# David J. N. Limebeer's research while affiliated with University of the Witwatersrand and other places

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## Publications (78)

In previous work a variational integration scheme for mechanical systems containing linear dissipation was developed [1]. The key idea is to use a transmission line as the solution to a power-balancing control problem-in the physics literature controllers of this type are known as ‘heat baths’. The power-balanced closed-loop system is thereby made...

The erection of the inverted pendulum is a classic control problem, which has appeared in several variants. One of the most challenging is the minimum-time erection of a pendulum that is mounted on a moving cart. The aim is to erect the pendulum from the ‘straight-down’ (stable equilibrium) to a ‘straight-up’ (unstable equilibrium) position in mini...

SIGNIFICANCE: To mark the centenary of the University of the Witwatersrand (Wits), we review several major engineering achievements made over the last century by South African citizens, or individuals educated in South Africa -several of these contributions were made by Wits graduates and academic staff members. Equally significant are some outstan...

There are significant advantages associated with the analysis of satellite trajectory control problems in the Hill's analysis framework. As with the circular restricted three‐body problem (CRTBP) equations, the Hill's equations support three‐dimensional “halo” orbits that require station‐keeping control. These orbits are typically in regions of spa...

The robust control of a satellite in a three-dimensional ‘halo’ obit is studied. These orbits focus typically on regions of space close to one of the three collinear Lagrange points. In most cases these orbits are unstable, with the flight dynamics further complicated by drag effects. In order to introduce simultaneously a quantifiable robust stabi...

State-of-the-art engine models are used to study the emissions production, and fuel consumption minimization, of a typical diesel-powered road car operating on a variable-gradient road. The engine models, which have been fit to measured test cell data, are used to represent both the performance and emissions generation characteristics of a typical...

This paper uses physical arguments to derive variational integration schemes for dissipative mechanical systems. These integration algorithms find utility in the solution of the equations of motion and optimal control problems for these systems. Engineers usually represent dissipation effects using phenomenological devices such as “dampers”. In the...

Optimal control calculations are used to study the effect of tyre wear on race car performance. This is achieved by solving a minimum lap time optimal control problem over multiple laps using a dynamical model of a Formula One car. A previously developed thermodynamic model is enhanced by adding an additional state for the carcass temperature of th...

The need to reduce development time whilst simultaneously improving engine performance has motivated this application of optimal control to product development processes for engines and powertrains. The optimisation of the fuel consumption is formulated as a constrained Optimal Control Problem (OCP) and solved using pseudospectral methods, giving t...

Dynamics and Optimal Control of Road Vehicles uniquely offers a unified treatment of tyre, car and motorcycle dynamics, and the application of nonlinear optimal control to vehicle-related problems within a single book. This is a comprehensive and accessible text that emphasises the theoretical aspects of vehicular modelling and control. The book fo...

The broad aim of this book is to provide a comprehensive coverage of the modelling and optimal control of both two‐ and four‐wheeled road vehicles. The first focus of this book is a review of classical mechanics and its use in building vehicle and tyre dynamic models. The second is nonlinear optimal control, which is used to solve a range of minimu...

Chapter 2 provides a comprehensive review of the classical mechanics required when building vehicle models. Both vector-based methods and the variational approach to classical mechanics are reviewed, and efforts to highlight the links between the two approaches have been made. A wide range of illustrative examples with a particular focus on non-hol...

Chapter 4 considers a number of precursory car and bicycle models, as well as an important oscillatory phenomenon commonly referred to as shimmy. The chapter begins by considering a classical single-track car model that will be used to analyse such things as yaw stability, the effects of acceleration and braking, and some of the influences of corne...

Chapter 6 dealswith road surfacemodelling and vehicle suspension systems, and their ride dynamics. A wide variety of car and motorcycle suspension configurations are now available. While most of these systems appear ‘very different’ from each other, many of their important properties can be analysed within a common ride-dynamics framework.The chapt...

Chapter 8 focuses on nonlinear optimal control and its applications. The chapter begins by introducing the fundamentals of optimal control and prototypical problem formulations. This is followed by the treatment of first-order necessary conditions including the Pontryagin minimum principle, dynamic programming, and the Hamilton–Jacobi–Bellman equat...

Chapter 5 provides a comprehensive review of the bicycle model presented inWhipple’s seminal 1899 paper.The set of nonlinear differential equations that describe the general motion of a bicycle and rider are derived and then linearized to study the smallmotions about a straight-running trim condition at a given constant speed.The stability of the W...

The broad aim of this book is to provide a comprehensive coverage of the modelling and optimal control of both two‐ and four‐wheeled road vehicles. The first focus of this book is a review of classical mechanics and its use in building vehicle and tyre dynamic models. The second is nonlinear optimal control, which is used to solve a range of minimu...

Chapter 3 focuses on modern tyre modelling. While classical two-dimensional nonholonomic- constraint models work reasonably well at very low speeds, these models are not acceptable in realistic applications. This chapter aims to explain the mechanisms and modelling issues related to the generation of tyre forces and moments. Physical models such as...

Chapter 9 deals with the solution of minimum-time and minimum-fuel vehicular optimal control problems. These problems are posed as fuel usage optimization problems under a time-of-arrival constraint, or minimum-time problems under a fuel usage constraint. The first example considers three variants of a simple fuel usage minimization problem under a...

In this study we will estimate the region of attraction (RoA) of the lateral dynamics of a nonlinear single-track vehicle model. The tyre forces are approximated using rational functions that are shown to capture the nonlinearities of tyre curves significantly better than polynomial functions. An existing sum-of-squares (SOS) programming algorithm...

An optimal control based methodology is proposed for minimising the combustible fuel consumption of a hybrid vehicle equipped with an internal combustion engine, a high-speed flywheel and a battery. The three-dimensionality of the road is recognised by the optimal control calculations. Fuel efficiency is achieved by optimally exploiting the primary...

The suppression or delay of flutter in long-span suspension bridges is investigated experimentally. Since the torsional stiffness of a bridge deck decreases with increasing length, an active flap flutter suppression system could potentially enable the construction of longer spans without costly increases in deck width and depth. An active flap flut...

Non-linear lateral stability of a vehicle is considered. The region of attraction is estimated using Sum of Squares (SOS) programming and compared to previous results from the literature. Parametric studies are then carried out to analyse their influence on the stability region. The vehicle in both cornering and straight line motion is considered.

The introduction of transient test cycles and the focus on real world driving emissions has increased the importance of ensuring the NOx and soot emissions are controlled during transient manoeuvres. At the same time, there is a drive to reduce the number of calibration variables used by engine control strategies to reduce development effort and co...

This paper describes experiments relating to the buffet response and control of a section of a long-span suspension bridge deck elastically mounted as part of a wind tunnel experiment. The bridge section is subject to grid generated flow turbulence. Two grids are used - one is a standard biplanar grid, while the second is a new design that provides...

The structural-aerodynamic modelling and dynamic stabilization of a three-dimensional suspension bridge model is considered. Our emphasis is on investigating the effectiveness of leading and trailing edge flaps in suppressing aeroelastic instabilities. The East Great Belt Bridge is chosen as a design example, and its aeroelastic limits are computed...

We experimentally investigate the suppression of flutter in long-span suspension bridges. A rigid sectional model of a long-span suspension bridge is mounted in a wind tunnel on a suspension system. Control surfaces, which are used to suppress flutter, are movable flaps that are fitted to the bridge section's leading and trailing edges. The flaps a...

The interactions between the multilink suspension system of an open-wheeled race car and the vehicle's aerodynamic performance are modeled and optimized. The closed-kinematic loops associated with the suspension mechanisms are modeled as a set of nonlinear equations that are derived using a first principle kinematic analysis. For the purpose of opt...

The application of optimal control to simulator motion cueing is examined. Existing motion cueing algorithms are hampered by the fact that they do not consider explicitly the optimal usage of simulator workspace. In this paper, numerical optimal control is used to minimize simulator platform acceleration errors, while explicitly recognizing the con...

The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, befor...

This paper conducts flutter and buffet suppression of a section of a long-span bridge deck mounted elastically across a wind tunnel working section. The incident stream turbulence for buffet tests is generated by a standard biplanar grid. The bridge deck response in its two major degrees of freedom, heave and pitch. A mechanical flutter control sys...

The minimum-lap-time optimal control problem for a Formula One race car is solved using direct transcription and nonlinear programming. Features of this work include significantly reduced full-lap solution times and the simultaneous optimisation of the driven line, the driver controls and multiple car set-up parameters. It is shown that significant...

Setting up engines to meet emissions limits often involves extensive steady-state calibration activities combined with ad-hoc strategies to compensate for transient operation. As engines become more complex and acceptable emissions levels ever lower, this task is becoming increasingly time consuming and expensive. The inclusion of models in the eng...

The identification of three-dimensional (3D) race track models from noisy measured GPS data is treated as a problem in the differential geometry of curves and surfaces. Curvilinear coordinates are adopted to facilitate the use of the track model in the solution of vehicular optimal control problems. Our proposal is to model race tracks using a gene...

The brachistochrone problem is usually solved in classical mechanics courses using the calculus of variations, although it is quintessentially an optimal control problem. In this paper, we address the classical brachistochrone problem and two vehicle-relevant generalisations from an optimal control perspective. We use optimal control arguments to d...

Burst oscillations occurring at high speed, and under firm acceleration, can be suppressed with a mechanical steering compensator. Burst instabilities in the subject racing motorcycle are the result of interactions between the wobble and weave modes under firm-acceleration at high speed. Under accelerating conditions, the wobble-mode frequency (of...

Motorcycles exhibit two potentially dangerous oscillatory modes known as ‘wobble’ and ‘weave’. The former is reminiscent of supermarket castor shimmy, while the latter is a low frequency ‘fish-tailing’ motion that involves a combination of rolling, yawing, steering and side-slipping motions. These unwanted dynamic features, which can occur when two...

The dynamic stabilization of a sectional model of a long-span suspension bridge is considered. Feedback control is achieved using leading- and trailing-edge flaps as actuators. While a wide variety of control systems is possible, we focus on compensation schemes that can be implemented using passive mechanical components such as springs, dampers, a...

We study the aerodynamic control of long-span suspension bridges and seek to raise the critical flutter wind speeds, while simultaneously suppressing buffeting. The control system design study is based on a simple flexible bridge section model that interacts with a constant-velocity air stream. A streamlined bridge deck is assumed and non-steady th...

The modeling, control, and dynamic stabilization of long-span suspension bridges are considered. By employing leading- and trailing-edge flaps in combination, we show that the critical wind speeds for flutter and torsional divergence can be increased significantly. The relatively less well known aerodynamic properties of leading-edge flaps will be...

This paper presents a design methodology for the suspension system of a novel aerodynamically efficient motorcycle. Since the machine’s layout and the rider’s seating position are unconventional, several aspects of the machine design, including the suspension, must be reviewed afresh. The design process is based on matrix inequalities that are used...

Burst oscillations occurring at high speed and under firm acceleration are suppressed with a mechanical steering compensator. Burst instabilities in the subject racing motorcycle are the result of interactions between the wobble and weave modes under high-speed cornering and firm-acceleration conditions. Under accelerating conditions the wobble-mod...

The purpose of this paper is to study the dynamics of the accelerating bicycle. It is shown that time-scale separation can be used to study the oscillatory characteristics of the accelerating machine using time-invariant models. These models are used to explain practically observed wobble-mode bursting oscillations that are associated most frequent...

An efficient numerical solution to the output feedback, discrete-time, H∞ preview control problem is presented. The first step of the solution comes from embedding the preview control problem in a two-degree-of-freedom generalised regulator framework, in which the augmented system state includes a segment of the (frequency weighted) previewed signa...

This paper studies the influence of road camber on the stability of single-track road vehicles. Road camber changes the magnitude and direction of the tire force and moment vectors relative to the wheels, as well as the combined-force limit one might obtain from the road tires. Camber-induced changes in the tire force and moment systems have knock-...

This paper investigates optimal trajectory tracking for air-to-surface missiles using bank-to-turn steering and equipped with Doppler-beam-sharpening seekers. The seeker generates a radar image of the terrain that is continually updated in flight. The guidance problem for this type of sensor is complex, because constraints must be imposed on the fl...

The purpose of this paper is to study the dynamics of the accelerating bicycle under straight-running and cornering conditions. If the bicycle is cornering at constant acceleration and roll angle, it is shown that for low values of acceleration (and braking), it follows closely a logarithmic spiral shaped trajectory. The studies presented are facil...

The approximation of high-order plant and controller models by models of lower-order is an integral part of control system
design and analysis. Until relatively recently model reduction was often based on physical intuition. For example, chemical
engineers often assume that mixing is instantaneous and that packed distillation columns may be modelle...

This paper introduces the idea of using mechanical steering compensators to improve the dynamic behavior of high-performance motorcycles. These compensators are seen as possible replacements for a conventional steering damper and comprise networks of springs, dampers, and a less familiar component called the inerter The inerter was recently introdu...

The establishment of damper settings that provide an optimal compromise between wobble- and weave-mode damping is discussed. The conventional steering damper is replaced with a network of interconnected mechanical components comprised of springs, dampers and inerters - that retain the virtue of the damper, while improving the weave-mode performance...

Modelling of the ride and handling dynamics of motorcycles using the symbolic mechanical multibody system package Autosim has been carried out since 1995. Motorcycles are principally of tree structure but their geometry is complex in relation to the tyre to road contact and tyre force and moment descriptions and to the chain drive system. They may...

A fixed-parameter active steering compensation scheme that is designed to improve the dynamic behaviour of high-performance
motorcycles is introduced. The design methodology is based on the Glover-McFarlane
H¥\mathcal{H}_\infty
loop-shaping procedure. The steering compensator so designed, is seen as a possible replacement for a conventional steeri...

The problem of optimizing the trajectory for an air-to-surface missile using bank-to-turn steering is addressed. The missile uses a radar imaging seeker to detect targets. A cruise missile scenario is considered that is characterized by low-level flight, followed by a climb and dive onto the target (a "bunt" trajectory). The use of a radar imaging...

A new approach to the modeling and control of tokamak fusion reactors is presented. A nonlinear model is derived using the classical arguments of Hamiltonian mechanics and a low-order linear model is derived from it. The modeling process used here addresses flux and energy conservation issues explicitly and self-consistently. The model is of partic...

In this paper we develop a controller reduction procedure for linear parameter-varying (LPV) systems. The method uses synthesis Riccati inequalities for the normalized robust stabilization problem as a basis for the approximation. The technique provides a priori error bounds which are used to obtain closed-loop stability conditions and performance...

The use of trajectory optimization techniques is presented for the terminal guidance of an air-to-surface missile using a Doppler beam sharpening (DBS) radar seeker. The terminal guidance problem is characterized by a stealthy terrain-following phase that is followed by a climb and dive onto the target (a "bunt" trajectory). The imaging properties...

A comprehensive study of the effects of acceleration and braking on motorcycle stability is presented. This work is based on a modified version of the dynamic model that was first presented in (Sharp & Limebeer, 2000), and is thought to be the most comprehensive motorcycle dynamic model in the public domain. Extensive use is made of both nonlinear...

The observed dynamic behaviour of motorcycles suggests that interesting and significant motions occur that are not currently understood. The most elaborate modelling exercise completed so far has produced results that need confirmation and extension. The construction of these models necessitates the use of automated methods and one such modelling m...

We describe the application of an H<sup>∞</sup> system
identification procedure to a Tokamak. The work is motivated by the need
to create linear models which are suitable for controller design and
which may be used to validate different models derived from physics
principles. The paper develops an H<sup>∞</sup> system
identification algorithm and d...

The aim of this paper is to show how Chebycheff approximation may be used in frequency weighted H∞ system identification and model reduction. We show how these techniques may be combined with known results from Hankel norm model reduction theory to reduce the error between the plant and the low-order approximation. In identification we also apply a...

Due to the model complexity, the manual derivation of the equations of motion of two-wheeled road vehicles is not practical, particularly if one wishes to study complex modes of operation such as certain cornering phenomena. We establish the feasibility of using multi-body model building software to study the straight running properties of a motorc...

A linear, straight running model of a motorcycle is developed using multibody simulation software. The stability properties as a function of forward speed obtained from this model exactly match well established results found by (Sharp 1994b), which validates the linearised equations of motion derived by the software. This model is used to obtain ti...

The manual derivation of the equations of motion of two wheeled
road vehicles is not practical if one wishes to study complex modes of
operation such as cornering phenomena. We establish the feasibility of
an automated approach to dynamic model generation using multi-body
simulation software. The straight running equations of motion are
derived usi...

The authors consider the problem of finding necessary and sufficient conditions for the existence of a Q with no more than k poles in the closed left-half plane such that the norm of a certain matrix <or=1. If solutions exist, a formula for all solutions is given. Special attention is given to the characterization of all optimal solutions. As an ap...

A representation formula for all controllers that satisfy an L∞-type constraint is derived for time-varying systems. It is now known that a formula based on two indefinite algebraic Riccati equations may be found for time-invariant systems over an infinite time support (see [J.C. Doyle et al., IEEE Trans. Automat, Control, AC-34 (1989), pp. 831-847...

Necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with H∞ control are derived using J-spectral factorization theory. The existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled J-spectral factorization pr...

A series of increasingly general model-matching problems that arise in the theory of H ∞ control is considered. Necessary and sufficient conditions for existence of solutions to the model-matching problems are given in terms of solutions of J-spectral factorization problems. First, the solution to the Nehari problem (given R, find Q such that ∥R+Q∥...

Adamjan et al. (1971) established an important equivalence between the optimal Hankel norm model reduction problem (the Schur-Takagi problem) and a classical interpolation problem due to Ahiezer. In two later papers, Ball (1983) and Ball and Helton (1985) investigated a vector version of this interpolation problem, using the sophisticated Ball-Helt...

The vector interpolation problem, posed by Ball and Helton [1,2], is the most general version of such Nevanlinna-Pick interpolation problems, containing the matrix interpolation problem as a special case. The problem has been solved by several methods, all of which rely on deep and abstract mathematics. Yet it is possible to solve the vector interp...

First Page of the Article

The concept of generalized block diagonal dominance is used to extend a number of recently developed robust stability results for multivariable systems to large-scale interconnected systems.

The 2D aerodynamic modelling of long-span suspension bridges is considered. We use thin airfoil theory from the aircraft industry and a sectional bridge model with an integrated controllable trailing-edge flap. The relatively less well known aerodynamic properties of leading-edge flap will be studied in detail. The optimal approximation of the clas...

## Citations

... In the case of dry friction, the higher amount of heat in tread is generated, which further cause increased wear of tires. 108 In the case of the wet road the water between road and tire behaves like lubricator, which reduces friction, 63 which means that the tire tread is protected and the wear is mild. ...

... This work was expanded in [79] to a dual-wheel configuration, which allowed for the assessment of the effects of gyro- scopic forces and axle width on shimmy behaviour of the NLG. Finally we remark that, although shimmy oscillations are a well documented phenomenon in aircraft ground dy- namics, as Shaw and Balachandran point out in the review paper [65], they may appear in mechanical systems outside of an aerospace context, for example in the dynamics of motorcycles [7,42]. These mechanical systems have also attracted research interest and examples of shimmy research here that employs numerical continuation techniques includes the studies of: an automobile under steady cornering conditions [17], the dy- namics of a pulled wheel system [73] and the active steering control of a car [10]. ...

... In each mode, fuel consumption was proven to depend on the instantaneous engine efficiency, approximated by two analytical functions instead of the commonly considered consumption map. Furthermore, the works [14][15][16][17][18] present the distinct approaches of instantaneous fuel consumption, which encompass vehicle characteristics, engine characteristics, as well as gear selection schemes. These models are further implemented in different types of passenger cars and their operations are examined by applying microscopic traffic simulation or artificial neural networks. ...

... With the improvement of the studies with artificial liberation points, future studies may generalize the work either to stronger kinds of control or to work with the 5 body problem or even the n-body problem [32,33]. ...

... Since then, in the following two decades, there has been several generalizations in an attempt to incorporate systems of more general nature, as for instance: coupled multi-body systems, field theories, electrical systems [4][5][6][7][8][9]; systems with more specific features (as symmetries) [10][11][12][13][14]; interactions with control theory [15][16][17][18][19]; higher-order and asynchronous techniques [20][21][22]. For more recent developments, see [23][24][25][26][27][28][29][30]. For surveys and reviews, see [31][32][33][34] and the references therein. ...

... which limits the maximum power on the road. The cost function of the OCP is given by (3), which is modified to include regularisation terms [23] ...

... For the last several decades, research and development of the vehicular lateral stability control (LSC) program have received a considerable amount of attention owing to its abilities to enhance vehicle handling and reduce roadway crashes [1], [2]. In a nutshell, making use of certain chassis sensors and actuators, the LSC system strives for reducing/preventing dangerous incidents involving understeer/oversteer, fishtailing, and spin-out [3], [4]. In particular, the sideslip angle, characterized as the angle between the vehicle's heading and the actual direction of travel, shall be maintained as small as possible to avoid the loss of vehicle stability [5]. ...

... A flywheel energy storage system (FESS) is a type of ESS that has the advantages of high efficiency, fast response, instantaneous high power, low maintenance, and long life [1][2][3]. Thus, FESSs have been applied in electric vehicles [4,5], hydraulic excavators [6], wind power generation [7,8], microgrids [9], photovoltaic generation [10], uninterruptible power supply (UPS) [11,12], and aerospace applications [13]. In an FESS, energy is stored in a high-speed flywheel rotor. ...

... There are many modeling studies on motorcycles. Gani et al. (1997) established motorcycle models through mathematical models and used programming analysis to calculate multi-body dynamics, which was the beginning of motorcycle modeling. Cossalter et al. (1998) used computer simulation to establish motorcycle models under different road conditions, such as steady circle and lane changing and through simulation results such as steering torque and roll angle to get the dynamic characteristics. ...

... In addition to the methods mentioned above, several challenging studies have been conducted to improve the aerodynamic performance of the bridge. For example, Li et al. [22] and Gouder et al. [23] proposed an active wind load feedback control method by sensors and actuator to change the wind flow near the bridge and improve the flutter resistance. ...