Exact methodology for testing linear system software using idempotent matrices and other closed-form analytic results

Abstract

We alert the reader here to a variety of structural properties associated with idempotent matrices that make them extremely useful in the verification/validation testing of general purpose control and estimation related software. A rigorous general methodology is provided here along with its rationale to justify use of idempotent matrices in conjunction with other tests (for expedient full functional coverage) as the basis of a coherent general strategy of software validation for these particular types of applications. The techniques espoused here are universal and independent of the constructs of particular computer languages and were honed from years of experience in crosschecking Kalman filter implementations in several diverse commercial and military applications. While standard Kalman filter implementation equations were originally derived by Rudolf E. Kalman in 1960 using the Projection Theorem in a Hubert Space context (with prescribed inner product related to expectations), there are now comparable Kalman filter results for systems described by partial differential equations (e.g., arising in some approaches to image restoration or with some distributed sensor situations for environmental toxic effluent monitoring) involving a type of Riccati-like PDE equation to be solved for the estimation error. The natural framework for such infinite dimensional PDE formulations is within a Banach Space (being norm-based) and there are generalizations of idempotent matrices similar to those offered herein for these spaces as well that allow closed-form test solutions for infinite dimensional linear systems to verify and confirm proper PDE implementations in S/W code. Other closed-form test case extensions discussed earlier by the author have been specifically tailored for S/W verification of multichannel maximum entropy power spectral estimation algorithms and of approximate nonlinear estimation implementations of Extended Kalman filtering and for Batch Least Squares (BLS) filters, respectively.