Thesis (Ph.D.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (ℓ.28-29)

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed January 5, 2010). PDF text: viii, 65 p. : ill. ; 507 K. UMI publication number: AAT 3360158. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph.D.)--Ohio University, June, 2005. / Title from PDF t.p. Includes bibliographical references (p. 72-74)

The process of quantifying the robustness of a given nonlinear system is not necessarily trivial. If the dynamics of the system in question are not sufficiently involved, then a tight estimate of a bound on system performance may be obtained. As the dynamics of the system concerned become more and more involved, it is often found that using the results existing in the literature provides a very conservative bound on system performance. Therefore, the motivation for this work is to develop a general method to obtain a less conservative estimate of a bound on system performance, compared to the results already available in literature. The scope of this work is limited to two dimensions at present. Note that working in a two dimensional space does not necessarily make the objective easily achievable. This is because quantifying the robustness of a general nonlinear system perturbed by disturbances can very easily become intractable, even on a space with dimension as low as two. The primary contribution of this work is a computational algorithm, the points generated by which are conjectured to lie on the boundary of the smallest robust forward invariant set for a given nonlinear system. A well known path-planning algorithm, available in existing literature, is leveraged to make the algorithm developed computationally efficient. If the system dynamics are not accurately known, then the above computed approximation of an invariant set may cease to be invariant over the given finite time interval for which the computed set is expected to be invariant. Therefore, the secondary contribution of this work is an algorithm monitoring a computed approximation of an invariant set. It is shown that for a certain type of systems, this secondary monitoring algorithm can be used to detect that a computed approximation of an invariant set is about to cease to be invariant, even if the primary algorithm computed the set based on an unsophisticated dynamical model of a system under consideration. The work related to computing approximations of invariant sets is tested mainly with the curve tracking problem in two dimensions. The algorithm monitoring whether a computed approximation of an invariant set is about to cease to be invariant is inspired by work related to detecting Lithium-ion (Li-ion) battery terminal voltage collapse detection.

Το αντικείμενο της παρούσας διπλωματικής εργασίας αφορά την ανάλυση και τον έλεγχο κυβερνοφυσικών δικτυωμένων δυναμικών συστημάτων. Ειδικότερα, μελετώνται γραμμικά χρονικά αμετάβλητα συστήματα μίας εισόδου με περιορισμούς τόσο στην είσοδο όσο και στο διάνυσμα καταστατικών μεταβλητών και προτείνονται στρατηγικές ανταγωνιστικού ελέγχου. Συγκεκριμένα, ένας βέλτιστος ελεγκτής εξασφαλίζει θετική αμεταβλητότητα ενός φραγμένου κυρτού πολυεδρικού συνόλου ως προς το σύστημα και ταυτόχρονα σύγκλιση της τροχιάς του διανύσματος κατάστασης στο μηδέν το συντομότερο δυνατό. Υποθέτουμε ότι ένας «ανταγωνιστής » ελεγκτής επιτυγχάνει κατά διαστήματα να αναλάβει τον έλεγχο του συστήματος και αποστέλλει μέσω του δικτύου εσφαλμένα σήματα εισόδου στοχεύοντας να οδηγήσει το διάνυσμα κατάστασης εκτός του πολυεδρικού συνόλου το συντομότερο δυνατό. Προσομοιώσεις καταδεικνύουν ότι το διακοπτικό φαινόμενο που ανακύπτει από τη διαδοχική εφαρμογή των δύο παραπάνω ελεγκτών προκαλεί μία περιοδική τροχιά του διανύσματος κατάστασης. Τα θεωρητικά εργαλεία που χρησιμοποιήθηκαν για την εξαγωγή των αποτελεσμάτων προέρχονται από τη θεωρία ευστάθειας Lyapunov και τη θεωρία αμετάβλητων συνόλων. / This thesis addresses the analysis and control of networked cyber-physical dynamical systems. We are mostly concerned with the study of linear, time-invariant systems with single input. The systems evolve in the discrete time and are subject to both state and input constraints and a certain number of adversary control strategies are proposed. A time-optimal control law guarantees positive invariance of a bounded convex polyhedral set with respect to the given system, while contracting the state space vector to the origin. At times, an adversary controller succeeds in gaining control of the system and sends false control commands attempting to lead the state vector outside the polyhedral set at the maximum admissible rate. Simulation studies highlight that consecutively applying the above control laws results in a periodic motion of the state vector. The theoretic tools used to obtain our conclusions associate with Lyapunov stability and set theory.

In the survival and reliability data analysis, parametric and nonparametric methods are used to estimate the hazard/risk rate and survival functions. A parametric approach is based on the assumption that the underlying survival distribution belongs to some specific family of closed form distributions (normal, Weibull, exponential, etc.). On the other hand, a nonparametric approach is centered around the best-fitting member of a class of survival distribution functions. Moreover, the Kaplan-Meier and Nelson-Aalen type nonparametric approach do not assume either distribution class or closed-form distributions. Historically, well-known time-to-event processes are death of living specie in populations and failure of component in engineering systems. Recently, the human mobility, electronic communications, technological changes, advancements in engineering, medical, and social sciences have further diversified the role and scope of time-to-event processes in cultural, epidemiological, financial, military, and social sciences. To incorporate extensions, generalizations and minimize scope of existing methods, we initiate an innovative alternative modeling approach for time-to-event dynamic processes. The innovative approach is composed of the following basic components: (1) development of continuous-time state of dynamic process, (2) introduction of discrete-time dynamic intervention process, (3) formulation of continuous and discrete-time interconnected dynamic system, (4) utilizing Euler-type discretized schemes, developing theoretical dynamic algorithms, and (5) introduction of conceptual and computational state and parameter estimation procedures. The presented approach is motivated by state and parameter estimation of time-to-event processes in biological, chemical, engineering, epidemiological, medical, military, multiple-markets and social dynamic processes under the influence of discrete-time intervention processes. We initiate (1) a time-to-event process to be a probabilistic dynamic process with unitary state. Action, normal, operational, radical, survival, susceptible, etc. and its complementary states, reaction, abnormal, nonoperational, non-radical, failure, infective and so on (quantitative and qualitative variables), are considered to be illustrations of a unitary state of time-to-event dynamic processes. A unitary state is measured by a probability distribution function. Employing Newtonian dynamic modeling approach and observing the definition of hazard rate as a specific rate, survival or failure probabilistic state dynamic model is developed. This dynamic model is further extended to incorporate internal or external discrete-time dynamic intervention processes acting on unitary state time-to-event processes (2). This further demanded a formulation and development of an interconnected continuous-discrete-time hybrid, and totally discrete-time dynamic models for time-to-event processes (3). Employing the developed hybrid model, Euler-type discretized schemes, a very general fundamental conceptual analytic algorithm is outlined (4). Using the developed theoretical computational procedure in (4), a general conceptual computational data organizational and simulation schemes are presented (5) for state and parameter estimation problems in unitary state time-to-event dynamic processes. The well-known theoretical existing results in the literature are exhibited as special cases in a systematic and unified manner (6). In fact, the Kaplan-Meier and Nelson-Aalen type nonparametric estimation approaches are systematically analyzed by the developed totally discrete-time hybrid dynamic modeling process. The developed approach is applied to two data sets. Moreover, this approach does not require a knowledge of either a closed-form solution distribution or a class of distributions functions. A hazard rate need not be constant. The procedure is dynamic. In the existing literature, the failure and survival distribution functions are treated to be evolving/progressing mutually exclusively with respect to corresponding to two mutually exclusive time varying events. We refer to these two functions (failure and survival) as cumulative distributions of two mutually disjoint state output processes with respect to two mutually exclusive time-varying complementary unitary states of a time-to-event processes in any discipline of interest (7). This kind of time-to-event process can be thought of as a Bernoulli-type of deterministic/stochastic process. Corresponding to these two complementary output processes of the Bernoulli-type of stochastic process, we associate two unitary dynamic states corresponding to a binary choice options/actions (8), namely, ({action, reaction}, {normal, abnormal}, {survival, failure}, {susceptible, infective}, {operational, nonoperational}, {radical, non-radical}, and so on.) Under this consideration, we extend unitary state time-to-event dynamic model to binary state time-to-event dynamic model. Using basic tools in mathematical sciences, we initiate a Newtonian-type dynamic approach for binary state time-to-event processes in the sciences, technologies, and engineering (9). Introducing an innovative concept of “survival state dynamic principle”, an innovative interconnected nonlinear non-stationary large-scale hybrid dynamic model for number of units/species and its unitary survival state corresponding to binary state time-to-event process is formulated (10). The developed model in (10) includes dynamic model (3) as a special case. The developed approach is directly applicable to binary state time-to-event dynamic processes in biological, chemical, engineering, financial, medical, physical, military, and social sciences in a coherent manner. A by-product of this is a transformed interconnected nonlinear hybrid dynamic model with a theoretical discrete-time conceptual computational dynamic process (11). Employing the transformed discrete-time conceptual computational dynamic process, we introduce notions of data coordination, state data decomposition and aggregation, theoretical conceptual iterative processes, conceptual and computational parameter estimation and simulation schemes, conceptual and computational state simulation schemes in a systematic way (12). The usefulness of the developed interconnected algorithm is validated by using three real world data sets (13). We note that the presented algorithm does not need a closed-form representation of distribution/likelihood function. In fact, it is free from any required assumptions of the “Classical Maximum Likelihood Function Approach” in the “Survival and Reliability Analysis.” The rapid electronic communication and human mobility processes have facilitated to transform information, knowledge, and ideas almost instantly around the globe. This indeed generates heterogeneity, and it causes to form nonlinear and non-stationary dynamic processes. Moreover, the heterogeneity, non-linearity, non-stationarity, further generates two types of uncertainties, namely, deterministic, and stochastic. In view of this, it is obvious that nothing is deterministic. In short, the 21st century problems are highly nonlinear, non-stationary and under the influence of internal and external random perturbations. Using tools in stochastic analysis, interconnected deterministic models in (3) and (10) are extended to interconnected stochastic hybrid dynamic model for binary state time-to-event processes (14). The developed model is described by a large-scale nonlinear and non-stationary stochastic differential equations. Moreover, a stochastic version of a survival function is also introduced (15). Analytical, computational, statistical, and simulation algorithms/procedures are also extended and analyzed in a systematic and unified way (16). The presented interconnected stochastic model is motivated to initiate conceptual computational parameter and state estimation schemes for time-to-event statistical data (17). Again, stochastic version of computational algorithms are validated in the context of three real world data sets. The obtained parameter and state estimates show that the algorithm is independent of the choice of nonlinear transformation (18). Utilizing the developed alternative innovative procedure and the recently modified deterministic version of Local Lagged Adapted Generalized Method of Moments (LLGMM) is also extended to stochastic version in a natural way (19). This approach provides a degree of measure of confidence, prediction, and planning assessments (20). In addition, it initiates a conceptual computational parameter and state estimation and simulation schemes that is suitable for the usage of mean square sub-optimal procedure (21). The usefulness and the significance of the approach is illustrated by applying to three data sets (22). The approach provides insight for investigating various type of invariant sets, namely, sustainable/unsustainable, survival/failure, reliable/unreliable (23), and qualitative properties such as sustainability versus unsustainability, reliability versus unreliability, etc. (24) Once again, the presented algorithm is independent of any form of survival distribution functions or data sets. Moreover, it does not require a closed form survival function distribution. We also note that the introduction of intervention processes provides a measure of influence and confidence for the usage of new tools/procedures/approaches in continuous-time binary state time-to-event dynamic process (25). Moreover, the presented dynamic modeling is more feasible for its usage of investigating a more complex time-to-event dynamic process (26). The developed procedure is dynamic and indeed non-parametric (27). The dynamic approach adapts with current changes and updates statistic process (28). The dynamic nature is natural rather than the existing static and single-shot techniques (29).

Historically human endeavors around the globe are in search of bilateral relationships. Knowledge and commerce has played a very significant role in increasing the ability for humans to connect for the betterment of the human species. As the means of communication improve, mutual benefits to the community as a whole also increase. Moreover, the benefits are filtered down to members of the overall community. Recent advancement in electronic communication technologies and in knowledge, in particular, physical, chemical, engineering and medical sciences and philosophies, have facilitated nearly instantaneous multi-cultural interactions. Local problems and solutions have become global. This has generated a need for cooperation, coordination, and co-management at local and global levels. This instant communication and easy access to almost all members of the global community, with minimal cost and effort, causes an increase in uncertainty and lack of clarity in communication and misunderstanding between the members of the community. This leads to a fuzzy and stochastic environment. In short, the 21st century has seen a significant increase in the need to consider all human endeavors as being subject to random environmental fluctuations. More precisely, currently the human species is highly mobile. In this work, an attempt is made (1) to balance communities working cooperatively and cohesively with one another while preserving member ability to retain individuality and fostering an environment of cultural state diversity. We develop (2) constructive analytic algorithms that provide tools to study qualitative and quantitative properties of multicultural diverse dynamic social networks. Under network parametric space/set conditions (3) cohesion and co-existence of members of multicultural dynamic network are insured. The parametric conditions (4) are algebraically simple, easy to verify, and robust. Moreover, the presented study of parametric representations of cohesion, co-existence and consensus attributes and robustness of multicultural dynamic networks provides a quantitative tool for planning, policy and performance of human mobility processes for members of a multicultural dynamic network. We develop and investigate (5) a deterministic dynamic multicultural network. To exhibit the significance of the analysis, ideas, the complexity and limitations, we present a specific prototype model. This serves to establish the framework for finding explicit sufficient conditions in terms of system parameters for studying a complex dynamic network. Further, we decompose the cultural state domain into invariant subsets (6) and consider the behavior of members within each cultural state subset. Moreover, we analyze the relative cultural affinity between individual members relative to the center of the social network. We then (7) outline the general method for investigating a multicultural cultural network. We also demonstrate the degree of conservatism of the estimates using Euler type numerical approximation schemes. We are then able to consider how changes in the various parametric effects are reflected on the dynamics of the network. The deterministic multicultural dynamic model and analysis is extended (8) to a multicultural dynamic network under random environmental perturbations. We present a detailed prototype model for the purpose of investigation. Introducing the concept of stochastic cohesion and consensus in the context of probabilistic modes of convergence (9), the explicit sufficient conditions in terms of system parameters are given to exhibit the cohesive property of the stochastic network. The effects of random fluctuations are characterized. We then extend the stochastic model (10) to a multicultural hybrid stochastic dynamic network model. By considering a hybrid dynamic, we can explore the properties of a multicultural dynamic under the influence of both continuous-time and discrete-time cultural dynamic systems. This model captures external influences and internal changes that may have an impact on the members and system parameters of the dynamic network. We extend the ideas of global cohesion and consensus to local cohesion and consensus (11). Again, the detailed study is focused on a prototype hybrid multicultural dynamic network. Using the ideas and tools developed from the stochastic network (12), we are able to establish conditions on the network parameters for which the cultural network is locally cohesive. Using Euler-Maruyama type numerical approximation schemes to model the network, we better understand to what extent the analytically developed estimates are feasible.

Since it is difficult or impossible to generate turbulent flow in a highly viscous fluid or a microfluidic system, efficient mixing becomes a challenge. However, it is possible in a laminar flow to generate chaotic particle trajectories (well-known as chaotic advection), that can lead to effective mixing. This dissertation studies mixing in flows with the limiting case of zero Reynolds numbers that are called Stokes flows and illustrates the practical use of different theories, namely the topological chaos theory, the set-oriented analysis and lobe dynamics in the analysis, design and optimization of different laminar-flow mixing systems. In a recent development, the topological chaos theory has been used to explain the chaos built in the flow only based on the topology of boundary motions. Without considering any details of the fluid dynamics, this novel method uses the Thurston-Nielsen (TN) classification theorem to predict and describe the stretching of material lines both qualitatively and quantitatively. The practical application of this theory toward design and optimization of a viscous-flow mixer and the important role of periodic orbits as "ghost rods" are studied. The relationship between stretching of material lines (chaos) and the homogenization of a scalar (mixing) in chaotic Stokes flows is examined in this work. This study helps determining the extent to which the stretching can represent real mixing. Using a set-oriented approach to describe the stirring in the flow, invariance or leakiness of the Almost Invariant Sets (AIS) playing the role of ghost rods is found to be in a direct relationship with the rate of homogenization of a scalar. The mixing caused by these AIS and the variations of their structure are explained from the point of view of geometric mechanics using transport through lobes. These lobes are made of segments of invariant manifolds of the periodic points that are generators of the ghost rods. A variety of the concentration-based measures, the important parameters of their calculation, and the implicit effect of diffusion are described. The studies, measures and methods of this dissertation help in the evaluation and understanding of chaotic mixing systems in nature and in industrial applications. They provide theoretical and numerical grounds for selection of the appropriate mixing protocol and design and optimization of mixing systems, examples of which can be seen throughout the dissertation. / Ph. D.

Cette thèse est dédiée à l’analyse du retard (de calcul ou induit par la communication), qui représente un des paramètres sensibles, et qui doit être pris en compte, pour la mise en œuvre de la Commande Prédictive en temps réel d’un processus dynamique. Dans la première partie, nous avons abordé le problème d’existence des ensembles D-invariants et avons fourni par la suite des conditions nécessaires et/ou suffisantes pour l’existence de ces ensembles. En outre, nous avons détaillé quelques nouvelles idées sur la construction des ensembles D-invariants en utilisant des algorithmes itératifs et d’autres algorithmes basés sur des techniques d’optimisation à deux niveaux. La seconde partie a été consacrée à l’étude du problème de robustesse des systèmes linéaires discrets affectés par un retard variable en boucle fermée avec un contrôleur affine par morceaux défini sur une partition polyédrale de l’espace d’état. L’étude a porté sur l’analyse de la fragilité d’une telle loi commande en présence du retard dans la boucle. Nous avons décrit les marges d’invariance robustes définies comme étant le plus grand sous-ensemble de l’incertitude paramétrique pour lequel l’invariance positive est garantie par rapport à la dynamique en boucle fermée en présence du retard. La dernière partie de cette thèse s’est articulée autour de la conception des lois de commande prédictives avec un attention particulière aux modèles linéaires discrets décrivant des dynamiques affectées par des contraintes en présence du retard. Nous avons proposé plusieurs méthodes offrant différentes solutions au problème de stabilisation locale sans contrainte. Afin d’assurer la stabilité et de garantir la satisfaction des contraintes, nous avons exploité le concept d’invariance et à l’aide du formalisme "ensemble terminal-coût terminal", un problème d’optimisation a été formulé où les états sont forcés d’atteindre l’ensemble maximal admissible d’états retardés/D-invariant à la fin de l’horizon de prédiction. Enfin, nous avons étudié le problème de stabilisation des systèmes continus commandés en réseau soumis à des retards incertains et éventuellement variant dans le temps. Nous avons montré que les ensembles λ-D-contractifs peuvent être utilisés comme ensembles cibles où la stratégie de commande consiste en un simple problème de programmation linéaire ’LP’ qui peut être résolu en ligne. / The research conducted in this thesis has been focusing on Model Predictive Control (MPC) and the implication of network induced time-varying delays. We have addressed, in the first part of this manuscript, the existence problem and the algorithmic computation of positive invariant sets in the state space of the original discrete delay difference equation. The second part of these thesis has been devoted to the study of the robustness problem for a specific class of dynamical systems, namely the piecewise affine systems, defined over a polyhedral partition of the state space in the presence of variable input delay. The starting point was the construction of a predictive control law which guarantees the existence of a non-empty robust positive invariant set with respect to the closed-loop dynamic. The variable delay inducing in fact a model uncertainty, the objective was to describe the robust invariance margins defined as the largest subset of the parametric uncertainty for which the positive invariance is guaranteed with respect to the closed-loop dynamics in the presence of small and large delays. The last part has been dedicated to Model Predictive Control design with a specific attention to linear discrete time-delay models affected by input/state constraints. The starting point in the analysis was the design of a local stabilizing control law using different feedback structures. We proposed several design methods offering different solutions to the local unconstrained stabilization problem. In order to ensure stability and guarantee input and state constraints satisfaction of the moving horizon controller, the concept of positive invariance related to time-delay systems was exploited. Using the "terminal setterminal cost" design, the states were forced to attain the maximal delayed-state admissible set at the end of the prediction horizon. Finally, we have investigated the stabilization problem of Networked Control Systems ’NCSs’ subject to uncertain, possibly time-varying, network-induced delays. We showed that λ-D-contractive sets can be used as a target sets in a set induced Lyapunov function control fashion where a simple Linear Programming ’LP’ problem is required to be solved at each sampling instance.

De nos jours, la plupart des assistances électriques pour vélos sont principalement basées sur des mesures de variables mécaniques. Cette thèse se concentre sur l'analyse du cycliste et du vélo dans son ensemble, afin de concevoir une loi de commande qui prenne en compte des contraintes physiologiques. En ce sens, la relation entre les indices physiologiques et les variables mécaniques est abordée. En particulier, dynamique d'échange de gaz pendant le pédalage est modélisée et analysée. Le modèle obtenu est linéaire à paramètres variables en temps discret. Les données mesurées de différents tests de cyclisme efectués au sein du GIPSA-Lab sont utilisées pour son calibration. La méthodologie d'identification paramétrique est développée comme solution à une séquence de problèmes d'optimisation non linéaire et non contraints. Le modèle dynamique d'échange de gaz fait référence à la détermination de la production excessive de dioxyde de carbone, qui est considérée comme un état physiologique indésirable du cycliste. En outre, un observateur de la production de dioxyde de carbone est mis en oeuvre en utilisant des mesures de la consommation d'oxygène et de la puissance de pédalage, ce qui fournit un intervalle déterministe contenant l'état réel. Enfin, une méthode est présentée pour trouver les conditions suffisantes sur la puissance de pédalage qui garantissent des niveaux admissibles de production de dioxyde de carbone. / Nowadays, most of the electrical assistances for bicycles are mainly based onmeasurements of mechanical variables. In this dissertation, cyclist and bicycle are considered as a whole energetic system in order to conceive physiology-aware control strategies. In this sense, the relationships between physiological indices and mechanical variables are studied. In particular, the gas exchange dynamics during cycling is modeled and analysed. The obtained model is discrete-time and linear parameter varying. Measured data of different cycling tests performed in GIPSA-Lab are used for its calibration. A methodology for parametric identification is developed as a solution of a sequence of non-linear unconstrained optimization problems. The gas exchange dynamical model allows the determination of excess carbon dioxide production, which is considered as an unsuitable physiological-state of the cyclist. Furthermore, a set-membership observer of carbon dioxide production is implemented using measurements of oxygen consumption and pedal power. It provides a deterministic interval which contains the real state. Finally, this thesis presents a method for finding the suficient conditions on the pedal power that guarantee admissible levels of carbon dioxide production.