4,962 research outputs found
Aggregation and Control of Populations of Thermostatically Controlled Loads by Formal Abstractions
This work discusses a two-step procedure, based on formal abstractions, to
generate a finite-space stochastic dynamical model as an aggregation of the
continuous temperature dynamics of a homogeneous population of Thermostatically
Controlled Loads (TCL). The temperature of a single TCL is described by a
stochastic difference equation and the TCL status (ON, OFF) by a deterministic
switching mechanism. The procedure is formal as it allows the exact
quantification of the error introduced by the abstraction -- as such it builds
and improves on a known, earlier approximation technique in the literature.
Further, the contribution discusses the extension to the case of a
heterogeneous population of TCL by means of two approaches resulting in the
notion of approximate abstractions. It moreover investigates the problem of
global (population-level) regulation and load balancing for the case of TCL
that are dependent on a control input. The procedure is tested on a case study
and benchmarked against the mentioned alternative approach in the literature.Comment: 40 pages, 21 figures; the paper generalizes the result of conference
publication: S. Esmaeil Zadeh Soudjani and A. Abate, "Aggregation of
Thermostatically Controlled Loads by Formal Abstractions," Proceedings of the
European Control Conference 2013, pp. 4232-4237. version 2: added references
for section
Inequalities for the ruin probability in a controlled discrete-time risk process
Ruin probabilities in a controlled discrete-time risk process with a Markov
chain interest are studied. To reduce the risk there is a possibility to reinsure a part or
the whole reserve. Recursive and integral equations for ruin probabilities are given.
Generalized Lundberg inequalities for the ruin probabilities are derived given a constant
stationary policy. The relationships between these inequalities are discussed. To
illustrate these results some numerical examples are included
Uniform polynomial rates of convergence for a class of L\'evy-driven controlled SDEs arising in multiclass many-server queues
We study the ergodic properties of a class of controlled stochastic
differential equations (SDEs) driven by -stable processes which arise
as the limiting equations of multiclass queueing models in the Halfin-Whitt
regime that have heavy-tailed arrival processes. When the safety staffing
parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a
polynomial rate of convergence to the invariant probability measure in total
variation, which is uniform over all stationary Markov controls resulting in a
locally Lipschitz continuous drift. We also derive a matching lower bound on
the rate of convergence (under no abandonment). On the other hand, when all
abandonment rates are positive, we show that the SDEs are exponentially ergodic
uniformly over the above-mentioned class of controls. Analogous results are
obtained for L\'evy-driven SDEs arising from multiclass many-server queues
under asymptotically negligible service interruptions. For these equations, we
show that the aforementioned ergodic properties are uniform over all stationary
Markov controls. We also extend a key functional central limit theorem
concerning diffusion approximations so as to make it applicable to the models
studied here
Sampling-based Approximations with Quantitative Performance for the Probabilistic Reach-Avoid Problem over General Markov Processes
This article deals with stochastic processes endowed with the Markov
(memoryless) property and evolving over general (uncountable) state spaces. The
models further depend on a non-deterministic quantity in the form of a control
input, which can be selected to affect the probabilistic dynamics. We address
the computation of maximal reach-avoid specifications, together with the
synthesis of the corresponding optimal controllers. The reach-avoid
specification deals with assessing the likelihood that any finite-horizon
trajectory of the model enters a given goal set, while avoiding a given set of
undesired states. This article newly provides an approximate computational
scheme for the reach-avoid specification based on the Fitted Value Iteration
algorithm, which hinges on random sample extractions, and gives a-priori
computable formal probabilistic bounds on the error made by the approximation
algorithm: as such, the output of the numerical scheme is quantitatively
assessed and thus meaningful for safety-critical applications. Furthermore, we
provide tighter probabilistic error bounds that are sample-based. The overall
computational scheme is put in relationship with alternative approximation
algorithms in the literature, and finally its performance is practically
assessed over a benchmark case study
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