A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone

Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These recommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: (i) breathing patterns affecting gas exchange in the conducting airways; (ii) the concentrations in the tracheo-bronchial lining fluid; (iii) the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real-time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases.Comment: 38 page

Fractional interval observers and initialization of fractional systems

International audienceIn this paper an interval observer is synthesized for fractional linear systems with additive noise and disturbances. The contribution of system whole past to future output is taken into account as an initialization function. Provided the initialization function is upper and lower bounded, it is shown in this paper that the fractional interval observer (FIO) allows to bound pseudo-state free responses by an upper and a lower trajectory. In case interval observers cannot be synthesized straightforwardly, so as to obtain a stable and non-negative estimation error, it is shown that a change of coordinates allows to overcome this problem. The proposed methodology allows to bound fractional systems trajectories when the whole past is unknown but can be bounded. Finally, a numerical example is given to show the effectiveness of the proposed methods on the initialization of fractional linear systems

Applications of equivalent representations of fractional- and integer-order linear time-invariant systems

Nicht-ganzzahlige - fraktionale - Ableitungsoperatoren beschreiben Prozesse mit Gedächtniseffekten, deshalb werden sie zur Modellierung verschiedenster Phänomene, z.B. viskoelastischen Verhaltens, genutzt. In der Regelungstechnik wird das Konzept vor allem wegen des erhöhten Freiheitsgrades im Frequenzbereich verwendet. Deshalb wurden in den vergangenen Dekaden neben einer Verallgemeinerung des PID-Reglers auch fortgeschrittenere Regelungskonzepte auf nicht-ganzzahlige Operatoren erweitert. Das Gedächtnis der nicht-ganzzahligen Ableitung ist zwar essentiell für die Modellbildung, hat jedoch Nachteile, wenn z.B. Zustände geschätzt oder Regler implementiert werden müssen: Das Gedächtnis führt zu einer langsamen, algebraischen Konvergenz der Transienten und da eine numerische Approximation ist speicherintensiv. Im Zentrum der Arbeit steht die Frage, mit welchen Maßnahmen sich das Konvergenzverhalten dieser nicht ganzzahligen Systeme beeinflussen lässt. Es wird vorgeschlagen, die Ordnung der nicht ganzzahligen Ableitung zu ändern. Zunächst werden Beobachter für verschiedene Klassen linearer zeitinvarianter Systeme entworfen. Die Entwurfsmethodik basiert dabei auf einer assoziierten Systemdarstellung, welche einen Differenzialoperator mit höherer Ordnung verwendet. Basierend auf dieser Systembeschreibung können Beobachter entworfen werden, welche das Gedächtnis besser mit einbeziehen und so schneller konvergieren. Anschließend werden ganzzahlige lineare zeitinvariante Systeme mit Hilfe nicht-ganzzahliger Operatoren dargestellt. Dies ermöglicht eine erhöhte Konvergenz im Zeitintervall direkt nach dem Anfangszeitpunkt auf Grund einer unbeschränkten ersten Ableitung. Die periodische Löschung des so eingeführten Gedächtnisses wird erzielt, indem die nicht ganzzahlige Dynamik periodisch zurückgesetzt wird. Damit wird der algebraischen Konvergenz entgegen gewirkt und exponentielle Stabilität erzielt. Der Reset reduziert den Speicherbedarf und induziert eine unterlagerte zeitdiskrete Dynamik. Diese bestimmt die Stabilität des hybriden nicht-ganzzahligen Systems und kann genutzt werden um den Frequenzgang für niedrige Frequenzen zu bestimmen. So lassen sich Beobachter und Regler für ganzzahlige System entwerfen. Im Rahmen des Reglerentwurfs können durch den Resets das Verhalten für niedrige und hohe Frequenzen in gewissen Grenzen getrennt voneinander entworfen werden.Non-integer, so-called fractional-order derivative operators allow to describe systems with infinite memory. Hence they are attractive to model various phenomena, e.g. viscoelastic deformation. In the field of control theory, both the higher degree of freedom in the frequency domain as well as the easy generalization of PID control have been the main motivation to extend various advanced control concepts to the fractional-order domain. The long term memory of these operators which helps to model real life phenomena, has, however, negative effects regarding the application as controllers or observers. Due to the infinite memory, the transients only decay algebraically and the implementation requires a lot of physical memory. The main focus of this thesis is the question of how to influence the convergence rates of these fractional-order systems by changing the type of convergence. The first part is concerned with the observer design for different classes of linear time-invariant fractional-order systems. We derive associated system representations with an increased order of differentiation. Based on these systems, the observers are designed to take the unknown memory into account and lead to higher convergence rates. The second part explores the representation of integer-order linear time-invariant systems in terms of fractional-order derivatives. The application of the fractional-order operator introduces an unbounded first-order derivative at the initial time. This accelerates the convergence for a short time interval. With periodic deletion of the memory - a reset of the fractional-order dynamics - the slow algebraic decay is avoided and exponential stability can be achieved despite the fractional-order terms. The periodic reset leads to a reduced implementation demand and also induces underlying discrete time dynamics which can be used to prove stability of the hybrid fractional-order system and to give an interpretation of the reset in the frequency domain for the low frequency signals. This concept of memory reset is applied to design an observer and improve fractional-order controllers for integer-order processes. For the controller design this gives us the possibility to design the high-frequency response independently from the behavior at lower frequencies within certain limits

Systems control theory applied to natural and synthetic musical sounds

Systems control theory is a far developped field which helps to study stability, estimation and control of dynamical systems. The physical behaviour of musical instruments, once described by dynamical systems, can then be controlled and numerically simulated for many purposes. The aim of this paper is twofold: first, to provide the theoretical background on linear system theory, both in continuous and discrete time, mainly in the case of a finite number of degrees of freedom ; second, to give illustrative examples on wind instruments, such as the vocal tract represented as a waveguide, and a sliding flute

Structure Identifiability of an NDS with LFT Parametrized Subsystems

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the NDS is structurally identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.Comment: 16 page

Routing Unmanned Vehicles in GPS-Denied Environments

Most of the routing algorithms for unmanned vehicles, that arise in data gathering and monitoring applications in the literature, rely on the Global Positioning System (GPS) information for localization. However, disruption of GPS signals either intentionally or unintentionally could potentially render these algorithms not applicable. In this article, we present a novel method to address this difficulty by combining methods from cooperative localization and routing. In particular, the article formulates a fundamental combinatorial optimization problem to plan routes for an unmanned vehicle in a GPS-restricted environment while enabling localization for the vehicle. We also develop algorithms to compute optimal paths for the vehicle using the proposed formulation. Extensive simulation results are also presented to corroborate the effectiveness and performance of the proposed formulation and algorithms.Comment: Publised in International Conference on Umanned Aerial System