By Shankar Sastry

This quantity surveys the foremost effects and methods of research within the box of adaptive keep watch over. targeting linear, non-stop time, single-input, single-output structures, the authors provide a transparent, conceptual presentation of adaptive equipment, permitting a severe evaluate of those thoughts and suggesting avenues of extra improvement. 1989 version

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By checking the nonsingularity of certain matrices. This is often an unpleasant chore without a digital computer. Since p[C] = p[C C7] one may choose to find det (C C7) to check controllability. But this determinant, because of numerical limitations, may end up with a value other than zero even when the system is uncontrollable. Note that a transformation T results in det (C, CJ) = (det (T))2 det (C C 7 ) and a particular representation may yield a large' or 'small' value for the determinant depending on 'scaling' etc.

We shall discuss them in a later Section. 1 Definitions A system is said to be state controllable if for every tOi x(t0), tx it is possible to find an unconstrained W(T), t0 ^ x

The common denominator of the right hand side is det (si — Ac). 27 If and only if (A, C) is unobservable there is a pole-zero cancellation in C(sl-A)\ The following points regarding controllability/observability are worth noting: (i) If the system is uncontrollable, we can find a transformation T to bring it to the form of eqn. 26. The p eigenvalues of Ac (n — p of Ad) are the controllable (uncontrollable) modes. -space [{n —p) space] spanned by the eigenvectors of Ac (Ad) is called the controllable (uncontrollable) subspace of the state space.

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