. Proceedings of the Physico-Mathematical Society of Japan, 1942, 24(3): 551-559.
[72] Ladde G S, Lakshmikantham V. On Flow-invariant Sets[J]. Pasific Journal of Mathematics, 1974, 51(1): 215-220.
[73] Terp C. Flow-invariant Sets[J]. Note di Matematica, 1989, 9(2): 221-227.
[74] Aubin J P, Cellina A. Differential Inclusions: Set-valued Maps and Viability Theory[M]. Spriger-Verlag, 1984.
[75] Bacciotti A, Rosier L. Liapunov Functions and Stability in Control Theory(2nd) [M]. NewYork, Springer-Verlag, 2005.
[76] Clarke F H. Nonsmooth Analysis in Control Theory: A Survey[J]. European Journal of Control, 2001, 7(3): 63-78.
[77] Clarke F H. Lyapunov Functions and Feedback in Nonlinear Control[C]// Lecture Notes in Control and Information Science. Springer-Verlag, 2004.
[78] Clarke F H, Ledyaev Y S, Stern R J. Asymptotic Stability and Smooth Lyapunov Functions[J]. Journal of Differential Equations, 1998, 149(11): 69-114.
[79] Rosier L. Smooth Lyapunov Functions for Discontinuous Stable Systems[J]. Set-Valued Analysis, 1999, 7(4): 375-405.
[80] Wu Q, Onyshko S, Sepehri N, et al. On Construction of Smooth Lyapunov Functions for Nonsmooth Systems[J]. International Journal of Control, 1998, 69(3): 442-457.
[81] Artstein Z. Stabilization with Relax Controls[J]. Nonlinear Analysis, Theory, Methods & Applications, 1983, 7(11): 1163-1173.
[82] Paden B, Sastry S. A Calculus for Computing Filippov's Differential Inclusion with Application to the Variable Structure Control of Robot Manipulators[J]. IEEE Transactionsm on Circuits Systems, 1987, 34(1): 73-82.
[83] Shevitz D, Paden B. Lyapunov Stability Theory of Nonsmooth Systems[J]. IEEE Transactions on Automatic Control, 1994, 39(9): 1910-1914.
[84] Bacciotti A, Ceragioli F. Stability and Stabilization of Discontnuous Systems and Nonsmooth Lyapunov Functions[J]. ESAIM: COCV, 1999, 4: 361-376.
[85] Ceragioli F. Some Remarks on Stabilization by Means of Discontinuous Feedbacks[J]. Systems & Control Letters, 2002, 45(4): 271-281.
[86] Guo Z Y, Huang L H. Generalized Lyapunov Method for Discontinuous Systems[J]. Nonlinear Analysis, 2009, 71(7-8): 3083-3092.
[87] Wu Q, Sepehri N. On Lyapunov's Stability Analysis of Non-smooth Systems with Applications to Control Engineering[J]. International Journal of Non- Linear Mechanics, 2001, 36(7): 1153-1161.
[88] Alvarez J, Orlov I, Acho L. An Invariance Principle for Discontinuous Dynamic Systems with Application to a Coulomb Friction Oscillator[J]. Journal of Dynamic Systems, Measurement, and Control, 2000, 122(4): 687-690.
[89] Cheng G, Mu X. LaSalle's Invariant Principle via Vector Lyapunov Functions of a Class of Discontinuous Systems[C]//Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai, China, 2009: 6317-6320.
[90] Ledyaev Y S, Sontag E D. A Lyapunov Characterization of Robust Stabilization [J]. Nonlinear Analysis, 1999, 37(7): 813-840.
[91] Clarke F H, Ledyaev Y S, Rifford L, et al. Feedback Stabilization and Lyapunov Functions[J]. SIAM Journal on Control and Optimization, 2000, 39(1): 25-48.
[92] Rifford L. Existence of Lipschitz and Semiconcave Control-Lyapunov Functions[J]. SIAM Journal on Control and Optimization, 2000, 39(4): 1043-1064.
[93] Rifford L. On the Existence of Nonsmooth Control-Lyapunov Functions in the Sense of Generalized Gradients[J]. ESAIM: COCV, 2001, 6: 539-611.
[94] Rifford L. Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks [J]. SIAM Journal on Control and Optimization, 2002, 41(3): 659-681.
[95] Rifford L. Singularities of Viscosity Solutions and the Stabilization Problem in the Plane[J]. Indiana University Mathematics Journal, 2003, 52(5): 1373-1396.
[96] Ancona F, Bressan A. Patchy Vector Fields and Asymptotic Stabilization[J]. ESAIM: COCV, 1999, 4: 445-472.
[97] Ancona F, Bressan A. Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization[J]. SIAM Journal on Control and Optimization, 2002, 41(5): 1455-1476.
[98] Ancona F, Bressan A. Stability Rates for Patchy Vector Fields[J]. ESAIM: COCV, 2004, 10: 168-200.
[99] Pogromsky A Y, Heemels W P M H, Nijmeijer H. On Solution Concepts and Well-posedness of Linear Relay Systems[J]. Automatica, 2003, 39(12): 2139-2147.
[100] Kim S J, Ha I J. On the Existence of Carathéodory Solutions in Nonlinear Systems with Discontinuous Switching Control Laws[J]. IEEE Transactions on Automatic Control, 2004, 49(7): 1167-1171.
[101] Bacciotti A, Ceragioli F. Nonpathological Lyapunov Functions and Discon- tinuous Carathéodory Systems[J]. Automatica, 2006, 42(3): 453-458.
[102] Bacciotti A. Genaralized Solustions of Differential Inclusions and Stability[J]. Italian Journal of Pure and Applied Mathematics, 2005, 17: 183-192.
[103] Shima H, Teel A R. Asymptotic Controllability and Observability Imply Semi- global Practical Asymptotic Stabilizability by Sample-data Output Feedback[J]. Automatica, 2003, 39(3): 441-454.
[104] Prieur C. Asymptotic Controllability and Robust Asymptotic Stabilizability[J]. SIAM Journal on Control and Optimization, 2005, 43(5): 1888-1912.
[105] Clarke F H, Vinter R B. Stability Analysis of Sliding-mode Feedback Control [J]. Journal of Control & Cybernetics, 2009, 38(4A): 1169-1192.
[106] Clarke F H. Control of Systems to Sets and Their Interior[J]. Journal of Mathematical Analysis and Applications, 1996, 88(1): 3-23.
[107] Nakakuki T, Tamura K. Control Design of Nonlinear Mechanical Systems with Friction[C]//Proceedings of 43rd IEEE Conference on Decision and Control. Atlantis, USA, 2004: 2966-2971.
[108] Akasaka D, Liu K Z. Nonlinear Output Feedback Control for Linear Systems with Input Saturation[C]//Proceedings of the IFAC World Congress. Seoul, Korea, 2008: 15154-15159.
[109] Wouw N V D, Leine R I. Attructivity of Equilibrium Sets of Systems with DryFriction[J]. Nonlinear Dynamics, 2004, 35(1):19-39.
[110] Adly S. Atractivity Theory for Second Order Nonsmooth Dynamical Systems with Application to Dry Friction[J]. Journal of Mathematical Analysis and Applications, 2006, 322(2): 1055-1070.
[111] Bacciotti A, Ceragioli F. L2-gain Stabilizability with Respect to Filippov Solutions[C]//Proceedings of the 41st IEEE Conference on Decision and Control. Las Vegas, USA, 2002: 713-716.
[112] Bacciotti A, Ceragioli F. Finite L2-gain with Nondifferentiable Storage Functions[J]. Nonlinear Differential Equations and Applications, 2006, 12(4): 401-418.
[113] Moulay E, Perruquetti W. Finite Time Stability of Differential Inclusions[J]. IMA Journal of Mathematical Control and Information, 2005, 22(4): 465-475.
[114] Cortés J. Finite-time Convergent Gradient Flows with Applications to Network Consensus[J]. Automatica, 2006, 42(11): 1993-2000.
[115] Ryan E P. On Brockett’s Condition for Smooth Stabilizability and Its Necessity in a Context of Nonsmooth Feedback[J]. SIAM Journal on Control and Optimization, 1994, 32(6): 1597-1604.
[116] Coron J M, Rosier L. A Relation between Continuous Time-varying and Discontinuous Feedback Stabilization[J]. Journal of Mathematical Systems Estimation, and Control, 1994, 4(1): 67-84.
[117] Frederick D K, Franklin G F. Design of Piecewise-linear Switching Functions for Relay Control Systems[J]. IEEE Transactions Automatic Control, 1967, 12(4): 380-387.
[118] Bacciotti A, Mazzi L. An Invariance Principle for Nonlinear Switched Systems [J]. Systems & Control Letters, 2005, 54(11): 1109-1119.
[119] Bacciotti A. Stabilization by Means of State Space Depending Switching Rules[J]. Systems & Control Letters, 2004, 53(3-4): 195-201.
[120] Bacciotti A, Ceragioli F. Closed Loop Stabilization of Planar Bilinear Switched Systems[J]. International Journal of Control, 2006, 79(1): 14-23.
[121] Ceragioli F. Finite Valued Feedback Laws and Piecewise Classical Solutions[J]. Nonlinear Analysis, 2006, 65(5): 984-998.
[122] Ceragioli F, Persis C D. Discontinuous Stabilization of Nonlinear Systems: Quantized and Switching Controls[J]. Systems & Control Letter, 2007, 56(7-8): 461-473.
[123] Orlov Y. Extended Invariance Principle for Nonautonomous Switched Systems [J]. IEEE Transactions on Automatic Control, 2003, 48(8): 1448-1452.
[124] Skafidas E, Evans R J, Savkin A V, et al. Stability Results for SwitchedController Systems[J]. Automatica, 1999, 35(4): 553-564.
[125]高为炳.变结构控制理论基础[M].中国科学技术出版社, 1990.
[126]胡跃明.变结构控制理论与应用[M].科学出版社, 2003.
[127]刘金琨,孙富春.滑模变结构控制理论及其算法研究与进展[J].控制理论与应用, 2007, 24(3): 407-418.
[128]冯勇,安澄全,李涛.采用双滑模平面减小一类非线性系统稳态误差[J].控制与决策, 2000, 15(3): 361-364.
[129] Yu X, Wu Y. Adaptive Terminal Sliding Mode Control of Uncertain Nonlinear Systems Backstepping Approach[J]. Control Theory and Applications, 1998, 15(6): 900-907.
[130] Wang W J, Lin H R. Fuzzy Control Design for the Trajectory Tracking on Uncertain Nonlinear Systems[J]. IEEE Transactions on Fuzzy Systems, 1999, 7(1): 53-62.
[131] Dotoli M. Fuzzy Sliding Mode Control with Piecewise Linear Switching Manifold[J]. Asian Journal of Control, 2003, 5(4): 528-542.
[132] Nakakuki T. Controller Design for a Class of Nonsmooth Systems and its Application to Mechanical Systems[D]. Sophia University, Japan, 2006.
[133] Nakakuki T, Shen T, Tamura K. Adaptive Control Design for a Class of Nonsmooth Nonlinear Systems with Matched and Linearly Parameterized Uncertainty[J]. International Journal of Robust and Nonlinear Control, 2008, 19(2): 243-255.
[134] Zhang J, Shen T. Functianal Differential Inclusion Based Approach to Control of Discontinuous Nonlinear System with Time Delay[C]//Proceedings of the 47th IEEE Conference on Decision and Control. Cancun, Mexico, 2008: 5300-5305.
[135] Zhang J, Shen T. Feedback Stabilization of a Class of Time-delay Systems with Discontinuity: A Functional Differential Inclusion-based Approach[C]//Inter- national Conference on Control, Automation and Systems. Seoul, Korea, 2008: 1240-1244.
[136] Tanner H G, Kyriakopoulos K J. Backstepping for Nonsmooth Systems[J]. Automatica, 2003, 39(7): 1259-1265.
[137] Zheng K, Shen T, He F, et al. Nonsmooth Backstepping Design for a Class of Parametric Strict-feedback Nonlinear Systems Based on Lipschitz Feedback[C] //American Control Conference. Baltimore, USA, 2010: 6351-6356.
[138] Zheng K, Zhang G, Du J, et al. Nonsmooth Backstepping Design for Motion Control of a Class of Euler-Lagrange Systems[C]//Proceedings of the 29th Chinese Control Conference. Beijing, China, 2010: 570-574.
[139] Zhou J, Wen C. Adaptive Backstepping Control of Uncertain Systems: Non- smooth Nonlinearities, Interactions or Time-Variations[M]. Springer, 2008.
[140] Zhou J, Wen C, Zhang Y. Adaptive Backstepping Control of a Class of Uncertain Nonlinear Systems with Unknown Backlash-like Hysteresis[J]. IEEE Transactions on Automatic Control, 2004, 49(10): 1751-1759.
[141] Lin W, Qian C. Adaptive Control of Nonlinearly Parameterized Systems: A Nonsmooth Feedback Framework[J]. IEEE Transactions on Automatic Control, 2002, 47(5): 757-774.
[142] Zhang J, Shen T, Jiao X. L2-gain Analysis and Feedback Design for Discon- tinuous Time-delay Systems Based on Functional Differential Inclusion[C]// Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai, China, 2009: 5114-5119.
[143] Soravia P. Feedback Stabilization and H∞Control of Nonlinear Systems Affected by Disturbance: the Differential Games Approach. Dynamics, Bifurcation, and Control[M]. London: Springer-Verlag, 2002.
[144] Kellett C M, Shim H, Teel A R. Robustness of Discontinuous Feedback via Sample and Hold Control[C]//Proceedings of American Control Conference. Anchorage, USA, 2002: 3512-3517.
[145] Apkarian P, Noll D. Nonsmooth H∞Synthesis[J]. IEEE Transactions on Automatic Control, 2006, 51(1): 71-86.
[146] Bompart V, Apkarian P, Noll D. Control Design in the Time and Frequency Domain Using Nonsmooth Techniques[J]. Systems & Control Letters, 2008, 57(3): 271-282.
[147] Apkarian P, Noll D. Nonsmooth Optimization for Multiband Frequency Domain Control Design[J]. Automatica, 2007, 43(4): 724-731.
[148] Fontes F A C C. A General Framework to Design Stabilizing Nonlinear Model Predictive Controller[J]. Systems & Control Letters, 2001, 42(2): 127-143.
[149] Fontes F A C C, Magni L. Min-max Model Predictive Control of Nonlinear Systems Using Discontinuous Feedbacks[J]. IEEE Transactions on Automatic Control, 2003, 48(10): 1750-1755.
[150] Messina M J, Tuna S E, Teel A R . Discrete-time Certainty Equivalence Output Feedback: Allowing Discontinuous Control Laws Including those from Model Predictive Control[J]. Automatica, 2005, 41(4): 617-628.
[151] Zhou X Y. Nonsmooth Analysis on Stochastic Controls: A Survey[C]// Proceedings of the 31th IEEE Conference on Decision and Control. Melbourne, USA, 1992: 2406-2411.
[152] Han J Q. From PID to Active Disturbance Rejection Control[J]. IEEE Trans-actions on Industrial Electronics, 2009, 56(3): 900-906.
[153] Burenkov V I. Sobolev Spaces on Domains[M]. B.G.Teubner Verlag, 1998.
[154] Dacorogna B. Introduction to the Calculus of Variations[M]. London: Imperial College Press, 2004.
[155] Simon L, Hungerbuhler N. Theorems on Regularity and Singularity of Energy Minimizing Maps[M]. Birkhauser, 1996.
[156] Neittaanmaki P, Rudnicki M, Savini A. Inverse Problems and Optimal Design in Electricity and Magnetism[M]. Oxford University Press, 1996.
[157] Delfour M C, Zolesio J P. Shapes and Geometries: Analysis, Differential Calculus and Optimization[M]. SIAM, 2001.
[158] Clarke F H, Ledyaev Y S. Qualitivie Properties of Trajectories of Control Systems: A Survey[J]. Journal of Dynamical and Control Systems, 1995, 1(1): 1-48.
[159] Aubin J P. A Survey of Viability Theory[J]. SIAM Journal on Control and Optimization, 1990, 28(4): 749-788.
[160] Khalil H K. Nonlinear Systems[M]. Prentice-Hall, 2002.
[161] Han J Q, Wang W, Huang Y. A Synthesis Method for Two-order Uncertain Systems: Auto-stable Region Approach[J]. Systems Science and Mathematical Sciences, 1997, 10(2), 152-159.
[162] Huang Y, Han J Q. The Self-stable Region Approach for Second Order Nonlinear Uncertain Systems[C]//Proceedings of 1999 IFAC World Congress, 1999, E: 135-140.
[163] Huang Y, Han J Q. A New Synthesis Method for Uncertain Systems: the Self-stable Region Approach[J]. International Journal of Systems Science, 1999, 30(1): 33-38.
[164]郑凯.基于Filippov微分包含解的非平滑控制系统设计方法研究[D].哈尔滨:哈尔滨工业大学控制科学与工程学科博士学位论文, 2009.
[165] Isidori A. Nonlinear Control Systems(3rd)[M]. New York: Springer, 1995.
[166] Curtis H D. Orbital Mechanics: for Engineering Students[M]. Elsevier, 2005.
[167] Gurfil P. Zero-Miss-Distance Guidance Law Based on Line-of-Sight Rate Measurement Only[J]. AIAA 2001-4277: 1-11.
[168] Hablani H B, Pearson D W. Determination of Guidance Parameters of Exo- atmospheric Interceptors via Miss Distance Error Analysis[J]. AIAA 2001-4279: 1-17.
[169] Hablani H B, Pearson D W. Miss Distance Error Analysis of Exoatmospheric Interceptors[J]. Journal of Guidance, Control, and Dynamics, 2004, 27(2): 283-289.
[170] Hablani H B. Endgame Guidance and Relative Navigation of Strategic Inter- ceptors with Delays[J]. Journal of Guidance, Control, and Dynamics. 2006, 29(1): 82-94.
[171] Hablani H B. Pulsed Guidance of Exoatmospheric Interceptors with Image Processing Delays in Angle Measurements[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Denver, USA, 2000: 1-15.
[172]程国采.战术导弹导引方法[M].北京:国防工业出版社, 1996.
[173]刘世勇,吴瑞林,周伯昭.大气层外拦截器末段轨控发动机开关规律[J].宇航学报, 2005, 26(sup): 106-109.
[174]张雅声,程国采,陈克俊.高空动能拦截器末制导导引方法设计与实现[J].现代防御技术, 2001, 29(2): 31-34.
[175]杨宝庆.基于预测控制的大气层外KKV制导控制规律研究[D].哈尔滨:哈尔滨工业大学控制科学与工程学科博士学位论文, 2009.
[176] Lawrence R V. Interceptor Line-of-Sight Rate Steering: Necessary Conditions for a Direct Hit[J]. Journal of Guidance, Control, and Dynamics, 1998, 21(3): 471-476.