Numerical Algorithms for Sensitivity Estimation in Control

小行星9633326 测量控制理论中各种问题的状态的标准方法，如解代数方程。 Riccati方程（ARE），将所有的灵敏度信息压缩到一个条件数中。 因此，在标准条件数不能准确反映溶液或溶液的特定条目的实际灵敏度的情况下，可能会发生信息丢失。 在这个项目中，我们将研究一种新的方法，克服这些和其他常见的缺陷。 新的程序措施的解决方案的输入数据中的小的随机变化的影响，并通过适当缩放的结果，获得计算的解决方案的每个条目的条件估计。这种方法，这是被称为小样本统计条件估计（SCE），适用于线性和非线性问题。 在前一种情况下（例如，当求解线性方程组或线性最小二乘问题时），计算量的显式弗雷歇导数是可用的。 因此，该方法是特别有效的，事实上，成本不超过标准normwise或componentwise估计。即使在非线性的情况下（例如，解决战神），相当大的效率时，迭代改进，说，牛顿的方法是可用的。SCE还具有相当大的灵活性。 例如，它可以很容易地适应对容许扰动的限制或与容许扰动相关的结构。该方法具有严格的统计理论，可用于条件估计的准确性概率。最后，它强调，SCE本身很容易直接执行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. ***