Standard Problem Formulation Research Articles

Constructing parsimonious analytic models for dynamic systems via symbolic regression

Developing mathematical models of dynamic systems is central to many disciplines of engineering and science. Models facilitate simulations, analysis of the system’s behavior, decision making and design of automatic control algorithms. Even inherently model-free control techniques such as reinforcement learning (RL) have been shown to benefit from the use of models, typically learned online. Any model construction method must address the tradeoff between the accuracy of the model and its complexity, which is difficult to strike. In this paper, we propose to employ symbolic regression (SR) to construct parsimonious process models described by analytic equations. We have equipped our method with two different state-of-the-art SR algorithms which automatically search for equations that fit the measured data: Single Node Genetic Programming (SNGP) and Multi-Gene Genetic Programming (MGGP). In addition to the standard problem formulation in the state-space domain, we show how the method can also be applied to input–output models of the NARX (nonlinear autoregressive with exogenous input) type. We present the approach on three simulated examples with up to 14-dimensional state space: an inverted pendulum, a mobile robot, and a bipedal walking robot. A comparison with deep neural networks and local linear regression shows that SR in most cases outperforms these commonly used alternative methods. We demonstrate on a real pendulum system that the analytic model found enables a RL controller to successfully perform the swing-up task, based on a model constructed from only 100 data samples.

Computationally Efficient Problem Reformulations for Capacitated Lot Sizing Problem

In this article, we propose novel reformulations for capacitated lot sizing problem. These reformulations are the result of reducing the number of variables (by eliminating the backorder variable) or increasing the number of constraints (time capacity constraints) in the standard problem formulation. These reformulations are expected to reduce the computational time complexity of the problem. Their computational efficiency is evaluated later in this article through numerical analysis on randomly generated problems.

Airfoil Design Under Uncertainty Using Non-Intrusive Polynomial Chaos Theory and Utility Functions

Fast and accurate airfoil design under uncertainty using non-intrusive polynomial chaos (NIPC) expansions and utility functions is proposed. The NIPC expansions provide a means to efficiently and accurately compute statistical information for a given set of input variables with associated probability distribution. Utility functions provide a way to rigorously formulate the design problem. In this work, these two methods are integrated for the design of airfoil shapes under uncertainty. The proposed approach is illustrated on a numerical example of lift-constrained airfoil drag minimization in transonic viscous flow using the Mach number as an uncertain variable. The results show that compared with the standard problem formulation the proposed approach yields more robust designs. In other words, the designs obtained by the proposed approach are less sensitive to variations in the uncertain variables than those obtained with the standard problem formulation.

Sparse Matrix Factorization Using Overlapped Localizing LOGOS Modes on a Shifted Grid

Different types of local global solution (LOGOS) modes have been proposed to develop fast direct solvers for electromagnetic wave problems. While radiating LOGOS modes are suitable for high frequency problems, this paper focuses on the localizing LOGOS modes, which are suitable for low-to-mid frequency problems. Factorizations relying on non-overlapped, localizing LOGOS (NL-LOGOS) modes are provably error controllable, but fail to provide acceptable asymptotic computational efficiencies. Factorizations using overlapped, localizing LOGOS (OL-LOGOS) modes provide improved asymptotic computational performance, but have been observed to suffer from significant overhead and, in some cases, poorer error control. This paper describes a new procedure for obtaining the OL-LOGOS modes. It is observed that the asymptotic efficiency is similar to previously reported OL-LOGOS schemes while providing stable error control for standard problem formulations. The overhead associated with computing the OL-LOGOS modes is also reduced significantly.

Introducing more choices (flexibility) in the upgrading of water distribution networks: the New York city tunnel network example

The New York city tunnel network problem is a classical case study for water distribution network rehabilitation or upgrading. However, standard problem formulations only offer single solutions based on fixed pressure head and demand requirements. This problem is re-examined through the use of multiple criteria optimization and a framework for interpreting these results is proposed. Results from previous studies were closely examined to highlight areas that can be improved. A 6-objective optimization problem was then formulated and results were compared with optimization using a single-objective genetic algorithm (GA) and 2-objectives optimization using a non-dominated sorted genetic algorithm-II. Although the initial runtime for the 6-objective optimization was long compared with single-objective GA, it provided solutions which span the whole spectrum of possibilities (in terms of pressure head requirements and cost). It is concluded that the proposed 6-objectives formulation can indeed offer more design choices, hence flexibility, for designers and decision makers.

Fault estimation—a standard problem approach

AbstractThis paper presents a range of optimization based approaches to fault diagnosis. A variety of fault diagnosis problems are reformulated in the so‐called standard problem set‐up introduced in the literature on robust control. Once the standard problem formulations are given, the fault diagnosis problems can be solved by standard optimization techniques. The proposed methods include (1) fault diagnosis (fault estimation, (FE)) for systems with model uncertainties; FE for systems with parametric faults, and FE for a class of nonlinear systems. Copyright © 2002 John Wiley & Sons, Ltd.

Finding consensus in speech recognition: word error minimization and other applications of confusion networks

We describe a new framework for distilling information from word lattices to improve the accuracy of the speech recognition output and obtain a more perspicuous representation of a set of alternative hypotheses. In the standard MAP decoding approach the recognizer outputs the string of words corresponding to the path with the highest posterior probability given the acoustics and a language model. However, even given optimal models, the MAP decoder does not necessarily minimize the commonly used performance metric, word error rate (WER). We describe a method for explicitly minimizing WER by extracting word hypotheses with the highest posterior probabilities from word lattices. We change the standard problem formulation by replacing global search over a large set of sentence hypotheses with local search over a small set of word candidates. In addition to improving the accuracy of the recognizer, our method produces a new representation of a set of candidate hypotheses that specifies the sequence of word-level confusions in a compact lattice format. We study the properties of confusion networks and examine their use for other tasks, such as lattice compression, word spotting, confidence annotation, and reevaluation of recognition hypotheses using higher-level knowledge sources.

Ride Quality of Civil Aircraft from a Probabilistic Viewpoint

A novel approach is proposed for the analysis and design of the ride quality augmentation system for a civil aircraft. In contrast to the standard problem formulation, where a mean acceleration level is minimized, the ride quality level is quantie ed in a probabilistic framework. A method is dee ned to limit the probability of exceeding a given vertical acceleration level in one or more points of the aircraft. The novelty of the proposed approach lies mainly in the dee nition of the control design in a probabilistic framework, which gives deep physical insight to the problem and more realistic results. An example is discussed to illustrate the proposed approach where the controller is obtained through an eigenstructure assignment procedure; the robustness of the controlled structure is guaranteed through a linear matrix inequality formulation.

Strategies for equilibrium‐stage separation calculations on parallel computers

AbstractWhen multicomponent, multistage separation problems are solved on parallel computers by successive linearization methods, the solution of a large sparse linear equation system becomes a computational bottleneck, since other parts of the calculation are more easily parallelized. When the standard problem formulation is used, this system has a block‐tridiagonal form. It is shown how this structure can be used in parallelizing the sparse matrix computation. By reformulating the problem so that it has a bordered‐block‐bidiagonal superstructure, it can be made even more amenable to parallezation. These strategies permit the use of a two‐level hierarchy of parallelism that provides substantial improvements in computational performance on parallel machines.