Analysis of systems with state delay: A comparison system framework

Time-delay is an important source which can dramatically limit the system performance and even destabilize the closed loop system. The subject of time-delay systems is interesting but difficult as the system is infinite dimensional. The previously existing stability analysis results were mainly based on the Lyapunov's Second Method and may be very, conservative. In this research we have developed a systematic approach, the comparison system framework, for analysis of linear time-delay systems (LTDS). This framework is based upon the concept of value set covering for delay elements . In particular, the value set of an irrational delay element can be covered with that of a related dynamical system with real or complex uncertainties. The delay element can then be removed from the LTDS, resulting in a delay-free comparison system with some uncertainties. The robust stability of the comparison system then guarantees the asymptotic stability of the original LTDS. In this dissertation, we will first investigate several common, existing Lyapunov-based criteria for LTDS stability under our comparison system framework, and demonstrate that they are, in fact, equivalent to the robust stability analysis of a comparison system with complex uncertainties via the scaled small gain lemma. Next, we will develop several new, less conservative, delay-dependent conditions by using less conservative covering sets, such as a shifted disk and/or a filter. These conditions are formulated in terms of linear matrix inequalities (LMIs), and hence can be solved very efficiently. Then we will establish an inner and outer inclusion relation for the value set of the irrational delay element via a parameter-dependent Pade approximation. This relation leads to a simple, sufficient stability condition which guarantees an a priori degree of conservatism upper bound. This upper bound depends only on the order of Pade approximation used and can be reduced to any desired degree by increasing the order of the Pade approximation. We then demonstrate that this result can be generalized for the systems with multiple delays as well. Finally, we will extend this approach to LTDS with external disturbances and dynamical uncertainties, and establish sufficient conditions for Hinfinity performance and robust stability, respectively.