Generalised absolute stability analysis and synthesis for Lur'e-type descriptor systems

Lur'e-type descriptor systems (LDS) described by a linear time-invariant descriptor system with feedback-connected nonlinearities are considered. Firstly, the notion of generalized absolute stability (GAS) is defined for LDS and a necessary and sufficient condition (NASC) is derived. It is shown that GAS for LDS implies that the nominal system is admissible and the overall system is globally exponentially stable. Then, a sufficient condition for a generalised absolutely stable LDS to be of index one and solvable is presented. Furthermore, a method to estimate the decay rate of the solutions to the system is obtained. Finally, the synthesis problem is addressed and an approach to designing a state-feedback controller that renders the closed-loop system is generalised absolutely stable with a given decay rate is given. A numerical example illustrates the approach.