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

By Shankar Sastry

This quantity surveys the main effects and methods of study within the box of adaptive keep an eye on. targeting linear, non-stop time, single-input, single-output structures, the authors provide a transparent, conceptual presentation of adaptive tools, allowing a severe overview of those strategies and suggesting avenues of extra improvement. 1989 variation

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The Gaussianity-whiteness GWx proposed herein reveals the underlying connection between the asymptotic variance (signal power) and entropy rate power (cf. 6) of an asymptotically stationary process. To be more specific, consider an asymptotically stationary process {xk } , xk ∈ R with asymptotic variance σx2 and entropy rate power N∞ (x). Then, σx2 GWx = N∞ (x) . 24) In the multivariate case, consider an asymptotically stationary process {xk } , xk ∈ Rm with asymptotic covariance matrix Σx and entropy rate power N∞ (x).

2. 2) where the infimum is taken over all possible densities pf (v) of the input process allowed for the channel with feedback. The difference between feedback blurredness and blurredness without feedback lies in that feedback blurredness allows the current input of the channel to depend on the past values of the output. Fig. 1 Definition 51 Fig. , Bf = B. ,k . In the inequality above, the equality holds if for i = 0, . . , the channel is without feedback. ,k = inf lim sup = B. p v k+1 k+1 k→∞ In other words, feedback does not decrease channel blurredness.

If {xk } , xk ∈ Rm is asymptotically stationary, then {yk } , yk = Axk , A ∈ m×m , yk ∈ Rm is also asymptotically stationary, where det A = 0. Moreover, we R have Φy (ω) = AΦx (ω) AT . It can then be shown that γy2 = γx2 . 22) In other words, the spectral flatness is invariant under linear transformation. We now introduce a new measure termed Gaussianity-whiteness, which is defined by combining negentropy rate and spectral flatness in a non-trivial way. 19 Consider an asymptotically stationary process {xk } , xk ∈ Rm with spectral flatness γx2 and negentropy rate J∞ (x).

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