Refined discretized Lyapunov functional method for systems with multiple delays

Abstract

SUMMARYThe previously proposed discretized Lyapunov functional method for systems with multiple delay is refined using variable elimination and Jensen inequality. The resulting new stability criterion is simpler. Numerical examples indicate that the new method is much less conservative for a given discretization mesh.

Stability of Time-Delay Systems

“…The simplification to the current form discussed here is based on Gu [90]. The expression of V(¢) in the proof of Proposition 7.8 can be generalized to The one used in the proof of Proposition 7.8 is the special case of the above by setting Z = W. For nonuniform mesh with large difference of mesh size, a stability criterion based on the above expression can be less conservative but much more complicated.…”

Structured invariant subspace and decomposition of systems with time delays and uncertainties

“…Definition 1. The transformation ( 14) and ( 15) is said to decompose the system described by ( 2) and (3) with the restriction (4) if (34) is satisfied in the resulting system described by (26) to (29) with feedback (18) and (19).…”

“…In order to demonstrate the effectiveness of this process, we will first compare the computational time for the DDE, CDDE‐RDC, and DCDDE for the cases of commensurate delays with $$$ {\tau}_2=2{\tau}_1 $$$ using appropriate versions of discretized Lyapunov–Krasovskii functional methods. Specifically, the method in Reference 26 is used for DDE, the method in Reference 14 is used for CDDE‐RDC and DCDDE. For the DDE case, $$$ {r}_1={\tau}_1 $$$ and $$$ {r}_2={\tau}_2-{\tau}_1 $$$ (which is equal to $$$ {r}_1 $$$ in this case) are evenly divided into three sections in discretization ($$$ {N}_{d1}={N}_{d2}=3 $$$).…”

“…In addition, we experimented the incommensurate delay case of $$$ {\tau}_2=\sqrt{3}{\tau}_1 $$$. The method in Reference 26 is still used for DDE, but the method in Reference 17 is used for CDDE‐RDC and DCDDE. In all cases, the interval $$$ {r}_1={\tau}_1 $$$ is evenly divided into three sections, and the interval $$$ {r}_2={\tau}_2-{\tau}_1 $$$ (which is equal to $$$ \left(\sqrt{3}-1\right){r}_1 $$$ in this case) is divided into two sections.…”

States observer design and stability analysis for networked control systems

“…Gu [4,5] proposed a discretized Lyapunov functional method for the stability analysis problem of multiple time-delay systems, which improved the restrict of time-delay to the stability mostly. Based on the method, Han [6,7] made a further improvement over the result of Gu and presented the sufficient condition of the stability criteria.…”