Control of Stochastic Systems

随机系统的控制

• 批准号: 1108884
• 负责人: Tyrone Duncan
• 金额: $ 33万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108884&HistoricalAwards=false
• 关键词: Control; Stochastic; Systems

The control of stochastic systems is the focus of this research and it provides a study that has wide applications to many areas of science and engineering as well as applications within mathematics. Based on empirical data from a wide variety of physical phenomena, the family of stochastic processes called fractional Brownian motions is appropriate for stochastic models of physical phenomena. However most of the members of this family of fractional Brownian motions do not possess the Markovian property or the semimartingale property that are usually assumed for stochastic models. Since most controlled stochastic systems in continuous time have been described with a noise modeled by a Brownian motion there is demand to study control problems for stochastic systems with other fractional Brownian motions. These control problems require significantly different methods for analysis than the well developed methods for the control of systems with Brownian motions. These major differences have resulted in having few results available for the optimal control of linear systems driven by an arbitrary fractional Brownian motion. The investigators for this grant have initiated a major study of these control problems for linear systems. This study should not only provide results for an arbitrary fractional Brownian motion but also provide results for the control of linear systems driven by processes from a general family of square integrable continuous stochastic processes. This work will develop further the investigators? initial work on finite time horizon, quadratic cost control of linear stochastic systems. It is planned to expand this study to other cost functionals and to other types of systems, both finite and infinite dimensional. Specifically this work is planned to determine explicit expressions for the optimal control and the optimal cost for an infinite time horizon ergodic (or long run average) quadratic cost functional. Furthermore it is planned to study the control of linear stochastic partial differential equations for both finite and infinite time horizon control problems with a quadratic cost functional. The noise stochastic process and the control for these equations are allowed to be restricted to the boundary of the domain. Typically the controlled stochastic systems have unknown parameters so it is necessary to identify the parameters and control the system simultaneously. This class of problems is called adaptive control and the investigators plan to extend their initial work on adaptive control for a scalar linear system to multidimensional linear systems with an arbitrary fractional Brownian motion. Many physical systems are controlled so the question of the determination of a best or optimal control arises. This area of research is called optimal control. Typically a mathematical model of a physical phenomena must account for perturbations of the system or unmodelled dynamics so a noise process is introduced in the model. These models are called stochastic systems. Given a controlled stochastic system and a cost for the use of control and the behavior of the system it is usually very difficult to obtain an explicit optimal control and the associated optimal cost. Furthermore the controlled stochastic systems in continuous time have been restricted to using one specific stochastic process, Brownian motion (white noise) as the noise. However empirical evidence from a wide variety of physical phenomena demonstrates a need for other noise processes, particularly for the family of fractional Brownian motions. The investigators for this grant have recently initiated a study of the control of linear systems with cost functionals that are quadratic in the system state and the control. They have obtained some results for explicit optimal controls for linear systems with an arbitrary fractional Brownian motion. These investigators plan to extend this work to more general linear systems and to infinite time horizon control problems with models that have an arbitrary fractional Brownian motion or a more general stochastic process. The optimal controls for these models can be used for control applications in many fields. The usefulness of fractional Brownian motions has been demonstrated in hydrology, telecommunications, turbulence, epilepsy and cognition as well as other areas. The optimal control results can be important in the use of effective controls for these physical phenomena. These results should have an important impact on many fields.

随机系统的控制是这项研究的重点，它提供了一项在科学和工程的许多领域以及数学领域应用中具有广泛应用的研究。 基于来自各种各样的物理现象的经验数据，称为分数布朗运动的随机过程家族适合于物理现象的随机模型。 然而，这个分数布朗运动族的大多数成员并不具有通常假设的随机模型的马尔可夫性质或半鞅性质。 由于大多数连续时间的受控随机系统都是用布朗运动模型的噪声来描述的，因此需要研究具有其他分数布朗运动的随机系统的控制问题。 这些控制问题需要显着不同的分析方法比发达国家的方法控制系统的布朗运动。 这些主要的差异导致了有几个结果可用于由任意分数布朗运动驱动的线性系统的最优控制。 该基金的研究人员已经开始了对这些线性系统控制问题的主要研究。 这项研究不仅提供了一个任意的分数布朗运动的结果，但也提供了结果的线性系统的控制过程从一般家庭的平方可积连续随机过程。 这项工作将进一步发展调查人员？线性随机系统有限时域二次成本控制初步研究 计划将这项研究扩展到其他成本泛函和其他类型的系统，包括有限维和无限维。 具体来说，这项工作计划确定显式表达式的最优控制和最优成本的无限时间范围遍历（或长期平均）的二次成本功能。 此外，它计划研究控制的线性随机偏微分方程的有限和无限的时间范围控制问题的二次成本功能。 允许噪声随机过程和对这些方程的控制被限制在域的边界上。 通常受控随机系统具有未知参数，因此需要同时辨识参数和控制系统。 这类问题被称为自适应控制和调查人员计划扩展他们的自适应控制的标量线性系统的多维线性系统的任意分数布朗运动的初步工作。许多物理系统是受控的，因此产生了确定最佳或最优控制的问题。 这个研究领域被称为最优控制。 通常，物理现象的数学模型必须考虑系统的扰动或未建模的动态，因此在模型中引入噪声过程。 这些模型被称为随机系统。 给定一个受控随机系统和使用控制的成本以及系统的行为，通常很难获得显式的最优控制和相关的最优成本。 此外，连续时间受控随机系统被限制为使用一个特定的随机过程，布朗运动（白色噪声）作为噪声。 然而，来自各种物理现象的经验证据表明需要其他噪声过程，特别是分数布朗运动家族。 这项研究的研究人员最近发起了一项研究的控制线性系统的成本泛函是二次的系统状态和控制。 他们得到了具有任意分数布朗运动的线性系统的显式最优控制的一些结果。 这些研究人员计划将这项工作扩展到更一般的线性系统和无限时域控制问题的模型，具有任意分数布朗运动或更一般的随机过程。 这些模型的最优控制可用于许多领域的控制应用。 分数布朗运动在水文学、通信、湍流、癫痫和认知等领域都有广泛的应用。 最优控制结果在对这些物理现象进行有效控制的过程中是非常重要的。 这些结果应该对许多领域产生重要影响。