Stochastic modeling in hydrology and reliability theory, and optimal control of dynamical systems

水文学和可靠性理论中的随机建模以及动力系统的最优控制

• 批准号: RGPIN-2014-05273
• 负责人: Lefebvre, Mario
• 金额: $ 1.75万
• 依托单位: École Polytechnique de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679996
• 关键词: Stochastic; modeling; hydrology; reliability; theory

We mainly study three interrelated problems for dynamical systems in engineering.**1) Filtered renewal processes in hydrology. We have successfully used filtered Poisson processes to model river flows. However, our model implies that the time between events that significantly increase the river flow is a random variable having an exponential distribution. In reality, this hypothesis is generally false. Consequently, in order to improve the hydrological forecasts obtained from this model, we will generalize it by assuming that the flow evolves according to a filtered renewal process. We will have to derive new formulas to estimate the river flow. Furthermore, with an appropriate response function, filtered renewal processes are related to models used in queuing theory. We will make use of this fact to propose models that should yield even more accurate forecasts of river flows. In the case when analytical expressions cannot be derived, we will resort to simulations. The results that we will obtain are of great importance to dam managers, in particular.**2) Reliability. A basic problem in reliability theory is the determination of the distribution of the lifetime of a given product. To do so realistically, when the system is not repairable, the model used must be such that the remaining lifetime decreases with time. We want to use two-dimensional diffusion processes for this remaining lifetime, defined in such a way that the first component of the vectors is a deterministic decreasing function of the second component, which is a diffusion process. We will also consider other models for which the remaining lifetime decreases with time, such as conditioned one-dimensional diffusion processes and processes for which there is a reflecting boundary. We want to determine, in particular, the mean time required to reach a given boundary, that is, the mean-time-to-failure (MTTF). To do so, we must solve partial differential equations under the appropriate boundary conditions. Companies are interested in the MTTF in order to estimate the cost of the warranties they offer on their products. In the case of repairable systems, filtered renewal processes similar to the ones that we will use in hydrology, but with a different response function, can be proposed.**3) Stochastic optimal control. It is sometimes possible to obtain the optimal control of continuous stochastic processes until a random final time by making use of a theorem due to Whittle that enables us to express this optimal control in terms of a mathematical expectation computed for the corresponding uncontrolled processes. However, in order to apply Whittle's theorem, a certain relation between the control and noise matrices must be satisfied, which is rarely true in the most interesting applications. We want to solve this type of problems, sometimes called LQG homing, in the case when the relation in question is not satisfied. This entails solving the appropriate dynamic programming equation. Depending on the sign of a parameter in the cost function, the aim can be to minimize the time spent by the controlled process in the continuation region, or to maximize survival time instead. An application of these problems consists in computing the control that enables an aircraft to land optimally. Moreover, we want to extend LQG homing problems to the discrete-time case, which is often more realistic for the applications considered. Then, we will have to deal with nonlinear difference equations. Finally, we will also consider the problem of optimally controlling the filtered renewal processes used in hydrology and in reliability theory. In particular, we will determine the control that enables a dam manager to optimally release some water when the risk of flooding becomes too high.

我们主要研究工程动力学系统的三个相互关联的问题。** 1）水文中过滤的更新过程。我们已经成功使用了过滤的泊松工艺来建模河流。但是，我们的模型暗示，大大增加河流的事件之间的时间是一个随机变量，具有指数分布。实际上，这个假设通常是错误的。因此，为了改善从该模型获得的水文预测，我们将通过假设流量根据过滤的更新过程而演变来概括它。我们将不得不推出新的公式来估计河流。此外，具有适当的响应函数，过滤的续订过程与排队理论中使用的模型有关。我们将利用这一事实提出的模型，该模型应产生更准确的河流预测。在无法得出分析表达式的情况下，我们将诉诸模拟。我们将获得的结果非常重要，特别是大坝经理。** 2）可靠性。可靠性理论的一个基本问题是确定给定产品寿命的分布。实际上，当系统无法修复时，所使用的模型必须使剩余的寿命随时间减少。我们希望在此剩余的寿命中使用二维扩散过程，以这样的方式定义，以使向量的第一个组件是第二个组件的确定性减小函数，这是扩散过程。我们还将考虑其余寿命随时间降低的其他模型，例如有条件的一维扩散过程和过程，其中存在反射边界。我们要确定达到给定边界所需的平均时间，即平均时间到失败（MTTF）。为此，我们必须在适当的边界条件下求解部分微分方程。公司对MTTF感兴趣，以估算其产品提供的保修成本。在可维修系统的情况下，经过过滤的续订过程类似于我们在水文学中使用的过程，但是可以提出不同的响应函数。** 3）随机最佳控制。有时可以通过使用质定理来获得对连续随机过程的最佳控制，直到随机最后一次，这使我们能够根据计算的数学期望来表达此最佳控制，以计算出相应的不受控制的过程。但是，为了应用Whittle定理，必须满足控制和噪声矩阵之间的一定关系，这在最有趣的应用中很少是正确的。在不满足的关系的情况下，我们想解决这种类型的问题，有时称为LQG归巢。这需要解决适当的动态编程方程。根据成本函数中参数的迹象，目的可以是最大程度地减少持续区域中受控过程所花费的时间，或者最大化生存时间。这些问题的应用包括计算能够最佳降落的控制控制。此外，我们希望将LQG归纳问题扩展到离散的时间案例，这对于所考虑的应用程序通常更现实。然后，我们将不得不处理非线性差异方程。最后，我们还将考虑最佳控制水文和可靠性理论中使用的过滤续订过程的问题。特别是，当洪水泛滥的风险太高时，我们将确定能够使大坝经理能够最佳释放一些水的控制。