Adaptive tracking with exponential stability and convolution bounds using vigilant estimation
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Adaptive tracking with exponential stability and convolution bounds using vigilant estimation Daniel E. Miller1
· Mohamad T. Shahab1
Received: 5 October 2019 / Accepted: 24 March 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract Classical discrete-time adaptive controllers provide asymptotic stabilization and tracking; neither exponential stabilization nor a bounded noise gain is typically proven. In our recent work, it is shown, in both the pole placement stability setting and the firstorder one-step-ahead tracking setting, that if the original, ideal, projection algorithm is used (subject to the common assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are restricted to that set) as part of the adaptive controller, then a linear-like convolution bound on the closed-loop behaviour can be proven; this immediately confers exponential stability and a bounded noise gain, and it can be leveraged to provide tolerance to unmodelled dynamics and plant parameter variation. In this paper, we solve the much harder problem of adaptive tracking; under classical assumptions on the set of unmodelled parameters, including the requirement that the plant be minimum phase, we are able to prove not only the linear-like properties discussed above, but also very desirable bounds on the tracking performance. We achieve this by using a modified version of the ideal projection algorithm, termed as vigilant estimator: it is equally alert when the plant state is large or small and is turned off when it is clear that the disturbance is overwhelming the estimation process. Keywords Adaptive control · Projection algorithm · Exponential stability · Bounded gain · Convolution bound
This research was supported by a grant from the Natural Sciences Research Council of Canada.
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Daniel E. Miller [email protected] Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada
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Mathematics of Control, Signals, and Systems
1 Introduction Adaptive control is an approach used to deal with systems with uncertain and/or timevarying parameters. In the classical approach to adaptive control, one combines a linear time-invariant (LTI) compensator together with a tuning mechanism to adjust the compensator parameters to match the plant. While adaptive control has been studied as far back as the 1950s, the first general proofs that parameter adaptive controllers work came around 1980, e.g. see [3,5,27,32,33]. However, the original controllers are typically not robust to unmodelled dynamics, do not tolerate time-variations well, have poor transient behaviour and do not handle noise (or disturbances) well, e.g. see [34]. During the following 2 decades, a good deal of research was carried out to alleviate these shortcomings. A number of small controller design changes were proposed, such as the use of signal normalization, deadzones and σ -modification, e.g. see [9,11,14,15,37]; arguably the simplest is that of using projection onto a co
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