A new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant (LTI) dynamics with multiple delays. These hypersurfaces, as they have very recently been introduced in a concept paper [R. Sipahi and N. Olgac, Automatica, 41 (2005), pp. 1413–1422], form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots (CTCR). In fact, these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine the kernel and offspring we use the extraordinary features of the “extended Kronecker summation” operation in this paper. The end result is that the infinite‐dimensional problem reduces to a finite‐dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper, we are able to shorten the computational time considerably in determining these hypersurfaces. We demonstrate these concepts via some example case studies. One of the examples treats a 3‐delay system. For this case another interesting perspective, called the “building block,” is also utilized to display the kernel in three‐dimensional space in the domain of “spectral delays.”

A trajectory-tracking problem is considered for a linear time invariant (LTI) dynamics with a fixed control law. However, the feedback line is affected by multiple time delays. The stability of the dynamics becomes a complex problem. It is well known that time-delayed LTI systems may exhibit multiple stable operating zones (which we call pockets) in the space of the delays. Our aim in this paper is to locate and experimentally validate these pockets. For the analytical determination of the pockets we utilize a new methodology, the cluster treatment of characteristic roots (CTCR). The study results in several interesting conclusions. (i) The systems may exhibit better control performance (for instance, faster disturbance rejection) for larger time delays. (ii) Consequently, we propose a unique and interesting utilization of the time delays as agents to enhance the control performance, the delay scheduling technique.

A novel, in situ, low-cost and facile method has been developed to fabricate flexible NO₂ sensors capable of operating at ambient temperature, addressing the urgent need for monitoring this toxic gas. This technique involves the synthesis of highly porous structures, as well as the specific development of laser-induced graphene (LIG) and its heterostructures with SnO₂, all through laser scribing. The morphology, phases, and compositions of the sensors were analyzed using scanning electron microscopy, X-ray diffraction, X-ray photoelectron spectroscopy and Raman spectroscopy. The effects of SnO₂ addition on structural and sensor properties were investigated. Gas-sensing measurements were conducted at room temperature with NO₂ concentrations ranging from 50 to 10 ppm. LIG and LIG/SnO₂ sensors exhibited distinct trends in response to NO₂, and the gas-sensing mechanism was elucidated. Overall, this study demonstrates the feasibility of utilizing LIG and LIG/SnO₂ heterostructures in gas-sensing applications at ambient temperatures, underscoring their broad potential across diverse fields.

The operable frequency range of the delayed resonators (DR) is known to be narrow due to stability issues. This study presents a novel approach for DR design with a combined feedback strategy that consists of a delayed velocity and nondelayed position feedback to extend the operable frequency range of the DR method. The nondelayed position feedback is used to alter the natural frequency of the DR artificially while delayed velocity feedback is employed to tune the frequency of DR matching with the undesired vibrations. The proposed method also introduces an optimization parameter that provides freedom for the designer to obtain fast vibration suppression while improving the stability range of the DR. An optimization approach is also provided within the scope of this study. Theoretical findings are verified over an experiment utilizing the active suspension system of the Quanser Company.