Applications of small-sample statistical condition estimation in control

Standard approaches to measuring the condition of various problems in numerical linear algebra 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. We describe 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. The method has a rigorous statistical theory available for the probability of accuracy of the condition estimates. The theory of this approach is described along with several illustrative examples related to solving linear equations, linear least squares problems, and eigenvalue/eigenvector problems. Applications to problems of specific interest to control engineers are also discussed, including the solution of matrix Lyapunov, Sylvester, and Riccati equations.