Stability Analysis and Stabilization of Linear Systems with Distributed Delays

Abstract

This thesis is devoted to the methods for the stability (dissipativity) analysis and stabilization of linear systems with non-trivial distributed delays based on the application of the Liapunov-Krasovskii functional (LKF) approach. We first propose methods for designing a dissipative state feedback controller for linear distributed delay systems, where the delay is known and the distributed delay kernels belong to a class of functions. The problem is solved by constructing a functional related to the distributed delay kernels via using a novel integral inequality. We subsequently extend the previous results to handle uncertain linear distributed delay systems, where the presence of linear fractional uncertainties is handled by a novel proposed lemma. Since integral inequalities play a vital role in utilizing the LKF approach, we then propose three general classes of novel integral inequalities, where relations and other properties are established in terms of inequality bound gaps. The proposed inequalities possess very general structures and generalize many existing integral inequalities in the delay-related literature. Next we propose a new method for the dissipativity and stability analysis of linear coupled differential-difference systems (CDDSs) with distributed delays of arbitrary L2 functions as its kernels. The distributed delay kernels are approximated by a class of functions including the option of Legendre polynomials. In addition, approximate errors are included by the resulting dissipativity (stability) condition via a matrix framework thanks to the application of a novel proposed integral inequality via the construction of an LKF. The previous results are then followed by a study of delay range analysis where the problem of dissipativity and stability analysis of a CDDS with distributed delays is considered with an unknown but bounded delay. By constructing a functional whose matrix parameters are dependent polynomially on the delay value, a dissipativity and stability condition can be derived in terms of sum-of-squares constraints. Finally, we present new methods for the dissipative synthesis of linear systems with non-trivial time-varying distributed delay terms where the value of the time-varying delay is only required to be bounded. The problem is solved by the LKF approach thanks to a novel proposed integral inequality.