经济管理中的博弈和控制方法及应用研究
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摘要
经济系统是一个开放的系统,必然会受到外部环境的影响。当经济系统中存在一定的参数摄动和不确定时,研究经济系统的鲁棒性问题是十分必要的。本文从解决实际经济问题的角度出发,建立了我国宏观经济系统模型,主要研究了将国际能源价格波动作为外部干扰的跳变时滞系统的鲁棒随机稳定性问题。
Markov随机跳变系统由于其深刻的实际背景近年来受到了广泛的关注。它虽然是一般线性系统在形式上的推广,但是由于它的结构更加的复杂,与一般线性系统有本质的区别。该系统可以描述众多的工程上的问题以及经济学问题,它的研究不仅具有理论意义,而且也具有实际的工程价值。这类复杂的跳变系统的状态向量中含有模态和状态两部分。模态表示系统的动态过程,是连续的变量,是由定义在有限状态空间上的连续Markov过程；系统的状态是离散的变量,每一模态下的系统状态是由一组微分方程描述。
本文首先以我国宏观经济系统为背景,研究了Markov跳变时滞系统的鲁棒随机稳定和保性能控制问题。为了解决随机系统的鲁棒随机稳定问题,基于时滞分解方法构造了一个全新的Lyapunov函数,应用随机微分算子的定义对其求微分,运用线性矩阵不等式(LMI)的性质给出了此类闭环系统保性能控制器存在的充分条件,设计出了鲁棒保性能控制器。通过数值算例说明了该方法的有效性。
然后,仍然以我国宏观经济系统模型为背景,研究了It6型跳变时滞系统的鲁棒随机稳定性问题。该系统中既有服从于布朗运动项又含有Markov跳变参数。通过添加自由权矩阵的方法本文得到了该系统鲁棒随机稳定的充分条件,将其化成求解一组线性矩阵不等式问题,并提出了设计状态反馈控制器的具体方法。该控制器保证了对于所有允许的不确定性,所得到的闭环系统都是鲁棒随机稳定的。
最后,本文将经济中的合作博弈理论应用到了鲁棒控制问题中,引入收益密度的概念,研究了以收益密度向量做为系统输出,对系统总收益进行利润分配的方法。基于核心的概念,提出了当核心存在时求解的公式以及当核心不存在时如何使得所有参与者满意的利润分配方法。本文不仅将核心这种合作博弈解的理论进行了完善,而且还将合作博弈与控制系统联系起来,使其更具有应用价值。
The economic system is an open system, it is easily affected by external environment. It is necessary to study on the problem of robust in the economic system when the economic system has some parameter perturbations and uncertain items. In order to solve the actual economic problem, the paper establishes the domestic macroeconomic system. Research the robust stochastic stabilization for jump system with time-delay. The price fluctuation of the international energy is regarded as the external disturbance in the system.
Markov stochastic jump system has been paid close extensive attention because of the profound practical background. Though it formally extends the general linear system, it has more complicated structure. There are essential differences between the general linear system and Markov jump system. The system could describe numerous engineering and economic problems. Not only does it have important theoretical significance but also it has practical engineering value. Jump systems are a kind of complicated systems with two components in their vector states:the modes and the states. The mode is the dynamic process and described by a continuous Markovian process with a finite state space. The state is discrete variable. The state in each mode is represented by a system of differential equations.
Firstly, this paper studies the problem of robust stochastic stability for the Markovian jump systems with time delay and guaranteed cost control in the domestic macroeconomic system. In order to solve the problem of robust stochastic stability of the stochastic system, a novel Lyapunov function is derived by delay partitioning approach. The paper uses the definition of the infinitesimal operator to differentiate Lyapunov function that has been derived along the system. Then applying the character of linear matrix inequalities (LMI), the sufficient condition of having the solution of the problem of the closed-loop system guaranteed cost control is given in the form of LMI and the robust guaranteed cost controller is designed. Illustrative example is provided to show the effectiveness.
Secondly, this paper studies the problem of robustly stochastic stability of Ito stochastic systems with time-varying delays, where the system is subjected to both Brownian motion and Markovian parameter jumping. By using the method of free-weighting matrices, we get the sufficient condition of robust stochastic stabilization. The sufficient condition is transformed into solving some linear matrix inequalities. Put forward the concrete methods to design a state feedback controller such that, for all admissible uncertainties, the closed-loop system is robustly stochastically stable.
Lastly, this paper applies the cooperate game theory in the economy to robust control theory. This article introduces the definition of income density and treat income density vector as system output, we study on the method of profit distribution in the system total income. Based on the definition of core, the author presents a formula for calculation when the core exists. When the core is an empty set, the paper also gives the method of how to get the profit distribution that can satisfy all the players. This article not only improves the theory of solving cooperate game but also contacts cooperate game with control system, so this theory has more practical value than before.
Markov随机跳变系统由于其深刻的实际背景近年来受到了广泛的关注。它虽然是一般线性系统在形式上的推广,但是由于它的结构更加的复杂,与一般线性系统有本质的区别。该系统可以描述众多的工程上的问题以及经济学问题,它的研究不仅具有理论意义,而且也具有实际的工程价值。这类复杂的跳变系统的状态向量中含有模态和状态两部分。模态表示系统的动态过程,是连续的变量,是由定义在有限状态空间上的连续Markov过程；系统的状态是离散的变量,每一模态下的系统状态是由一组微分方程描述。
本文首先以我国宏观经济系统为背景,研究了Markov跳变时滞系统的鲁棒随机稳定和保性能控制问题。为了解决随机系统的鲁棒随机稳定问题,基于时滞分解方法构造了一个全新的Lyapunov函数,应用随机微分算子的定义对其求微分,运用线性矩阵不等式(LMI)的性质给出了此类闭环系统保性能控制器存在的充分条件,设计出了鲁棒保性能控制器。通过数值算例说明了该方法的有效性。
然后,仍然以我国宏观经济系统模型为背景,研究了It6型跳变时滞系统的鲁棒随机稳定性问题。该系统中既有服从于布朗运动项又含有Markov跳变参数。通过添加自由权矩阵的方法本文得到了该系统鲁棒随机稳定的充分条件,将其化成求解一组线性矩阵不等式问题,并提出了设计状态反馈控制器的具体方法。该控制器保证了对于所有允许的不确定性,所得到的闭环系统都是鲁棒随机稳定的。
最后,本文将经济中的合作博弈理论应用到了鲁棒控制问题中,引入收益密度的概念,研究了以收益密度向量做为系统输出,对系统总收益进行利润分配的方法。基于核心的概念,提出了当核心存在时求解的公式以及当核心不存在时如何使得所有参与者满意的利润分配方法。本文不仅将核心这种合作博弈解的理论进行了完善,而且还将合作博弈与控制系统联系起来,使其更具有应用价值。
The economic system is an open system, it is easily affected by external environment. It is necessary to study on the problem of robust in the economic system when the economic system has some parameter perturbations and uncertain items. In order to solve the actual economic problem, the paper establishes the domestic macroeconomic system. Research the robust stochastic stabilization for jump system with time-delay. The price fluctuation of the international energy is regarded as the external disturbance in the system.
Markov stochastic jump system has been paid close extensive attention because of the profound practical background. Though it formally extends the general linear system, it has more complicated structure. There are essential differences between the general linear system and Markov jump system. The system could describe numerous engineering and economic problems. Not only does it have important theoretical significance but also it has practical engineering value. Jump systems are a kind of complicated systems with two components in their vector states:the modes and the states. The mode is the dynamic process and described by a continuous Markovian process with a finite state space. The state is discrete variable. The state in each mode is represented by a system of differential equations.
Firstly, this paper studies the problem of robust stochastic stability for the Markovian jump systems with time delay and guaranteed cost control in the domestic macroeconomic system. In order to solve the problem of robust stochastic stability of the stochastic system, a novel Lyapunov function is derived by delay partitioning approach. The paper uses the definition of the infinitesimal operator to differentiate Lyapunov function that has been derived along the system. Then applying the character of linear matrix inequalities (LMI), the sufficient condition of having the solution of the problem of the closed-loop system guaranteed cost control is given in the form of LMI and the robust guaranteed cost controller is designed. Illustrative example is provided to show the effectiveness.
Secondly, this paper studies the problem of robustly stochastic stability of Ito stochastic systems with time-varying delays, where the system is subjected to both Brownian motion and Markovian parameter jumping. By using the method of free-weighting matrices, we get the sufficient condition of robust stochastic stabilization. The sufficient condition is transformed into solving some linear matrix inequalities. Put forward the concrete methods to design a state feedback controller such that, for all admissible uncertainties, the closed-loop system is robustly stochastically stable.
Lastly, this paper applies the cooperate game theory in the economy to robust control theory. This article introduces the definition of income density and treat income density vector as system output, we study on the method of profit distribution in the system total income. Based on the definition of core, the author presents a formula for calculation when the core exists. When the core is an empty set, the paper also gives the method of how to get the profit distribution that can satisfy all the players. This article not only improves the theory of solving cooperate game but also contacts cooperate game with control system, so this theory has more practical value than before.
引文
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[2]高莹.金融系统鲁棒优化[M].北京：经济科学出版社,2008,1-27.
[3]谢胜智.数理经济学[M].成都：西南财经大学出版社,2004,12：148-177.
[4]H. Markowitz. Portfolio Selection:Efficient Diversification of Investments[M]. New York:Wiley,1959.
[5]Wei Chen, Shaohua Tan. Robust portfolio selection based on asymmetric measures of variability of stock returns[J]. Journal of Computational and Applied Mathematics, 2009,232(2):295-304.
[6]Roll R.. A mean-variance analysis of tracking error[J]. Journal of Portfolio Management,1992,19(4):13-22.
[7]Anna Grazia Quaranta, Alberto Zaffaroni. Robust optimization of conditional value at risk and portfolio selection[J]. Journal of Banking & Finance,2008,32(10):2046-2056.
[8]Dashan Huang, Shu-Shang Zhu, Frank J. F., etc. Porfolio selection with uncertain exit time:A robust CVaR approach[J]. Journal of Economic Dynamics & Control,2008,32:594-623.
[9]郑立辉.基于鲁棒控制的期权定价方法[J].管理科学学报,2000,3(3)：60-64.
[10]刘海龙,吴冲锋.基于鲁棒控制的期权套期保值策略[J].控制与决策,2001,16(6)：974-976.
[11]黄小原.H∞控制方法在证券组合问题中的应用[J].控制与决策,1998,13(1)：49-53.
[12]黄小原,钟麦英.辽宁省宏观经济模型及其H∞控制[J].控制理论与应用,2000,17(5)：781-788.
[13]黄小原,钟麦英.具有不等式约束的H∞控制及其在证券投资问题中的应用[J].信息与控制,1999,28(6)：475-478.
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