Discussion on: “Parameter-Dependent Lyapunov Function Approach to Stability Analysis and Design for Uncertain Systems with Time-Varying Delay”

From the paper by Cao and Xue that combines several techniques for contributing to LMI methods for uncertain time-delay systems, we choose to discuss the use of the so-called ”slack variables”. This technique has been extensively used these last six years since [9] and has proved to be quite useful. Among papers related to this technique we may point out those that clarify the involved mechanisms. In [13] the additional variables are proved to be related to the elimination lemma technique which is a corollary of the Finsler lemma. In [3] the additional variables are seen as Lagrange multipliers. In [5] they are obtained as a special case of parameterdependent multipliers involved in a new robust stability condition. In [11] they are proved to define a virtual system with identical properties as the one tested. In [8] they are proved to be multipliers involved in the generalized S-procedure. All these results help the comprehension of the technique and are good guides for possible extensions. Without anticipating the extensions, the discussion will concentrate on two limitations of the ”slack variables” technique as they appear to us currently.