296 research outputs found

    Dynamic Management of Portfolios with Transaction Costs under Tychastic Uncertainty.

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    We use in this chapter the viability/capturability approach for studying the problem of dynamic valuation and management of a portfolio with transaction costs in the framework of tychastic control systems (or dynamical games against nature) instead of stochastic control systems. Indeed, the very definition of the guaranteed valuation set can be formulated directly in terms of guaranteed viable-capture basin of a dynamical game. Hence, we shall “compute” the guaranteed viable-capture basin and find a formula for the valuation function involving an underlying criterion, use the tangential properties of such basins for proving that the valuation function is a solution to Hamilton-Jacobi-Isaacs partial differential equations. We then derive a dynamical feedback providing an adjustment law regulating the evolution of the portfolios obeying viability constraints until it achieves the given objective in finite time. We shall show that the Pujal—Saint-Pierre viability/capturability algorithm applied to this specific case provides both the valuation function and the associated portfolios.dynamic games; dynamic valuation; tychastic control systems; management of portfolio;

    Differential games through viability theory : old and recent results.

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    This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas : games with hard constraints, stochastic differential games, and hybrid differential games. We also discuss several applications.Game theory; Differential game; viability algorithm;

    Guaranteed Inertia Functions in Dynamical Games.

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    This paper deals with inertia functions in control theory introduced in Aubin, Bernardo and Saint-Pierre (2004, 2005) and their adaptation to dynamical games. The inertia function associates with any initial state-control pair the smallest of the worst norms over time of the velocities of the controls regulating viable evolutions. For tychastic systems (parameterized systems where the parameters are tyches, disturbances, perturbations, etc.), the palicinesia of a tyche measure the worst norm over time of the velocities of the tyches. The palicinesia function is the largest palicinesia threshold c such that all evolutions with palicinesia smaller than or equal to c are viable. For dynamical games where one parameter is the control and the other one is a tyche (games against nature or robust control), we define the guaranteed inertia function associated with any initial state-control-tyche triple the best of the worst of the norms of the velocities of the controls and of the tyches and study their properties. Viability Characterizations and Hamilton-Jacobi equations of which these inertia and palicinesia functions are solutions are provided.Viability; dynamical games; inertia function; Tychastic systems; palicinesia;

    Viability-based computation of spatially constrained minimum time trajectories for an autonomous underwater vehicle: implementation and experiments.

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    A viability algorithm is developed to compute the constrained minimum time function for general dynamical systems. The algorithm is instantiated for a speciïŹc dynamics(Dubin’s vehicle forced by a ïŹ‚ow ïŹeld) in order to numerically solve the minimum time problem. With the speciïŹc dynamics considered, the framework of hybrid systems enables us to solve the problem efïŹciently. The algorithm is implemented in C using epigraphical techniques to reduce the dimension of the problem. The feasibility of this optimal trajectory algorithm is tested in an experiment with a Light Autonomous Underwater Vehicle (LAUV) system. The hydrodynamics of the LAUV are analyzed in order to develop a low-dimension vehicle model. Deployment results from experiments performed in the Sacramento River in California are presented, which show good performance of the algorithm.trajectories; underwater vehicle; viability algorithm; hybrid systems; implementation;

    Static analysis of SPDIs for state-space reduction

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    Polygonal hybrid systems (SPDI) are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. The reachability problem as well as the computation of certain objects of the phase portrait, namely the viability, controllability and invariance kernels, for such systems is decidable. In this paper we show how to compute another object of an SPDI phase portrait, namely semi-separatrix curves and show how the phase portrait can be used for reducing the state-space for optimizing the reachability analysis.peer-reviewe

    Improving polygonal hybrid systems reachability analysis through the use of the phase portrait

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    Polygonal hybrid systems (SPDI) are a subclass of planar hybrid automata which can be represented by piecewise constant dierential inclusions. The computation of certain objects of the phase portrait of an SPDI, namely the viability, controllability, invariance kernels and semi-separatrix curves have been shown to be eciently decidable. On the other hand, although the reachability problem for SPDIs is known to be decidable, its complexity makes it unfeasible on large systems. We summarise our recent results on the use of the SPDI phase portraits for improving reachability analysis by (i) state-space reduction and (ii) decomposition techniques of the state space, enabling compositional parallelisation of the analysis. Both techniques contribute to increasing the feasability of reachability analysis on large SPDI systems.peer-reviewe

    An Interval Constraint Programming Approach for Quasi Capture Tube Validation

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    Proving that the state of a controlled nonlinear system always stays inside a time moving bubble (or capture tube) amounts to proving the inconsistency of a set of nonlinear inequalities in the time-state space. In practice however, even with a good intuition, it is difficult for a human to find such a capture tube except for simple examples. In 2014, Jaulin et al. established properties that support a new interval approach for validating a quasi capture tube, i.e. a candidate tube (with a simple form) from which the mobile system can escape, but into which it enters again before a given time. A quasi capture tube is easy to find in practice for a controlled system. Merging the trajectories originated from the candidate tube yields the smallest capture tube enclosing it. This paper proposes an interval constraint programming solver dedicated to the quasi capture tube validation. The problem is viewed as a differential CSP where the functional variables correspond to the state variables of the system and the constraints define system trajectories that escape from the candidate tube "for ever". The solver performs a branch and contract procedure for computing the trajectories that escape from the candidate tube. If no solution is found, the quasi capture tube is validated and, as a side effect, a corrected smallest capture tube enclosing the quasi one is computed. The approach is experimentally validated on several examples having 2 to 5 degrees of freedom

    Viability in State-Action Space: Connecting Morphology, Control, and Learning

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    Wie können wir Robotern ermöglichen, modellfrei und direkt auf der Hardware zu lernen? Das maschinelle Lernen nimmt als Standardwerkzeug im Arsenal des Robotikers seinen Platz ein. Es gibt jedoch einige offene Fragen, wie man die Kontrolle ĂŒber physikalische Systeme lernen kann. Diese Arbeit gibt zwei Antworten auf diese motivierende Frage. Das erste ist ein formales Mittel, um die inhĂ€rente Robustheit eines gegebenen Systemdesigns zu quantifizieren, bevor der Controller oder das Lernverfahren entworfen wird. Dies unterstreicht die Notwendigkeit, sowohl das Hardals auch das Software-Design eines Roboters zu berĂŒcksichtigen, da beide Aspekte in der Systemdynamik untrennbar miteinander verbunden sind. Die zweite ist die Formalisierung einer Sicherheitsmass, die modellfrei erlernt werden kann. Intuitiv zeigt diese Mass an, wie leicht ein Roboter FehlschlĂ€ge vermeiden kann. Auf diese Weise können Roboter unbekannte Umgebungen erkunden und gleichzeitig AusfĂ€lle vermeiden. Die wichtigsten BeitrĂ€ge dieser Dissertation basieren sich auf der ViabilitĂ€tstheorie. ViabilitĂ€t bietet eine alternative Sichtweise auf dynamische Systeme: Anstatt sich auf die Konvergenzeigenschaften eines Systems in Richtung Gleichgewichte zu konzentrieren, wird der Fokus auf Menge von FehlerzustĂ€nden und die FĂ€higkeit des Systems, diese zu vermeiden, verlagert. Diese Sichtweise eignet sich besonders gut fĂŒr das Studium der Lernkontrolle an Robotern, da StabilitĂ€t im Sinne einer Konvergenz wĂ€hrend des Lernprozesses selten gewĂ€hrleistet werden kann. Der Begriff der ViabilitĂ€t wird formal auf den Zustand-Aktion-Raum erweitert, mit ViabilitĂ€tsmengen von Staat-Aktionspaaren. Eine ĂŒber diese Mengen definierte Mass ermöglicht eine quantifizierte Bewertung der Robustheit, die fĂŒr die Familie aller fehlervermeidenden Regler gilt, und ebnet den Weg fĂŒr ein sicheres, modellfreies Lernen. Die Arbeit beinhaltet auch zwei kleinere BeitrĂ€ge. Der erste kleine Beitrag ist eine empirische Demonstration der Shaping durch ausschliessliche Modifikation der Systemdynamik. Diese Demonstration verdeutlicht die Bedeutung der Robustheit gegenĂŒber Fehlern fĂŒr die Lernkontrolle: AusfĂ€lle können nicht nur SchĂ€den verursachen, sondern liefern in der Regel auch keine nĂŒtzlichen Gradienteninformationen fĂŒr den Lernprozess. Der zweite kleine Beitrag ist eine Studie ĂŒber die Wahl der Zustandsinitialisierungen. Entgegen der Intuition und der ĂŒblichen Praxis zeigt diese Studie, dass es zuverlĂ€ssiger sein kann, das System gelegentlich aus einem Zustand zu initialisieren, der bekanntermassen unkontrollierbar ist.How can we enable robots to learn control model-free and directly on hardware? Machine learning is taking its place as a standard tool in the roboticist’s arsenal. However, there are several open questions on how to learn control for physical systems. This thesis provides two answers to this motivating question. The first is a formal means to quantify the inherent robustness of a given system design, prior to designing the controller or learning agent. This emphasizes the need to consider both the hardware and software design of a robot, which are inseparably intertwined in the system dynamics. The second is the formalization of a safety-measure, which can be learned model-free. Intuitively, this measure indicates how easily a robot can avoid failure, and enables robots to explore unknown environments while avoiding failures. The main contributions of this dissertation are based on viability theory. Viability theory provides a slightly unconventional view of dynamical systems: instead of focusing on a system’s convergence properties towards equilibria, the focus is shifted towards sets of failure states and the system’s ability to avoid these sets. This view is particularly well suited to studying learning control in robots, since stability in the sense of convergence can rarely be guaranteed during the learning process. The notion of viability is formally extended to state-action space, with viable sets of state-action pairs. A measure defined over these sets allows a quantified evaluation of robustness valid for the family of all failure-avoiding control policies, and also paves the way for enabling safe model-free learning. The thesis also includes two minor contributions. The first minor contribution is an empirical demonstration of shaping by exclusively modifying the system dynamics. This demonstration highlights the importance of robustness to failures for learning control: not only can failures cause damage, but they typically do not provide useful gradient information for the learning process. The second minor contribution is a study on the choice of state initializations. Counter to intuition and common practice, this study shows it can be more reliable to occasionally initialize the system from a state that is known to be uncontrollable

    Multi-Objective Hybrid Optimal Control for Multiple-Flyby Interplanetary Mission Design Using Chemical Propulsion

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    Preliminary design of high-thrust interplanetary missions is a highly complex process. The mission designer must choose discrete parameters such as the number of flybys and the bodies at which those flybys are performed. For some missions, such as surveys of small bodies, the mission designer also contributes to target selection. In addition, real-valued decision variables, such as launch epoch, flight times, maneuver and flyby epochs, and flyby altitudes must be chosen. There are often many thousands of possible trajectories to be evaluated. The customer who commissions a trajectory design is not usually interested in a point solution, but rather the exploration of the trade space of trajectories between several different objective functions. This can be a very expensive process in terms of the number of human analyst hours required. An automated approach is therefore very desirable. This work presents such an approach by posing the impulsive mission design problem as a multiobjective hybrid optimal control problem. The method is demonstrated on several real-world problems
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