Numerical sensitivity of Linear Matrix Inequalities for shorter sampling periods

The numerical sensitivity of Linear Matrix Inequalities (LMIs) arising in the H∞ norm computation in discrete time is analyzed. Rapid sampling scenarios are examined comparing both shift and delta operator formulations of the equations. The shift operator formulation is shown in general to be arbitrarily poorly conditioned as the sampling rate increases. The delta operator formulation includes both recentering (to avoid cancellation problems) and rescaling, and avoids these difficulties. However, it is also shown that rescaling of the shift operator formulation gives substantial improvements in numerical conditioning, whilst recentering is of more limited benefit

Numerical Sensitivity of Linear Matrix Inequalities Using Shift and Delta Operators

The numerical sensitivity of linear matrix inequalities (LMIs) arising from discrete-time control with short sampling periods is analyzed using shift and delta operators. The delta operator avoids cancellation problems for short sampling periods, and it includes a system scaling proportional to the inverse of the sampling period. The numerical sensitivity of both these mechanisms is investigated analytically, and verified by numerical examples. The conclusion is that the scaling procedure is (somewhat surprisingly) much more essential for shorter sampling periods than avoiding the cancellation problem

Commitment and Dispatch of Heat and Power Units via Affinely Adjustable Robust Optimization

The joint management of heat and power systems is believed to be key to the integration of renewables into energy systems with a large penetration of district heating. Determining the day-ahead unit commitment and production schedules for these systems is an optimization problem subject to uncertainty stemming from the unpredictability of demand and prices for heat and electricity. Furthermore, owing to the dynamic features of production and heat storage units as well as to the length and granularity of the optimization horizon (e.g., one whole day with hourly resolution), this problem is in essence a multi-stage one. We propose a formulation based on robust optimization where recourse decisions are approximated as linear or piecewise-linear functions of the uncertain parameters. This approach allows for a rigorous modeling of the uncertainty in multi-stage decision-making without compromising computational tractability. We perform an extensive numerical study based on data from the Copenhagen area in Denmark, which highlights important features of the proposed model. Firstly, we illustrate commitment and dispatch choices that increase conservativeness in the robust optimization approach. Secondly, we appraise the gain obtained by switching from linear to piecewise-linear decision rules within robust optimization. Furthermore, we give directions for selecting the parameters defining the uncertainty set (size, budget) and assess the resulting trade-off between average profit and conservativeness of the solution. Finally, we perform a thorough comparison with competing models based on deterministic optimization and stochastic programming.Comment: 31 page

Harmonic Scheduling and Control Co-Design

Harmonic task scheduling has many attractive properties, including a utilization bound of 100% under rate-monotonic scheduling and reduced jitter. At the same time, it places a severe constraint on the task period assignment for any application. In this paper, we explore the use of harmonic task scheduling for applications with multiple feedback control tasks. We present an algorithm for finding harmonic task periods that minimizes the distance from an initial set of non-harmonic periods. We apply the algorithm in a scheduling and control co-design procedure, where the goal is to optimize the total performance of a number of control tasks that share a common computing platform. The procedure is evaluated in simulated randomized examples, where it is shown that, in general, harmonic scheduling combined with release offsets gives better control performance than standard, non-harmonic scheduling

Statistical Inference for Computable General Equilibrium Models with Application to a Model of the Moroccan Economy

We study the problem of measuring the uncertainty of CGE (or RBC)-type model simulations associated with parameter uncertainty. We describe two approaches for building confidence sets on model endogenous variables. The first one uses a standard Wald-type statistic. The second approach assumes that a confidence set (sampling or Bayesian) is available for the free parameters, from which confidence sets are derived by a projection technique. The latter has two advantages: first, confidence set validity is not affected by model nonlinearities; second, we can easily build simultaneous confidence intervals for an unlimited number of variables. We study conditions under which these confidence sets take the form of intervals and show they can be implemented using standard methods for solving CGE models. We present an application to a CGE model of the Moroccan economy to study the effects of policy-induced increases of transfers from Moroccan expatriates.Nous étudions le problème de la mesure de l’incertitude des simulations de modèles d’équilibre général calculable (MEGC). Nous décrivons deux approches pour construire des régions de confiance pour les variables endogènes de ces modèles. La première utilise une statistique standard de type Wald. La seconde approche suppose l’existence, pour les paramètres libres du modèle, d’une région de confiance (échantillonnale ou bayesienne) à partir de laquelle des régions de confiance, pour les variables endogènes, sont déduites par une technique de projection. Cette dernière méthode a deux avantages: premièrement, la validité des régions de confiance construites n’est pas affectée par la non-linéarité du modèle; deuxièmement, on peut facilement construire des intervalles de confiance pour un nombre illimité de variables. Nous étudions aussi les conditions sous lesquelles ces régions de confiance prennent la forme d’intervalles et nous montrons que ces méthodes peuvent facilement être utilisées au moyen de méthodes standard de résolution des MEGC. Nous présentons une application sur un modèle de l’économie marocaine qui étudie les effets visant à faire augmenter les rapatriements de capitaux par les résidents marocains à l’étranger