Numerical Algorithms for Sensitivity Estimation in Control

9633326 Laub控制理论中测量各种问题条件的标准方法，例如求解代数Riccati方程（ARE），将所有灵敏度信息压缩到单个条件数中。因此，当标准条件数不能准确反映溶液的实际灵敏度或溶液的特定项时，就会发生信息丢失。在这个项目中，我们将研究一种克服这些和其他常见缺陷的新方法。新程序测量输入数据中的小随机变化对解的影响，并通过适当缩放结果，获得计算解的每个条目的条件估计。这种方法被称为小样本统计条件估计（SCE），它既适用于线性问题，也适用于非线性问题。在前一种情况下（例如，当解决线性方程组或线性最小二乘问题时），计算量的显式Frechet导数是可用的。因此，该方法特别有效，实际上，成本不超过标准的正态或组件估计。即使在非线性情况下（例如，求解AREs），当通过牛顿方法进行迭代改进时，也可以获得相当大的效率。SCE还具有相当大的灵活性。例如，它很容易适应允许摄动的限制或与之相关的结构。该方法对状态估计的准确度概率有严格的统计理论依据。最后，强调SCE本身很容易直接实现到许多现有的计算机辅助控制系统设计软件包。＊＊＊

9633326 Laub Standard approaches to measuring the condition of various problems in control theory, such as solving an algebraic Riccati equation (ARE), compress all sensitivity information into a single condition number. Thus, a loss of information can occur in situations in which this standard condition number does not accurately reflect the actual sensitivity of a solution or particular entries of a solution. In this project we shall investigate a new method that overcomes these and other common deficiencies. The new procedure measures the effects on the solution of small random changes in the input data and, by properly scaling the results, obtains condition estimates for each entry of a computed solution. This approach, which is referred to as small-sample statistical condition estimation (SCE), applies to both linear and nonlinear problems. In the former case (for example, when solving a system of linear equations or a linear least squares problem), an explicit Frechet derivative of the computed quantity is available. Thus the method is especially efficient and, in fact, costs no more than standard normwise or componentwise estimates. Even in the nonlinear case (for example, solving AREs), considerable efficiencies are gained when iterative improvement by, say, Newton's method is available. SCE also has the advantage of considerable flexibility. For example, it easily accommodates restrictions on, or structure associated with, allowable perturbations. The method has a rigorous statistical theory available for the probability of accuracy of the condition estimates. Finally,it is emphasized that SCE lends itself readily to straightforward implementation into many existing computer- aided control system design software packages. ***