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The robust control of a satellite in a three-dimensional ‘halo’ obit is studied. In order to introduce a quantifiable robust stability margin, as well as maintain a pre-selected orbit, a two-degree...

Optimal control is used to study the management of tyre performance for a Formula One car. Tyre friction is compromised by accumulated wear and operation outside the tyre design temperature window ...

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 diesel-fuelled car engine. A simple example is used to highlight the impact of elevation changes on the main structural features of fuel optimal control problems (OCPs). A typical semiurban test route with legislated speed limits and enforced stops is used for performance evaluation purposes. The optimal functioning of a discrete-gear automatic transmission system, as opposed to a simple continuously variable transmission, is studied in detail. The main focus of this paper is to evaluate the importance of 3-D road influences (gradient and curvature), preimposed time-of-arrival constraints, enforced stops, and emissions constraints on the fuel consumption and optimal driving of typical diesel-powered road vehicles. This paper proposes the use of multiple-phase optimal control to elicit a better understanding of “real” driving situations and motivates a move away from standardized drive cycles.

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 the tyres. The tyre grip is modelled as a function of the tyres’ wear and temperature. Grip reduces when tyres get worn, or the tyres are not operated within their optimal temperature window. Overheating the tyres can accelerate wear, which in turn, degrades performance. The optimal control problem solver needs to ‘manage’ the state of tyres throughout a race (not just a single lap) to ensure that optimal race performance is achieved. At some point during a race a pit stop may be required to change worn tyres so that tyre grip can be restored. It is essential to understand the wear characteristics of various tyre compounds in order to determine the point when the time needed for a pit stop is justified in terms of subsequent racing performance.

The primary aim of this work is to validate dynamic minimum-lap-time optimal control simulations for a sports motorcycle on a 3D track. The racing line and rider inputs are optimised, while recogni...

The paper begins with a survey of advances in state-of-the-art minimum-time simulation for road vehicles. The techniques covered include both quasi-steady-state and transient vehicle models, which ...

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 work presented here we replace these dampers with a lossless transmissionline in order that the equations of motion are derivable from a variational principle. The associated system Lagrangian can then be discretized and used to develop low-order variational integration schemes that inherit the advantageous features of their conservative counterparts. The properties of a lossless spring-inerter based transmission system are analyzed in detail, with the resulting variational integration schemes shown to have excellent numerical properties. The paper concludes with the analysis of a dissipative variant of the classical Kepler central force problem.

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 s...

Variations in track temperature, surface conditions and layout have led tyre manufacturers to produce a range of rubber compounds for race events. Each compound has unique friction and durability characteristics. Efficient tyre management over a full race distance is a crucial component of a competitive race strategy. A minimum lap time optimal control calculation and a thermodynamic tyre wear model are used to establish optimal tyre warming and tyre usage strategies. Lap time sensitivities demonstrate that relatively small changes in control strategy can lead to significant reductions in the associated wear metrics. The illustrated methodology shows how vehicle setup parameters can be optimised for minimum tyre usage.

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 costs. In this paper, a control orientated combustion model, [1], and model predictive control strategy, [2], that were developed in simulation and reported in earlier papers, are applied to a Diesel engine and demonstrated in a test vehicle. The paper describes how the control approach developed in simulation was implemented in embedded hardware, using an FPGA to accelerate the emissions calculations. The development of the predictive controller includes the application of a simplified optimisation algorithm to enable a real-time calculation in the test vehicle. The test vehicle was calibrated over a constant speed step manoeuvre and then demonstrated over the standard US06 drive cycle, where it was found to reduce NOₓ and CO₂ emissions. The paper concludes with a discussion on the merits of an emissions based engine control strategy, where the controller seeks to control emissions directly rather than an indirect quantity such as boost pressure.

Summary 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 using both thin aerofoil theory and flutter derivatives. The problem is cast in an efficient reduced size finite element formulation with aerodynamic forces expressed in the Laplace domain by use of a high-fidelity rational function approximation. Circulatory aerodynamic forces are modelled using a feedback loop for every element, and the problem is expressed in a form suitable for implementation of modern control techniques. The structure's full multimodal response is considered, and numerical predictions show very good agreement against experimental data from the literature. In order to account for modelling errors and uncertainties while designing the controller, elements from robust control theory are invoked. The stability and robustness of the bridge when fitted with flaps controlled by optimal and suboptimal H∞ controllers are discussed for varying lengths of control surfaces along the suspended span as the optimum configuration for aerodynamic performance is investigated. Copyright © 2016 John Wiley & Sons, Ltd.

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 time-of-arrival constraint. The first variant is worked out theoretically, and serves to highlight several of the structural features of these problems; the other two more complicated variants are solved numerically.The second example is also a multi-stage fuel usage minimization problem under a timeof- arrival constraint.More complicated track and vehicle models are then employed; the problem is solved numerically. The third problem is a lap time minimization problem taken from Formula One and features a thermoelectric hybrid powertrain. The fourth and final problem is a minimum-time closed-circuit racing problem featuring a racing motorcycle and rider.

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-holonomic systems are studied, including Chaplygin’s sleigh, rolling balls, and rolling discs. Equilibria and stability, and the connection between timereversal symmetry and dissipation, are also briefly discussed.

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 Whipple model is discussed and some of the handling characteristics of bicycles are analysed using control-theoretic ideas. Extensions relating to accelerating and braking vehicles, tyred vehicles, and vehicles with flexible frames are also considered.

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 minimum‐time, minimum‐fuel, and track curvature reconstruction problems. As is known classically, all thismaterial is bound together by the calculus of variations and stationary principles. The treatment of this material is supplemented with a large number of examples that highlight obscurities and subtleties in the theory. A particular strength of the book is its unified treatment of tyre, car, and motorcycle dynamics and the application of nonlinear optimal control to vehicle‐related problems within a single text. These topics are usually treated independently, and can only be found in disparate texts and journal articles. It is our contention that presentday vehicle dynamicists should be familiar with all of these topic areas. The aim in writing this book is to provide a comprehensive and yet accessible text that emphasizes particularly the theoretical aspects of vehicular modelling and control.

The aim of this chapter is to discuss some of themodelling techniques used to replicate the important features of bicycle, car, motorcycle, and driver dynamics. These more ‘advanced’ vehicle models will include a number of features that were neglected in models described in Chapters 4 and 5, including suspensions with anti-dive and/or anti-squat geometries, environmental influences such as aerodynamic effects, threedimensional road geometries, and tyre models of the type described in Chapter 3.

Chapter 1 is almost entirely discursive and covers the early history of road vehicles, outlining some of the important technological achievements that underpin the development of modern road vehicular transport. The focus is on bicycles, motorcycles, and cars; the history of steering mechanisms for four-wheeled vehicles is considered early on. Several early engine, suspension, and tyre developments are also discussed.

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 cornering. After that, the Timoshenko–Young bicycle model that has steering and rolling freedoms is considered. Bicycle stability is then considered with the help of a point-mass bicycle model. Shimmy, including gyroscopic shimmy, is discussed in the concluding part of the chapter. The reader will also be introduced to the fundamental notions of oversteer and understeer.

Vehicular optimal control problems have been studied extensively since the early part of the 20th century. Progress in solving these problems has been driven primarily by applications in space and atmospheric flight, including launch vehicles, Earth-based and interplanetary space orbital transfer, and high-performance supersonic aircraft. In all of these applications, the ability to solve increasingly complex optimal control problems has been made possible by advances in high-speed computing. The mathematical and computing techniques being developed are now so diverse, and the range of applications of optimal control so broad, that a comprehensive review of the entire scope of vehicular optimal control is an impossible task. Instead, we will attempt only to provide a flavor of the scope and variety of these problems.

This study presents a system based on passively controlled leading- and trailing-edge flaps that is designed to suppress wind-induced instabilities such as flutter and torsional divergence. The utility of the approach is demonstrated on a three-dimensional bridge model. Particular emphasis is placed on the early stages of the deck erection process when the bridge is particularly vulnerable to flutter. The flaps are activated by the deck's movements though passive phase-compensating mechanisms comprising of springs, dampers and inerters. It is demonstrated that optimised compensator parameters, and optimum hinge locations, result in a substantially improved deck aerodynamic performance. Particular importance is given to ensuring that the controlled system has good closed-loop ‘robustness’ properties, or in other words, that the controlled system has a high tolerance to parameter variations and uncertainties in the system dynamics. The practical use of a nonlinear optimisation algorithm with a FE bridge aeroelastic model, which includes the flap dynamics, necessitates the use of reduced-order models. A novel model reduction procedure that is based on the retention of dominant poles is introduced into the aeroelastic modelling framework. Multimodal interactions are observed at the various erection stages and conclusions are drawn with regard to the contributions of various modes of vibration to aeroelastic instabilities. The main advantage of this approach lies on the passive system's simplicity and its ability to simultaneously increase the flutter and torsional divergence boundaries. The Humber Bridge in the U.K. is chosen as a study example for numerical simulations.

A method for evaluating driving technique and engine operation, with emissions constraints, on a real driving route, is presented. A single degree of freedom vehicle model with simple resistive forces is driven along the pre-planned route. Instead of the velocity trajectory being determined a priori, the optimisation can choose the optimal speed along the route as well as the engine operating point. When constraints on the total emissions are imposed, the engine no longer operates on the minimum fuel locus; the power delivery changes as the emissions constraints are tightened. The driving technique changes from a series of bang-bang switches, into a smoother application of the engine power. The route is driven in the absence of stops and traffic restrictions.

An optimal control based methodology is proposed for minimizing 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 recognized by the optimal control calculations. Fuel efficiency is achieved by optimally exploiting the primary and secondary energy sources and controlling the vehicle so that the fuel consumption is minimized for a given but arbitrary three-dimensional route. A time-of-arrival constraint rather than a driving cycle is used. The benefits of using multiple auxiliary storage systems are demonstrated and a lower-bound estimate of the fuel consumption is presented.

We study the modelling and flutter suppression of a suspended-span bridge section using aerodynamic control means. The aerodynamic actuators are controllable leading- and trailing-edge flaps fitted to both edges of the bridge deck. The modelling is based on thin aerofoil theory which describes the bridge deck - air stream interactions. The model also consider the mass and inertia effects of the flaps. Passive mechanical controllers are proposed that sense the vertical velocity of the leading- and trailing-edge flap pivots and use these signals to produce control torques and adjust the flap angles accordingly. The control system is insensitive to the wind direction in the case of identical leading- and trailing-edge mechanical controllers. The Akashi-Kaikyo bridge is used as a working example for the numerical simulation evaluation of the control system performance.

This paper explores the application of numerical optimal control (NOC) to the synthesis of motion cueing algorithms for race-car simulators. These techniques are used to design washout filters that address the limitations of the commonly-employed linear quadratic Guassian (LQG) regulator and classical frequency shaping strategies. The primary disadvantage of the LQG and classical tuning methods is that they do not recognise explicitly the hardware limitations of the platform and thus have to rely on an iterative process to address workspace limitations. In two new algorithms a kinematic model of the platform is used to constrain explicitly the platform's motion within the available workspace. The first algorithm is directed to the optimisation of the parameters of a linear washout filter, while in the second approach the platform motion is governed by an open-loop optimal control. The results of the two new algorithms are tested on race-car minimum lap-time simulations, and then compared with a linear-optimal-control based solution. The race-car scenario presents different challenges from the passenger car and aircraft contexts; these differences are discussed.

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 the optimum heat release and injection profiles in the presence of cylinder pressure rate and cylinder pressure constraints. The technique is applied to an engine design problem and used to reduce fuel consumption by optimising compression ratio within a cylinder pressure limit, also providing new insights into the combustion processes.

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 derive closed-form solutions for both the optimal trajectory and the minimum achievable transit time for these generalisations. We then study optimal control problems involving a steerable disc rolling between prescribed points on the interior surface of a hemisphere. The effects of boundary and control constraints are examined. For three-dimensional problems of this type, which involve rolling bodies and nonholonomic constraints, numerical solutions are used. © 2013 Taylor and Francis.

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 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 studied in detail prior to their utilization in aeroelastic stability and control system design studies. The optimal approximation of the classical Theodorsen circulation function will be studied as part of the bridge section model building exercise. 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, gearboxes, and levers. A single-loop control system that controls the leading- and trailing-edge flaps by sensing the main deck pitch angle is investigated. The key finding is that the critical wind speeds for flutter and torsional divergence of the sectional model of the bridge can be greatly increased, with good robustness characteristics, through passive feedback control. Static winglets are shown to be relatively ineffective.

The control of Internal Combustion (IC) engines is dominated by conventional controllers that regulate the engine operating condition to set-points, which are usually derived from steady-state testing. A significant issue is that IC engines can spend much of their time operating under transient conditions, which take the engine away from their optimized steady-state operating conditions. Currently, a variety of ad-hoc methodologies are used to compensate for transient operation, by modifying set-points, gains and feed-forward values for example. It is proposed that Model Predictive Control be used for the online optimization of the combustion processes as a more structured way of controlling transient engine operation. This type of optimization requires a combustion model that captures the response of the combustion system to changes in the control inputs. This paper describes a model that is detailed enough to predict emissions trends, yet is suitable for implementation in a future real-time automotive control unit. © 2012 IFAC.

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 an ℒ∞-norm variant of Levy's method (Levy, 1959). © 1997 Elsevier Science Ltd.

Hankel norm and Nehari-type approximation problems arise in model reduction and /spl Hscr//sub /spl infin//-control theory. Existing solutions to the discrete-time version of these problems may be derived using a standard state-space framework, but the resulting solution formulas require an invertible A-matrix. As a further complication, the D-matrix in the representation formula for all solutions becomes unbounded in the optimal case. The aim of this paper is to show that both these complications may be removed by analyzing these problems in a descriptor framework.

The purpose of this paper is to provide a set of synthesis and design tools for a wide class of H2 preview control systems. A generic preview design problem, which features both previewable and nonpreviewable disturbances, is embedded in a standard generalized regulator framework. Preview regulation is accomplished by a two-degrees-of-freedom output-feedback controller. A number of theoretical issues are studied, including the efficient solution of the standard H2 full-information Riccati equation and the efficient evaluation of the full-information preview gain matrices. The full-information problem is then extended to include the efficient implementation of the output-feedback controller. The synthesis of feedforward controllers with preview is analyzed as a special case—this problem is of interest to designers who wish to introduce preview as a separate part of a system design. The way in which preview reduces the H2-norm of the closed-loop system is analyzed in detail. Closed-loop norm reduction formulas provide a systematic way of establishing how much preview is required to solve a particular problem, and determine when extending the preview horizon will not produce worthwhile benefits. The paper concludes with a summary of the main features of preview control, as well as some controller design insights. New application examples are introduced by reference.

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 reductions in the driven lap time can be obtained from track-specific set-up parameter optimisation. Reduced computing times are achieved using a combination of a track description based on curvilinear coordinates, analytical derivatives and model non- dimensionalisation. The curvature of the track centre line is found by solving an auxiliary optimal control problem that negates the difficulties associated with integration drift and trajectory closure. © 2013 Taylor and Francis.

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 confines of the workspace. Actuator constraints are included and the impact of restricted actuator performance is thereby facilitated. The solution of open-loop optimal control calculations are also used as a baseline against which to compare the commonly employed linear quadratic Gaussian (LQG) and model predictive control-based techniques. The limitations of these methods are identified and two additional modules are introduced to the LQG algorithm to improve its performance.