Stability Analysis of Linear Time-Varying System via Adjustable Parameter-Based Functions

In recent years, researchers have focused on investigating the slack matrices-based function using appropriate polynomials. This approach has proven to be valuable in deriving tractable stability conditions expressed in terms of linear matrix inequalities. However, an open problem remains in addressing the inherent complexity of slack matrices, which has not been adequately considered thus far. To address this issue, a new approach is proposed in this paper, which involves the use of adjustable parameter-based functions (APFs) constructed using flexible polynomials. The introduction of APFs allows for the relaxation of not only the complex dimensions of slack matrices but also higher order time delays and zero components that exist in previous works. By leveraging APFs, two representative stability criteria are derived for linear delay systems. These stability conditions offer flexibility in selecting various permutations and combinations of flexible vectors within the APFs. The paper provides two illustrative examples to demonstrate the effectiveness of the obtained results.