Convoluted Double Trouble

The Inversion Problem Our standard modeling and control concepts and techniques include linear convolution models and various transform techniques (the Laplace transform, the Z-transform, the bilateral Laplace transform, the bilateral Z-transform, the continuous Fourier transform, and the discrete Fourier transform). In fact, these techniques form the basis of the broad areas of circuits, communications, control, signals, and systems [1]-[5]. It is obviously beneficial for electrical engineering curriculum purposes to be able to treat these subjects in a unified framework. In communications and signal processing, doubly-infinite time and spatial axis signal descriptions are standard and convenient to use in many applications. It is for these reasons that doubly-infinite time axis, and the mathematically equivalent doubly-infinite spatial axis, formulations of estimation, filtering, and control problems are appealing [2], [5]-[10]. Unfortunately, there are some not so obvious pitfalls to be dealt with. Specifically, the standard mathematical machinery for modeling an unstable plant (and/or controller) as an unstable convolution operator or as an unstable transfer function runs into serious technical (mathematical) trouble in the full time axis case. That is, the basic model y Gu = , relating the plant output to the plant input via a time domain convolution or transfer domain product, is inconvenient to use for stabilization studies in the doubly-infinite time axis case and in the standard singly-infinite time axis case for noncausal systems. This means that the approach we usually teach in our basic control courses, which is pervasive in many basic and advanced textbooks in control, has limitations in those signal and system settings that dominate communications and signal processing. Would it perhaps be better to concentrate more on modeling and control concepts and techniques that generalize without difficulty to the signal and system settings that are standard in signal processing and communications? Erroneous treatments of various doubly-infinite time axis control problems using the y Gu = plant model continue to appear in the literature. Mixing unstable convolution operators (and their transfer functions) in the plant and/or controller modeling prohibits the use of standard rules for allowing the change of order of summation in multiple sums and integrals, thus making the analysis of full time axis problems with such operators almost intractable. It seems especially easy to make an erroneous treatment using transform techniques, as often due regard is not given to regions of convergence of the transform. There are cases where there is no region of convergence, and thus totally nonsensical results can be obtained if formal algebraic transform domain operations are used in the usual manner. For our purposes, it suffices to discuss discrete-time, first-order convolution systems. Consider the strictly causal, noise-corrupted convolution system