Gasoline fuelled partially premixed compression ignition in a light duty multi cylinder engine: A study of low load and low speed operation

The objective of this study was to examine the operating characteristics of a light duty multi cylinder compression ignition engine with regular gasoline fuel at low engine speed and load. The effects of fuel stratification by means of multiple injections as well as the sensitivity of auto-ignition and burn rate to intake pressure and temperature are presented. The measurements used in this study included gaseous emissions, filter smoke opacity and in-cylinder indicated information. It was found that stable, low emission operation was possible with raised intake manifold pressure and temperature, and that fuel stratification can lead to an increase in stability and a reduced reliance on increased temperature and pressure. It was also found that the auto-ignition delay sensitivity of gasoline to intake temperature and pressure was low within the operating window considered in this study. Nevertheless, the requirement for an increase of pressure, temperature and stratification in order to achieve auto-ignition time scales small enough for combustion in the engine was clear, using pump gasoline. Copyright © 2009 SAE International.

Study of steady state and transient EGR behaviour of a medium duty diesel engine

It is well known that accurate EGR control is paramount to controlling engine out emissions during steady state and transient operation of a diesel engine. The direct measurement of EGR is however non-trivial and especially difficult in engines with no external EGR control where the intake manifold CO2 levels can be measured more readily. This work studies the EGR behaviour in a medium duty diesel engine with a passive EGR rebreathing strategy for steady state and transient operation. High speed (response time ∼1ms) in-cylinder sampling using modified GDI valves is coupled with high frequency response analysers to measure the cyclic in-cylinder CO2, from which the EGR rate is deduced. It was found that controlling the EGR using the passive rebreathing strategy during certain combined speed and load transients is challenging, causing high smoke and NO emissions. The in-cylinder sampling method coupled with fast CO2 measurement (time constant ∼8ms) in the exhaust port gave insights about the EGR rate during these transients. The complex interaction of the manifold pressures, turbo-charger operation and trapped charge composition from the previous cycle simply can cause high dilution and therefore high smoke levels. The steady state variation of NO emissions with respect to EGR is also studied using a fast NO analyzer (time constant ∼2ms) in the exhaust port. Cyclic variation was found to be up to ±5% at some load conditions. © 2008 SAE International.

A simple diesel engine air-path model to predict the cylinder charge during transients: Strategies for reducing transient emissions spikes

Simple air-path models for modern (VGT/EGR equipped) diesel engines are in common use, and have been reported in the literature. This paper addresses some of the shortcomings of control-oriented models to allow better prediction of the cylinder charge properties. A fast response CO2 analyzer is used to validate the model by comparing the recorded and predicted CO2 concentrations in both the intake port and exhaust manifold of one of the cylinders. Data showing the recorded NOx emissions and exhaust gas opacity during a step change in engine load illustrate the spikes in both NOx and smoke seen during transient conditions. The predicted cylinder charge properties from the model are examined and compared with the measured NOx and opacity. Together, the emissions data and charge properties paint a consistent picture of the phenomena occurring during the transient. Alternative strategies for the fueling and cylinder charge during these load transients are investigated and discussed. Experimental results are presented showing that spikes in both NOx and smoke can be avoided at the expense of some loss in torque response. Even if the torque response must be maintained, it is demonstrated that it is still possible to eliminate spikes in NOx emissions for the transient situation being examined. Copyright © 2006 SAE International.

A numerical simulation of AFR switch of SI engines

A novel mechanical method of achieving a rapid switch between stoichiometric and lean conditions for SI engines is explored. Two and three throttle configurations, a switch strategy which employs a standard intake manifold and an assembly of pipes and throttle(s), are investigated numerically by using a one-dimensional engine simulation program based on the method of characteristics. The results indicate that it is possible to achieve rapid AFR switch without a torque jump, i.e. unperceptible to the driver. © 1998 Society of Automotive Engineers, Inc.

Time resolved measurement of cold start hc concentration using the fast FID

Understanding mixture formation phenomena during the first few cycles of an engine cold start is extremely important for achieving the minimum engine-out emission levels at the time when the catalytic converter is not yet operational. Of special importance is the structure of the charge (film, droplets and vapour) which enters the cylinder during this time interval as well as its concentration profile. However, direct experimental studies of the fuel behaviour in the inlet port have so far been less than fully successful due to the brevity of the process and lack of a suitable experimental technique. We present measurements of the hydrocarbon (HC) concentration in the manifold and port of a production SI engine using the Fast Response Flame Ionisation Detector (FRFID). It has been widely reported in the past few years how the FRFID can be used to study the exhaust and in-cylinder HC concentrations with a time resolution of a few degrees of crank angle, and the device has contributed significantly to the understanding of unburned HC emissions. Using the FRFID in the inlet manifold is difficult because of the presence of liquid droplets, and the low and fluctuating pressure levels, which leads to significant changes in the response time of the instrument. However, using recently developed procedures to correct for the errors caused by these effects, the concentration at the sampling point can be reconstructed to align the FRFID signal with actual events in the engine. © 1996 Society of Automotive Engineers, Inc.

Diesel smoke transient control using a real-time smoke sensor

A novel real time smoke sensor is described, which is mounted in the exhaust manifold and detects the smoke by virtue of the natural electrical charge which is carried on the smoke. The somewhat obscure origin of the charge on the smoke is briefly considered, as well as the operation of the sensor itself. The use of the sensor as part of a feedback control shows that it can be very effective in reducing smoke puffs. Copyright © 1987 Society of Automotive Engineers, Inc.

Linear regression under fixed-rank constraints: A Riemannian approach

In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop efficient line-search algorithms. The proposed algorithms have many applications, scale to high-dimensional problems, enjoy local convergence properties and confer a geometric basis to recent contributions on learning fixed-rank matrices. Numerical experiments on benchmarks suggest that the proposed algorithms compete with the state-of-the-art, and that manifold optimization offers a versatile framework for the design of rank-constrained machine learning algorithms. Copyright 2011 by the author(s)/owner(s).

Delayed decision-making in bistable models

Switching between two modes of operation is a common property of biological systems. In continuous-time differential equation models, this is often realised by bistability, i.e. the existence of two asymptotically stable steadystates. Several biological models are shown to exhibit delayed switching, with a pronounced transient phase, in particular for near-threshold perturbations. This study shows that this delay in switching from one mode to the other in response to a transient input is reflected in local properties of an unstable saddle point, which has a one dimensional unstable manifold with a significantly slower eigenvalue than the stable ones. Thus, the trajectories first approximatively converge to the saddle point, then linger along the saddle's unstable manifold before quickly approaching one of the stable equilibria. ©2010 IEEE.

Low-rank optimization on the cone of positive semidefinite matrices

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y Y T leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis. © 2010 Society for Industrial and Applied Mathematics.

Consensus on homogeneous manifolds

The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative state according to an interconnection graph. The paper first formalizes the consensus problem for synchronization (i.e. maximizing the consensus) and balancing (i.e. minimizing the consensus); it thereby introduces the induced arithmetic mean, an easily computable mean position on CCH manifolds. Then it proposes and analyzes various consensus algorithms on manifolds: natural gradient algorithms which reach local consensus equilibria; an adaptation using auxiliary variables for almost-global synchronization or balancing; and a stochastic gossip setting for global synchronization. It closes by investigating the dependence of synchronization properties on the attraction function between interacting agents on the circle. The theory is also illustrated on SO(n) and on the Grassmann manifolds. ©2009 IEEE.

Consensus optimization on manifolds

The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO (n) and the Grassmann manifold Grass (p, n) are treated as original examples. A link is also drawn with the many existing results on the circle. © 2009 Society for Industrial and Applied Mathematics.

Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in ℝn. In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.

A Newton algorithm for invariant subspace computation with large basins of attraction

We study the global behaviour of a Newton algorithm on the Grassmann manifold for invariant subspace computation. It is shown that the basins of attraction of the invariant subspaces may collapse in case of small eigenvalue gaps. A Levenberg-Marquardt-like modification of the algorithm with low numerical cost is proposed. A simple strategy for choosing the parameter is shown to dramatically enlarge the basins of attraction of the invariant subspaces while preserving the fast local convergence.

A Grassman-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh Quotient Iteration (RQI) computes a 1-dimensional invariant subspace of a symmetric matrix A with cubic convergence. We propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. The geometry of the algorithm on the Grassmann manifold Gr(p,n) is developed to show cubic convergence and to draw connections with recently proposed Newton algorithms on Riemannian manifolds.

Geometric perspectives on MDO and MDO architectures

Two main perspectives have been developed within the Multidisciplinary Design Optimization (MDO) literature for classifying and comparing MDO architectures: a numerical point of view and a formulation/data flow point of view. Although significant work has been done here, these perspectives have not provided much in the way of a priori information or predictive power about architecture performance. In this report, we outline a new perspective, called the geometric perspective, which we believe will be able to provide such predictive power. Using tools from differential geometry, we take several prominent architectures and describe mathematically how each constructs the space through which it moves. We then consider how the architecture moves through the space which it has constructed. Taken together, these investigations show how each architecture relates to the original feasible design manifold, how the architectures relate to each other, and how each architecture deals with the design coupling inherent to the original system. This in turn lays the groundwork for further theoretical comparisons between and analyses of MDO architectures and their behaviour using tools and techniques derived from differential geometry. © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.