Systems & Control Letters 39 (2000) 295–299 www.elsevier.com/locate/sysconle Vertex results for parametric H ∞ problems of weighted interval feedback systems Sen-Jian An a ; ∗ , Lin Huang b , Enping Wang c a Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, People’s Republic of China b Department of Mechanics & Engineering Science, Peking University, Beijing 100871, People’s Republic of China c Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China Received 12 September 1997; received in revised form 25 October 1999 Abstract This paper deals with the parametric H∞ problems of weighted interval feedback systems. It will be shown, by an example, that the maximal H∞ norm of the rst-order weighted sensitivity function does not enjoy the extremality property generally, but it is achieved at one of the 16 edge plants and this result can be generalized to the case of some high-order weighting functions. The conditions under which vertex results hold will be investigated nally. c  2000 Elsevier Science B.V. All rights reserved. Keywords: Vertex result; Weighting function; Sensitivity and complementary sensitivity function 1. Introduction Consider the following interval plant: G I : =  P(s): P(s)= N (s) D(s) ;N (s) ∈ N I ;D(s) ∈ D I  where N I and D I are interval polynomial families (m6n): N I : =  N (s)= m  i=0 b i s i ;b i ∈ [b - i ;b + i ]; 06i6m  ; D I : =  D(s)= s n + n-1  i=0 a i s i ;a i ∈ [a - i ;a + i ]; 06i6n − 1  : * Corresponding author. E-mail address: sjan@263.net (S. An) Let N K and D K denote the Kharitonov polynomial families (see [7]) associated with N I and D I , respec- tively, N K : = {N 1 (s);N 2 (s);N 3 (s);N 4 (s)}; D K : = {D 1 (s);D 2 (s);D 3 (s);D 4 (s)} where N 1 (s) : = b + 0 + b + 1 s + b - 2 s 2 + b - 3 s 3 + b + 4 s 4 +b + 5 s 5 + ··· ; N 2 (s) : = b - 0 + b - 1 s + b + 2 s 2 + b + 3 s 3 + b - 4 s 4 +b - 5 s 5 + ··· ; N 3 (s) : = b - 0 + b + 1 s + b + 2 s 2 + b - 3 s 3 + b - 4 s 4 +b + 5 s 5 + ··· ; N 4 (s) : = b + 0 + b - 1 s + b - 2 s 2 + b + 3 s 3 + b + 4 s 4 +b - 5 s 5 + ··· ; 0167-6911/00/$ - see front matter c  2000 Elsevier Science B.V. All rights reserved. PII:S0167-6911(99)00107-3