By Torsten Soderstrom, Petre Stoica, P. G. Stoica
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Example text
E . 4. 4. 17) Clearly R is symmetric and at least nonnegative definite. Conditions for R to be positive definite (and hence nonsingular) are basically that the model is not overparameterized ( i . e . A5 holds) and that the input is persistently exciting. 1. 9), becomes 0 = E~(t)v(t) which apparently is true due to Assumption A7. 28 I t should be pointed out that this IV variant, although of theoretical interest as we w i l l see l a t e r , cannot be used in a straightforward way, The reason is of course that T(t) is not known.
Input signal the probability for rank R : ne is one. 2). Similarly, for given f i l t e r s K(q-1), C(q-1)/D(q "I) and B*(q-1)/A*(q - I ) we can randomly choose the parameters of an ARMA input process and s t i l l have rank R = ne with probability one. 2). As mentioned previously there exist though cases where the matrix R does not have ful] rank. We now il]ustrate this fact with the following example. 13). Assume that the general assumptions apply, and let K(q-I) = I, na = na* = 2, nb= nb*= I.
13a). 15). 1 that S(-C,D) and sT(-B*,A*) are nonsingular. 8 that the middle matrix P is nonsingular i f i ) , and I) or II) are satisfied. This completes the proof. • Ramaut~ I. Note that i f K(q"I) = I, na* = I, nd = I then the positive realness condition is always f u l f i l l e d . This is seen as follows. -la*I)(1-~d) > 0 (1+(dl Remo~k2. 8 that the result of the theorem remains true also i f u(t) is an AR process of an order not exceeding na-na*+nb. Rama~k 3. Part (II) suggests a way of choosing the f i l t e r s .
