Implementation of a stability test of 1-D discrete system based on Schussler's theorem and some consequent coefficient conditions

✍ Scribed by V. Ramachandran; C.S. Gargour

• Publisher: Elsevier Science
• Year: 1984
• Tongue: English
• Weight: 845 KB
• Volume: 317
• Category: Article
• ISSN: 0016-0032
• DOI: 10.1016/0016-0032(84)90020-6

✦ Synopsis

Based on Schussler's theorem, some new properties ofpolynomials containing zeros inside the unit circle are obtained. These properties give rise to (i) a new stability test of 1-D discrete systems, and (ii) some necessary coeficient conditions that have to be satisfied by the denominator polynomial of a stable

1-D discrete system. Nomenclature f(j)(x) = z = jth derivative off(x) with respect to x f (j)( 1) = jth derivative off(x) evaluated at x = 1 T,(p) = nth order Chebyshev polynomial of first kind Lemma 1 If D(z) contains a factor d,(z) = z2 +a,z+ 1,O < Ia11 < 2, then both F,(z) and F2(z) also contain d,(z). ProoJ: Let D(z) = d,(z) D,(z). By rewriting d,(z) as zzdl(z-'), it is easily verified that both F,(z) and F,(z) contain d,(z).

Lemma 1 essentially means that any complex zero on the unit circle contained in D(z) will also be contained in both the polynomials F,(z) and F2(z).