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We focus on the proportional-integral-derivative (PID) controller design for linear time-delay systems. All the controller gains ($k_P$ , $k_I$ , and $k_D$) and the delay ($\tau$) are treated as free parameters and no particular constraints are imposed on the controlled plants. Such a problem (involving totally four free parameters) is of theoretical as well as practical importance, but, to the best of the authors’ knowledge, it has not been fully explored. First, we will develop an algebraic algorithm to solve the complete stability problem w.r.t. $\tau$. Consequently, for any given PID controller vector $(k_P, k_I, k_D)$ , the distribution of ${NU}(\tau)$ (${NU}(\tau)$ denotes the number of characteristic roots in the right-half plane, as a function of $\tau$) can be accurately obtained and the exhaustive stability range of $\tau$ may be automatically calculated. Next, a global understanding of the distribution of ${NU}(\tau)$ over the whole $(k_P, k_I, k_D)$ -space may be achieved and all structural changes regarding the ${NU}(\tau)$ distribution can be analytically determined. To achieve such a goal, a complete positive real root classification (for some appropriate auxiliary characteristic equation) will be explicitly proposed. Finally, we will give a new methodology, a new parameter-space approach, for determining the stability set in the $(k_P, k_I, k_D, \tau)$ -space. [ABSTRACT FROM AUTHOR]

This note focuses on the stability of some class of delay systems, including uncertainty in the delays. More precisely, we are interested in guaranteeing the stability of perturbed delay systems by assuming the stability of the nominal system. If the delay perturbation is constant, necessary and sufficient conditions are derived in terms of generalized eigenvalue distribution of some (finite-dimensional) constant matrix pencil. If the delay perturbation is time-varying, some sufficient stability conditions are derived using "exact" Lyapunov-Krasovskii functionals. Illustrative examples are also included. [ABSTRACT FROM AUTHOR]

Identifies the sufficient conditions for a linear system with point or distributed delay in the state to be passive. Clarification on passivity based on memoryless stabilization of linear delay systems; Lyapunov-Krasovskii functional construction.

Provides information on a ...memoryless control problem with supplementary ...-stability constraints for a class of linear systems with delayed state and disturbance inputs. Proposal of a linear matrix inequality (LMI) optimization for the considered problem; Conclusions reached.

We study stabilization problems for time-varying linear systems with constant input delays. Our reduction method ensures input-to-state stability with respect to additive uncertainties, under arbitrarily long delays. It applies to rapidly time-varying systems, and gives a lower bound on the admissible rapidness parameters. We also cover slowly time-varying systems, including upper bounds on the allowable slowness parameters. We illustrate our work using a pendulum model. [ABSTRACT FROM AUTHOR]