Nonlinear Systems and Numerical Methods for HJB Equations

HJB 方程的非线性系统和数值方法

• 批准号: 9971546
• 负责人: William McEneaney
• 金额: $ 9.2万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971546&HistoricalAwards=false
• 关键词: Nonlinear; Systems; Numerical; Methods; HJB

9971546McEneaneyThe research is proceeding along two directions within the general area of control and estimation for nonlinear systems. The first direction involves a new class of numerical methods for such problems. Specifically, note that one of the most persistent difficulties with control and estimation of significantly nonlinear systems has been computational cost. Dynamic programming methods for H-infinity control and risk sensitive stochastic control lead to nonlinear PDEs (typically Hamilton-Jacobi-Bellman equations) which must be solved over some region of the state-space. Since the number of state variables can be quite high, the dimension of the regions over which one needs to solve these PDEs increases. This restricts the set of problems for which one can compute such controls to those with very low state-space dimension. By changes in the underlying definitions of addition and multiplication certain classes of nonlinear Hamilton-Jacobi-Bellman equations are transformed into problems with linear semi-groups. This property has led us to new methods that exploit this linearity. For instance, certain steady-state problems lead to spectral methods for linear problems (over the new algebras). It is hoped that the possible improvements in speed with these methods will lead to an ability to solve a wider range of problems. Some simple examples are already being tested. The second direction involves risk averse limits in nonlinear control systems. It is now well known that risk averse limits of risk sensitive controllers lead to robust/H-infinity controllers. However, the classes of systems for which this has actually been proven is rather small, and does not include most typical problems. We have recently obtained some uniqueness results that should allow one to extend the risk sensitive limit results for infinite time-horizon problems to a reasonably wide class. Analogous work on risk sensitive limits in the estimation problem is beginning as well.Many modern systems, from active automotive hydro-suspension systems to fighter aircraft, behave in a significantly nonlinear manner. These systems require active control in order to perform at increasingly competitive levels. There is a serious technical challenge in that the design of nonlinear controllers requires extremely heavy computational loads. In fact, the computation of controllers where the number of system state variables is more than two or three has been unattainable in general. The difficulty is that these computations require the solution of a nonlinear partial differential equation over a space whose dimension is that of the number of state variables. A recent advance is the observation that, despite the nonlinearity, the propagation of these solutions is often still linear IF one switches to the max-plus algebra in place of the traditional definitions of addition and subtraction. In the max-plus algebra, traditional addition is replaced by maximization, and traditional multiplication is replaced by addition. One of the goals of this project is to exploit this max-plus linearity to extend the class of nonlinear systems for which we can design active controllers.

9971546 McEneaney的研究正在进行沿着两个方向内的一般区域的控制和估计的非线性系统。第一个方向涉及一类新的数值方法，这样的问题。具体地说，要注意的是，控制和估计显着的非线性系统的最持久的困难之一是计算成本。H ∞控制和风险敏感随机控制的动态规划方法导致非线性偏微分方程（通常是Hamilton-Jacobi-Bellman方程），必须在状态空间的某些区域上求解。由于状态变量的数量可能相当高，因此需要求解这些偏微分方程的区域的维数会增加。这限制了一组问题，其中一个可以计算这样的控制，那些具有非常低的状态空间维度。通过改变加法和乘法的基本定义，将某些非线性Hamilton-Jacobi-Bellman方程转化为线性半群问题。这一特性使我们找到了利用这种线性的新方法。例如，某些稳态问题导致线性问题的谱方法（在新代数上）。希望这些方法在速度上的可能改进将导致解决更广泛问题的能力。一些简单的例子已经在测试中。第二个方向涉及非线性控制系统中的风险规避限制。风险敏感控制器的风险规避极限导致鲁棒/H ∞控制器。然而，实际上已经证明这一点的系统类别相当小，并且不包括大多数典型问题。我们最近得到了一些唯一性的结果，应该允许一个扩展的风险敏感极限结果无限的时间跨度问题的一个合理的广泛的类。在估计问题中的风险敏感极限的类似工作也开始了。许多现代系统，从主动汽车液压悬架系统到战斗机，都表现为显著的非线性方式。这些系统需要主动控制，以便在日益具有竞争力的水平上运行。存在严重的技术挑战，因为非线性控制器的设计需要极其繁重的计算负荷。实际上，当系统状态变量的数目大于两个或三个时，控制器的计算一般是无法实现的。困难的是，这些计算需要解决的非线性偏微分方程的空间，其尺寸是状态变量的数量。最近的一个进展是观察到，尽管非线性，这些解决方案的传播往往仍然是线性的，如果一个开关的最大代数的地方，传统的定义的加法和减法。在极大代数中，用极大化代替了传统的加法，用加法代替了传统的乘法。这个项目的目标之一是利用这种最大加线性扩展类的非线性系统，我们可以设计主动控制器。