The general problem of multi-agent persistent monitoring finds applications in a variety of domains ranging from meter to kilometer-scale systems, such as surveillance or environmental monitoring, down to nano-scale systems such as tracking biological macromolecules for studying basic biology and disease. The problem can be cast as moving the agents between targets, acquiring information from or in some fashion controlling the states of the targets. Under this formulation, at least two questions need to be addressed. The first is the design of motion trajectories for the agents as they move among the spatially distributed targets and jointly optimize a given cost function that describes some desired application. The second is the design of the controller that an agent will use at a target to steer the target's state as desired. The first question can be viewed in at least two ways: first, as an optimal control problem with the domain of the targets described as a continuous space, and second as a discrete scheduling task. In this work we focus on the second approach, which formulates the target dynamics as a hybrid automaton, and the geometry of the targets as a graph. We show how to find solutions by translating the scheduling problem into a search for the optimal route. With a route specifying the visiting sequence in place, we derive the optimal time the agent spends at each target analytically. The second question, namely that of steering the target's state, can be formulated from the perspective of the target, rather than the agent. The mobile nature of the agents leads to intermittencontrol, such that the controller is assumed to be disconnected when no agent is at the target. The design of the visiting schedule of agents to one target can affect the reachability (controllability) of this target's control system and the design of any specific controller. Existing test techniques for reachability are combined with the idea of lifting to provide conditions on systems such that reachability is maintained in the presence of periodic disconnections from the controller. While considering an intermittently connected control with constraints on the control authority and in the presence of a disturbance, the concept of 'degree of controllability' is introduced. The degree is measured by a region of states that can be brought back to the origin in a given finite time. The size of this region is estimated to evaluate the performance of a given sequence.

Un problème fondamental à résoudre en Automatique réside dans la commande des systèmes incertains qui présentent des contraintes sur les variables de l’entrée, de l’état ou la sortie. Ce problème peut être théoriquement résolu au moyen d’une commande optimale. Cependant la commande optimale par principe n’est pas une commande par retour d’état ou retour de sortie et offre seulement une trajectoire optimale le plus souvent par le biais d’une solution numérique.Par conséquent, dans la pratique, le problème peut être approché par de nombreuses méthodes, tels que”commande over-ride” et ”anti-windup”. Une autre solution, devenu populaire au cours des dernières décennies est la commande prédictive. Selon cette méthode, un problème de la commande optimale est résolu à chaque instant d’échantillonnage, et le composant du vecteur de commande destiné à l’échelon curant est appliquée. En dépit de la montée en puissance des architecture de calcul temps-réel, la commande prédictive est à l’heure actuelle principalement approprié lorsque l’ordre est faible, bien connu, et souvent pour des systèmes linéaires. La version robuste de la commande prédictive est conservatrice et compliquée à mettre en œuvre, tandis que la version explicite de la commande prédictive donnant une solution affine par morceaux implique une compartimentation de l’état-espace en cellules polyédrales, très compliquée.Dans cette thèse, une solution élégante et peu coûteuse en temps de calcul est présentée pour des systèmes linéaire, variant dans le temps ou incertains. Les développements se concentre sur les dynamiques en temps discret avec contraintes polyédriques sur l’entrée et l’état (ou la sortie) des vecteurs, dont les perturbations sont bornées. Cette solution est basée sur l’interpolation entre un correcteur pour la région extérieure qui respecte les contraintes sur l’entrée et de l’état, et un autre pour la région intérieure, ce dernier plus agressif, conçue par n’importe quelle méthode classique, ayant un ensemble robuste positivement invariant associé à l’intérieur des contraintes. Une simple fonction de Lyapunov est utilisée afin d’apporter la preuve de la stabilité en boucle fermée. / A fundamental problem in automatic control is the control of uncertain plants in the presence of input and state or output constraints. An elegant and theoretically most satisfying framework is represented by optimal control policies which, however, rarely gives an analytical feedback solution, and oftentimes builds on numerical solutions (approximations).Therefore, in practice, the problem has seen many ad-hoc solutions, such as override control, anti-windup, as well as modern techniques developed during the last decades usually based on state space models. One of the popular example is Model Predictive Control (MPC) where an optimal control problem is solved at each sampling instant, and the element of the control vector meant for the nearest sampling interval is applied. In spite of the increased computational power of control computers, MPC is at present mainly suitable for low-order, nominally linear systems. The robust version of MPC is conservative and computationally complicated, while the explicit version of MPC that gives an affine state feedback solution involves a very complicated division of the state space into polyhedral cells.In this thesis a novel and computationally cheap solution is presented for linear, time-varying or uncertain, discrete-time systems with polytopic bounded control and state (or output) vectors, with bounded disturbances. The approach is based on the interpolation between a stabilizing, outer controller that respects the control and state constraints, and an inner, more aggressive controller, designed by any method that has a robustly positively invariant set within the constraints. A simple Lyapunov function is used for the proof of closed loop stability.In contrast to MPC, the new interpolation based controller is not necessarily employing an optimization criterion inspired by performance. In its explicit form, the cell partitioning is simpler that the MPC counterpart. For the implicit version, the on-line computational demand can be restricted to the solution of one linear program or quadratic program. Several simulation examples are given, including uncertain linear systems with output feedback and disturbances. Some examples are compared with MPC. The control of a laboratory ball-and-plate system is also demonstrated. It is believed that the new controller might see wide-spread use in industry, including the automotive industry, also for the control of fast, high-order systems with constraints.

Ensuring constraint satisfaction in large-scale systems with hard constraints is vital in many safety critical systems. The challenge is to design controllers that are efficiently synthesized offline, easily implementable online, and provide formal correctness guarantees. We take a divide and conquer approach to design controllers for reachability and infinite-time/finite-time constraint satisfaction control problems given large-scale interconnected linear systems with polyhedral constraints on states, controls, and disturbances. Such systems are made of small subsystems with coupled dynamics. Our goals are to design controllers that are i) fully compositional and ii) decentralized, such that online implementation requires only local state information. We treat the couplings among the subsystems as additional disturbances and use assume-guarantee (AG) contracts to characterize these disturbance sets. For each subsystem, we design and implement a robust controller locally, subject to its own constraints and contracts. Our main contribution is a method to derive the contracts via a novel parameterization, and a corresponding potential function that characterizes the distance to the correct composition of controllers and contracts, where all contracts are held. We show that the potential function is convex in the contract parameters. This enables the subsystems to negotiate the contracts with the gradient information from the dual of their local synthesis optimization problems in a distributed way, facilitating compositional control synthesis that scales to large systems. We then incorporate Signal Temporal Logic (STL) specifications into our formulation. We develop a decentralized control method for a network of perturbed linear systems with dynamical couplings subject to STL specifications. We first transform the STL requirements into set containment problems, then we develop controllers to solve these problems. The set containment requirements and parameterized contracts are added to the subsystems’ constraints. We introduce a centralized optimization problem to derive the contracts, reachability tubes, and decentralized closed-loop control laws. We show that, when the STL formula is separable with respect to the subsystems, the centralized optimization problem can be solved in a distributed way, which scales to large systems. We present formal theoretical guarantees on robustness of STL satisfaction. We present numerical examples, including scalability studies on systems with tens of thousands of dimensions, and case studies on applying our method to a distributed Model Predictive Control (MPC) problem in a power system. / 2024-01-16T00:00:00Z

Uncertainties and constraints are present in most control systems. For example, robot motion planning and building climate regulation can be modeled as uncertain constrained systems. In this thesis, we develop mathematical and computational tools to analyze and synthesize controllers for such systems. As our first contribution, we characterize when a set is a probabilistic controlled invariant set and we develop tools to compute such sets. A probabilistic controlled invariantset is a set within which the controller is able to keep the system state with a certainprobability. It is a natural complement to the existing notion of robust controlled invariantsets. We provide iterative algorithms to compute a probabilistic controlled invariantset within a given set based on stochastic backward reachability. We prove that thesealgorithms are computationally tractable and converge in a finite number of iterations. The computational tools are demonstrated on examples of motion planning, climate regulation, and model predictive control. As our second contribution, we address the control design problem for uncertain constrained systems with aperiodic sensing and actuation. Firstly, we propose a stochastic self-triggered model predictive control algorithm for linear systems subject to exogenous disturbances and probabilistic constraints. We prove that probabilistic constraint satisfaction, recursive feasibility, and closed-loop stability can be guaranteed. The control algorithm is computationally tractable as we are able to reformulate the problem into a quadratic program. Secondly, we develop a robust self-triggered control algorithm for time-varying and uncertain systems with constraints based on reachability analysis. In the particular case when there is no uncertainty, the design leads to a control system requiring minimum number of samples over finite time horizon. Furthermore, when the plant is linear and the constraints are polyhedral, we prove that the previous algorithms can be reformulated as mixed integer linear programs. The method is applied to a motion planning problem with temporal constraints. / <p>QC 20181016</p>

Fluid mixing is a challenging problem in laminar flow systems. Chaotic advection can play an important role in enhancing mixing in such flow. In this thesis, different approaches are used to enhance fluid mixing in two laminar flow systems. In the first system, chaos is generated in a flow between two closely spaced parallel circular plates by pulsed operation of fluid extraction and reinjection through singularities in the domain. A singularity through which fluid is injected (or extracted) is called a source (or a sink). In a bounded domain, one source and one sink with equal strength operate together as a source-sink pair to conserve the fluid volume. Fluid flow between two closely spaced parallel plates is modeled as Hele-Shaw flow with the depth averaged velocity proportional to the gradient of the pressure. So, with the depth-averaged velocity, the flow between the parallel plates can effectively be modeled as two-dimensional potential flow. This thesis discusses pulsed source-sink systems with two source-sink pairs operating alternately to generate zig-zag trajectories of fluid particles in the domain. For reinjection purpose, fluid extracted through a sink-type singularity can either be relocated to a source-type one, or the same sink-type singularity can be activated as a source to reinject it without relocation. Relocation of fluid can be accomplished using either "first out first in" or "last out first in" scheme. Both relocation methods add delay to the pulse time of the system. This thesis analyzes mixing in pulsed source-sink systems both with and without fluid relocation. It is shown that a pulsed source-sink system with "first out first in" scheme generates comparatively complex fluid flow than pulsed source-sink systems with "last out first in" scheme. It is also shown that a pulsed source-sink system without fluid relocation can generate complex fluid flow. In the second system, mixing and transport is analyzed in a two-dimensional Stokes flow system. Appropriate periodic motions of three rods or periodic points in a two-dimensional flow are determined using the Thurston-Nielsen Classification Theorem (TNCT), which also predicts a lower bound on the complexity generated in the fluid flow. This thesis extends the TNCT -based framework by demonstrating that, in a perturbed system with no lower order fixed points, almost invariant sets are natural objects on which to apply the TNCT. In addition, a method is presented to compute line stretching by tracking appropriate motion of finite size rods. This method accounts for the effect of the rod size in computing the complexity generated in the fluid flow. The last section verifies the existence of almost invariant sets in a two-dimensional flow at finite Reynolds number. The almost invariant set structures move with appropriate periodic motion validating the application of the TNCT to predict a lower bound on the complexity generated in the fluid flow. / Ph. D.

The technological changes and educational expansion have created the heterogeneity in the human species. Clearly, this heterogeneity generates a structure in the population dynamics, namely: citizen, permanent resident, visitor, and etc. Furthermore, as the heterogeneity in the population increases, the human mobility between meta-populations patches also increases. Depending on spatial scales, a meta-population patch can be decomposed into sub-patches, for examples: homes, neighborhoods, towns, etc. The dynamics of human mobility in a heterogeneous and scaled structured population is still its infancy level. We develop and investigate (1) an algorithmic two scale human mobility dynamic model for a meta-population. Moreover,the two scale human mobility dynamic model can be extended to multi-scales by applying the algorithm. The subregions and regions are interlinked via intra-and inter regional transport network systems. Under various types of growth order assumptions on the intra and interregional residence times of the residents of a sub region, different patterns of static behavior of the mobility process are studied. Furthermore, the human mobility dynamic model is applied to a two-scale population dynamic exhibiting a special real life human transportation network pattern. The static evolution of all categories of residents of a given site ( homesite, visiting sites within the region, and visiting sites in other regions) over continuous changes in the intra and inter-regional visiting times is also analyzed. The development of the two scale human mobility dynamic model provides a suitable approach to undertake the study of the non-uniform global spread of emergent infectious diseases of humans in a systematic and unified way. In view of this, we derive (2) a SIRS stochastic epidemic dynamic process in a two scale structured population. By defining a positively self invariant set for the dynamic model the stochastic asymptotic stability results of the disease free equilibrium are developed(2). Furthermore, the significance of the stability results are illustrated in a simple real life scenario that is under controlled quarantine disease strategy. In addition, the epidemic dynamic model (2) is applied to a SIR influenza epidemic in a two scale population that is under the influence of a special real life human mobility pattern. The simulated trajectories for the different states (susceptible, Infective, Removal) with respect to current location in the two-scale population structure are presented. The simulated findings reveal comparative evolution patterns for the different states and current locations over time. The SIRS stochastic epidemic dynamic model (2) is extended to a SIR delayed stochastic epidemic dynamic model(3). The delay effects in the dynamic model (3) is temporary and account for natural or infection acquired immunity conferred by the disease after disease recovery. Again, we justify the model validation as a prerequisite for the dynamic modeling. Moreover, we also exhibit the real life scenario under controlled quarantine disease strategy.In addition, the developed delayed SIR dynamic model is also applied to SIR influenza epidemic with temporary immunity to an influenza disease strain. The simulated results reveal an oscillatory effect in the trajectory of the naturally immune population. Moreover, the oscillations are more significant at the homesite. We further extended the stochastic temporary delayed epidemic dynamic model (3) into a stochastic delayed epidemic dynamic model with varying immunity period(4). The varying immunity period accounts for the varying time lengths of natural immunity against the infectious agent exhibited within the naturally immune population. Obviously, the stochastic dynamic model with varying immunity period generalizes the SIR temporary delayed dynamic.

The purpose of this thesis is to generalize the growing theory of iterated function systems (IFSs). Earlier, hyperbolic IFSs with finitely many functions have been studied extensively. Also, hyperbolic IFSs with infinitely many functions have been studied. In this thesis, more general IFSs are studied. The Hausdorff pseudometric is studied. This is a generalization of the Hausdorff metric. Wide and narrow limit sets are studied. These are two types of limits of sequences of sets in a complete pseudometric space. Stable Iterated Function Systems, a kind of generalization of hyperbolic IFSs, are defined. Some different, but closely related, types of stability for the IFSs are considered. It is proved that the IFSs with the most general type of stability have unique attractors. Also, invariant sets, addressing, and periodic points for stable IFSs are studied. Hutchinson’s metric (also called Vaserhstein’s metric) is generalized from being defined on a space of probability measures, into a class of norms, the £-norms, on a space of real measures (on certain metric spaces). Under rather general conditions, it is proved that these norms, when they are restricted to positive measures, give rise to complete metric spaces with the metric topology coinciding with the weak*-topology. Then, IFSs with probabilities (IFSPs) are studied, in particular, stable IFSPs. The £-norm-results are used to prove that, as in the case of hyperbolic IFSPs, IFSPs with the most general kind of stability have unique invariant measures. These measures are ”attractive”. Also, an invariant measure is constructed by first ”lifting” the IFSP to the code space. Finally, it is proved that the Random Iteration Algorithm in a sense will ”work” for some stable IFSPs. / <p>Diss. Umeå : Umeå universitet, 1992</p> / digitalisering@umu

Dans cette thèse de doctorat, nous apportons quelques contributions à la stabilité de Lyapunov non-régulière des inclusions différentielles de premier ordre avec opérateurs monotones maximaux, dans un cadre Hilbertien de dimension infini. Nous fournissons des caractérisations explicites, primales et/ou duales, des paires de Lyapunov faibles et fortes, dont les fonctions sont semi-continues inférieurement à valeurs réelles étendues, et associées à des inclusions différentielles dont la partie de droite est gouvernée par des perturbations Lipschitziennes des opérateurs dits Cusco F, ou des opérateurs monotones maximaux A, ou les deux à la fois x(t) ∈ F(x(t}} A(x(t}} t ≥ 0, x(0) ∈ domA. De manière équivalente, nous étudions l'invariance faible et forte des ensembles fermés pour ces inclusions différentielles. Comme dans L'approche classique de Lyapunov à la stabilité des équations différentielles, les résultats présentés dans cette thèse n'utilisent que les données du système différentiel; c'est-à-dire, l'opérateur A et la multifonction F, et donc pas besoin de connaître les solutions, ni les semi-groupes générés par les opérateurs monotones en question. Parce que les paires de Lyapunov sont formées par des fonctions qui sont simplement semi-continues inférieurement, et les ensembles invariants ne sont que ensembles fermés, nous faisons usage dans cette thèse à des outils de l'analyse non-lisse, afin de fournir des critères du premier ordre, utilisant des sous-différentiels généraux et des cônes normaux. Nous fournissons une analyse similaire pour les inclusions différentielles gouvernées par le cône normal proximal à des ensembles prox-réguliers. Notre analyse ci-dessus, nous a permis de présenter ces systèmes prox-réguliers d’apparence plus générale, comme des inclusions différentielles avec opérateurs monotones maximaux. Nous utilisons aussi nos résultats pour étudier la géométrie des opérateurs monotones maximaux, et plus précisément, la caractérisation de la frontière des valeurs de ces opérateurs seulement au moyen des valeurs situées à proximité, distinctes du point de référence. Ce résultat a des applications dans la stabilité des problèmes de la programmation semi-infinie. Nous utilisons également nos résultats sur les paires de Lyapunov et les ensembles invariants pour établir une étude systématique des observateurs de type Luenberger pour des inclusions différentielles avec des cônes normaux à des ensembles prox-réguliers. / In this PhD thesis, we make some contributions to nonsmooth Lyapunov stability of first-order differential inclusions with maximal monotone operators, in the setting of infinite-dimensional Hilbert spaces. We provide primal and dual explicit characterizations for parameterized weak and strong Lyapunov pairs of lower semicontinuous extended-real-valued functions, referred to as a-Lyapunov pairs, associated to differential inclusions with right-hand-sides governed by Lipschitz or Cusco perturbationsF of maximal monotone operators A, x(t) ∈ F(x(t}} A(x(t}} t ≥ 0, x(0) ∈ domA. Equivalently, we study the weak and strong invariance of sets with respect to such differential inclusions. As in the classical Lyapunov approach to the stability of differential equations, the presented results make use of only the data of the differential system; that is, the operator A and the multifunction F, and so no need to know about the solutions, nor the semi-groups generated by the monotone operators. Because our Lyapunov pairs and invariant sets candidates are just lower semicontinuous and closed, respectively, we make use of nonsmooth analysis to provide first-order-like criteria using general subdifferentials and normal cones. We provide similar analysis to non-convex differential inclusions governed by proximal normal cones to prox-regular sets. Our analysis above allowed to prove that such apparently more general systems can be easily coined into our convex setting. We also use our results to study the geometry of maximal monotone operators, and specifically, the characterization of the boundary of the values of such operators by means only of the values at nearby points, which are distinct of the reference point. This result has its application in the stability of semi-infinite programming problems. We also use our results on Lyapunov pairs and invariant sets to provide a systematic study of Luenberger-like observers design for differential inclusions with normal cones to prox-regular sets.

Le problème de l'efficacité énergétique dans le domaine des transports a comme principal défi savoir comment utiliser la source d'énergie pour que l'efficacité énergétique puisse être maximisée, c'est-à-dire comment le véhicule doit être conduit de telle sorte que la quantité minimale d’énergie est utilisée. Ce problème est le principal problème considéré dans cette thèse. Le véhicule est un prototype impliqué dans la course européenne Shell Eco-Marathon. La dynamique du véhicule est d'abord obtenu par l'identification expérimentale des paramètres. Une stratégie en boucle ouverte de conduite optimale en termes de consommation électrique est calculée. Plusieurs approches ont été étudiées pour le suivi de la référence optimale (stratégie de conduite optimale). Ces approches doivent prendre en compte les ressources limitées en taille mémoire et capacité de calcul. Une commande prédictive (MPC) basée sur la dynamique linéarisée est tout d'abord synthétisée. Le problème de poursuite nécessite une MPC avec contraintes variant dans le temps. La stabilité et la convergence de la commande prédictive sont prouvées à l'aide du formalisme des ensembles invariants. En troisième partie, à partie du modèle LPV, une adaptation de techniques standards basées sur des fonctions de Lyapunov quadratiques et à paramètres variants avec calculs hors-ligne est proposée. Elle est implémentée sur un banc de test. Enfin, une technique adaptative robuste avec identification en ligne de la dynamique est proposée et implémentée dans le véhicule. Cette technique a été testée et validée en course. Les résultats expérimentaux obtenus montrent de bonnes performances de la stratégie de conduite / In the field of transportation, the research on energy efficiency has been carried out for few decades by the automotive industry, where one of the main objectives is to reduce the energetic consumption. This particular problem can be rephrased as how the vehicle must be driven so that the minimum quantity of energy is used. This is the optimal driving strategy. In this project, a suitable model of the Vir'volt electric vehicle involved in the European Shell Eco-Marathon is obtained. The unknown parameters involved in the vehicle dynamics are estimated using Parameter identification from experimental data. The identified dynamics is used to derive an optimal driving strategy that is intended to be tracked on-line during the driving task. The tracking task is subject to time-varying polytopic constraint on the input and/or the state. A MPC-based tracking strategy that uses an homothetic transformation as a suitable time-varying invariant set is used. The time-varying invariant set guarantees the asymptotic stability of the control law. The problem of the MPC tracking for Linear Parametric Varying (LPV) systems is introduced. A new explicit MPC strategy for LPV systems is developed. This strategy uses a Parameter dependent Lyapunov Function (PDLF) to involve explicitly the time-varying parameter in the control law and so it reduces conservatism. A benchmark is used to test the performances of the optimal driving strategy and the explicit MPC tracking strategy. Finally, a robust adaptive technique with on-line identification of the dynamics is has been proposed and tested in the race showing good performances of the adaptive driving strategy