Ismail Ilker Delice

Delay-Independent Stability Test for Systems With Multiple Time-Delays

Delay-independent stability (DIS) of a general class of linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in the entire delay-parameter space. Stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays. We build an analytical approach that requires the inspection of the roots of finite number of single-variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays, excluding infinite delays. The approach enables to test the necessary and sufficient conditions of DIS of LTI-MTDS, technically known as weak DIS, as well as the robust stability of single-delay systems against all variations in delay ratios. The proposed approach, which does not require any parameter sweeping and graphical display, becomes possible by establishing a link between the infinite spectrum and algebraic geometry. Case studies are provided to show the effectiveness of the approach.

Advanced Clustering With Frequency Sweeping Methodology for the Stability Analysis of Multiple Time-Delay Systems

A new methodology, advanced clustering with frequency sweeping (ACFS), is introduced for the stability analysis of the most general class of linear time-invariant time-delay systems with multiple delays. ACFS is a cross fertilization of algebraic geometry, frequency sweeping technique and the root clustering paradigm, and it can directly extract the 2-D cross sections of the stability views in any two-delay domain. In this domain, ACFS can also test the delay-independent stability based on necessary and sufficient conditions, and can compute, for the delay-dependent case, the precise lower and upper bounds of the only parameter, the frequency, that ACFS sweeps. Case studies that are prohibitive to analyze with the existing analytical methods are presented to demonstrate the strengths of the method.

Controller Design for Delay-Independent Stability of Multiple Time-Delay Systems via DéScirctes's Rule of Signs

Abstract A general class of multi-input linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in order to obtain a control law which stabilizes the LTI-MTDS independently of all the delays. The method commences by reformulating the infinite-dimensional analysis as a finite-dimensional algebraic one without any sacrifice of accuracy and exactness. After this step, iterated discriminant allows one to construct a single-variable polynomial, coefficients of which are the controller gains. This crucial step succinctly formulates the delay-independent stability (DIS) condition of the controlled MTDS based on the roots of the single-variable polynomial. Implementation of the DeScirctess rule of signs then reveals, without computing these roots, the sufficient conditions on the controller gains to make the LTI-MTDS delay-independent stable. Case studies are provided to demonstrate the effectiveness of the proposed methodology.

Advanced clustering with frequency sweeping (ACFS) methodology for the stability analysis of multiple time-delay systems

A novel methodology, Advanced Clustering with Frequency Sweeping (ACFS), is introduced for the stability analysis of the most general class of linear time-invariant (LTI) time-delay systems (TDS) with multiple delays. Different from the literature, ACFS does not impose any restrictions in system order, the number of delays and the ranks of the system matrices in the LTI-TDS considered. An unmatched strength of ACFS is that it can achieve to directly extract the 2D cross-sections of the stability views in the domain of any of the two delays. ACFS owes this superiority to an elegant way of cross-fertilizing the resultant theory, frequency sweeping technique, and the root clustering paradigms. ACFS also proposes a new formula that can compute the precise lower and upper bounds of the only parameter, the frequency, that it sweeps. Case studies that are prohibitive to analyze with the existing methods are presented to demonstrate the strengths of ACFS.

Exact Upper and Lower Bounds of Crossing Frequency Set and Delay Independent Stability Test for Multiple Time Delayed Systems

Abstract A general class of linear time invariant (LTI) multiple time-delay system is investigated in order to (i) obtain exact lower and upper bounds of crossing frequency set (CFS), and (ii) test the necessary and sufficient conditions of delay independent stability (DIS). The method commences by deploying Rekasius substitution for the transcendental terms in the characteristic function, reducing it into a finite dimensional one. After substitution, utilization of elimination theory allows one to achieve the two nontrivial objectives (i)-(ii). The approach is new and novel as it does not require any parameter sweeping and graphical display; it is exact and can test necessary and sufficient conditions of DIS over only a single variable polynomial. A case study is provided to show the effectiveness of the proposed method.

Inventory Dynamics Models of Supply Chains with Delays; System-Level Connection & Stability

This paper surveys some deterministic continuous-time inventory dynamics of supply chains which inevitably function with delays due to the presence of manufacturing lead times, decision making, transportation and information flows. Availability of such models opens up new research directions where operations research and the field of time-delay systems can be merged primarily for characterizing inventory oscillations, developing new forecasting techniques and regulating inventories under uncertainties in consumers’ buying trends, stochastic nature of delays, market behavior and possible price competitions across enterprizes. This paper also offers one of the first connections between the disciplines via stability maps. We show how stability maps can prove to be useful in maintaining the stable nature of inventory oscillations. In a case study we demonstrate that the supply chain manager, by inspecting the stability maps, can be aware of which delay combinations in the supply chain may cause (un)desirable inventory behaviors.

Supply Network Dynamics and Delays; Performance, Synchronization, Stability

Recent research results in operations research specifically emphasize the critical role of delays in the functionality of supply networks. Delays arise due to the time needed for material deliveries, information flow and human perception towards adjusting to new decisions. Delays contaminate decision-making and lead to poor performance, synchronization problems and fluctuations in inventory levels resulting in major economical losses. This paper surveys continuous time deterministic mathematical models developed at system-level for supply network dynamics along with standard delay models pertaining to material deliveries, information flows and human perception. Next, the analogy between such delay models arising in supply networks and other real-life applications is pointed out. It is foreseen that complexity of the problem requires multi-disciplinary research bridging operations research, business, systems and control engineering, and mathematics. The paper concludes with an illustrative example and discussions of specific challenges anticipated in future research.

Supply Chain Dynamics With Decision Making and Production Delays: Stability Analysis and Optimum Controller Selection

Delays in supply chains due to decision making (of managers) and lead time (in production) cause undesirable effects to inventory levels. Via system’s thinking, key mechanisms leading to such effects are investigated by analyzing the stability of an inventory, departing from a commonly studied model. On this model, two distinct delays and a PI controller to remove offset in inventory, are incorporated. On the plane of delays, stable and unstable operational behavior of inventory level are identified on so-called stability maps. Next, effects of PI controller to stability are revealed and this controller is optimized for avoiding oscillatory nature of inventory level. Case studies are provided to demonstrate counter-intuitive effects of delay presence to supply chain management and how stability maps may serve as decision-making tools for managers in achieving better control on inventory behavior.Copyright