Design of delay-based output-feedback controllers optimizing a quadratic cost function via the delay Lyapunov matrix

Characterization and optimization of the smoothed spectral abscissa for time‐delay systems

“…We provide explicit formulae for the computation of U γ (0, p ) and its derivatives. Unlike the formulae introduced in the work of Gomez et al for the computation of the sensitivity of U γ (0, p ), it is worth mentioning that, here, no integral term is explicitly required and the computation of both the delay Lyapunov matrix and its sensitivity can be carried out in only one stage. We use the semianalytic approach, which consists of solving a delay‐free boundary value problem explicitly in terms of matrix exponentials, deduced from , and (see section 3.4 in the book of Kharitonov).…”

“…The smoothness property allows us to solve the optimization of the smoothed spectral abscissa problem within a gradient‐based optimization framework, where the delay Lyapunov matrix plays a key role since it enables us to obtain explicit expressions for the gradient computation. Unlike previous formulae of the sensitivity of the delay Lyapunov matrix with respect to the system parameters introduced in Gomez et al, here, we present a simpler characterization that avoids the numerical computation of some integral terms. It is worthy mentioning that the optimization of the smoothed spectral abscissa can be solved with standard optimization techniques, in contrast to the minimization of the spectral and pseudospectral abscissa, which are in general nonsmooth functions of the system parameters…”

“…Unlike previous formulae of the sensitivity of the delay Lyapunov matrix with respect to the system parameters introduced in Gomez et al, 18,19 here, we present a simpler characterization that avoids the numerical computation of some integral terms. It is worthy mentioning that the optimization of the smoothed spectral abscissa can be solved with standard optimization techniques, in contrast to the minimization of the spectral and pseudospectral abscissa, which are in general nonsmooth functions of the system parameters.…”

“…We illustrate the theoretical results applied to the stabilization of a triple integrator with a delay‐based feedback controller in Section 5, and we end the contribution with some final comments in Section 6. A preliminary version of this paper is presented in the work of Gomez and Michiels, where the proofs of Lemma , Theorem , and Theorem are only sketched, and the computation of the derivatives of the delay Lyapunov matrix with respect to the system parameters is carried out in line with the schemes proposed in the work of Gomez et al, which are different from the proposed scheme in Section 4, as previously explained.…”

Improved Results on Delay-Dependent Robust H ∞ Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

“…By applying a H ∞ state-feedback controller in [23], a neutral-type time-delay system was studied. And in another paper [24], the H ∞ feedback controller was used to obtain the results.…”