Positive realness and absolute stability problem of descriptor systems

This paper considers a class of nonlinear descriptor systems described by a linear time-invariant descriptor system with feedback-connected sector-constrained nonlinearities. First, we discuss the positive realness problem of descriptor systems and present a new version of positive real lemma. Second, we define the notion of strongly absolute stability (SAB) which is equivalent to the linear part is regular and impulsive-free and the overall feedback system is exponential stable and a SAB criteria in frequency domain is derived. Then, we address the problem of designing a state feedback controller such that the closed-loop feedback-connected system is SAB. To achieve this, we give a linear matrix inequality (LMI)-based SAB criteria, and the above problem is converted into an LMI feasibility problem. Finally, some numerical examples are given to illustrate our approach.