基于并行的切换最优控制与混杂动力系统辨识
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摘要
近年来,一类既含有连续动态又含有离散事件的系统——混杂系统,由于其广泛的应用和本身的复杂性吸引了研究者的广泛关注,成为一个新的研究热点.针对两类需要解决的混杂系统问题,本文一是以切换系统为研究对象,在不预先指定切换序列和切换次数的前提下,考虑了一类切换系统的最忧控制问题；二是以微生物发酵生产1,3-丙二醇为背景,研究了一类非线性混杂系统的辨识问题.在两项国家自然科学基金项目的资助下,本文的研究工作既扩展了最优控制理论以及优化算法的研究,同时为实际应用提供重要参考,具有一定的理论意义和实际应用价值.
1.在切换序列和切换次数未知的情况下,考虑了一类由s(s≥2)个切换子系统组成的一般凸紧约束的切换最优控制问题.构造了嵌入系统、松弛系统,并讨论了它们与切换系统轨线之间的关系,给出了切换控制次优解的构造方法.基于极大值原理,讨论了嵌入系统最优解的必要性条件,以及切换系统最优解或近似次优解的判别条件,构造了切换最优控制最优解(或次优解)的求解方法.基于并行原理,构造了一个并行优化算法,利用联想深腾1800集群对算例进行了数值计算,数值结果验证了算法的有效性.
2.考虑到约束问题的特殊性,在不预先指定切换顺序和切换次数的情况下,研究了带有状态约束、控制约束和终端约束的切换最优控制问题.重新构造了嵌入系统,讨论了状态边界约束条件,给出了切换系统、切换子系统、嵌入系统的控制域,以及嵌入系统有效控制域的定义.基于这些控制域,讨论了带有约束的嵌入系统与切换系统轨线的关系,给出了切换系统最优控制次优解的构造方法.在约束条件下,讨论了嵌入系统最优解的必要性条件,以及嵌入系统与切换系统最优解之间的关系,并通过限定嵌入系统在有效控制域内取值,将带有状态约束的问题转化为一般的凸紧约束,得到了约束问题切换最优控制的求解方法.数值结果表明,算法是有效的.
3.针对微生物连续发酵生产1,3-丙二醇的过程中,代谢机理不清的问题,综合考虑底物和产物可能的跨膜运输方式,以及中间代谢产物对细胞本身以及胞内酶物质的不同抑制作用,组合不同方式,建立了一个由72条可能的代谢路径,8维微分方程,既含有连续变量又含有离散变量的非线性混杂动力系统模型.针对该模型,探讨了解的性质.在缺少胞内实验数据的情况下,利用生物系统鲁棒性,结合测得的细胞外物质实验数据,首次建立了一个由43848个连续变量,1152个离散变量组成的复杂代谢系统路径辨识模型.考虑到动力系统向量场不可微,求解规模巨大,本文构造了一个并行粒子群路径辨识算法(PPSO-PIA).应用该算法,在联想深腾1800集群上进行了数值计算,共进行了77172480次微分方程数值计算,获得的最优路径合理地描述了发酵过程,为推断生物发酵机理提供了重要参考.
In recent years, a class of systems which contains both continuous dynamics and discrete events have attracted many researchers due to its wide range of applications and inherent com-plexity with the result that the research has been a new topic for it. These systems are called as hybrid systems. Under the framework of hybrid systems, two classes of problems that need to be worked out are studied in this dissertation. For one thing, a class of optimal control problem-s for switching systems is studied without imposing restrictions on the mode sequence or the number of mode switchings. For another, this dissertation investigates a class of nonlinear hy-brid systems and their system identification based on the background of microbial production of1,3-propanediol. Under the support of the two National Natural Science Foundations of China, this research can not only develop the optimal control theory and optimization algorithm, but also provide certain reference for the practical applications. Therefore, it is very interesting both in theory and in practice. The main contributions obtained in this dissertation are summarized as follows.
1. Under the conditions that the mode sequence and the number of mode switchings are not pre-specified, this dissertation considers a class of optimal switching control problems with the general convex compact constraints involving a switched system composed of s (s≥2) subsystems. We formulate an embedded system, and further construct a relaxed system. The relationships between trajectories of these systems and the switched system are discussed by a constructive method, and the suboptimal solutions of switched optimal control are constructed. According to the Maximum Principle, a few conditions are developed for judging optimal or suboptimal solutions of the switched system based on the necessary conditions for optimality of the embedded system. On the basis of the conditions, a solving method for optimal or suboptimal solutions is given. Furthermore, we provide a numerical algorithm for determining optimal or suboptimal solutions on the basis of parallel principle. The examples are computed applying the proposed algorithm on a Lenovo DeepComp1800PC-cluster Server, and the numerical results show the effectiveness of the proposed algorithm.
2. Taking the particular nature of constraint problems into consideration, this dissertation studies a class of switched optimal control problems with the state, control and terminal con-straints, in which the mode sequence or the number of mode switchings are not also pre-specified as well. Similar to the preceding, we reconstruct an embedded system, and the boundary condi-tions for the state constraint are discussed. The control regions for the switched system, switched subsystems and embedded system combining with the effective control region for the embedded system are defined. On the basis of the control regions, the relationships between trajectories of the switched system and embedded system with the constraint conditions are discussed, and therefore a construction method for the suboptimal solutions of switched optimal control prob-lem is given. Under the constraint conditions, the necessary conditions for optimality of the embedded system and the relationships between the optimal solutions of switched system and embedded system are discussed. We transform the problem with state constraints into the one with the general convex compact constraints by limiting control inputs of the embedded sys-tem in the effective control region so that a solving method with constraint is obtained. The illustrations show the effectiveness of the proposed algorithm.
3. Based on the microbial continuous culture, since the metabolic mechanisms are still unclear in the process of fermentation of glycerol to1,3-propanediol, we consider the possible ways that the substrate and production pass the membrane and the possible ways that the in-termediate metabolic production inhibits both the cells and intracellular enzymes. Combining various possible transport and inhibition mechanisms, we give a nonlinear hybrid dynamical system model composed of72possible metabolic pathways which are described by an eight-dimensional differential equation involving discrete and continuous variables. The properties of solutions to the system are explored. Under the lack of intracellular experimental results, a pathway identification model is proposed for the complex metabolic system using the biological robustness of the intracellular substances and the measured experimental data of the extracel-lular substances. The model includes43848continuous variables and1152discrete variables. Considering the non-differentiable vector field and the large scale of problem, this dissertation provides a parallel particle swarm optimization-pathway identification algorithm (PPSO-PIA). Applying the proposed algorithm, we obtain an optimal pathway and the parameters, in which77172480numerical computations of differential equations are involved on Lenovo DeepComp1800PC-cluster Server. The obtained optimal pathway can describe the continuous fermen- tation reasonably, and the results provide a important reference for conjecturing fermentation mechanisms.
1.在切换序列和切换次数未知的情况下,考虑了一类由s(s≥2)个切换子系统组成的一般凸紧约束的切换最优控制问题.构造了嵌入系统、松弛系统,并讨论了它们与切换系统轨线之间的关系,给出了切换控制次优解的构造方法.基于极大值原理,讨论了嵌入系统最优解的必要性条件,以及切换系统最优解或近似次优解的判别条件,构造了切换最优控制最优解(或次优解)的求解方法.基于并行原理,构造了一个并行优化算法,利用联想深腾1800集群对算例进行了数值计算,数值结果验证了算法的有效性.
2.考虑到约束问题的特殊性,在不预先指定切换顺序和切换次数的情况下,研究了带有状态约束、控制约束和终端约束的切换最优控制问题.重新构造了嵌入系统,讨论了状态边界约束条件,给出了切换系统、切换子系统、嵌入系统的控制域,以及嵌入系统有效控制域的定义.基于这些控制域,讨论了带有约束的嵌入系统与切换系统轨线的关系,给出了切换系统最优控制次优解的构造方法.在约束条件下,讨论了嵌入系统最优解的必要性条件,以及嵌入系统与切换系统最优解之间的关系,并通过限定嵌入系统在有效控制域内取值,将带有状态约束的问题转化为一般的凸紧约束,得到了约束问题切换最优控制的求解方法.数值结果表明,算法是有效的.
3.针对微生物连续发酵生产1,3-丙二醇的过程中,代谢机理不清的问题,综合考虑底物和产物可能的跨膜运输方式,以及中间代谢产物对细胞本身以及胞内酶物质的不同抑制作用,组合不同方式,建立了一个由72条可能的代谢路径,8维微分方程,既含有连续变量又含有离散变量的非线性混杂动力系统模型.针对该模型,探讨了解的性质.在缺少胞内实验数据的情况下,利用生物系统鲁棒性,结合测得的细胞外物质实验数据,首次建立了一个由43848个连续变量,1152个离散变量组成的复杂代谢系统路径辨识模型.考虑到动力系统向量场不可微,求解规模巨大,本文构造了一个并行粒子群路径辨识算法(PPSO-PIA).应用该算法,在联想深腾1800集群上进行了数值计算,共进行了77172480次微分方程数值计算,获得的最优路径合理地描述了发酵过程,为推断生物发酵机理提供了重要参考.
In recent years, a class of systems which contains both continuous dynamics and discrete events have attracted many researchers due to its wide range of applications and inherent com-plexity with the result that the research has been a new topic for it. These systems are called as hybrid systems. Under the framework of hybrid systems, two classes of problems that need to be worked out are studied in this dissertation. For one thing, a class of optimal control problem-s for switching systems is studied without imposing restrictions on the mode sequence or the number of mode switchings. For another, this dissertation investigates a class of nonlinear hy-brid systems and their system identification based on the background of microbial production of1,3-propanediol. Under the support of the two National Natural Science Foundations of China, this research can not only develop the optimal control theory and optimization algorithm, but also provide certain reference for the practical applications. Therefore, it is very interesting both in theory and in practice. The main contributions obtained in this dissertation are summarized as follows.
1. Under the conditions that the mode sequence and the number of mode switchings are not pre-specified, this dissertation considers a class of optimal switching control problems with the general convex compact constraints involving a switched system composed of s (s≥2) subsystems. We formulate an embedded system, and further construct a relaxed system. The relationships between trajectories of these systems and the switched system are discussed by a constructive method, and the suboptimal solutions of switched optimal control are constructed. According to the Maximum Principle, a few conditions are developed for judging optimal or suboptimal solutions of the switched system based on the necessary conditions for optimality of the embedded system. On the basis of the conditions, a solving method for optimal or suboptimal solutions is given. Furthermore, we provide a numerical algorithm for determining optimal or suboptimal solutions on the basis of parallel principle. The examples are computed applying the proposed algorithm on a Lenovo DeepComp1800PC-cluster Server, and the numerical results show the effectiveness of the proposed algorithm.
2. Taking the particular nature of constraint problems into consideration, this dissertation studies a class of switched optimal control problems with the state, control and terminal con-straints, in which the mode sequence or the number of mode switchings are not also pre-specified as well. Similar to the preceding, we reconstruct an embedded system, and the boundary condi-tions for the state constraint are discussed. The control regions for the switched system, switched subsystems and embedded system combining with the effective control region for the embedded system are defined. On the basis of the control regions, the relationships between trajectories of the switched system and embedded system with the constraint conditions are discussed, and therefore a construction method for the suboptimal solutions of switched optimal control prob-lem is given. Under the constraint conditions, the necessary conditions for optimality of the embedded system and the relationships between the optimal solutions of switched system and embedded system are discussed. We transform the problem with state constraints into the one with the general convex compact constraints by limiting control inputs of the embedded sys-tem in the effective control region so that a solving method with constraint is obtained. The illustrations show the effectiveness of the proposed algorithm.
3. Based on the microbial continuous culture, since the metabolic mechanisms are still unclear in the process of fermentation of glycerol to1,3-propanediol, we consider the possible ways that the substrate and production pass the membrane and the possible ways that the in-termediate metabolic production inhibits both the cells and intracellular enzymes. Combining various possible transport and inhibition mechanisms, we give a nonlinear hybrid dynamical system model composed of72possible metabolic pathways which are described by an eight-dimensional differential equation involving discrete and continuous variables. The properties of solutions to the system are explored. Under the lack of intracellular experimental results, a pathway identification model is proposed for the complex metabolic system using the biological robustness of the intracellular substances and the measured experimental data of the extracel-lular substances. The model includes43848continuous variables and1152discrete variables. Considering the non-differentiable vector field and the large scale of problem, this dissertation provides a parallel particle swarm optimization-pathway identification algorithm (PPSO-PIA). Applying the proposed algorithm, we obtain an optimal pathway and the parameters, in which77172480numerical computations of differential equations are involved on Lenovo DeepComp1800PC-cluster Server. The obtained optimal pathway can describe the continuous fermen- tation reasonably, and the results provide a important reference for conjecturing fermentation mechanisms.
引文
[1]WITSENHAUSEN H. A class of hybrid-state continuous-time dynamic systems[J]. IEEE Transactions on Automatic Control,1966,11(2):161-167.
[2]PAVLIDIS T. Stability of systems described by differential equations containing impulses[J]. IEEE Transactions on Automatic Control,1967,12(1):43-45.
[3]VAN DER SCHAFT A J, SCHUMACHER J M. An introduction to hybrid dynamical systems[M]. Vol.251, Springer London,2000.
[4]BALLUCHI A, BENVENUTI L, Di BENEDETTO M D, et al. Automotive engine control and hybrid systems: Challenges and opportunities[J]. Proceedings of the IEEE,2000,88(7):888-912.
[5]YAMAGUCHI T, NUMASATO H, HIRAI H. A mode-switching control for motion control and its applica-tion to disk drives:Design of optimal mode-switching conditions[J]. IEEE/ASME Transactions on Mechatronics,1998,3(3):202-209.
[6]BISHOP B E, SPONG M W. Control of redundant manipulators using logic-based switching[C]//Pro-ceedings of the 37th IEEE Conference on Decision and Control,1998,2:1488-1493.
[7]ENGELL S, KOWALEWSKI S, SCHULZ C, et al. Continuous-discrete interactions in chemical processing plants[J]. Proceedings of the IEEE,2000,88(7):1050-1068.
[8]KLEBAN J. Switched Systems[M]. In-Teh, Vukovar,2009.
[9]LIBERZON D. Switching in systems and control [M]. Springer,2003.
[10]LI Z, SOH Y, WEN C. Switched and impulsive systems:analysis, design and applications[M]. Vol.313, Springer,2005.
[II]DE SCHUTTER B, HEEMELS W, LUNZE J, et al. Survey of modelling, analysis and control of hybrid systems[J]. HYCON Handbook of Hybrid Systems Control Theory-Tools-Applications,2009:31-55.
[12]LIBERZON D, MORSE A S. Basic problems in stability and design of switched systems[J]. IEEE Control Systems,1999,19(5):59-70.
[13]CHENG D. Stabilization of planar switched systems[J]. Systems & Control Letters,2004,51(2):79-88.
[14]GONCALVES J M, MEGRETSKI A, DAHLEH M A. Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions[J]. IEEE Transactions on Automatic Control,2003,48(12): 2089-2106.
[15]BRANICKY M S. Stability of switched and hybrid systems[C]//Proceedings of the 33rd IEEE Conference on Decision and Control,1994,4:3498-3503.
[16]BRANICKY M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid system-s[J]. IEEE Transactions on Automatic Control,1998,43(4):475-482.
[17]EZZINE J, HADDAD A. On the controllability and observability of hybrid systems[C]//1988 American Control Conference,1988:41-46.
[18]BLONDEL V D, TSITSIKLIS J N. Complexity of stability and controllability of elementary hybrid system-s[J]. Automatica,1999,35(3):479-489.
[19]GE S, SUN Z, LEE T. Reachability and controllability of switched linear discrete-time systems[J]. IEEE Transactions on Automatic Control,2001,46(9):1437-1441.
[20]XIE G, WANG L. Controllability and stabilizability of switched linear-systems[J]. Systems & Control Letters,2003,48(2):135-155.
[21]JOHANSSON R, RANTZER A. Nonlinear and hybrid systems in automotive control[M]. Springer London, 2003.
[22]RINEHART M, DAHLEH M, REED D, et al. Suboptimal control of switched systems with an application to the disc engine[J]. IEEE Transactions on Control Systems Technology,2008,16(2):189-201.
[23]HOWLETT P G, PUDNEY P J, Vu X. Local energy minimization in optimal train control[J]. Automatica, 2009,45(11):2692-2698.
[24]CASSANDRAS C G, PEPYNE D L, WARDI Y. Optimal control of a class of hybrid systems[J]. IEEE Transactions on Automatic Control,2001,46(3):398-415.
[25]GEYER T, PAPAFOTIOU G, MORARI M. On the optimal control of switch-mode dc-dc converters [J]. Hybrid systems:Computation and control,2004:77-85.
[26]LOXTON R, TEO K L, REHBOCK V, et al. Optimal switching instants for a switched-capacitor DC/DC power converter [J]. Automatica,2009,45(4):973-980.
[27]GHOSH R, TOMLIN C. Symbolic reachable set computation of piecewise affIne hybrid automata and its application to biological modelling:Delta-Notch protein signalling[J]. Systems Biology,2004,1(1): 170-183.
[28]LIU C, FENG E. Optimal control of switched autonomous systems in microbial fed-batch cultures[J]. International Journal of Computer Mathematics,2011,88(2):396-407.
[29]LIU C, GONG Z, FENG E, et al. Optimal switching control of a fed-batch fermentation process[J]. Journal of Global Optimization,2012,52:265-280.
[30]YE J, ZHANG Y, FENG E, et al. Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and pH logic control [J]. Applied Mathematical Modelling, 2012,36(1):357-369.
[31]PARDALOS P M, YATSENKO V A. Optimization and control of bilinear systems:Theory, algorithms, and applications[M]. Vol.11, Springer,2008.
[32]DOLCETTA I C, EVANS L. Optimal switching for ordinary differential equations[J]. SIAM journal on control and optimization,1984,22(1):143-161.
[33]YONG J. Systems governed by ordinary differential equations with continuous, switching and impulse controls[J]. Applied mathematics & optimization,1989,20(1):223-235.
[34]SEIDMAN T I. Optimal control for switching systems[C]//Proceedings of the 21st Annual Conference on Information sciences and Systems,1987:485-489.
[35]BRANICKY M S, BORKAR V S, MITTER S K. A unified framework for hybrid control:Model and optimal control theory[J]. IEEE Transactions on Automatic Control,1998,43(1):31-45.
[36]SUSSMANN H J. A maximum principle for hybrid optimal control problems[C]//Proceedings of the 38th IEEE Conference on Decision and Control,1999,1:425-430.
[37]PICCOLI B. Necessary conditions for hybrid optimization[C]//Proceedings of the 38th IEEE Conference on Decision and Control,1999,1:410-415.
[38]EGERSTEDT M, OGREN P, SHAKERNIA O, et al. Toward optimal control of switched linear system-s[C]//Proceedings of the 39th IEEE Conference on Decision and Control,2000,1:587-592.
[39]VERRIEST E, DELMOTTE F, EGERSTEDT M. Optimal impulsive control for point delay systems with re-fractory period[C]//Proceedings of the 5th IFAC workshop on time delay systems, Leuven, Belgium, 2004.
[40]HEDLUND S, RANTZER A. Optimal control of hybrid systems[C]//Proceedings of the 38th IEEE Confer-ence on Decision and Control,1999,4:3972-3977.
[41]HEDLUND S, RANTZER A. Convex dynamic programming for hybrid systems[J]. IEEE Transactions on Automatic Control,2002,47(9):1536-1540.
[42]BORRELLI F, BAOTIC M, BEMPORAD A, et al. Dynamic programming for constrained optimal control of discrete-time linear hybrid systems[J]. Automatica,2005,41(10):1709-1721.
[43]SEATZU C, CORONA D, GIUA A, et al. Optimal control of continuous-time switched affine systems[J]. IEEE Transactions on Automatic Control,2006,51(5):726-741.
[44]CHUDOUNG J, BECK C. The minimum principle for deterministic impulsive control system-s[C]//Proceedings of the 40th IEEE Conference on Decision and Control,2001,4:3569-3574.
[45]RIEDINGER P, IUNG C, KRATZ F. An optimal control approach for hybrid systems[J]. European Journal of Control,2003,9(5):449-458.
[46]ALAMIR M, ATTIA S A. An efficient algorithm to solve optimal control problems for nonlinear switched hybrid systems [C]//Proceedings of 6th IFAC symposium on nonlinear control systems,2004:417-422.
[47]ALAMTR M, ATTIA S A. On solving optimal control problems for switched hybrid nonlinear systems by strong variations algorithms[C]//Proceedings of 6th IFAC symposium on nonlinear control systems, 2004:558-563.
[48]GARAVELLO M, PICCOLI B. Hybrid necessary principle[J]. SIAM Journal on Control and Optimization, 2005,43(5):1867-1887.
[49]SHAIKH M S, CAINES P E. On the hybrid optimal control problem:Theory and algorithms[J]. IEEE Transactions on Automatic Control,2007,52(9):1587-1603.
[50]EGERSTEDT M, WARDI Y, AXELSSON H. Transition-time optimization for switched-mode dynamical sys-tems[J]. IEEE Transactions on Automatic Control,2006,51(1):110-115.
[51]GONZALEZ H, VASUDEVAN R, KAMGARPOUR M, et al. A descent algorithm for the optimal control of con-strained nonlinear switched dynamical systems[C]//Proceedings of the 13th ACM international confer-ence on Hybrid systems:computation and control,2010:51-60.
[52]AZHMYAKOV V, BASIN M V, RAISCH J. A proximal point based approach to optimal control of affine switched systems[J]. Discrete Event Dynamic Systems,2012,22(1):61-81.
[53]KAMGARPOUR M, TOMLIN C. On optimal control of non-autonomous switched systems with a fixed mode sequence[J]. Automatica,2012,48:1177-1181.
[54]GIUA A, SEATZU C, VAN DER MEE C. Optimal control of switched autonomous linear system- s[C]//Proceedings of the 40th IEEE Conference on Decision and Control,2001,3:2472-2477.
[55]GIUA A, SEATZU C, VAN DER MEE C. Optimal control of autonomous linear systems switched with a pre-assigned finite sequence[C]//Proceedings of the 2001 IEEE International Symposium on Intelligent Control (ISIC'01),2001:144-149.
[56]Xu X, ANTSAKLIS P J. Optimal control of switched autonomous systems[C]//Proceedings of the 41st IEEE Conference on Decision and Control,2002,4:4401-4406.
[57]EGERSTEDT M, WARDI Y, DELMOTTE F. Optimal control of switching times in switched dynamical sys-tems[C]//Proceedings of the 42nd IEEE Conference on Decision and Control,2003,3:2138-2143.
[58]Xu X, ANTSAKLIS P J. Optimal control of switched systems:New results and open problem-s[C]//Proceedings of the 2000 American Control Conference,2000,4:2683-2687.
[59]Xu X, ANTSAKLIS P J. An approach for solving general switched linear quadratic optimal control problems[C]//Proceedings of the 40th IEEE Conference on Decision and Control,2001,3:2478-2483.
[60]Xu X, ANTSAKLIS P J. Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions[J]. International Journal of Control,2002,75(16-17):1406-1426.
[61]Xu X, ANTSAKLIS P. Results and perspectives on computational methods for optimal control of switched systems[J]. Hybrid Systems:Computation and Control,2003:540-555.
[62]Xu X, ANTSAKLIS P J. Optimal control of switched systems based on parameterization of the switching instants[J]. IEEE Transactions on Automatic Control,2004,49(1):2-16.
[63]SHAIKH M, CAINES P. On the optimal control of hybrid systems:Optimization of trajectories, switching times, and location schedules[J]. Hybrid systems:Computation and control,2003:466-481.
[64]AXELSSON H, WARDI Y, EGERSTEDT M, et al. Gradient descent approach to optimal mode scheduling in hybrid dynamical systems[J]. Journal of Optimization Theory and Applications,2008,136(2):167-186.
[65]TEO K, GOH C, KH W. A unified computational approach to optimal control problems[M]. Longman New York,1991.
[66]TEO K, JENNINGS L, LEE H, et al. The control parameterization enhancing transform for constrained optimal control problems[J]. The Journal of the Australian Mathematical Society, Series B:Applied Mathematics,1999,40(03):314-335.
[67]LI R, TEO K, WONG K, et al. Control parameterization enhancing transform for optimal control of switched systems[J]. Mathematical and computer modelling,2006,43(11):1393-1403.
[68]LI R, FENG Z, TEO K, et al. Optimal piecewise state feedback control for impulsive switched systems[J]. Mathematical and Computer Modelling,2008,48(3):468-479.
[69]Wu C, TEO K. Global impulsive optimal control computation[J]. Journal of Industrial and Management Optimization,2006,2(4):435-450.
[70]LOXTON R, TEO K, REHBOCK V. Optimal control problems with multiple characteristic time points in the objective and constraints[J]. Automatica,2008,44(11):2923-2929.
[71]LOXTON R, TEO K L, REHBOCK V, et al. Optimal control problems with a continuous inequality constraint on the state and the control[J]. Automatica,2009,45(10):2250-2257.
[72]FARHADINIA B, TEO K, LOXTON R. A computational method for a class of non-standard time optimal control problems involving multiple time horizons[J]. Mathematical and Computer Modelling,2009, 49(7):1682-1691.
[73]LOXTON R C, TEO K L, REHBOCK V. Computational method for a class of switched system optimal control problems[J]. IEEE Transactions on Automatic Control,2009,54(10):2455-2460.
[74]LIN Q, LOXTON R, TEO K L, et al. Optimal control computation for nonlinear systems with state-dependent stopping criteria[J]. Automatica,2012,48:2116-2129.
[75]BENGEA S C, DECARLO R A. Optimal and suboptimal control of switching systems[C]//Proceedings of the 42nd IEEE Conference on Decision and Control,2003,5:5295-5300.
[76]BENGEA S C, DECARLO R A. Optimal control of switching systems[J]. automatica,2005,41(1):11-27.
[77]HOMANN T, TAG C, BIEBL H, et al. Fermentation of glycerol to 1,3-propanediol by Klebsiella and Cit-robacter strains[J]. Applied Microbiology and Biotechnology,1990,33(2):121-126.
[78]BIEBL H, MENZEL K, ZENG A P, et al. Microbial production of 1,3-propanediol[J]. Applied Microbiology and Biotechnology,1999,52(3):289-297.
[79]BIEBL H, MARTEN S, HIPPE H, et al. Glycerol conversion to 1,3-propanediol by newly isolated clostridi-a[J]. Applied Microbiology and Biotechnology,1992,36(5):592-597.
[80]MENZEL K, ZENG A P, DECKWER W D. High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae[J]. Enzyme and Microbial Technology, 1997,20(2):82-86.
[81]REIMANN A, BIEBL H, DECLWER W D. Production of 1,3-propanediol by Clostridium butyricum in continuous culture with cell recycling[J]. Applied microbiology and biotechnology,1998,49(4):359-363.
[82]ZEMG A P, BIEBL H. Bulk chemicals from biotechnology:The case of 1,3-propanediol production and the new trends[J]. Tools and Applications of Biochemical Engineering Science,2002:239-259.
[83]NAKAMURA C E, WHITED G M. Metabolic engineering for the microbial production of 1,3-propanediol[J]. Current opinion in biotechnology,2003,14(5):454-459.
[84]LAFFEND L A, NAGARAJAN V, NAKAMURA C E. Bioconversion of a fermentable carbon source to 1,3-propanediol by a single microorganism[P]. Google Patents,1997. US Patent 5,686,276.
[85]ZENG A P, Ross A, BIEBL H, et al. Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation[J]. Biotechnology and bioengineering, 1994,44(8):902-911.
[86]修志龙,曾安平,安利佳.甘油生物歧化过程动力学数学模拟和多稳态研究[J].大连理工大学学报,2000,40(4)：428-433.
[87]修志龙,曾安平,安利佳等.甘油连续生物歧化过程的过渡行为及其数学模拟[J].高校化学工程学报,2000,14(1)：53-58.
[88]GAO C, FENG E, WANG Z, et al. Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures[J]. Applied mathematics and computation,2005,169(1):476-484.
[89]高彩霞,王宗涛,冯恩民等.微生物间歇发酵生产1,3-丙二醇过程的辨识与优化[J].大连理工大学 学报,2006,46(5)：771-774.
[90]马永峰,孙丽华,修志龙.微生物连续培养过程中振荡的理论分析[J].工程数学学报,2003,20(1)：1-6.
[91]LIAN H, FENG E, LI X, et al. Oscillatory behavior in microbial continuous culture with discrete time delay[J]. Nonlinear Analysis:Real World Applications,2009,10(5):2749-2757.
[92]李晓红,冯恩民,修志龙.微生物连续培养过程平衡点的稳定分析[J].高校应用数学学报B辑,2005,20(4)：377-383.
[93]李晓红,冯恩民,修志龙.微生物间歇发酵非线性动力系统的性质及最优控制[J].运筹学学报,2005,9(4)：89-96.
[94]Li X, FENG E, XIU Z. Stability and optimal control of microorganisms in continuous culture[J]. Journal of Applied Mathematics and Computing,2006,22(1):425-434.
[95]GAO C, LANG Y, FENG E, et al. Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control[J]. Journal of Applied Mathematics and Computing,2005,19(1):203-214.
[96]GAO C, FENGE, WANG Z, et al. Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls[J]. Journal of Industrial and Management Optimization,2005, 1(3):377.
[97]YE J, FENG E, YIN H, et al. Modelling and well-posedness of a nonlinear hybrid system in fed-batch pro-duction of 1,3-propanediol with open loop glycerol input and pH logic control[J]. Nonlinear Analysis: Real World Applications,2011,12(1):364-376.
[98]LIU C, GONG Z, FENG E, et al. Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture[J]. Journal of Industrial and Management Optimization,2009, 5(4):835-850.
[99]WANGJ, SUN Q, FENG E. Modelling and properties of a nonlinear autonomous switching system in fed-batch culture of glycerol [J]. Communications in Nonlinear Science and Numerical Simulation,2012, 11:4446-4454.
[100]WANG L, FENG E, YE J, et al. Modeling and properties of nonlinear stochastic dynamical system of continuous culture[J]. Complex Sciences,2009:458-466.
[101]WANG L, FENG E, XIU Z. Nonlinear stochastic dynamical system of bio-dissimilation of glycerol to 1,3-propanediol in batch culture and its viable set[C]//Proceedings of the 3rd International Conference on Bioinformatics and Biomedical Engineering (ICBBE 2009),2009:1-4.
[102]WANGL, XIU Z, FENG E. A stochastic model of microbial bioconversion process in batch culture[J]. International Journal of Chemical Reactor Engineering,2011,9(1).
[103]沈邦玉.批式与连续发酵的非光滑动力系统辨识与优化控制[D].大连大连理工大学,2012.
[104]高井贵.一类微生物发酵的非线性混杂动力系统辨识及控制[D].大连大连理工大学,2012.
[105]SUN Y Q, QI W T, TENG H, et al. Mathematical modeling of glycerol fermentation by Klebsiella pneu-moniae:Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane[J]. Biochemical Engineering Journal,2008,38(1):22-32.
[106]SHEN B, LIU C, YE J, et al. Parameter identification and optimization algorithm in microbial continuous culture[J]. Applied Mathematical Modelling,2012,36(2):585-595.
[107]ZHANG Y, FENGE, XIU Z. Robust analysis of hybrid dynamical systems for 1,3-propanediol transport mechanisms in microbial continuous fermentation[J]. Mathematical and Computer Modelling,2011, 54(11):3164-3171.
[108]WANG J, YE J, FENG E, et al. Complex metabolic network of glycerol fermentation by Klebsiella pneu-moniae and its system identification via biological robustness[J]. Nonlinear Analysis:Hybrid Systems, 2011,5(1):102-112.
[109]陈国良.并行算法的设计与分析(修订版)[M].北京：高等教育出版社,2002.
[110]陈国良.并行算法—结构·算法·编程[M].北京：高等教育出版社,2003.
[111]陈国良,安虹,陈峻等.并行算法实践[M].北京：高等教育出版社,2004.
[112]王鹏,吕爽,聂治等.并行计算应用及实战[M].北京：机械工业出版社,2008.
[113]陈国良.并行算法研究进展[R].中国计算机学会通信,2005. http://www.ccf.org.cn/web/ resource/newspic/2005/9/20/bingxin.pdf.
[114]REIF J H. An optimal parallel algorithm for integer sorting[C]//Proceedings of the 26th Annual Symposium on Foundations of Computer Science,1985:496-504.
[115]GOLDBERG M, SPENCER T. A new parallel algorithm for the maximal independent set problem[J]. SIAM Journal on Computing,1989,18(2):419-427.
[116]ANDERSON R J, MILLER G L. A simple randomized parallel algorithm for list-ranking [J]. Information Processing Letters,1990,33(5):269-273.
[117]DONGARRA J J, SORENSEN D C. A fully parallel algorithm for the symmetric eigenvalue problem[J]. SIAM Journal on Scientific and Statistical Computing,1987,8(2):s139-s154.
[118]SORENSEN D C, TANG P T P. On the orthogonality of eigenvectors computed by divide-and-conquer techniques[J]. SIAM Journal on Numerical Analysis,1991,28(6):1752-1775.
[119]PARDALOS P. Parallel search algorithms in global optimization[J]. Applied Mathematics and Computa-tion,1989,29(3):219-229.
[120]CHEN Z, FEI P. A parallel algorithm for constrained optimization problems[J]. Journal of computational and applied mathematics,1995,61(2):225-230.
[121]CONFORTI D, MUSMANNO R. Parallel algorithm for unconstrained optimization based on decomposition techniques[J]. Journal of optimization theory and applications,1997,95(3):531-544.
[122]LIU C S, TSENG C H. Parallel synchronous and asynchronous space-decomposition algorithms for large-scale minimization problems[J]. Computational Optimization and Applications,2000,17(1):85-107.
[123]MIGDALAS A, TORALDO G, KUMAR V. Nonlinear optimization and parallel computing[J]. Parallel Com-puting,2003,29(4):375-391.
[124]SCHUTTE J, REINBOLT J, FREGLY B, et al. Parallel global optimization with the particle swarm algorithm[J]. International Journal for Numerical Methods in Engineering,2004,61(13):2296-2315.
[125]KOH B I, GEORGE A D, HAFTKA R T, et al. Parallel asynchronous particle swarm optimization[J]. Inter-national Journal for Numerical Methods in Engineering,2006,67(4):578-595.
[126]OZCAN E, ONBASIOGLU E. Memetic algorithms for parallel code optimization[J]. International Journal of Parallel Programming,2007,35(1):33-61.
[127]MAHDAD B, SRAIRI K, BOUKTIR T, et al. Fuzzy controlled parallel PSO to solving large practical e-conomic dispatch[C]//IECON 2010-36th Annual Conference on IEEE Industrial Electronics Society, 2010:2695-2701.
[128]CAO Y, SUN D. A Parallel Computing Framework for Large-Scale Air Traffic Flow Optimization[J]. IEEE Transactions on Intelligent Transportation Systems,2012,13(4):1855-1864.
[129]ZHANG Y, GALLIPOLI D, AUGARDE C. Parameter identification for elasto-plastic modelling of unsaturated soils from pressuremeter tests by parallel modified particle swarm optimization [J]. Computers and Geotechnics,2013,48(3):293-303.
[130]KONDOV I. Protein structure prediction using distributed parallel particle swarm optimization[J]. Natural Computing,2013,12:29-41.
[131]LUONG T, MELAB N, TALBI E. GPU computing for parallel local search metaheuristic algorithms[J]. IEEE Transactions on Computers,2013,62(1):173-185.
[132]KITANO H. Biological robustness[J]. Nature Reviews Genetics,2004,5(11):826-837.
[133]KITANO H. Towards a theory of biological robustness [J]. Molecular systems biology,2007,3(1).
[134]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京：科学出版社,2001.
[135]CLARKE F H, LEDYAEV Y S, STERN R J, et al. Nonsmooth analysis and control theory[M]. Vol.178, Springer New York,1998.
[136]HSIEH P F, SIBUYA Y. Basic theory of ordinary differential equations[M]. Springer,1999.
[137]BERKOVITZ L D. Optimal control theory[M]. Springer New York,1974.
[138]KIRK D E. Optimal control theory:An introduction[M]. Dover New York,1998.
[139]LUUS R, CHEN Y Q. Optimal switching control via direct search optimization [J]. Asian Journal of Control,2004,6(2):302-306.
[140]PAPAMICHAIL I, ADJIMAN C S. A rigorous global optimization algorithm for problems with ordinary differential equations[J]. Journal of Global Optimization,2002,24(1):1-33.
[141]SPINELLI W, BOLZERN P, COLANERI P. A note on optimal control of autonomous switched systems on a finite time interval[C]//2006 American Control Conference,2006:5947-5952.
[142]ZHANG L, CHEN Y, CUI P. Stabilization for a class of second-order switched systems[J]. Nonlinear Analysis:Theory, Methods & Applications,2005,62(8):1527-1535.
[143]BEMPORAD A, MORARI M. Control of systems integrating logic, dynamics, and constraints[J]. Automat-ica,1999,35(3):407-427.
[144]BUSKENS C, MAURER H. SQP-methods for solving optimal control problems with control and state constraints:adjoint variables, sensitivity analysis and real-time control [J]. Journal of computational and applied mathematics,2000,120(1):85-108.
[145]GERDTS M, KUNKEL M. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints[J]. Journal of Industrial and Management Optimization,2008,4(2):247-270.
[146]AZHMYAKOV V, RAISCH J. Convex control systems and convex optimal control problems with con-straints[J]. IEEE Transactions on Automatic Control,2008,53(4):993-998.
[147]ATTIA S, ALAMIR M, CANUDAS DE WIT C. Sub optimal control of switched nonlinear systems under location and switching constraints [C]//Proceedings of the 16th IFAC world congress, Prague, Czech Republick,2005.
[148]GAO R, LIU X, YANG J. On optimal control problems of a class of impulsive switching systems with terminal states constraints[J]. Nonlinear Analysis:Theory, Methods & Applications,2010,73(7): 1940-1951.
[149]HARTL R F, SETHI S P, VICKSON R G. A survey of the maximum principles for optimal control problems with state constraints[J]. SIAM review,1995,37(2):181-218.
[150]NOCEDAL J, WRIGHT S J. Numerical optimization[M]. Springer-Verlag New York,2006.
[151]ZENG A P, DECKWER W D. A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions[J]. Biotechnology progress,1995,11(1):71-79.
[152]ZENG A P. A kinetic model for product formation of microbial and mammalian cells[J]. Biotechnology and bioengineering,2004,46(4):314-324.
[153]YE J, FENG E, LIAN H, et al. Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures[J]. Applied Mathematics and Computation,2009,207(2): 307-318.
[154]WANG J, YE J, FENG E, et al. Modeling and identification of a nonlinear hybrid dynamical system in batch fermentation of glycerol[J]. Mathematical and Computer Modelling,2011,54(1):618-624.
[155]MA Y F, XIU Z L, SUN L H, et al. Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay[J]. International Journal of Nonlinear Sciences and Numerical Simulation,2006, 7(3):305-308.
[156]LI X, GUO J, FENG E, et al. Discrete optimal control model and bound error for microbial continuous fermentation[J]. Nonlinear Analysis:Real World Applications,2010,11(1):131-138.
[157]HAO J, LIN R, ZHENG Z, et al.3-Hydroxypropionaldehyde guided glycerol feeding strategy in aero-bic 1,3-propanediol production by Klebsiella pneumoniae[J]. Journal of industrial microbiology & biotechnology,2008,35(12):1615-1624.
[158]WANG S, FENG E. Stability of nonlinear microbial bioconversion system concerning glycerol's active transport and 1,3-PD's passive transport[J]. Nonlinear Analysis:Real World Applications,2010,11(5): 3501-3511.
[159]KENNEDY J, EBERHART R. Particle swarm optimization [C]//Proceedings of IEEE International Confer-ence on Neural Networks,1995,4:1942-1948.
[160]EBERHART R, SHI Y. Particle swarm optimization:Developments, applications and re-sources[C]//Proceedings of the 2001 Congress on Evolutionary Computation,2001,1:81-86.
[161]KARAKUZU C, et al. Fuzzy controller training using particle swarm optimization for nonlinear system control[J]. ISA transactions,2008,47(2):229-239.
[162]LIU L, LIU W, CARTES D A. Particle swarm optimization-based parameter identification applied to permanent magnet synchronous motors[J]. Engineering Applications of Artificial Intelligence,2008, 21(7):1092-1100.
[163]王凌,刘波.微粒群优化与调度算法[M].北京：清华大学出版社,2008.
[164]李丽,牛奔.粒子群优化算法[M].北京：冶金工业出版社,2009.
[165]Xu L, ZHANG F. Parallel particle swarm optimization for attribute reduction[C]//SNPD 2007 Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Par-allel/Distributed Computing,2007,1:770-775.
[2]PAVLIDIS T. Stability of systems described by differential equations containing impulses[J]. IEEE Transactions on Automatic Control,1967,12(1):43-45.
[3]VAN DER SCHAFT A J, SCHUMACHER J M. An introduction to hybrid dynamical systems[M]. Vol.251, Springer London,2000.
[4]BALLUCHI A, BENVENUTI L, Di BENEDETTO M D, et al. Automotive engine control and hybrid systems: Challenges and opportunities[J]. Proceedings of the IEEE,2000,88(7):888-912.
[5]YAMAGUCHI T, NUMASATO H, HIRAI H. A mode-switching control for motion control and its applica-tion to disk drives:Design of optimal mode-switching conditions[J]. IEEE/ASME Transactions on Mechatronics,1998,3(3):202-209.
[6]BISHOP B E, SPONG M W. Control of redundant manipulators using logic-based switching[C]//Pro-ceedings of the 37th IEEE Conference on Decision and Control,1998,2:1488-1493.
[7]ENGELL S, KOWALEWSKI S, SCHULZ C, et al. Continuous-discrete interactions in chemical processing plants[J]. Proceedings of the IEEE,2000,88(7):1050-1068.
[8]KLEBAN J. Switched Systems[M]. In-Teh, Vukovar,2009.
[9]LIBERZON D. Switching in systems and control [M]. Springer,2003.
[10]LI Z, SOH Y, WEN C. Switched and impulsive systems:analysis, design and applications[M]. Vol.313, Springer,2005.
[II]DE SCHUTTER B, HEEMELS W, LUNZE J, et al. Survey of modelling, analysis and control of hybrid systems[J]. HYCON Handbook of Hybrid Systems Control Theory-Tools-Applications,2009:31-55.
[12]LIBERZON D, MORSE A S. Basic problems in stability and design of switched systems[J]. IEEE Control Systems,1999,19(5):59-70.
[13]CHENG D. Stabilization of planar switched systems[J]. Systems & Control Letters,2004,51(2):79-88.
[14]GONCALVES J M, MEGRETSKI A, DAHLEH M A. Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions[J]. IEEE Transactions on Automatic Control,2003,48(12): 2089-2106.
[15]BRANICKY M S. Stability of switched and hybrid systems[C]//Proceedings of the 33rd IEEE Conference on Decision and Control,1994,4:3498-3503.
[16]BRANICKY M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid system-s[J]. IEEE Transactions on Automatic Control,1998,43(4):475-482.
[17]EZZINE J, HADDAD A. On the controllability and observability of hybrid systems[C]//1988 American Control Conference,1988:41-46.
[18]BLONDEL V D, TSITSIKLIS J N. Complexity of stability and controllability of elementary hybrid system-s[J]. Automatica,1999,35(3):479-489.
[19]GE S, SUN Z, LEE T. Reachability and controllability of switched linear discrete-time systems[J]. IEEE Transactions on Automatic Control,2001,46(9):1437-1441.
[20]XIE G, WANG L. Controllability and stabilizability of switched linear-systems[J]. Systems & Control Letters,2003,48(2):135-155.
[21]JOHANSSON R, RANTZER A. Nonlinear and hybrid systems in automotive control[M]. Springer London, 2003.
[22]RINEHART M, DAHLEH M, REED D, et al. Suboptimal control of switched systems with an application to the disc engine[J]. IEEE Transactions on Control Systems Technology,2008,16(2):189-201.
[23]HOWLETT P G, PUDNEY P J, Vu X. Local energy minimization in optimal train control[J]. Automatica, 2009,45(11):2692-2698.
[24]CASSANDRAS C G, PEPYNE D L, WARDI Y. Optimal control of a class of hybrid systems[J]. IEEE Transactions on Automatic Control,2001,46(3):398-415.
[25]GEYER T, PAPAFOTIOU G, MORARI M. On the optimal control of switch-mode dc-dc converters [J]. Hybrid systems:Computation and control,2004:77-85.
[26]LOXTON R, TEO K L, REHBOCK V, et al. Optimal switching instants for a switched-capacitor DC/DC power converter [J]. Automatica,2009,45(4):973-980.
[27]GHOSH R, TOMLIN C. Symbolic reachable set computation of piecewise affIne hybrid automata and its application to biological modelling:Delta-Notch protein signalling[J]. Systems Biology,2004,1(1): 170-183.
[28]LIU C, FENG E. Optimal control of switched autonomous systems in microbial fed-batch cultures[J]. International Journal of Computer Mathematics,2011,88(2):396-407.
[29]LIU C, GONG Z, FENG E, et al. Optimal switching control of a fed-batch fermentation process[J]. Journal of Global Optimization,2012,52:265-280.
[30]YE J, ZHANG Y, FENG E, et al. Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and pH logic control [J]. Applied Mathematical Modelling, 2012,36(1):357-369.
[31]PARDALOS P M, YATSENKO V A. Optimization and control of bilinear systems:Theory, algorithms, and applications[M]. Vol.11, Springer,2008.
[32]DOLCETTA I C, EVANS L. Optimal switching for ordinary differential equations[J]. SIAM journal on control and optimization,1984,22(1):143-161.
[33]YONG J. Systems governed by ordinary differential equations with continuous, switching and impulse controls[J]. Applied mathematics & optimization,1989,20(1):223-235.
[34]SEIDMAN T I. Optimal control for switching systems[C]//Proceedings of the 21st Annual Conference on Information sciences and Systems,1987:485-489.
[35]BRANICKY M S, BORKAR V S, MITTER S K. A unified framework for hybrid control:Model and optimal control theory[J]. IEEE Transactions on Automatic Control,1998,43(1):31-45.
[36]SUSSMANN H J. A maximum principle for hybrid optimal control problems[C]//Proceedings of the 38th IEEE Conference on Decision and Control,1999,1:425-430.
[37]PICCOLI B. Necessary conditions for hybrid optimization[C]//Proceedings of the 38th IEEE Conference on Decision and Control,1999,1:410-415.
[38]EGERSTEDT M, OGREN P, SHAKERNIA O, et al. Toward optimal control of switched linear system-s[C]//Proceedings of the 39th IEEE Conference on Decision and Control,2000,1:587-592.
[39]VERRIEST E, DELMOTTE F, EGERSTEDT M. Optimal impulsive control for point delay systems with re-fractory period[C]//Proceedings of the 5th IFAC workshop on time delay systems, Leuven, Belgium, 2004.
[40]HEDLUND S, RANTZER A. Optimal control of hybrid systems[C]//Proceedings of the 38th IEEE Confer-ence on Decision and Control,1999,4:3972-3977.
[41]HEDLUND S, RANTZER A. Convex dynamic programming for hybrid systems[J]. IEEE Transactions on Automatic Control,2002,47(9):1536-1540.
[42]BORRELLI F, BAOTIC M, BEMPORAD A, et al. Dynamic programming for constrained optimal control of discrete-time linear hybrid systems[J]. Automatica,2005,41(10):1709-1721.
[43]SEATZU C, CORONA D, GIUA A, et al. Optimal control of continuous-time switched affine systems[J]. IEEE Transactions on Automatic Control,2006,51(5):726-741.
[44]CHUDOUNG J, BECK C. The minimum principle for deterministic impulsive control system-s[C]//Proceedings of the 40th IEEE Conference on Decision and Control,2001,4:3569-3574.
[45]RIEDINGER P, IUNG C, KRATZ F. An optimal control approach for hybrid systems[J]. European Journal of Control,2003,9(5):449-458.
[46]ALAMIR M, ATTIA S A. An efficient algorithm to solve optimal control problems for nonlinear switched hybrid systems [C]//Proceedings of 6th IFAC symposium on nonlinear control systems,2004:417-422.
[47]ALAMTR M, ATTIA S A. On solving optimal control problems for switched hybrid nonlinear systems by strong variations algorithms[C]//Proceedings of 6th IFAC symposium on nonlinear control systems, 2004:558-563.
[48]GARAVELLO M, PICCOLI B. Hybrid necessary principle[J]. SIAM Journal on Control and Optimization, 2005,43(5):1867-1887.
[49]SHAIKH M S, CAINES P E. On the hybrid optimal control problem:Theory and algorithms[J]. IEEE Transactions on Automatic Control,2007,52(9):1587-1603.
[50]EGERSTEDT M, WARDI Y, AXELSSON H. Transition-time optimization for switched-mode dynamical sys-tems[J]. IEEE Transactions on Automatic Control,2006,51(1):110-115.
[51]GONZALEZ H, VASUDEVAN R, KAMGARPOUR M, et al. A descent algorithm for the optimal control of con-strained nonlinear switched dynamical systems[C]//Proceedings of the 13th ACM international confer-ence on Hybrid systems:computation and control,2010:51-60.
[52]AZHMYAKOV V, BASIN M V, RAISCH J. A proximal point based approach to optimal control of affine switched systems[J]. Discrete Event Dynamic Systems,2012,22(1):61-81.
[53]KAMGARPOUR M, TOMLIN C. On optimal control of non-autonomous switched systems with a fixed mode sequence[J]. Automatica,2012,48:1177-1181.
[54]GIUA A, SEATZU C, VAN DER MEE C. Optimal control of switched autonomous linear system- s[C]//Proceedings of the 40th IEEE Conference on Decision and Control,2001,3:2472-2477.
[55]GIUA A, SEATZU C, VAN DER MEE C. Optimal control of autonomous linear systems switched with a pre-assigned finite sequence[C]//Proceedings of the 2001 IEEE International Symposium on Intelligent Control (ISIC'01),2001:144-149.
[56]Xu X, ANTSAKLIS P J. Optimal control of switched autonomous systems[C]//Proceedings of the 41st IEEE Conference on Decision and Control,2002,4:4401-4406.
[57]EGERSTEDT M, WARDI Y, DELMOTTE F. Optimal control of switching times in switched dynamical sys-tems[C]//Proceedings of the 42nd IEEE Conference on Decision and Control,2003,3:2138-2143.
[58]Xu X, ANTSAKLIS P J. Optimal control of switched systems:New results and open problem-s[C]//Proceedings of the 2000 American Control Conference,2000,4:2683-2687.
[59]Xu X, ANTSAKLIS P J. An approach for solving general switched linear quadratic optimal control problems[C]//Proceedings of the 40th IEEE Conference on Decision and Control,2001,3:2478-2483.
[60]Xu X, ANTSAKLIS P J. Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions[J]. International Journal of Control,2002,75(16-17):1406-1426.
[61]Xu X, ANTSAKLIS P. Results and perspectives on computational methods for optimal control of switched systems[J]. Hybrid Systems:Computation and Control,2003:540-555.
[62]Xu X, ANTSAKLIS P J. Optimal control of switched systems based on parameterization of the switching instants[J]. IEEE Transactions on Automatic Control,2004,49(1):2-16.
[63]SHAIKH M, CAINES P. On the optimal control of hybrid systems:Optimization of trajectories, switching times, and location schedules[J]. Hybrid systems:Computation and control,2003:466-481.
[64]AXELSSON H, WARDI Y, EGERSTEDT M, et al. Gradient descent approach to optimal mode scheduling in hybrid dynamical systems[J]. Journal of Optimization Theory and Applications,2008,136(2):167-186.
[65]TEO K, GOH C, KH W. A unified computational approach to optimal control problems[M]. Longman New York,1991.
[66]TEO K, JENNINGS L, LEE H, et al. The control parameterization enhancing transform for constrained optimal control problems[J]. The Journal of the Australian Mathematical Society, Series B:Applied Mathematics,1999,40(03):314-335.
[67]LI R, TEO K, WONG K, et al. Control parameterization enhancing transform for optimal control of switched systems[J]. Mathematical and computer modelling,2006,43(11):1393-1403.
[68]LI R, FENG Z, TEO K, et al. Optimal piecewise state feedback control for impulsive switched systems[J]. Mathematical and Computer Modelling,2008,48(3):468-479.
[69]Wu C, TEO K. Global impulsive optimal control computation[J]. Journal of Industrial and Management Optimization,2006,2(4):435-450.
[70]LOXTON R, TEO K, REHBOCK V. Optimal control problems with multiple characteristic time points in the objective and constraints[J]. Automatica,2008,44(11):2923-2929.
[71]LOXTON R, TEO K L, REHBOCK V, et al. Optimal control problems with a continuous inequality constraint on the state and the control[J]. Automatica,2009,45(10):2250-2257.
[72]FARHADINIA B, TEO K, LOXTON R. A computational method for a class of non-standard time optimal control problems involving multiple time horizons[J]. Mathematical and Computer Modelling,2009, 49(7):1682-1691.
[73]LOXTON R C, TEO K L, REHBOCK V. Computational method for a class of switched system optimal control problems[J]. IEEE Transactions on Automatic Control,2009,54(10):2455-2460.
[74]LIN Q, LOXTON R, TEO K L, et al. Optimal control computation for nonlinear systems with state-dependent stopping criteria[J]. Automatica,2012,48:2116-2129.
[75]BENGEA S C, DECARLO R A. Optimal and suboptimal control of switching systems[C]//Proceedings of the 42nd IEEE Conference on Decision and Control,2003,5:5295-5300.
[76]BENGEA S C, DECARLO R A. Optimal control of switching systems[J]. automatica,2005,41(1):11-27.
[77]HOMANN T, TAG C, BIEBL H, et al. Fermentation of glycerol to 1,3-propanediol by Klebsiella and Cit-robacter strains[J]. Applied Microbiology and Biotechnology,1990,33(2):121-126.
[78]BIEBL H, MENZEL K, ZENG A P, et al. Microbial production of 1,3-propanediol[J]. Applied Microbiology and Biotechnology,1999,52(3):289-297.
[79]BIEBL H, MARTEN S, HIPPE H, et al. Glycerol conversion to 1,3-propanediol by newly isolated clostridi-a[J]. Applied Microbiology and Biotechnology,1992,36(5):592-597.
[80]MENZEL K, ZENG A P, DECKWER W D. High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae[J]. Enzyme and Microbial Technology, 1997,20(2):82-86.
[81]REIMANN A, BIEBL H, DECLWER W D. Production of 1,3-propanediol by Clostridium butyricum in continuous culture with cell recycling[J]. Applied microbiology and biotechnology,1998,49(4):359-363.
[82]ZEMG A P, BIEBL H. Bulk chemicals from biotechnology:The case of 1,3-propanediol production and the new trends[J]. Tools and Applications of Biochemical Engineering Science,2002:239-259.
[83]NAKAMURA C E, WHITED G M. Metabolic engineering for the microbial production of 1,3-propanediol[J]. Current opinion in biotechnology,2003,14(5):454-459.
[84]LAFFEND L A, NAGARAJAN V, NAKAMURA C E. Bioconversion of a fermentable carbon source to 1,3-propanediol by a single microorganism[P]. Google Patents,1997. US Patent 5,686,276.
[85]ZENG A P, Ross A, BIEBL H, et al. Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation[J]. Biotechnology and bioengineering, 1994,44(8):902-911.
[86]修志龙,曾安平,安利佳.甘油生物歧化过程动力学数学模拟和多稳态研究[J].大连理工大学学报,2000,40(4)：428-433.
[87]修志龙,曾安平,安利佳等.甘油连续生物歧化过程的过渡行为及其数学模拟[J].高校化学工程学报,2000,14(1)：53-58.
[88]GAO C, FENG E, WANG Z, et al. Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures[J]. Applied mathematics and computation,2005,169(1):476-484.
[89]高彩霞,王宗涛,冯恩民等.微生物间歇发酵生产1,3-丙二醇过程的辨识与优化[J].大连理工大学 学报,2006,46(5)：771-774.
[90]马永峰,孙丽华,修志龙.微生物连续培养过程中振荡的理论分析[J].工程数学学报,2003,20(1)：1-6.
[91]LIAN H, FENG E, LI X, et al. Oscillatory behavior in microbial continuous culture with discrete time delay[J]. Nonlinear Analysis:Real World Applications,2009,10(5):2749-2757.
[92]李晓红,冯恩民,修志龙.微生物连续培养过程平衡点的稳定分析[J].高校应用数学学报B辑,2005,20(4)：377-383.
[93]李晓红,冯恩民,修志龙.微生物间歇发酵非线性动力系统的性质及最优控制[J].运筹学学报,2005,9(4)：89-96.
[94]Li X, FENG E, XIU Z. Stability and optimal control of microorganisms in continuous culture[J]. Journal of Applied Mathematics and Computing,2006,22(1):425-434.
[95]GAO C, LANG Y, FENG E, et al. Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control[J]. Journal of Applied Mathematics and Computing,2005,19(1):203-214.
[96]GAO C, FENGE, WANG Z, et al. Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls[J]. Journal of Industrial and Management Optimization,2005, 1(3):377.
[97]YE J, FENG E, YIN H, et al. Modelling and well-posedness of a nonlinear hybrid system in fed-batch pro-duction of 1,3-propanediol with open loop glycerol input and pH logic control[J]. Nonlinear Analysis: Real World Applications,2011,12(1):364-376.
[98]LIU C, GONG Z, FENG E, et al. Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture[J]. Journal of Industrial and Management Optimization,2009, 5(4):835-850.
[99]WANGJ, SUN Q, FENG E. Modelling and properties of a nonlinear autonomous switching system in fed-batch culture of glycerol [J]. Communications in Nonlinear Science and Numerical Simulation,2012, 11:4446-4454.
[100]WANG L, FENG E, YE J, et al. Modeling and properties of nonlinear stochastic dynamical system of continuous culture[J]. Complex Sciences,2009:458-466.
[101]WANG L, FENG E, XIU Z. Nonlinear stochastic dynamical system of bio-dissimilation of glycerol to 1,3-propanediol in batch culture and its viable set[C]//Proceedings of the 3rd International Conference on Bioinformatics and Biomedical Engineering (ICBBE 2009),2009:1-4.
[102]WANGL, XIU Z, FENG E. A stochastic model of microbial bioconversion process in batch culture[J]. International Journal of Chemical Reactor Engineering,2011,9(1).
[103]沈邦玉.批式与连续发酵的非光滑动力系统辨识与优化控制[D].大连大连理工大学,2012.
[104]高井贵.一类微生物发酵的非线性混杂动力系统辨识及控制[D].大连大连理工大学,2012.
[105]SUN Y Q, QI W T, TENG H, et al. Mathematical modeling of glycerol fermentation by Klebsiella pneu-moniae:Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane[J]. Biochemical Engineering Journal,2008,38(1):22-32.
[106]SHEN B, LIU C, YE J, et al. Parameter identification and optimization algorithm in microbial continuous culture[J]. Applied Mathematical Modelling,2012,36(2):585-595.
[107]ZHANG Y, FENGE, XIU Z. Robust analysis of hybrid dynamical systems for 1,3-propanediol transport mechanisms in microbial continuous fermentation[J]. Mathematical and Computer Modelling,2011, 54(11):3164-3171.
[108]WANG J, YE J, FENG E, et al. Complex metabolic network of glycerol fermentation by Klebsiella pneu-moniae and its system identification via biological robustness[J]. Nonlinear Analysis:Hybrid Systems, 2011,5(1):102-112.
[109]陈国良.并行算法的设计与分析(修订版)[M].北京：高等教育出版社,2002.
[110]陈国良.并行算法—结构·算法·编程[M].北京：高等教育出版社,2003.
[111]陈国良,安虹,陈峻等.并行算法实践[M].北京：高等教育出版社,2004.
[112]王鹏,吕爽,聂治等.并行计算应用及实战[M].北京：机械工业出版社,2008.
[113]陈国良.并行算法研究进展[R].中国计算机学会通信,2005. http://www.ccf.org.cn/web/ resource/newspic/2005/9/20/bingxin.pdf.
[114]REIF J H. An optimal parallel algorithm for integer sorting[C]//Proceedings of the 26th Annual Symposium on Foundations of Computer Science,1985:496-504.
[115]GOLDBERG M, SPENCER T. A new parallel algorithm for the maximal independent set problem[J]. SIAM Journal on Computing,1989,18(2):419-427.
[116]ANDERSON R J, MILLER G L. A simple randomized parallel algorithm for list-ranking [J]. Information Processing Letters,1990,33(5):269-273.
[117]DONGARRA J J, SORENSEN D C. A fully parallel algorithm for the symmetric eigenvalue problem[J]. SIAM Journal on Scientific and Statistical Computing,1987,8(2):s139-s154.
[118]SORENSEN D C, TANG P T P. On the orthogonality of eigenvectors computed by divide-and-conquer techniques[J]. SIAM Journal on Numerical Analysis,1991,28(6):1752-1775.
[119]PARDALOS P. Parallel search algorithms in global optimization[J]. Applied Mathematics and Computa-tion,1989,29(3):219-229.
[120]CHEN Z, FEI P. A parallel algorithm for constrained optimization problems[J]. Journal of computational and applied mathematics,1995,61(2):225-230.
[121]CONFORTI D, MUSMANNO R. Parallel algorithm for unconstrained optimization based on decomposition techniques[J]. Journal of optimization theory and applications,1997,95(3):531-544.
[122]LIU C S, TSENG C H. Parallel synchronous and asynchronous space-decomposition algorithms for large-scale minimization problems[J]. Computational Optimization and Applications,2000,17(1):85-107.
[123]MIGDALAS A, TORALDO G, KUMAR V. Nonlinear optimization and parallel computing[J]. Parallel Com-puting,2003,29(4):375-391.
[124]SCHUTTE J, REINBOLT J, FREGLY B, et al. Parallel global optimization with the particle swarm algorithm[J]. International Journal for Numerical Methods in Engineering,2004,61(13):2296-2315.
[125]KOH B I, GEORGE A D, HAFTKA R T, et al. Parallel asynchronous particle swarm optimization[J]. Inter-national Journal for Numerical Methods in Engineering,2006,67(4):578-595.
[126]OZCAN E, ONBASIOGLU E. Memetic algorithms for parallel code optimization[J]. International Journal of Parallel Programming,2007,35(1):33-61.
[127]MAHDAD B, SRAIRI K, BOUKTIR T, et al. Fuzzy controlled parallel PSO to solving large practical e-conomic dispatch[C]//IECON 2010-36th Annual Conference on IEEE Industrial Electronics Society, 2010:2695-2701.
[128]CAO Y, SUN D. A Parallel Computing Framework for Large-Scale Air Traffic Flow Optimization[J]. IEEE Transactions on Intelligent Transportation Systems,2012,13(4):1855-1864.
[129]ZHANG Y, GALLIPOLI D, AUGARDE C. Parameter identification for elasto-plastic modelling of unsaturated soils from pressuremeter tests by parallel modified particle swarm optimization [J]. Computers and Geotechnics,2013,48(3):293-303.
[130]KONDOV I. Protein structure prediction using distributed parallel particle swarm optimization[J]. Natural Computing,2013,12:29-41.
[131]LUONG T, MELAB N, TALBI E. GPU computing for parallel local search metaheuristic algorithms[J]. IEEE Transactions on Computers,2013,62(1):173-185.
[132]KITANO H. Biological robustness[J]. Nature Reviews Genetics,2004,5(11):826-837.
[133]KITANO H. Towards a theory of biological robustness [J]. Molecular systems biology,2007,3(1).
[134]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京：科学出版社,2001.
[135]CLARKE F H, LEDYAEV Y S, STERN R J, et al. Nonsmooth analysis and control theory[M]. Vol.178, Springer New York,1998.
[136]HSIEH P F, SIBUYA Y. Basic theory of ordinary differential equations[M]. Springer,1999.
[137]BERKOVITZ L D. Optimal control theory[M]. Springer New York,1974.
[138]KIRK D E. Optimal control theory:An introduction[M]. Dover New York,1998.
[139]LUUS R, CHEN Y Q. Optimal switching control via direct search optimization [J]. Asian Journal of Control,2004,6(2):302-306.
[140]PAPAMICHAIL I, ADJIMAN C S. A rigorous global optimization algorithm for problems with ordinary differential equations[J]. Journal of Global Optimization,2002,24(1):1-33.
[141]SPINELLI W, BOLZERN P, COLANERI P. A note on optimal control of autonomous switched systems on a finite time interval[C]//2006 American Control Conference,2006:5947-5952.
[142]ZHANG L, CHEN Y, CUI P. Stabilization for a class of second-order switched systems[J]. Nonlinear Analysis:Theory, Methods & Applications,2005,62(8):1527-1535.
[143]BEMPORAD A, MORARI M. Control of systems integrating logic, dynamics, and constraints[J]. Automat-ica,1999,35(3):407-427.
[144]BUSKENS C, MAURER H. SQP-methods for solving optimal control problems with control and state constraints:adjoint variables, sensitivity analysis and real-time control [J]. Journal of computational and applied mathematics,2000,120(1):85-108.
[145]GERDTS M, KUNKEL M. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints[J]. Journal of Industrial and Management Optimization,2008,4(2):247-270.
[146]AZHMYAKOV V, RAISCH J. Convex control systems and convex optimal control problems with con-straints[J]. IEEE Transactions on Automatic Control,2008,53(4):993-998.
[147]ATTIA S, ALAMIR M, CANUDAS DE WIT C. Sub optimal control of switched nonlinear systems under location and switching constraints [C]//Proceedings of the 16th IFAC world congress, Prague, Czech Republick,2005.
[148]GAO R, LIU X, YANG J. On optimal control problems of a class of impulsive switching systems with terminal states constraints[J]. Nonlinear Analysis:Theory, Methods & Applications,2010,73(7): 1940-1951.
[149]HARTL R F, SETHI S P, VICKSON R G. A survey of the maximum principles for optimal control problems with state constraints[J]. SIAM review,1995,37(2):181-218.
[150]NOCEDAL J, WRIGHT S J. Numerical optimization[M]. Springer-Verlag New York,2006.
[151]ZENG A P, DECKWER W D. A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions[J]. Biotechnology progress,1995,11(1):71-79.
[152]ZENG A P. A kinetic model for product formation of microbial and mammalian cells[J]. Biotechnology and bioengineering,2004,46(4):314-324.
[153]YE J, FENG E, LIAN H, et al. Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures[J]. Applied Mathematics and Computation,2009,207(2): 307-318.
[154]WANG J, YE J, FENG E, et al. Modeling and identification of a nonlinear hybrid dynamical system in batch fermentation of glycerol[J]. Mathematical and Computer Modelling,2011,54(1):618-624.
[155]MA Y F, XIU Z L, SUN L H, et al. Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay[J]. International Journal of Nonlinear Sciences and Numerical Simulation,2006, 7(3):305-308.
[156]LI X, GUO J, FENG E, et al. Discrete optimal control model and bound error for microbial continuous fermentation[J]. Nonlinear Analysis:Real World Applications,2010,11(1):131-138.
[157]HAO J, LIN R, ZHENG Z, et al.3-Hydroxypropionaldehyde guided glycerol feeding strategy in aero-bic 1,3-propanediol production by Klebsiella pneumoniae[J]. Journal of industrial microbiology & biotechnology,2008,35(12):1615-1624.
[158]WANG S, FENG E. Stability of nonlinear microbial bioconversion system concerning glycerol's active transport and 1,3-PD's passive transport[J]. Nonlinear Analysis:Real World Applications,2010,11(5): 3501-3511.
[159]KENNEDY J, EBERHART R. Particle swarm optimization [C]//Proceedings of IEEE International Confer-ence on Neural Networks,1995,4:1942-1948.
[160]EBERHART R, SHI Y. Particle swarm optimization:Developments, applications and re-sources[C]//Proceedings of the 2001 Congress on Evolutionary Computation,2001,1:81-86.
[161]KARAKUZU C, et al. Fuzzy controller training using particle swarm optimization for nonlinear system control[J]. ISA transactions,2008,47(2):229-239.
[162]LIU L, LIU W, CARTES D A. Particle swarm optimization-based parameter identification applied to permanent magnet synchronous motors[J]. Engineering Applications of Artificial Intelligence,2008, 21(7):1092-1100.
[163]王凌,刘波.微粒群优化与调度算法[M].北京：清华大学出版社,2008.
[164]李丽,牛奔.粒子群优化算法[M].北京：冶金工业出版社,2009.
[165]Xu L, ZHANG F. Parallel particle swarm optimization for attribute reduction[C]//SNPD 2007 Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Par-allel/Distributed Computing,2007,1:770-775.
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