An improved method of analysis and design offers the potential to enhance the performances of adaptive feedforward tracking systems. The method can, in principle, make it possible to increase such performance measures as rates of convergence and noise-cancellation ratios, without need for additional hardware. The method is applicable to communication systems, adaptive equalizers, adaptive systems for canceling noise from machinery, advanced vibration-suppressing systems, and echo-canceling systems. The method is particularly significant for overparameterized adaptive feedforward systems, which are very important in practice; they are used to ensure high-performance "tuning" of key signals in a wide range of terrestrial and outer-space applications, including the ones listed above.

It is necessary to explain some mathematical terms as a prerequisite to a summary of the improved method. In some respects, this explanation resembles the one in the preceding article, "LTI Approach to Adaptive Narrow-Band Feedforward Control" (NPO-20184); however, there are some differences, making it necessary to give a separate explanation here.

As before, 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 below. The regressor vector is given by x = Xc(*t*), where X is an *N*-by-*n*matrix and c(*t*) is as described below. A system of the type to which the present analysis applies is one for which *N* > *n*; this is what is meant by "overparameterized" in the present context.

One constructs a suitable stable time-domain filter operator *F*(*p*), where *p *denotes the differential operator (replacing the Laplace *s *for all time-domain interpretations). The regressor vector x is filtered through *F*(*p*) to obtain *x̃*. The filtering operation is represented by

*x̃*= *F*(*p*)[x].

The basic equation of the adaptation algorithm is

w = *µ*Γ(*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.

The vector c(*t*) is not the same as in the preceding article. Instead, c(*t*) in this case is an *n*-dimensional vector of piecewise-continuous signals that satisfies a condition called the "persistent excitation" (PE) condition, defined as follows: Let there be positive constants β_{1}, β_{2}, and *T*_{0}, and a suitably dimensioned identity matrix, I, such that

for all *t* __>__. If any signal vector c(t) satisfies these criteria, it is said to be persistently exciting (PE) with bounds {β_{1},*β*_{2},*T*_{0}}. This completes the explanation of terms.

In the theoretical analysis on which the improved method is based, attention is focused on two matrix products; the confluence matrix (defined as X^{T}X) and the autocorrelation matrix (defined as XX^{T}). This analysis reveals that the older methods based on the autocorrelation matrix and associated PE conditions are overly stringent for ensuring exponential convergence of the tracking error in an overparameterized adaptive feedforward system. Indeed, when an adaptive system is overparameterized, the PE condition is never satisfied, and therefore the older methods do not provide any clue as to the performance of the final adaptive design, much less provide guidelines on how to improve performance.

The analysis reveals, further, that if the confluence matrix is positive definite, then the adaptive feedforward operator *H* from the error *e* to the estimate is input-output equivalent to an adaptive system with a PE regressor; this implies that the tracking-error convergence is exponential for a large class of overparameterized adaptive feedforward systems. Thus, the positive-definiteness condition for the confluence matrix, which is easily satisfied, supplants the positive-definiteness condition for the autocorrelation matrix, and thus constitutes the theoretical basis of the improved method. The analysis also shows that the only penalties for overparameterization are that (1) the optimal exponential rate of convergence of tracking error is degraded by the condition number of the confluence matrix and (2) the parameter errors converge exponentially on a reduced subspace rather than over the entire space.

*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. 120* on the TSP Order Card in this issue to receive a copy by mail ($5 charge). NPO-20183*

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

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