Adaptive control : stability, convergence, and robustness by Shankar Sastry

By Shankar Sastry

This quantity surveys the most important effects and methods of research within the box of adaptive keep watch over. concentrating on linear, non-stop time, single-input, single-output platforms, the authors provide a transparent, conceptual presentation of adaptive tools, permitting a serious evaluate of those innovations and suggesting avenues of extra improvement. 1989 variation

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If strict inequality holds, measuring RV Z decreases our uncertainty about the unknown RV X. 40) The PDF fX/Z (x/z) is normal with conditional mean X/Z and conditional covariance PX/Z . It is important to note that X/Z depends on the value of Z, but that covariance PX/Z is independent of the value of Z. Thus, PX/Z can be computed before we take any measurements. It is not difficult to redo this example for the case of nonorthogonal X and V. See the problems. 41) a. Express PX/Z and X/Z in terms of K.

Sequence σk is decreasing, indicating that each measurement ˆk . 2 31 Sequential Maximum-Likelihood Estimation Let us generalize our results by deriving a sequential maximum-likelihood estimator. 102) where X ∈ Rn , Z ∈ Rp , V ∈ Rp , V ∼ N (0, R), and |R| = 0. Recall that “maximum likelihood” implies ignorance of any statistics of X. 102 explicitly in terms of the components of vectors as       hT1 v1 z1  T v2   z2   h 2         ..  =  . X +  ..   .   ..    zp vp hTp where hTi is the ith row of H and suppose that R = diag{σi2 }.

Thus, X is deterministic and Z is stochastic. pdf 20/7/2007 12:35 20 Optimal and Robust Estimation The conditional PDF of Z given the unknown X, fZ/X (z/X), contains information about X, and if it can be computed, then X may be estimated according to the maximum-likelihood estimation criterion, which can be stated ˆ ML is as follows: Given a measurement Z, the maximum-likelihood estimate X the value of X which maximizes fZ/X , the likelihood that X resulted in the observed Z. The PDF fZ/X is a likelihood function.

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