B.D.O. Anderson

Stability theorems for the relaxation of the strictly positive real condition in hyperstable adaptive schemes

The hyperstability theorems of Popov have played an important role in establishing the convergence of adaptive schemes, notably adaptive output error identification and adaptive control. The error system of these schemes has the form of a feedback loop with a time-invariant forward path and a passive time-varying feedback path. The strict positive realness of the forward path suffices to establish asymptotic stability of the feedback loop and therefore establishes convergence of the adaptive scheme. In this paper we study conditions which preserve the asymptotic stability but permit relaxation of the strict positive real condition at high frequencies, subject to restrictions on algorithm gain parameters and frequency content of the input signal. These theorems are important for the design of robust adaptive methods.

Caution in Iterative Modeling and Control Design 1

Abstract The Vinnicombe metric defining the gap between two plants, or two controllers, and its related robust stability results, are used as a tool to understand the need for cautious iterations (i.e. small controller modifications and, possibly, small model adjustments) that has been observed to be useful in iterative identification and control design. By the same token, these gap metric results allow one to compute controller updates that are smaller than would result from the optimal design based on the nominal design, and that guarantee stability of the actual closed loop system.

Exponential convergence of recursive least squares with exponential forgetting factor

This paper demonstrates that, provided the system input is persistently exciting, the recursive least squares estimation algorithm with exponential forgetting factor is exponentially convergent. Further, it is shown that the incorporation of the exponential forgetting factor is necessary to attain this convergence and that the persistence of excitation is virtually necessary. The result holds for stable finite-dimensional, linear, time-invariant systems but has its chief implications to the robustness of the parameter estimator when these conditions fail.

Adaptive robust control: online learning

A method of online adaptation and learning is proposed which makes use of a probing signal whose frequency content is concentrated at the bandwidth of the current controller. As the plant is learned the procedure naturally increases the learning bandwidth.<<ETX>>

Brief Recursive identification algorithms for continuous-time nonlinear plants operating in closed loop

A family of algorithms for the identification of continuous-time nonlinear plants operating in closed loop is presented. An adjustable closed-loop output error-type predictor parameterized in terms of the existing controller and the estimated plant model is used. The algorithms are derived from stability considerations in the absence of noise and assuming that the plant model is in the model set. Some convergence results based on passivity concepts are presented. Subsequently, the algorithms are analyzed in the presence of noise and when the plant model is not in the model set.

Local stable minima of the Sato recursive identification scheme

A common recursive identification scheme used in a class of adaptive systems problems involving blind channel equalization (but potentially usable elsewhere) is an algorithm due to Sato. The authors study the convergence properties of the Sato blind algorithm by characterizing the mean cost surface. The results show the important feature that the equalizer parameters may converge to parameter settings which fail to achieve the ideal objective which is to approximate the inverse with sufficient accuracy. A proof that a well-posed Sato algorithm can misbehave is presented. Examples are used to illustrate the results.<<ETX>>

Easily Testable Sufficient Conditions for the Robust Stability of Systems with Multiaffine Parameter Dependence

A number of robust stability problems take the following form: A polynomial has real coefficients which are multiaffine in real parameters that are confined to a box in parameter space. An efficient method is required for checking the stability of this set of polynomials. We present two sufficient conditions in this paper. They involve checking certain properties at the corners and edges of the parameter space box.

STOCHASTIC DYNAMICS OF BLIND DECISION FEEDBACK EQUALIZER ADAPTATION

Abstract An outline of the analysis of the stochastic dynamics of the blind adaptation of decision feedback equalizers (DFEs) operating on noiseless, binary communication channels is presented. The general mechanism whereby the blind adaptation is flawed in the sense that there exist attraction points in parameter space which lead to unequalized systems is defined and clarified with an example. This behaviour is predicted by defining the implicit error function whose gradient defines the direction of the drift of the adapting parameters. This analysis is complemented by an averaging analysis which also graphically illustrates the drift mechanism.

On a nonlinear generalization of the /spl nu/-gap metric

Presents a nonlinear extension of the /spl nu/-gap metric introduced in Vinnicombe (1993). Indeed, we present an input-output version of the generalized stability margin and the /spl nu/-gap metric. The notion of image representation as presented in van der Schaft (1996) allows one to use these alternative definitions in a nonlinear context for the derivation of a robust stability theorem.

Positive Real Conditions for Adaptive Control are Not Really Necessary

Abstract Model reference adaptive control with constrained complexity controllers is studied and stability results obtained. The use and advantages of regression vector filtering are also explained in connection with the stability argument.