Numerical Algorithms for Sensitivity Estimation in Control

9633326 Laub标准方法用于测量控制理论中各种问题的条件，例如求解代数黎卡提方程(ARE)，将所有敏感性信息压缩到一个条件数中。因此，在此标准条件数不能准确反映解决方案或解决方案的特定条目的实际敏感性的情况下，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来克服这些和其他常见的缺陷。新的程序测量输入数据中的微小随机变化对解的影响，并通过适当地缩放结果，获得计算解的每一项的条件估计。这种方法被称为小样本统计条件估计，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，在解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，这种方法特别有效，而且，实际上，成本不超过标准的常态或分量估计。即使在非线性情况下(例如，求解Ares)，当通过例如牛顿方法进行迭代改进时，也获得了相当大的效率。姐妹会还具有相当大的灵活性的优势。例如，它很容易适应对允许扰动的限制或与之相关的结构。该方法对状态估计的准确率有严格的统计理论可用。最后强调指出，在许多现有的计算机辅助控制系统设计软件包中，该软件易于直接实现。***

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. ***