Robustness analysis of control systems with disc uncertainties Zhizhen WANG 1 , Long WANG 2 , Wensheng YU 3 ( 1 . Department of Applied Mathematics , Shanghai Normal University ,Shanghai 200234 , China ; 2 . Department of Mechanics and Engineering Sciences , Peking University ,Beijing 100871 , China ; 3 . Laboratory of Complex Systems and Intelligence Sciences ,Institute of Automation , Chinese Academy of Sciences ,Beijing 100080 , China) Abstract : For a class of control systems with disc uncertainties , robust performance analysis is developed. On the strictly positive realness and H-infinity- norm of uncertain systems ,from the geometric point of view , two new sufficient and necessary conditions are given. The largest H-infinity- norm bound , containing the coefficients of only stable polynomials and centered at a nominal stable point in the coecient space is found. The results obtained in the paper are tractable and concise ,which is illustrated by some numerical examples. Keywor ds : Robustness ; Disc polynomial ; Strictly positive realness ; Hurwitz stability; H-infinity- norm 1 Introduction In practice , a single nominal plant model never exactly matches the true behavior of the plant. This is due to the model uncertainty or plant parameter variations. The former is called the unstructured uncertainty and the latter is called the structured uncertainty , i. e. the parametric uncertainty. Much attention is paid to the subject of analysis and synthesis of control systems under uncertainties , which is in the robust control framework. In recent years , rapid and spectacular developments have taken place in robust control[ 1 11 ] ,such as the Kharitonov Theorem[ 1 ] ,the Edge Theorem[ 2 ] and the Box Theorem[ 3 , 4 ]. As an alternative model of uncertainty , the disc polynomials correspond to transfer functions with complex coefficients. Such transfer functions make sense in itself right ,such as the model given in references [ 5 9 ] and references therein. It is important to have computational methods available to verify the stability of a family of polynomials with uncertainties [ 6 14 ]. With regarding the mapping of the uncertain parameters from their domains to the complex plane , some results have been developed to calculate the robust stability margin or to analyze robust performance of control systems with uncertainties [ 6 9 ]. Chapellat et al. give an equivalent condition for the robust stability of disc polynomials[ 10 ] ,and Polyak et al. provide a robust Popov criterion for disc polynomials[ 11 ]. Here , by means of mapping the whole family into the complex plane , we consider the robust strictly positive realness and H - norm of disc polynomials. Moreover , the largest weighted l radius in the parameter space centered at a specied stable point is given. Some examples are presented. 2 Notations and preliminaries In this paper , 1) R , C , R + denote the real , complex and positive real number sets , respectively ; 2) Re( ) and Im( ) correspond to the real ,image parts of ( ) ,respectively. deg( ) denotes the degree of ( ) ; 3) The standard H -norm of function f ( s) which is bounded and analytic in the complex open right half plane , is f = sup R f( j ) . And some definitions and several lemmas are introduced. Definition 1 A polynomial is called Hurwitz if all its roots belong to the left half plane. A family of polynomials is Hurwitz if every polynomial of them is Hurwitz. Definition 2 A rational function g(s) = n(s) d(s) is proper if deg (n(s)) deg (d(s)). Definition 3 A transfer function g(s) = n(s) d(s) is Received 14 March 2005 ; revised 14 August 2005. The research work was supported by National Natural Science Foundation of China (No. 60204006 , 69925307). Journal of Control Theory and Applications 4 ( 2005) 327 - 333