| In recent years development of best-basis representations has allowed advances in both signal compression and classification. This work introduces the concept to the field of optimal control and estimation theory. Our objective is to demonstrate a significant reduction in the computational resources required to solve optimal control and estimation problems. Via a best-basis approach we show how to rapidly search an exponential number of linear combinations of system states and controls to find those combinations that are "most significant" to the estimation and control processes. We then reduce the dimensionality of the representation by discarding those combinations with little impact. Computational efficiency is achieved by solving the control or estimation problem in the reduced dimensional space.; Although applicable to a wider class of control problems, we integrate the best-basis idea with control theory using the linear-quadratic controller as a platform. The discussion proceeds by providing a new solution to the linear-quadratic controller in a subspace of the original control space. By concentrating on control space dimensionality, we hope to evoke consideration of a new set of system designs that initially employ a large number of controls. Best-bases subsequently show how to make such designs computationally tractable.; In the case of estimation, we specifically target the discrete Kalman filter. Again using a best-basis approach, we construct an adaptation of the filter for which the state and potentially the measurement spaces are compressed. As illustrated with an example, dimensionality reduction of both the control and estimation problems can be quite significant.; Motivated by Kalman estimation, we develop a new methodology for the implementation of fast matrix algorithms using the non-standard form of best-basis representations. This development is a natural extension of that using wavelet bases. With best-bases, however, a more general class of matrices may be considered. We provide examples for matrix-matrix multiplication and generalized inversion using Schultz iteration. We also apply the techniques to Kalman estimation, offering an alternative solution to the use of best-bases to compress the state space. |
