A unified theory of linear time-invariant (LTI) representations has been developed for use in the analysis and design of adaptive, narrow-band, feedforward controllers for systems characterized by harmonic regressors (the term "harmonic regressor" is defined below). Such controllers are used, for example, to suppress harmonic noise and vibrations; the basic idea is to track disturbance tones actively and "notch" them by use of high-gain feedback to achieve large attenuation. The LTI approach embodied in the unified theory makes it possible to apply established LTI theoretical concepts to the adaptive-feedforward-control design problem, and thus to design stable, high-performance controllers while avoiding the complexities and risks associated with nonlinear and/or time-varying control loops.

It is necessary to explain some mathematical terms as a prerequisite to a summary of the unified theory. The figure schematically illustrates the relationships among the terms.

As part of the control problem, one seeks to obtain an estimate, *ŷ*, of some signal *y*(*t*), where (*t*) signifies dependence on time. One constructs *ŷ* as a linear combination of the elements of a regressor vector x(*t*); namely, *ŷ* = w(*t*)^{T}x(*t*), where T denotes the matrix transpose and w(*t*) is an *N*-dimensional parameter vector that is "tuned" in real time via an adaptation algorithm, the basic equation of which is described in the paragraph after the next one.

A system of the type to which this analysis applies is characterized by a harmonic regressor, among other things. A harmonic regressor is given by x = Xc(*t*), where X is an *N*-by-2*m* matrix and c(*t*) is a 2*m*-dimensional vector of paired sinusoids and cosinusoids of arbitrary frequencies; namely,

c(*t*) = [sin(ω_{1}*t*),cos(*w*_{1}*t*), . . . , sin(ω* _{m}t*), cos(ω

*)]*

_{m}t^{T}.

The regressor vector x is filtered through *F*(*p*) to obtain x̃. The filtering operation is represented by

x̃ = *F*(*p*)[x].

where *p *denotes the differential operator (replacing the Laplace *s*). The basic equation of the algorithm is

w = *µ*G(*p*)[x̃(*t*)*e*(*t*)],

where *µ *> 0 is an adaptation gain, *e*(*t*) is an error signal, Γ(*p*)[·] denotes a filtering operation in the same way that *F*(*p*)[·] above denotes a filtering operation, and Γ(*p*) is a multivariable LTI transfer function of the designer's choosing, representing the desired adaptation law.

Taken together, the foregoing equations define an open-loop mapping from the error signal *e* to the estimated output *ŷ*. The mapping is represented by *ŷ* = *H*[*e*], where *H* denotes the mapping operator. In general, *H* is a linear time-varying (LTV) operator, and this is where the unified theory makes a major contribution: The theory includes a theorem that provides a necessary and sufficient condition under which the overall effect of *H* can be represented exactly by an LTI operator, even though *H* may include time-varying elements. The condition is a result of the modulation and demodulation properties of products of the sinusoids contained in the harmonic regressors. The condition, called the "X-orthogonality" (XO) condition, is that the elements of X must be chosen so that X^{T}X is a block-diagonal matrix consisting of 2 × 2 matrices proportional to positive identity matrices. When the XO condition is satisfied, the LTI representation is specified by a closed-form analytic expression.

The theory is characterized as "unified" because it produces, as special cases, all known instances of LTI adaptive feedforward systems previously reported in the literature. The theory generalizes the previous results by showing this class of LTI adaptive systems to be considerably larger than it was known to be. The theory has also been specialized to adaptive systems, the regressors of which are formed by filtering signals through tapped delay lines (TDLs). TDLs can be implemented easily and are used in diverse adaptive signal-processing applications. It has been shown (under mild conditions) that an adaptive system becomes asymptotically LTI as the number of taps in the TDL is increased.

* This work was done by David S. Bayard of Caltech for *NASA's Jet Propulsion Laboratory.* For further information,* *access the Technical Support Package (TSP) **free on-line at www.techbriefs.com** under the Mathematics and Information Sciences category, *or circle no. 147* on the TSP Order Card in this issue to receive a copy by mail ($5 charge). NPO-20184*

##### NASA Tech Briefs Magazine

This article first appeared in the March, 1998 issue of *NASA Tech Briefs* Magazine.

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