Some extended Wirtinger׳s inequalities and distributed proportional-spatial integral control of distributed parameter systems with multi-time delays

In this paper, the framework of Lyapunov–Krasovskii functional for dealing with the problem of stability analysis and synthesis for time-delay systems is extended to address the problem of the time-delay distributed proportional-spatial integral (P-sI) control design for a class of linear distributed parameter systems with multi-time delays described by partial differential-integral equations (PDIEs). Initially, some extended Wirtinger׳s inequalities are presented based on the classic scalar Wirtinger׳s ones. Both delay-independent and delay-dependent design methods are subsequently developed by constructing integral types of Lyapunov–Krasovskii functional candidates and using the extended Wirtinger׳s inequality, and presented in terms of standard linear matrix inequality (LMI). Finally, the effectiveness and merit of the proposed control methodologies is demonstrated by a numerical example.