Constantinos Mavroidis, Nader Jalili, Surendra M. Gupta
Date of Award
Doctor of Philosophy
Department or Academic Unit
College of Engineering, Department of Mechanical and Industrial Engineering
mechanical engineering, delay-independent stability analysis, Iterated discriminants, multiple time delay systems, resultant theory, stability analysis, supply chain management
Time-Delay Systems (TDS) arise in many applications from diverse areas such as economy, biology, population dynamics, traffic flow and communication systems. Asymptotic stability analysis of even linear time-invariant time-delay systems is a notoriously complex task due to the NP-hard nature of the stability problem. Additionally, consideration of multiple delays totally hampers the existing stability analyses which are limited to less than three delays. There is still no comprehensive treatment for the most general time-delay systems where the system order, the number of delays or the rank conditions of the system matrices are not limited. All the existing techniques are case-specific and derived only for lower order time-delay systems. The main goal of this dissertation is to develop a stability analysis procedure for the most general linear time-invariant multiple time-delay systems, relaxing all the mentioned limitations.
A novel methodology, Advanced Clustering with Frequency Sweeping (ACFS), is introduced for the delay-dependent stability (DDS) 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. ACFS owes these superiorities to an elegant way of cross-fertilizing the resultant theory, frequency sweeping technique and the root clustering paradigms. ACFS can achieve to directly extract the 2D cross-sections of the stability views in the domain of any of the two delays. ACFS reveals the complexity measures of the stability views as a function of system properties and a new formula that can compute the precise lower and upper bounds of the only parameter, the frequency, that ACFS sweeps. Furthermore, another stability technique is developed for the treatment of sub-class of the general LTI-TDS.
Delay-independent stability (DIS) of a general class of LTI multiple time-delay system (MTDS) is then investigated in the entire delay-parameter space. Stability of MTDS may change only if their spectrum lies on the imaginary axis for some delays. An analytical approach, which requires the inspection of the roots of finite number of single-variable polynomials, is built 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 the delay-independent stability of LTI-MTDS, technically known as weak DIS, as well as the robust stability of single-delay systems against all variations in delay ratios. Moreover, general class LTI-MTDS is investigated in order to obtain a control law which stabilizes the LTI-MTDS independently of all the delays.
Delays exist in supply chains due to decision-making, production lead-time, transportation times and lags in flow of information. Finally, developed techniques are applied to inventory regulation problem. The presence of delays may cause poor management in the supply chains which eventually leads to undesirable behavior (i.e. oscillations) of inventory levels. These behaviors are examined via new developed methodologies, which can characterize the inventory oscillations as a parameter of multiple delays in supply chains. A generalized supply chain model is developed departing from a commonly studied simpler one based on fluid-flow dynamics. In the generalized model, in order to eliminate drift, deficit or surfeit of stocks in the inventory levels, a proportional-integral (PI) decision-maker is implemented. Inventory oscillations are then characterized with respect to the parameters of the PI and some parameters inherent to the supply network. New stability techniques, combined with the generalized supply network model, could provide both thorough insight into better controlling the inventories in supply chains as well as managerial interpretations. Hence, a novel ordering policy design with which the inventory variations can be rendered insensitive to detrimental effects of delays is presented.
Ismail Ilker Delice
Delice, Ismail Ilker, "Stability analysis of multiple time-delay systems with applications to supply chain management" (2011). Mechanical Engineering Dissertations. Paper 12. http://hdl.handle.net/2047/d20000981
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