Columns subjected to time varying axial load may exhibit dynamic instability due to parametric resonance. This type of instability is inherent in structures; it is not due to material or geometrical imperfections, and can occur even in perfectly constructed structures. This characteristic makes parametric resonance a very difficult to predict and therefore dangerous phenomenon. In this thesis the stability of a structural column under bounded noise axial load is studied by use of Lyapunov exponents. Bounded noise is especially useful as a loading because it may be used to represent both wide and narrow band processes, making the stability equations developed general enough to handle a wide variety of real world probabilistic loadings. The equation of motion of the first mode of vibration for this system is a second-order nonlinear stochastic ordinary differential equation. The nonlinearity makes the system exhibit bifurcating behaviour where stability shifts from the trivial solution to a non-zero mean stationary solution. The stability of the trivial and non-trivial solutions is important in obtaining a complete picture of the dynamical behaviour of the system. The effect that damping, the amplitude of noise, and the level of nonlinearity have on the stability of a structural column is studied using both analytical and numerical approaches. The largest Lyapunov exponent of the trivial solution is determined analytically by using time averaged versions of the original equation of motion. The validity of the analytical time averaged equation of motion is also verified with Monte Carlo simulations. Due to the mathematical complexity the largest Lyapunov exponent of the non-trivial stationary solutions is obtained using Monte Carlo simulation only.

Columns subjected to time varying axial load may exhibit dynamic instability due to parametric resonance. This type of instability is inherent in structures; it is not due to material or geometrical imperfections, and can occur even in perfectly constructed structures. This characteristic makes parametric resonance a very difficult to predict and therefore dangerous phenomenon. In this thesis the stability of a structural column under bounded noise axial load is studied by use of Lyapunov exponents. Bounded noise is especially useful as a loading because it may be used to represent both wide and narrow band processes, making the stability equations developed general enough to handle a wide variety of real world probabilistic loadings. The equation of motion of the first mode of vibration for this system is a second-order nonlinear stochastic ordinary differential equation. The nonlinearity makes the system exhibit bifurcating behaviour where stability shifts from the trivial solution to a non-zero mean stationary solution. The stability of the trivial and non-trivial solutions is important in obtaining a complete picture of the dynamical behaviour of the system. The effect that damping, the amplitude of noise, and the level of nonlinearity have on the stability of a structural column is studied using both analytical and numerical approaches. The largest Lyapunov exponent of the trivial solution is determined analytically by using time averaged versions of the original equation of motion. The validity of the analytical time averaged equation of motion is also verified with Monte Carlo simulations. Due to the mathematical complexity the largest Lyapunov exponent of the non-trivial stationary solutions is obtained using Monte Carlo simulation only.

Flow-induced structural vibration is experienced in many engineering applications, such as aerospace industry and civil engineering infrastructures. One of the main mechanisms of flow-induced vibration is instability which can be triggered by parametric excitations or fluid-elastic forces. Experiments show that turbulence has a significant impact on the stability of structures. The objective of this research is to bridge the gap between flow-induced vibration and stochastic stability of structures. The flow-induced vibration of a spring-supported circular cylinder is studied in this research. The equations of motion for the cylinder placed in a cross-flow are set up, in which the vortex force is modeled by a bounded noise because of its narrow-band characteristics. Since the vibration in the lift direction is more prominent in the lock-in region, the system is reduced to one degree-of-freedom, i.e., only the vibration of the cylinder in the lift direction is considered. The equation of motion for the cylinder can be generalized as a two-dimensional system excited by a bounded noise. Stochastic analysis is used to determine the moment Lyapunov exponents and Lyapunov exponents for the generalized system. The results are then applied to study the parametric instability of a cylinder in the lock-in region. Fluidelastic instability can occur when the cylinder is placed in a shear flow. The equations of motion are established by using the quasi-steady theory to model the fluid-elastic forces. To study the turbulence effect on the stability of the cylinder, a real noise or an Ornstein-Uhlenbeck process is used to model the grid-generated turbulence. The equations of motion are randomized resulting in a four-dimensional system excited by a real noise. The stability of the stochastic system is studied by determining the moment Lyapunov exponents and Lyapunov exponents. Parameters of the system and the noise are varied to investigate their effects on the stability. It is found that the grid-generated turbulence can stabilize the system when the parameters take certain values, which agrees with the experimental observations. Many flow-induced vibration problems can be modeled by a two degrees-of-freedom system parametically excited by a narrow-band process modeled by a bounded noise. The system can be in subharmonic resonance, combination (additive or differential) resonance, or both if the central frequency of the bounded noise takes an appropriate value. The method for a single degree-of-freedom system is extended to study the stochastic stability of the two degrees-of-freedom system. The moment Lyapunov exponents and Lyapunov exponents for the three cases are obtained using a perturbation method. The effect of noise on various types of parametric resonance, such as subharmonic resonance, combination additive resonance, and combined subharmonic and combination additive resonance, is investigated. The main contributions of this thesis are stochastic stability analysis of one-degree-of-freedom systems and two-degree-of-freedom systems. Stability analysis for systems under the excitation of real noise and bounded noise is carried out by determining the moment Lyapunov exponents and Lyapunov exponents. Good agreement is obtained between analytical results and those obtained from Monte Carlo simulations. In the two degrees-of-freedom case, the effect of free stream turbulence on cylinder vibration and its stability is examined.

Flow-induced structural vibration is experienced in many engineering applications, such as aerospace industry and civil engineering infrastructures. One of the main mechanisms of flow-induced vibration is instability which can be triggered by parametric excitations or fluid-elastic forces. Experiments show that turbulence has a significant impact on the stability of structures. The objective of this research is to bridge the gap between flow-induced vibration and stochastic stability of structures. The flow-induced vibration of a spring-supported circular cylinder is studied in this research. The equations of motion for the cylinder placed in a cross-flow are set up, in which the vortex force is modeled by a bounded noise because of its narrow-band characteristics. Since the vibration in the lift direction is more prominent in the lock-in region, the system is reduced to one degree-of-freedom, i.e., only the vibration of the cylinder in the lift direction is considered. The equation of motion for the cylinder can be generalized as a two-dimensional system excited by a bounded noise. Stochastic analysis is used to determine the moment Lyapunov exponents and Lyapunov exponents for the generalized system. The results are then applied to study the parametric instability of a cylinder in the lock-in region. Fluidelastic instability can occur when the cylinder is placed in a shear flow. The equations of motion are established by using the quasi-steady theory to model the fluid-elastic forces. To study the turbulence effect on the stability of the cylinder, a real noise or an Ornstein-Uhlenbeck process is used to model the grid-generated turbulence. The equations of motion are randomized resulting in a four-dimensional system excited by a real noise. The stability of the stochastic system is studied by determining the moment Lyapunov exponents and Lyapunov exponents. Parameters of the system and the noise are varied to investigate their effects on the stability. It is found that the grid-generated turbulence can stabilize the system when the parameters take certain values, which agrees with the experimental observations. Many flow-induced vibration problems can be modeled by a two degrees-of-freedom system parametically excited by a narrow-band process modeled by a bounded noise. The system can be in subharmonic resonance, combination (additive or differential) resonance, or both if the central frequency of the bounded noise takes an appropriate value. The method for a single degree-of-freedom system is extended to study the stochastic stability of the two degrees-of-freedom system. The moment Lyapunov exponents and Lyapunov exponents for the three cases are obtained using a perturbation method. The effect of noise on various types of parametric resonance, such as subharmonic resonance, combination additive resonance, and combined subharmonic and combination additive resonance, is investigated. The main contributions of this thesis are stochastic stability analysis of one-degree-of-freedom systems and two-degree-of-freedom systems. Stability analysis for systems under the excitation of real noise and bounded noise is carried out by determining the moment Lyapunov exponents and Lyapunov exponents. Good agreement is obtained between analytical results and those obtained from Monte Carlo simulations. In the two degrees-of-freedom case, the effect of free stream turbulence on cylinder vibration and its stability is examined.

Doctor of Philosophy / Department of Electrical and Computer Engineering / Balasubramaniam Natarajan / A cyber physical system (CPS) is an intelligent integration of computation and communication infrastructure for monitoring and/or control of an underlying physical system. In this dissertation, we consider a specific class of CPS architectures where state of the system is spatially distributed in physical space. Examples that fit this category of CPS include, smart distribution gird, smart highway/transportation network etc. We study state estimation and control process in such systems where, (1) multiple sensors and actuators are arbitrarily deployed to jointly sense and control the system; (2) sensors directly communicate their observations to a central estimation and control unit (ECU) over communication links; and, (3) the ECU, on computing the control action, communicates control actions to actuators over communication links. Since communication links are susceptible to random failures, the overall estimation and control process is subjected to: (1) partial observation updates in estimation process; and (2) partial actuator actions in control process. We analyze stochastic stability of estimation and control process, in this scenario by establishing the conditions under which estimation accuracy and deviation from desired state trajectory is bounded. Our key contribution is the derivation of a new fundamental result on bounds for critical probabilities of individual communication link failure to maintain stability of overall system. The overall analysis illustrates that there is trade-off between stability of estimation and control process and quality of underlying communication network. In order to demonstrate practical implication of our work, we also present a case study in smart distribution grid as a system example of spatially distributed CPSs. Voltage/VAR support via distributed generators is studied in a stochastic nonlinear control framework.

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads. The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies. The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents. Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed. The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads. The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies. The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents. Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed. The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.

We present methods for distributed self-assembly that utilize simple rule-of-thumb control and communication schemes providing probabilistic performance guarantees. These methods represents a staunch departure from existing approaches that require more sophisticated control and communication, but provide deterministic guarantees. In particular, we show that even under severe communication restrictions, any assembly described by an acyclic weighted graph can be assembled with a rule set that is linear in the number of nodes contained in the desired assembly graph. We introduce the concept of stochastic stability to the self-assembly problem and show that stochastic stability of desirable configurations can be exploited to provide probabilistic performance guarantees for the process. Relaxation of the communication restrictions allows simple approaches giving deterministic guarantees. We establish a clear relationship between availability of communication and convergence properties. We consider Self-assembly tasks for the cases of many and few agents as well as large and small assembly goals. We analyze sensitivity of the presented process to communication errors as well as ill-intentioned agents. We discuss convergence rates of the presented process and directions for improving them.

Cette thèse étudie quelques aspects de la modélisation stochastique des carnets d'ordres. Nous analysons dans la première partie un modèle dans lequel les temps d'arrivées des ordres sont Poissoniens indépendants. Nous démontrons que le carnet d'ordres est stable (au sens des chaines de Markov) et qu'il converge vers sa distribution stationnaire exponentiellement vite. Nous en déduisons que le prix engendré dans ce cadre converge vers un mouvement Brownien aux grandes échelles de temps. Nous illustrons les résultats numériquement et les comparons aux données de marché en soulignant les succès du modèle et ses limites. Dans une deuxième partie, nous généralisons les résultats à un cadre où les temps d'arrivés sont régis par des processus auto et mutuellement existants, moyennant des hypothèses sur la mémoire de ces processus. La dernière partie est plus appliquée et traite de l'identification d'un modèle réaliste multivarié à partir des flux des ordres. Nous détaillons deux approches : la première par maximisation de la vraisemblance et la seconde à partir de la densité de covariance, et réussissons à avoir une concordance remarquable avec les données. Nous appliquons le modèle ainsi estimé à deux problèmes concrets de trading algorithmique, à savoir la mesure de la probabilité d'exécution et le coût d'un ordre limite. / This thesis presents some aspects of stochastic order book modelling. In the first part, we analyze a model in which order arrivals are independent Poisson. We show that the order book is stable (in the sense of Markov chains) and that it converges to its stationary state exponentially fast. We deduce that the price generated in this setting converges to a Brownian motion at large time scales. We illustrate the results numerically and compare them to market data. In the second part, we generalize the results to a setting in which arrival times are governed by self and mutually existing processes. The last part is more applied and deals with the identification of a realistic multivariate model from the order flow. We describe two approaches: the first based on maximum likelihood estimation and the second on the covariance density function, and obtain a remarkable agreement with the data. We apply the estimated model to two specific algorithmic trading problems, namely the measurement of the execution probability of a limit order and its cost.

This thesis consists of five papers, presented as separate chapters within three parts: Industrial Organization, Evolutionary Game Theory and Game Theory. The common basis of these parts is research in the field of game theory and more specifically, equilibrium selection in different frameworks. The first part, Industrial Organization, consists of one paper co-authored with Prajit Dutta and Jörgen Weibull. Forward-looking consumers are analysed in a Bertrand framework. It is assumed that if firms can anticipate a price war and act accordingly, so can consumers. The second part, Evolutionary Game Theory, contains three chapters. All models in these papers are based on Young’s (1993, 1998) approach. In Chapter 2, the Saez Marti and Weibull’s (1999) model is generalized from the Nash Demand Game to generic two-player games. In Chapter 3, co-authored with Jens Josephson, a special set of stochastically stable states is introduced, minimal construction, which is the long-run prediction under imitation behavior in normal form games. In Chapter 4, best reply and imitation rules are considered on extensive form games with perfect information. / Diss. Stockholm : Handelshögsk., 2001