Abstract
After explaining what a dynamic response is, and how it is described, methods that have been suggested for the determination of the dynamic response of systems from normal operating data are reviewed. All are found to employ arbitrary constraint conditions which (i) are difficult to support as physically significant, (ii) are unsuitable for generalisation, (iii) cannot be tested for compatibility, and (iv) require the derivation of functions which cannot yet be adequately estimated from experimental records. Objections (i), (iii) and (iv) above make these methods suspect. Objections (i), (ii) and (iii) have been removed in this investigation by the adoption of 'least-squares' constraints, similar to, but more widely applicable than, those used by Wiener (1949) for the synthesis of filter and prediction operators. Since mathematicians, and others, are currently attempting to remove objection (iv) (which has occurred as a difficulty of many recent investigations) this has been considered as a subsidiary, and temporary,difficulty only. Least squares methods are developed for the selection of optimum linear, time-invariant, descriptions of n-port system dynamics. Examples are included demonstrating the application of these methods, and showing that linear dependence of input variables (such as occur in linear passive closed loop systems) present special difficulties. These difficulties are studied in detail, and methods of resolving them are suggested. A final chapter is devoted to a study of finite memory systems. The methods there developed are not subject to objection (iv) above. These methods, like those of the previous chapters, reveal that mathematical solutions of little or no value may result if the mathematical results are extremely sensitive to small changes (errors) in observed and estimated quantities. In such cases it is found that the data (not the method) are at fault. Such data are unsuitable for deriving the required information, and must be supplemented by the introduction of test disturbances before a realistic description of the dynamics can be obtained.