Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio

Identification of quasi-optimal regions in the design space using surrogate modeling

The use of Surrogate Based Optimization (SBO) is widely spread in engineering design to find optimal performance characteristics of expensive simulations (forward analysis: from input to optimal output). However, often the practitioner knows a priori the desired performance and is interested in finding the associated input parameters (reverse analysis: from desired output to input). A popular method to solve such reverse (inverse) problems is to minimize the error between the simulated performance and the desired goal. However, there might be multiple quasi-optimal solutions to the problem. In this paper, the authors propose a novel method to efficiently solve inverse problems and to sample Quasi-Optimal Regions (QORs) in the input (design) space more densely. The development of this technique, based on the probability of improvement criterion and kriging models, is driven by a real-life problem from bio-mechanics, i.e., determining the elasticity of the (rabbit) tympanic membrane, a membrane that converts acoustic sound wave into vibrations of the middle ear ossicular bones

A Transfer Operator Methodology for Optimal Sensor Placement Accounting for Uncertainty

Sensors in buildings are used for a wide variety of applications such as monitoring air quality, contaminants, indoor temperature, and relative humidity. These are used for accessing and ensuring indoor air quality, and also for ensuring safety in the event of chemical and biological attacks. It follows that optimal placement of sensors become important to accurately monitor contaminant levels in the indoor environment. However, contaminant transport inside the indoor environment is governed by the indoor flow conditions which are affected by various uncertainties associated with the building systems including occupancy and boundary fluxes. Therefore, it is important to account for all associated uncertainties while designing the sensor layout. The transfer operator based framework provides an effective way to identify optimal placement of sensors. Previous work has been limited to sensor placements under deterministic scenarios. In this work we extend the transfer operator based approach for optimal sensor placement while accounting for building systems uncertainties. The methodology provides a probabilistic metric to gauge coverage under uncertain conditions. We illustrate the capabilities of the framework with examples exhibiting boundary flux uncertainty

Fitness sharing and niching methods revisited

Interest in multimodal optimization function is expanding rapidly since real-world optimization problems often require the location of multiple optima in the search space. In this context, fitness sharing has been used widely to maintain population diversity and permit the investigation of many peaks in the feasible domain. This paper reviews various strategies of sharing and proposes new recombination schemes to improve its efficiency. Some empirical results are presented for high and a limited number of fitness function evaluations. Finally, the study compares the sharing method with other niching techniques

Optimization Under Uncertainty Using the Generalized Inverse Distribution Function

A framework for robust optimization under uncertainty based on the use of the generalized inverse distribution function (GIDF), also called quantile function, is here proposed. Compared to more classical approaches that rely on the usage of statistical moments as deterministic attributes that define the objectives of the optimization process, the inverse cumulative distribution function allows for the use of all the possible information available in the probabilistic domain. Furthermore, the use of a quantile based approach leads naturally to a multi-objective methodology which allows an a-posteriori selection of the candidate design based on risk/opportunity criteria defined by the designer. Finally, the error on the estimation of the objectives due to the resolution of the GIDF will be proven to be quantifiableComment: 20 pages, 25 figure