The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05]. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

Numerous examples arise in fields ranging from mechanics to biology where disappearance of Chaos can be detrimental. Preventing such transient nature of chaos has been proven to be quite challenging. The utility of Reinforcement Learning (RL), which is a specific class of machine learning techniques, in discovering effective control mechanisms in this regard is shown. The autonomous control algorithm is able to prevent the disappearance of chaos in the Lorenz system exhibiting meta-stable chaos, without requiring any a-priori knowledge about the underlying dynamics. The autonomous decisions taken by the RL algorithm are analyzed to understand how the system’s dynamics are impacted. Learning from this analysis, a simple control-law capable of restoring chaotic behavior is formulated. The reverse-engineering approach adopted in this work underlines the immense potential of the techniques used here to discover effective control strategies in complex dynamical systems. The autonomous nature of the learning algorithm makes it applicable to a diverse variety of non-linear systems, and highlights the potential of RLenabled control for regulating other transient-chaos like catastrophic events. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

Les méthodologies de l'automatique ont joué au cours des dernières décennies un ´r^ole essentiel au sein de nombreux secteurs technologiques avancées. Cependant, de nombreuse questions restent ouvertes. Parmi celles-ci, celles concernant la stabilité et la stabilisation de systèmes non linéaires sont d'intérêt primordial. Afin de stabilizer un système (physique ou non), il est nécessaire de capter et interpreter en temps réel les informations hétérogènes caractérisant son fonctionnement afin intervenir efficacement. Actuellement ces informations ne sont pas captées en temps continu, mais de façon synchrone ou asynchrone et ceci est valable aussi pour les actuateurs. De façon très naturelle, on définit donc un système hybride, caractérisé par des dynamiques à la fois discrètes et continues. Dans ce contexte, cette thèse est orientée au développement de nouvelles méthodologies pour la stabilisation de systèmes échantillonnés non linéaires en se focalisant sur la stabilisation de formes cascades qui se retrouvent dans de nombreuse situations concretes. Pour cela, on étudiera l'effet de l'échantillonnage sur les propriétés de la dynamique continue et l'on proposera des méthodologies pour la conception de lois de commande qui ne requièrent pas d'assumptions supplémentaires au cas continu.Enfin, on étudiera l'effet de l'échantillonnage sur des systèmes présentant de retards sur les entrées. On développera des lois de commande stabilisantes exploitant la structure en cascade induite par l'échantillonnage. Des exemples académiques illustreront les calcules des solutions et leur performances tout au long du manuscript. / Over the last decades the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model.Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system. Academic examples will illustrate the computational aspects together with their performances throughout the whole manuscript.

The most suitable model that idealizes random sequences of shock and impacts on vibratory systems is that of a random train of pulses (or impulses), whose arrivals are characterized in terms of stochastic point processes. Most of the existing methods of stochastic dynamics are relevant to random impulsive excitations driven by Poisson processes and there exist some methods for Erlang renewal-driven impulse processes. Herein, two classes of random impulse processes are considered. The first one is the train of impulses whose interarrival timesare driven by an Erlang renewal process. The second class is obtained by selecting some impulses from the train driven by an Erlang renewal process. The selection is performed with the aid of the jump, zero-one, stochastic process governed by the stochastic differential equation driven by the independent Erlang renewal processes. The underlying counting process, driving the arrival times of the impulses, is fully characterized. The expressions for the probability density functions of the first and second waiting times are derived and by means of these functions it is proved that the underlying counting process is a renewal (non-Erlang) process. The probability density functions of the interarrival times are evaluated for four different cases of the driving process and the results obtained for some example sets of parameters are shown graphically. The advantage of modeling the interarrival times using the class of non-Erlang renewal processes analyzed in the present dissertation, rather than the Poisson or Erlang distributions is that it is possible to deal with a broader class of the interarrival probability density functions. The non-Erlang renewal processes considered herein, obtained from two independent Erlang renewal processes, are characterized by four parameters that can be chosen to fit more closely the actual data on the distribution of the interarrival times. As the renewal counting process is not the one with independent increments, the state vector of the dynamic system under a renewal impulse process excitation is not a Markov process. The non-Markov problem may be then converted into a Markov one at the expense of augmenting the state vector by auxiliary discrete stochastic variables driven by a Poisson process. Other than the existing in literature (Iwankiewicz and Nielsen), a novel technique of conversion is devised here, where the auxiliary variables are all zero-one processes. In a considered class of non-Erlang renewal impulse processes each of the driving Erlang processes is recast in terms of the Poisson process, the augmented state vector driven by two independent Poisson processes becomes a non-diffusive Markov process. For a linear oscillator, under a considered class of non-Erlang renewal impulse process, the equations for response moments are obtained from the generalized Ito’s differential rule and the mean value and variance of the response are evaluated and shown graphically for some selected sets of parameters. For a non-linear oscillator under both Erlang renewal-driven impulses and the considered class of non-Erlang renewal impulse processes, the technique of equations for moments together with a modified closure technique is devised. The specific physical properties of an impulsive load process allow to modify the classical cumulant-neglect closure scheme and to develop a more efficient technique for the class of excitations considered. The joint probability density of the augmented state vector is expressed as sum of contributions conditioned on the ‘on’ and ‘off’ states of the auxiliary variables. A discrete part of the joint probability density function accounts for the fact that there is a finite probability of the system being in a deterministic state (for example at rest) from the initial time to the occurrence of the first impulse. The continuous part, which is the conditional probability given that the first impulse has occurred, can be expressed in terms of functions of the displacement and velocity of the system. These functions can be viewed as unknown probability densities of a bi-variate stochastic process, each of which originates a set of ‘conditional moments’. The set of relationships between unconditional and conditional moments is derived. The ordinary cumulant neglect closure is then performed on the conditional moments pertinent to the continuous part only. The closure scheme is then formulated by expressing the ‘unconditional’ moments of order greater then the order of closure, in terms of unconditional moments of lower order. The stochastic analysis of a Duffing oscillator under the the random train of impulses driven by an Erlang renewal processes and a non-Erlang renewal process R(t), is performed by applying the second order ordinary cumulant neglect closure and the modified second order closure approximation and the approximate analytical results are verified against direct Monte Carlo simulation. The modified closure scheme proves to give better results for highly non-Gaussian train of impulses, characterized by low mean arrival rate.

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Indiana University-Purdue University Indianapolis (IUPUI) / The inner structural characteristics of non-minimum phase DC-DC converters pose a severe limitation in direct regulation of voltage when addressed from a control perspective. This constraint is reflected by the presence of right half plane zeros or the unstable zero dynamics of the output voltage of these converters. The existing controllers make use of one-to-one correspondence between the voltage and current equilibriums of the non-minimum phase converters and exploit the property that when the average output of these converters is the inductor current, the system dynamics are stable and hence they indirectly regulate the voltage. As a result, the system performance is susceptible to circuit parameter and load variation and require additional controllers, which in turn increase the system complexity. In this thesis, a novel approach to this problem is proposed for second order non-minimum phase converters such as Boost and Buck-Boost Converter. Different solutions have been suggested to the problem based on whether the converter is modeled as a linear system or as a nonlinear system. For the converter modeled as a linear system, the non-minimum phase part of the system is decoupled and its transfer function is converted to minimum phase using a parallel compensator. Then the control action is achieved by using a simple proportional gain controller. This method accelerates the transient response of the converter, reduces the initial undershoot in the response, and considerably reduces the oscillations in the transient response. Simulation results demonstrate the effectiveness of the proposed approach. When the converter is modeled as a bilinear system, it preserves the stabilizing nonlinearities of the system. Hence, a more effective control approach is adopted by using Passivity properties. In this approach, the non-minimum phase converter system is viewed from an energy-based perspective and the property of passivity is used to achieve stable zero dynamics of the output voltage. A system is passive if its rate of energy storage is less than the supply rate i.e. the system dissipates more energy than stores. As a result, the energy storage function of the system is less than the supply rate function. Non-minimum phase systems are not passive, and passivation of non-minimum phase power converters is an attractive solution to the posed problem. Stability of non-minimum phase systems can also be investigated by defining the passivity indices. This research approaches the problem by characterizing the degree of passivity i.e. the amount of damping in the system, from passivity indices. Thus, the problem is viewed from a system level rather than from a circuit level description. This method uses feed-forward passivation to compensate for the shortage of passivity in the non-minimum phase converter and makes use of a parallel interconnection to the open-loop system to attain exponentially stable zero dynamics of the output voltage. Detailed analytical analysis regarding the control structure and passivation process is performed on a buck-boost converter. Simulation and experimental results carried out on the test bed validate the effectiveness of the proposed method.

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A significant challenge in the design of multidisciplinary systems (e.g., airplanes, robots, cell phones) is to predict the effects of design decisions at the time these decisions are being made early in the design process. These predictions are used to choose among design options and to validate design decisions. System behavioral models, which predict a system's response to stimulus, provide an analytical method for evaluating a system's behavior. Because multidisciplinary systems contain many different types of components that have diverse interactions, system behavioral models are difficult to develop early in system design and are challenging to maintain as designs are refined. This research develops methods to create, verify, and maintain multidisciplinary system models developed from models that are already part of system design. First, this research introduces a system model formulation that enables virtually any existing engineering model to become part of a large, trusted population of component models from which system behavioral models can be developed. Second, it creates a new algorithm to efficiently quantify the feasible domain over which the system model can be used. Finally, it quantifies system model accuracy early in system design before system measurements are available so that system models can be used to validate system design decisions. The results of this research are enabling system designers to evaluate the effects of design decisions early in system design, improving the predictability of the system design process, and enabling exploration of system designs that differ greatly from existing solutions.

This dissertation concerns the formation problems for multiple reduced attitudes, which are extensively utilized in many pointing applications and under-actuated scenarios for attitude maneuvers. In contrast to most existing methodologies on formation control, the proposed method does not need to contain any formation errors in the protocol. Instead, the constructed formation is attributed to geometric properties of the configuration space and the designed connection topology. We refer to this type of formation control as intrinsic formation control. Besides, the control protocols proposed in this work are designed directly in space S2, avoiding to use any attitude parameterisations. At last but not least, along the studies, some elementary tools for reduced attitudes control are developed.In paper A, a continuous control law is provided for a reduced attitude systems, by which a regular tetrahedron formation can achieve asymptotic stability under a quite large family of gain functions in the control. Then, with a further restriction on the control gain, almost global stability of the tetrahedron formation is also obtained. In this work, we introduce a novel coordinates transformation that represents the relative reduced attitudes be-tween the agents. The proposed method is an intrinsic formation control that does not need to involve any information of the desired formation before-hand. Another virtue of the method proposed is that only relative attitude measurement is required.Paper B further concerns the formation control of all regular polyhedral configurations (also called Platonic solids) for reduced attitudes. According to the symmetries possessed by regular polyhedra, a unified framework is proposed for their formations. Via using the coordinates transformation previously proposed, it is shown that the stability of the desired formations can be provided by stabilizing a constrained nonlinear system. Then, a methodology to investigate the stability of this type of constrained systems is also presented. Paper C considers the problem of tracking and encircling a moving target by agents in 3-dimensional space. By this work, we show that similar design techniques proposed for reduced attitudes formations can also be applied to the formation control for point mass systems. Therein, a group of agents are driven to some desired formation on a spherical surface and simultaneously keep the center of this spherical formation coinciding with the target to be tracked. By properly designing communication topology, the agents constitute a cyclic formation along the equator of an encircling sphere. / <p>QC 20170302</p>