Parametric polynomial spectral factorization using the sum of roots and its application to a control design problem

Computer algebra for guaranteed accuracy. How does it help?

“…It is straightforward to show that the ideal of spectral factorization is O-dimensional and that the number of its zeros with multiplicities counted is 2 n . Furthermore, it can be shown [40] that b n -1 is generically a separating element [43]. These facts indicate that (t#) has a special Grabner basis.…”

“…(2) In the following, it is assumed that a2k E lR for k = 0, 1, ... ,n for the brevity of the exposition, but the approach can be generalized to the parametric case where a2k is some polynomial in parameters [40]. Assume without loss of generality that a2n > O.…”

“…Also important is that, of 2 n roots of Sj(b n -1 ) , the largest real root !J yields the spectral factor g(s) [40,42] and that no extra effort is required to choose from a set of possible solutions. In regard to computation efficacy, what is needed is basis conversion from one Grabner basis to another Grabner basis.…”

“…Efficient algorithms, e.g., [43,44], are available and one does not have to resort to a generally expensive Grabner basis computation method based on, e.g., Buchberger's algorithm. Lastly, this approach can be extended to the parametric case [40] due to the fact that all the manipulation required is algebraic operations.…”

Multiple Model Adaptive Control Design A Computer Algebra Approach

“…could also be shown to be a rational function of and Furthermore, because of the fact that the characteristic polynomial of is equivalent to ( ) ( ) ( ), and along with the characteristics of the ideal of spectral factorization, the can become an expression of polynomials in only [8]. Consequently, tend to be an expression of rational functions in .…”