the control handbook control system applications second edition electrical engineering handbook_

xix SECTION I Automotive 1 Linear Parameter-Varying Control of Nonlinear Systems with Applications to Automotive and Aerospace Controls.. Linear Parameter-Varying Control of Nonlinear Sy

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Library of Congress Cataloging-in-Publication Data

Control system applications / edited by William S Levine 2nd ed.

p cm (The electrical engineering handbook series) Includes bibliographical references and index.

Preface to the Second Edition xi

Acknowledgments xiii

Editorial Board xv

Editor xvii

Contributors xix

SECTION I Automotive

1 Linear Parameter-Varying Control of Nonlinear Systems with Applications to Automotive and Aerospace Controls .1-1

Hans P Geering

2 Powertrain Control .2-1

Davor Hrovat, Mrdjan Jankovic, Ilya Kolmanovsky, Stephen Magner, and Diana Yanakiev

3 Vehicle Controls .3-1

Davor Hrovat, Hongtei E Tseng, Jianbo Lu, Josko Deur, Francis Assadian, Francesco Borrelli, and Paolo Falcone

4 Model-Based Supervisory Control for Energy Optimization of Hybrid-Electric Vehicles .4-1

Lino Guzzella and Antonio Sciarretta

5 Purge Scheduling for Dead-Ended Anode Operation of PEM Fuel Cells .5-1

Jason B Siegel, Anna G Stefanopoulou, Giulio Ripaccioli, and Stefano Di Cairano

SECTION II Aerospace

6 Aerospace Real-Time Control System and Software .6-1

Rongsheng (Ken) Li and Michael Santina

7 Stochastic Decision Making and Aerial Surveillance Control Strategies for Teams of Unmanned Aerial Vehicles .7-1

Raymond W Holsapple, John J Baker, and Amir J Matlock

8 Control Allocation .8-1

Michael W Oppenheimer, David B Doman, and Michael A Bolender

9 Swarm Stability 9-1

Veysel Gazi and Kevin M Passino

SECTION III Industrial

10 Control of Machine Tools and Machining Processes .10-1

Jaspreet S Dhupia and A Galip Ulsoy

11 Process Control in Semiconductor Manufacturing .11-1

Thomas F Edgar

12 Control of Polymerization Processes 12-1

Babatunde Ogunnaike, Grégory François, Masoud Soroush, and Dominique Bonvin

13 Multiscale Modeling and Control of Porous Thin Film Growth 13-1

Gangshi Hu, Xinyu Zhang, Gerassimos Orkoulas, and Panagiotis D Christofides

14 Control of Particulate Processes .14-1

Mingheng Li and Panagiotis D Christofides

15 Nonlinear Model Predictive Control for Batch Processes .15-1

Zoltan K Nagy and Richard D Braatz

16 The Use of Multivariate Statistics in Process Control 16-1

Michael J Piovoso and Karlene A Hoo

17 Plantwide Control .17-1

Karlene A Hoo

18 Automation and Control Solutions for Flat Strip Metal Processing .18-1

Francesco Alessandro Cuzzola and Thomas Parisini

SECTION IV Biological and Medical

19 Model-Based Control of Biochemical Reactors .19-1

Michael A Henson

20 Robotic Surgery .20-1

Rajesh Kumar

21 Stochastic Gene Expression: Modeling, Analysis, and Identification 21-1

Mustafa Khammash and Brian Munsky

22 Modeling the Human Body as a Dynamical System: Applications

to Drug Discovery and Development .22-1

M Vidyasagar

SECTION V Electronics

23 Control of Brushless DC Motors .23-1

Farhad Aghili

24 Hybrid Model Predictive Control of the Boost Converter 24-1

Raymond A DeCarlo, Jason C Neely, and Steven D Pekarek

SECTION VI Networks

25 The SNR Approach to Networked Control .25-1

Eduardo I Silva, Juan C Agüero, Graham C Goodwin, Katrina Lau, and Meng Wang

26 Optimization and Control of Communication Networks .26-1

Srinivas Shakkottai and Atilla Eryilmaz

SECTION VII Special Applications

27 Advanced Motion Control Design .27-1

Maarten Steinbuch, Roel J E Merry, Matthijs L G Boerlage, Michael J C Ronde, and Marinus J G van de Molengraft

28 Color Controls: An Advanced Feedback System .28-1

Lalit K Mestha and Alvaro E Gil

29 The Construction of Portfolios of Financial Assets: An Application

of Optimal Stochastic Control .29-1

Charles E Rohrs and Melanie B Rudoy

30 Earthquake Response Control for Civil Structures 30-1

Jeff T Scruggs and Henri P Gavin

31 Quantum Estimation and Control 31-1

Matthew R James and Robert L Kosut

32 Motion Control of Marine Craft 32-1

Tristan Perez and Thor I Fossen

33 Control of Unstable Oscillations in Flows .33-1

Anuradha M Annaswamy and Seunghyuck Hong

34 Modeling and Control of Air Conditioning and Refrigeration Systems 34-1

Andrew Alleyne, Vikas Chandan, Neera Jain, Bin Li, and Rich Otten

Index .Index-1

Preface to the Second Edition

As you may know, the first edition of The Control Handbook was very well received Many copies were

sold and a gratifying number of people took the time to tell me that they found it useful To the publisher,these are all reasons to do a second edition To the editor of the first edition, these same facts are a modestdisincentive The risk that a second edition will not be as good as the first one is real and worrisome Ihave tried very hard to insure that the second edition is at least as good as the first one was I hope youagree that I have succeeded

I have made two major changes in the second edition The first is that all the Applications chapters

are new It is simply a fact of life in engineering that once a problem is solved, people are no longer asinterested in it as they were when it was unsolved I have tried to find especially inspiring and excitingapplications for this second edition

Secondly, it has become clear to me that organizing the Applications book by academic discipline is

no longer sensible Most control applications are interdisciplinary For example, an automotive controlsystem that involves sensors to convert mechanical signals into electrical ones, actuators that convertelectrical signals into mechanical ones, several computers and a communication network to link sensorsand actuators to the computers does not belong solely to any specific academic area You will notice thatthe applications are now organized broadly by application areas, such as automotive and aerospace

One aspect of this new organization has created a minor and, I think, amusing problem Severalwonderful applications did not fit into my new taxonomy I originally grouped them under the titleMiscellaneous Several authors objected to the slightly pejorative nature of the term “miscellaneous.”

I agreed with them and, after some thinking, consulting with literate friends and with some of thelibrary resources, I have renamed that section “Special Applications.” Regardless of the name, they areall interesting and important and I hope you will read those articles as well as the ones that did fit myorganizational scheme

There has also been considerable progress in the areas covered in the Advanced Methods book This

is reflected in the roughly two dozen articles in this second edition that are completely new Some ofthese are in two new sections, “Analysis and Design of Hybrid Systems” and “Networks and NetworkedControls.”

There have even been a few changes in the Fundamentals Primarily, there is greater emphasis on

sampling and discretization This is because most control systems are now implemented digitally

I have enjoyed editing this second edition and learned a great deal while I was doing it I hope that youwill enjoy reading it and learn a great deal from doing so

William S Levine

MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc For productinformation, please contact:

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The people who were most crucial to the second edition were the authors of the articles It took a greatdeal of work to write each of these articles and I doubt that I will ever be able to repay the authors fortheir efforts I do thank them very much

The members of the advisory/editorial board for the second edition were a very great help in choosingtopics and finding authors I thank them all Two of them were especially helpful Davor Hrovat tookresponsibility for the automotive applications and Richard Braatz was crucial in selecting the applications

to industrial process control

It is a great pleasure to be able to provide some recognition and to thank the people who helped

bring this second edition of The Control Handbook into being Nora Konopka, publisher of engineering

and environmental sciences for Taylor & Francis/CRC Press, began encouraging me to create a secondedition quite some time ago Although it was not easy, she finally convinced me Jessica Vakili and KariBudyk, the project coordinators, were an enormous help in keeping track of potential authors as well

as those who had committed to write an article Syed Mohamad Shajahan, senior project executive atTechset, very capably handled all phases of production, while Richard Tressider, project editor for Taylor

& Francis/CRC Press, provided direction, oversight, and quality control Without all of them and theirassistants, the second edition would probably never have appeared and, if it had, it would have been farinferior to what it is

Most importantly, I thank my wife Shirley Johannesen Levine for everything she has done for me overthe many years we have been married It would not be possible to enumerate all the ways in which shehas contributed to each and everything I have done, not just editing this second edition

William S Levine

Tamer Ba¸sar

Department of Electrical andComputer EngineeringUniversity of Illinois at Urbana–ChampaignUrbana, Illinois

Richard Braatz

Department of Chemical EngineeringMassachusetts Institute of TechnologyCambridge, Massachusetts

Masayoshi Tomizuka

Department of MechanicalEngineering

University of California, BerkeleyBerkeley, California

Mathukumalli Vidyasagar

Department of BioengineeringThe University of Texas at DallasRichardson, Texas

William S Levinereceived B.S., M.S., and Ph.D degrees from the Massachusetts Institute of Technology

He then joined the faculty of the University of Maryland, College Park where he is currently a researchprofessor in the Department of Electrical and Computer Engineering Throughout his career he hasspecialized in the design and analysis of control systems and related problems in estimation, filtering, andsystem modeling Motivated by the desire to understand a collection of interesting controller designs,

he has done a great deal of research on mammalian control of movement in collaboration with severalneurophysiologists

He is co-author of Using MATLAB to Analyze and Design Control Systems, March 1992 Second Edition, March 1995 He is the coeditor of The Handbook of Networked and Embedded Control Systems, published

by Birkhauser in 2005 He is the editor of a series on control engineering for Birkhauser He has beenpresident of the IEEE Control Systems Society and the American Control Council He is presently thechairman of the SIAM special interest group in control theory and its applications

He is a fellow of the IEEE, a distinguished member of the IEEE Control Systems Society, and arecipient of the IEEE Third Millennium Medal He and his collaborators received the Schroers Awardfor outstanding rotorcraft research in 1998 He and another group of collaborators received the award

for outstanding paper in the IEEE Transactions on Automatic Control, entitled “Discrete-Time Point

Processes in Urban Traffic Queue Estimation.”

Francesco Borrelli

Department of Mechanical EngineeringUniversity of California, BerkelyBerkeley, California

Richard D Braatz

Department of Chemical EngineeringUniversity of Illinois at Urbana–ChampaignUrbana, Illinois

Francesco Alessandro Cuzzola

Danieli AutomationButtrio, Italy

Raymond A DeCarlo

Department of Electrical andComputer EngineeringPurdue UniversityWest Lafayette, Indiana

Josko Deur

Mechanical Engineering and Naval ArchitectureUniversity of Zagreb

Zagreb, Croatia

Jaspreet S Dhupia

School of Mechanical andAerospace EngineeringNanyang Technological UniversitySingapore

Atilla Eryilmaz

Electrical and ComputerEngineering DepartmentThe Ohio State UniversityColumbus, Ohio

Paolo Falcone

Signals and Systems DepartmentChalmers University of TechnologyGoteborg, Sweden

Trondheim, Norway

Grégory François

Automatic Control LaboratorySwiss Federal Institute of Technology inLausanne

Lausanne, Switzerland

Henri P Gavin

Department of Civil andEnvironmental EngineeringDuke University

Durham, North Carolina

Raymond W Holsapple

Control Science Center of ExcellenceAir Force Research LaboratoryWright-Patterson Air Force Base, Ohio

Seunghyuck Hong

Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridge, Massachusetts

Gangshi Hu

Department of Chemical andBiomolecular EngineeringUniversity of California, Los AngelesLos Angeles, California

University of California, Santa BarbaraSanta Barbara, California

Rajesh Kumar

Department of Computer ScienceJohns Hopkins UniversityBaltimore, Maryland

Roel J.E Merry

Department of Mechanical EngineeringEindhoven University of TechnologyEindhoven, the Netherlands

Marinus J van de Molengraft

Department of Mechanical EngineeringEindhoven University of TechnologyEindhoven, the Netherlands

Babatunde Ogunnaike

Chemical EngineeringUniversity of DelawareNewark, Delaware

Tristan Perez

School of EngineeringThe University of NewcastleCallaghan, New South Wales, Australiaand

Centre for Ships andOcean StructuresNorwegian University of Science andTechnology

Michael J C Ronde

Department of Mechanical EngineeringEindhoven University of TechnologyEindhoven, the Netherlands

Melanie B Rudoy

Department of Electrical andComputer ScienceMassachusetts Institute of TechnologyCambridge, Massachusetts

Masoud Soroush

Department of Chemical and BiologicalEngineering

Drexel UniversityPhiladelphia, Pennsylvania

Meng Wang

School of Electrical Engineeringand Computer ScienceThe University of NewcastleCallaghan, New South Wales, Australia

Automotive

Linear Parameter-Varying Control of Nonlinear

Systems with Applications to Automotive and

Feedback Fuel Control • Feedforward Fuel Control

continuous-time controller with the system matrices F(θ), G(θ), and H(θ) of its state–space model.

In Section 1.3, the control problem is formulated as an H∞ problem using the mixed sensitivity

approach The shaping weights We(θ, s), Wu(θ, s), and Wy(θ, s) are allowed to be parameter-varying Themost appealing feature of this approach is that it yields a parameter-varying bandwidthωc(θ) of the robustcontrol system Choosing appropriate shaping weights is described in Section 1.4 For more details aboutthe design methodology, the reader is referred to [1–6]

∗ Parts reprinted from H P Geering, Proceedings of the IEEE International Symposium on Industrial Electronics—ISIE 2005,

Dubrovnik, Croatia, June 20–23, 2005, pp 241–246, © 2005 IEEE With permission.

In Section 1.5, it is shown, how parameter-varying time-delays in the plant dynamics can be handled

in the framework proposed in Sections 1.3 and 1.4 For more details, consult [5,7]

In Section 1.6, two applications in the area of automotive engine control are discussed In the firstapplication [4,5,8], the design of an LPV feedback controller for the fuel injection is shown, which issuitable over the whole operating envelope of engine

In the second application [9,10], the philosophy of designing an LPV feedback controller is carriedover to the problem of designing an additional LPV feedforward controller compensating the parameter-varying wall-wetting dynamics in the intake manifold of the port-injected gasoline engine

In Section 1.7, the problem of LPV control of the short-period motion of an aircraft is discussed

1.2 Statement of the Control Problem

We consider the following nonlinear time-invariant dynamic system (“plant”) with the unconstrained

input vector U(t) ∈ R m , the state vector X(t) ∈ R n , and the output vector Y (t) ∈ R p:

˙X(t) = f (X(t), U(t)),

Y (t) = g(X(t)), where f and g are fairly “smooth” continuously differentiable functions.

Let us assume that we have found a reasonable or even optimal open-loop control strategy Unom(t) for

a rather large time interval t ∈ [0, T] (perhaps T = ∞), which theoretically generates the nominal state and output trajectories Xnom(t) and Ynom(t), respectively.

In order to ensure that the actual state and output trajectories X(t) and Y (t) stay close to the nominal ones at all times, we augment the open-loop control Unom(t) with a (correcting) feedback part u(t) Thus,

the combined open-closed-loop input vector becomes

U(t) = Unom(t) + u(t).

Assuming that the errors

x(t) = X(t) − Xnom(t) and y(t) = Y(t) − Ynom(t)

of the state and output trajectories, respectively, can be kept minimum with small closed-loop corrections

u(t), allows us to design a linear (parameter-varying) output feedback controller based on the linearized

dynamics of the plant:

˙x(t) = A(θ)x(t) + B(θ)u(t),

y(t) = C(θ)x(t), where A(θ), B(θ), and C(θ) symbolically denote the following Jacobi matrices:

respectively, but it may also contain additional “exogenous” signals influencing the parameters of the

r e

FIGURE 1.1 Schematic representation of the feedback control system.

nonlinear equations describing the dynamics of the plant (e.g., a temperature, which is not included inthe model as a state variable)

By using the symbolθ rather than θ(t), we indicate that we base the design of the feedback controller

on a time-invariant linearized plant at every instant t (“frozen linearized dynamics”).

This leads us to posing the following problem of designing an LPV controller:

For all of the attainable values of the parameter vectorθ, design a robust dynamic controller (with a

suitable order nc) with the state–space representation

specifica-In Section 1.3, this rather general problem statement will be narrowed down to a suitable and

trans-parent setting of H∞control and the solution will be presented

Furthermore, we assume that the input us, the state xs, and the output ysare suitably scaled, such that

the singular values of the frequency response matrix Gs(jω) = Cs[jωI − As]−1B

sare not spread too wideapart

For the design of the LPV time-invariant controller K(θ) depicted in Figure 1.1, we use the H∞

method [1,2] As a novel feature, we use parameter-dependent weights W• θ, s) This allows in particular

that we can adapt the bandwidthωc(θ) of the closed-loop control system to the parameter-dependentproperties of the plant!

Figure 1.2 shows the abstract schematic of the generic H∞control system Again, K(θ) is the controller, which we want to design and G(θ, s) is the so-called augmented plant The goal of the design is finding a compensator K(θ, s), such that the H∞norm from the auxiliary input w to the auxiliary output z is less

w

z e K

FIGURE 1.2 Schematic representation of the H∞ control system.

FIGURE 1.3 S / KS / T weighting scheme.

thanγ (γ ≤ 1), that is,

Tzw(θ, s)∞<γ ≤ 1for all of the attainable values of the constant parameter vectorθ

For the H∞design we choose the mixed-sensitivity approach This allows us to shape the singular

values of the sensitivity matrix S( jω) and of the complementary sensitivity matrix T( jω) of our control

system (Figure 1.1), where

S( θ, s) = [I + Gs(θ, s)K(θ, s)]−1

T( θ, s) = Gs(θ, s)K(θ, s)[I + Gs(θ, s)K(θ, s)]−1

= Gs(θ, s)K(θ, s)S(θ, s).

Thus, we choose the standard S / KS / T weighting scheme as depicted in Figure 1.3 This yields the

following transfer matrix:

In general, the four subsystems Gs(θ, s), We(θ, s), Wu(θ, s), and Wy(θ, s) are LPV time-invariant systems

By concatenating their individual state vectors into one state vector x, we can describe the dynamics of

the augmented plant by the following state–space model:

˙x(t) = A(θ)x(t) +B1(θ) B2(θ)

u s(t)

z(t) e(t)

The following conditions are necessary for the existence of a solution to the H∞ control designproblem∗:

1 The weights We, Wu, and Wyare asymptotically stable

2 The plant[As, Bs] is stabilizable.

3 The plant[As, Cs] is detectable.

4 The maximal singular value of D11is sufficiently small:σ(D11)<γ

5 Rank(D12)= ms, that is, there is a full feedthrough from us to z.

6 Rank(D21)= ps = ms, that is, there is a full feedthrough to e from w.

7 The system[A, B1] has no uncontrollable poles on the imaginary axis

8 The system[A, C1] has no undetectable poles on the imaginary axis

Remarks

Condition 6 is automatically satisfied (seeFigure 1.4) Condition 7 demands that the plant Gshas no poles

on the imaginary axis Condition 5 can be most easily satisfied by choosing Wuas a static system with a

small, square feedthrough: Wu(s) ≡ εI.

In order to present the solution to the H∞problem in a reasonably esthetic way, it is useful to introducethe following substitutions [3,4]:

positive-semidefinite Remember: all of these matrices are functions of the frozen parameter vectorθ

∗ In order to prevent cumbersome notation, the dependence on the parameter vectorθ is dropped in the development of the results.

The solution of the H∞problem is described in the following.

The controller K(s) has the following state–space description:

the attainable values of the parameter vectorθ

For a parameter-independent SISO plant Gs(s), a control engineer knows how to choose the bandwidth

ωcof the control system (Figure 1.1), that is, the cross-over frequency of the loop gain|Gs( jω)K( jω)|,

after inspecting the magnitude plot|Gs( jω)| of the plant.

He also knows how to choose quantitative specifications for performance via the sensitivity|S( jω)| in

the passband and for robustness via the complementary sensitivity|T( jω)| in the rejection band and the

peak values of|S( jω)| (e.g., 3 dB ≈ 1.4) and |T( jω)| (e.g., 1 dB ≈ 1.12) in the crossover region.

Since the weights We and Wy shape the sensitivity S and the complementary sensitivity T, respectively, the weights are chosen in the following way: (1) choose the functions S(ω) and T(ω) bounding the

quantitative specifications for|S( jω)| and |T( jω)|, respectively, from above, at all frequencies; (2) choose the weights We and Wy as the inverses of S and T, respectively, such that |We( jω)| ≡ 1 / S(ω) and

|Wy( jω)| ≡ 1 / T(ω) hold

Example:

For the sake of simplicity, let S( ω) correspond to a lead-lag element with S(0) = Smin= 0.01, S(∞) =

Smax= 10, and the corner frequencies Sminωc and Smaxωc ; and let T (ω) correspond to a lag-lead

element with T (0) = Tmax= 10, T(∞) = Tmin= 10−3, and the corner frequenciesκωc /Tmax and

κωc /Tminwithκ in the range 0.7 < κ ≤ 2 10.

For a parameter-dependent SISO plant Gs(θ, s), for each value of the parameter vector θ, we proceed as described above in a coherent way By coherent, we mean that we should “stress” the plant about equally

strongly in each of the attainable operating points Thus, we should probably have a pretty constant ratioωc(θ)/ωn(θ) of the chosen bandwidth of the closed-loop control system and the natural bandwidth ωn(θ)

of the plant Gs(θ, s).

For a SISO plant, we obtain the following weights in the example described above:

For Wethe parameter-varying lag-lead element

In the case of a parameter-varying MIMO plant, the weights We , Wu, and Wy are square matrices

because e, us, and ysare vectors In general, we use diagonal matrices with identical diagonal elements

w e(θ, s), wu(θ, s), and wy(θ, s), respectively

If constant parameters Smin, Smax, Tmax, Tmin, andκ can be used, this is a good indication that theparameter dependence of the cross-over frequencyωc(θ) has been chosen in a coherent way

1.5 Handling PV Time Delays

In many applications, modeling the linearized plant by a rational transfer matrix Gs(θ, s) only is not

sufficient because the dynamics of the plant include significant, possibly parameter-varying, time delays

For choosing the order N of the approximation, we proceed along the following lines of thought [4]:

• The above Padé approximation is very good at low frequencies The frequencyω∗, where the phase

error isπ/6 (30◦), isω∗≈2N

T

• In order to suffer only a small loss of phase margin due to the Padé approximation error, we put

ω∗way out into the rejection band We proposeω∗= αωcwithα = 30

• This leads to the following choice for the approximation order N:

N(θ) =α

2ωc(θ)T(θ).Obviously, choosingωc(θ) ∼ 1/ T( θ) would yield a parameter-independent approximation order N.

1.6 Applications in Automotive Engine Control

In this section, the LPV H∞methodology for the design of a LPV controller is applied to two problems

of fuel control for a 4-stroke, spark-ignited, port-injected gasoline engine: In Section 1.6.1, we discuss themodel-based feedback control, and in Section 1.6.2, the model-based feedforward control

As usual, in control, the requirements for the accuracy of the mathematical models upon which thecontrol designs are based differ significantly between feedback control and feedforward control

For the design of a robust feedback control for an asymptotically stable plant, once we have chosen

a controller, glibly speaking, it suffices to know the following data with good precision: the crossoverfrequencyωc, the (sufficiently large) phase marginϕ, and the (acceptable) direction of the tangent to theNyquist curve atωc In other words, a rather crude model of the plant just satisfying these requirementswill do

Conversely, for the design of a feedforward control, a rather precise model of the plant is neededbecause, essentially, the feedforward controller is supposed to invert the dynamics of the plant Thefeedback part of the control scheme will then be mainly responsible for stability and robustness, besidesfurther improving the command tracking performance

1.6.1 Feedback Fuel Control

In this example, we want to design a feedback controller Its task is keeping the air to fuel ratio of themixture in the cylinders stoichiometric

The fuel injection is governed by the control law ti = β(n, m)U Here, tiis the duration of the injection

impulse, n the engine speed, and m the mass of air in a cylinder (calculated by the input manifold observer).

The functionβ(n, m) is defined in such a way that the dimensionless control variable U is nominally 1 at every static operating point: Unom(n, m)≡ 1

With a wide-rangeλ-sensor, the resulting air to fuel ratio is measured in the exhaust manifold of theengine Its signalΛ is proportional to the air to fuel ratio and scaled such that Λ = 1 corresponds tostoichiometric Hence,Λnom≡ 1

We want to find a robust compensator K(θ, s), such that small changes us(t) in the control U(t)=

Unom+ us(t) will keep the errorsλs(t)= Λ(t) − Λnomminimum even in an arbitrary transient operation

of the engine Obviously, the parameter vector describing the operating point isθ(t) = [n(t), m(t)].

For modeling the linearized dynamics of the fuel path of the engine, the following phenomena must

be considered: the wall-wetting dynamics in the intake manifold, turbulent mixing of the gas in theexhaust manifold, the dynamics of theλ-sensor, and (last but most important) the time delay betweenthe injection of the fuel and the arrival of the corresponding mixture at the position of theλ-sensor Thesimplest model we can get away with successfully is

G s(θ, s) = −e −sT(θ) 1

τ(θ)s + 1.

The design of the controller proceeds in the following steps:

• Identify the functions T(θ) for the time delay and τ(θ) for the time constant over the full operating

envelope of the engine This is done by measuring the step response of the engine to a step change

in the amount of injected fuel over a sufficiently fine mesh of the parametersθ = [n, m].

• Choose a functionωc(T,τ) for the parameter-varying bandwidth of the control system (Pleasenote the change of variables!)

• Choose the weighting functions We (T, τ, s), Wu(s), and Wy(T, τ, s).

• Choose an approximation order N(T) for the Padé approximation of e −sT This yields a rationalapproximate transfer function G s(T, τ, s) of the plant.

• Solve the H∞problem for every pair (T,τ)

• Reduce the order of the resulting compensator K(T, τ, s) by one successively, watching the resulting

Nyquist curves and stop before the Nyquist curve deforms significantly For practical purposes, the

reduced order of the resulting final compensators should be constant over T andτ

• Find a structurally suitable representation for the reduced-order transfer function K(T, τ, s), so that its parameters, say ki(T, τ), can continuously and robustly be mapped over T and τ.

The engine control operates as follows in real time: At each control instant, that is, for each upcoming

cylinder requesting its injection signal ti, the instantaneous engine speed n and air mass m in the cylinder

are available and the following steps are taken:

• Calculate T(n, m) and τ(n, m).

• Calculate the parameters ki(T,τ) of the continuous-time controller with the transfer function

K(T, τ, s).

• Discretize the controller to discrete-time

• Process one time step of the discrete-time controller and output the corresponding signal ti.

Remark

Note the fine point here: As the engine speed changes, the time increment of the discrete-time controllerchanges

For a BMW 1.8-liter 4-cylinder engine, the time delay T and the time constantτ were found to be in

the ranges T= 0.02 .1.0 s andτ = 0.01 .0.5 s over the full operating envelope of the engine In [4],the bandwidth was chosen asωc(T,τ) = π/ 6T With α = 30, this resulted in the constant order N = 8

for the Padé approximation of the time delay

For more details about these LPV feedback fuel control schemes, the reader is referred to [4,5,8,11],and [12, ch 4.2.2]

1.6.2 Feedforward Fuel Control

In this example, we want to design a feedforward controller Its task is inverting the wall-wetting dynamics

of the intake manifold, such that, theoretically, the air to fuel ratioΛ(t) never deviates from its nominal

valueΛnom= 1 in dynamic operation of the engine

In 1981, Aquino published an empirical model for the wall-wetting dynamics [13], that is, for the

dynamic mismatch between the mass mFi of fuel injected and the mass mFoof fuel reaching the cylinder:

Therefore, Aquino’s model lends itself to the model-based design of a feedforward fuel controller Butits parameters should be derived using first physical principles.

Mathematical models derived by this approach and the corresponding designs of parameter-varyingfeedforward fuel controllers have been published in [9,10,23–25] A summary of the model can be found

in [12, ch 2.4.2]

1.7 Application in Aircraft Flight Control

In this section, the LPV H∞methodology for the design of a LPV controller is applied to the control ofthe short-period motion of a small unmanned airplane

For controlling an aircraft in a vertical plane, the following two physical control variables are available:

The thrust F (for control in the “forward” direction) and the elevator angleδe(for angular control aroundthe pitch axis)

In most cases, the airplane’s pitch dynamics can be separated into a fast mode (from the elevator angle

δe to the angle of attack α) and a slow mode (from the angle of attack α to the flight path angle γ)

Therefore, for flight control, it is more useful to use the angle of attack as a control variable (instead ofδe)

In this case, the fast dynamics fromδetoα should be considered in the flight control design as actuatordynamics from the commanded angle of attackαcom to the actual angle of attack of the aircraft in asuitable way These “actuator dynamics” are usually associated with the notion of “short-period motion”

because a step in the elevator angle produces a damped oscillatory response of the angle of attack

Thus, we get the following control problem for the fast inner control loop of the overall control scheme,that is, the problem of controlling the short-period motion:

For the under-critically damped second-order parameter-varying system with the inputδe, the output

α, and the transfer function Gαδe(θ, s), find a parameter-varying controller with the transfer function

K( θ, s), such that the command-following system

α(s) = K( θ, s)Gαδe(θ, s)

1+ K(θ, s)Gαδe(θ, s)αcom(s)

is robust and performs in a satisfactory way over the full operating envelope described by the parametervectorθ = (v, h) with the velocity v = vmin vmaxand the altitude h = hmin hmax

In [14,15], a small unmanned airplane with a takeoff mass of 28 kg, and a wingspan of 3.1 m, and an

operating envelope for the velocity of v= 20 100 m/s and for the altitude of h= 0 .800 m has beeninvestigated for robust, as well as fail-safe flight control

The transfer function Gαδe(θ, s) can be written in the form

Gαδe(θ, s) = Gαδe(θ, 0) s1(θ)s2(θ)

(s − s1(θ))(s − s2(θ)).

Over the full flight envelope, the poles s1(θ), s2(θ) (in rad/s) and the steady-state gain Gαδe(θ, 0) can be

parametrized with very high precision as follows:

c4= −3.08 × 10−6rad/m2

c5= 1.20

c6= −7.47 × 10−5m−1.For manually piloting the unmanned airplane, it is desirable that the dynamic and static charac-teristics of the command following control fromαcom toα be parameter-independent over the full

flight envelope This can easily be achieved by using the S / KS / T weighting scheme (Figure 1.3) with

parameter-independent weights We , Wu, and Wy.

Smin(θ) ≡ 0.01

Smax(θ) ≡ 10

Tmax(θ) ≡ 100

Tmin(θ) ≡ 0.001ε(θ) ≡ 10−4,

a suitable unit step response fromαcomtoα with a rise time of about one second and no overshoot results

for all v= 20 100 m/s and h= 0 .800 m

1.8 Conclusions

In this chapter, some concepts of LPV H∞control for a LPV plant A(θ), B(θ), C(θ) have been presented

in detail The salient feature is choosing a parameter-varying specificationωc(θ) for the bandwidth of the

control system Furthermore, parameter-varying weighting functions W• θ, s) have been stipulated.

Applying these concepts has been discussed briefly for both LPV feedback and LPV feedforward control

in the area of fuel control of an automotive engine, as well as for LPV control of the short-period motion

in aircraft flight control

In this chapter, the parameterθ has been assumed to be “frozen”, that is, its truly time-varying nature,

θ(t), has been neglected Naturally, the question arises whether such an LVP control system will be

asymptotically stable and sufficiently robust even during rapid changes of the parameter vector Presently,there is a lot of research activity worldwide addressing the question of time-varying parameters, see[16–22] for instance Suffice it to say here that in our examples about fuel injection of an automotiveengine and about the control of the short-period motion of an airplane, the bandwidth of the dynamics

ofθ(t) is at least an order of magnitude smaller than the bandwidth of our control systems, even in severe

transient operation of the engine or the airplane, respectively!

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H∞control problems, IEEE Transactions on Automatic Control, vol 34, pp 831–847, 1989.

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dissertation no 15700, Swiss Federal Institute of Technology, Zurich, Switzerland, 2004

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engines, in SAE SP-1830: Modeling of Spark Ignition Engines, March 2004, pp 285–290.

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disserta-tion no 16586, Swiss Federal Institute of Technology, Zurich, Switzerland, 2006

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ETH dissertation no 17505, Swiss Federal Institute of Technology, Zurich, Switzerland, 2007

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on Automatic Control, vol 35, pp 898–907, 1990.

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Transactions on Automatic Control, vol 38, pp 1021–1039, 1993.

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in Ford vehicles during the 1970s and early 1980s were all used for on-board engine controls (Powers,1993) The first chassis/suspension and vehicle control applications then followed in the mid-1980s.

Early powertrain control applications were driven by U.S regulatory requirements for improved fueleconomy and reduced emissions Additional benefits included improved functionality, performance,drivability, reliability, and reduced time-to-market as facilitated by the inherent flexibility of comput-ers and associated software Thus, in the mid-1970s, American automotive manufacturers introducedmicroprocessor-based engine control systems to meet the sometimes conflicting demands of high fueleconomy and low emissions

Present-day engine control systems contain many inputs (e.g., pressures, temperatures, rotationalspeeds, exhaust gas characteristics) and outputs (e.g., spark timing, exhaust gas recirculation, fuel-injectorpulse widths, throttle position, valve/cam timing) The unique aspect of the automotive control is therequirement to develop systems that are relatively low in cost, which will be applied to several hundredthousand units in the field, that must work on automobiles with inherent manufacturing variability, whichwill be used by a spectrum of human operators, and are subject to irregular maintenance and varyingoperating conditions This should be contrasted with the aircraft/spacecraft control problem, for whichmany of the sophisticated control techniques have been developed In this case, we can enumerate nearly

an opposite set of conditions

∗ Figures from this chapter are available in color athttp://www.crcpress.com/product/isbn/9781420073607

The software structure of the (embedded) controllers that have been developed to date is much likethose in the other areas (i.e., aircraft controllers, process controllers) in that there exists an “outer-loop”

operational mode structure, in this case, typically provided by a driver whose commands—for example,gas pedal position—are then interpreted as commands or reference signals for subsequent control agents

The latter typically consist of feedforward–feedback-adaptive (learning) modules Traditionally, the forward or “open-loop” portion has been by far the most dominant with numerous logical constructssuch as “if–then–else” contingency statements and related 2D and 3D tables, myriads of parameters,and online models of underlying physics and devices More precisely, the feedforward component mayinclude inverse models of relevant components and related physics such as nonlinear static formulas forflow across the throttle actuator, for example

feed-Assuming the existence of the microcomputer module with given chronometric and memory bilities, the structured/disciplined approach to developing a total embedded control system typicallyinvolves the following major steps: (1) development of requirements; (2) development of appropriatelinear and nonlinear plant models; (3) preliminary design with linear digital control system methods;

capa-(4) nonlinear simulation/controller design; and (5) hardware-in-the-loop/real-time simulation ity for identification, calibration, and verification including confirmation that the above chronometricand memory constraints have not been violated This includes appropriate dynamometer and vehicletesting with the help of rapid prototyping, autocoding, and data collection and manipulation tools, asneeded

capabil-Typical powertrain control strategies contain several hundred thousand lines of C code and sands of associated parameters and calibration variables requiring thousands of man-hours to calibrate,although the ongoing efforts in “self-calibrating” approaches and tools may substantially reduce thistime-consuming task In addition, different sensors and actuators—such as Electronic Throttle Con-trol (ETC), Variable Cam Timing (VCT), and Universal Exhaust Gas Oxygen (UEGO) sensors—areconstantly added and upgraded So are the new functions and requirements When computer memoryand/or chronometric capabilities have been exhausted and new or improved functionality is needed, thiscan lead to development of new requirements for the next generation of engine computer modules Forexample, in the period 1977–1982, Ford Motor Company introduced four generations of Engine ControlComputers (EEC) of ever increasing capabilities This evolution was needed to address more demand-ing fuel economy and pollution requirements, while at the same time for introducing new and/or moresophisticated functionality and for improved performance (Powers, 1993)

thou-The five control strategy development steps mentioned above may not always be sequential and some ofthe steps can be omitted Moreover, in practice, there are implied iteration loops between different steps

For example, one starts Step 1 with the best available requirements at the time, which can subsequently

be refined as one progresses through Steps 2 through 5 Similarly, the models from Step 2 can be furtherrefined as a result of Step 5 hardware implementation and testing

In some cases, the detailed nonlinear models of a plant or component may already exist and can beused (with possible simplifications) to design a nonlinear plant-based controller directly Alternatively, thedetailed nonlinear model can be used to extract simplified linearized models with details that are relevantfor the linear control system design within the bandwidth of interest The model development, Step 2, mayitself include models based on first principles (physics-based models) or semiempirical and identification-based models (“gray” and “black” boxes) Each approach has its advantages and disadvantages The blackand gray box models rely heavily on actual experimental data so that by default they are “validated” andtypically require lesser time to develop On the other hand, physically based models allow for (at leastpreliminary) controller design even before an actual hardware/plant is built and in the case of open-loopunstable systems they are needed to devise an initial stabilizing controller Moreover, they can be an

invaluable source of insight and critical information about the plant modus operandi, especially when

dealing with sometimes elusive dynamic effects In this context, they can influence the overall systemdesign—both hardware and software—in a true symbiotic way

Before we focus on concrete applications of powertrain controls, it is first important to enumerate what

2.2 Powertrain Controls and Related Attributes

Modern powertrains must satisfy numerous often competing requirements so that the design of a typicalpowertrain control system involves tradeoffs among a number of attributes (Hrovat and Powers, 1988,1990) When viewed in a control theory context, the various attributes are categorized quantitatively

as follows:

• Emissions: A set of terminal or final time inequality constraints (e.g., in case of gasoline engine, this

would apply to key pollution components: NOx, CO, and HC over certification drive cycles)

• Fuel consumption: A scalar quantity to be minimized over a drive cycle is usually the objective

function to be minimized

• Driveability: Expressed as constraints on key characteristic variables such as the damping ratio of

dominant vibration modes or as one or more state variable inequality constraints, which must besatisfied at every instant on the time interval (e.g., wheel torque or vehicle acceleration should bewithin a certain prescribed band)

• Performance: Either part of the objective function or an intermediate point constraint, for example,

achieve a specified 0–60 mph acceleration time

• Reliability: As a part of the emission control system, the components in the computer control

system (sensors, actuators, and computers) have up to 150,000 mile or 15-year warranty for PartialZero-Emission Vehicle (PZEV)-certified vehicles In the design process, reliability can enter as asensitivity or robustness condition, for example, location of roots in the complex plane, or moreexplicitly as uncertainty bounds and weighted sensitivity/complimentary sensitivity bounds in the

context of H∞orμ-synthesis and analysis methodology

• Cost: The effects of cost are problem dependent Typical ways that costs enter the problem

quanti-tatively are increased weights on control variables in quadratic performance indices (which impliesrelatively lower-cost actuators) and output instead of state feedback (which implies fewer sensorsbut more software)

• Packing: Networking of computers and/or smart sensors and actuators requires distributed control

theory and tradeoffs among data rates, task partitioning, and redundancy, among others

• Electromagnetic interference: This is mainly a hardware problem, which is rarely treated explicitly

in the analytic control design process

• Tamper-proof: This is one of the reasons for computer control, and leads to adaptive/self-calibrating

systems so that dealer adjustments are not required as the powertrain ages or changes

To illustrate how control theoretic techniques are employed in the design of powertrain control systems,the examples of typical engine, transmission, and driveline controls will be reviewed along with someimportant related considerations such as drivability and diagnostics This chapter concludes with adiscussion of current trends in on-board computer control diagnostics

To improve fuel economy and performance, the automakers have added new engine devices and

hardware Additional devices such as variable valve lift, variable displacement (also called cylinder vation or displacement-on-demand), charge motion control valves, intake manifold tuning, turbocharg-ers, superchargers, and so on are also used in production applications.

deacti-Each device operation is computer controlled Finding the steady-state set-point combinations thatachieve the best tradeoff between fuel economy, peak torque/power, and emissions is the subject of theengine mapping and optimization process Maintaining the device output to the desired set-point typicallyrequires development of a local feedback control system for each one Engines spend a significant fraction

of time in transients Because the optimization devices may interact in unexpected or undesirable ways,

it is important to control and synchronize their transient behavior

Figure 2.1 shows the view of an engine as the system to be controlled The prominent feature isthat the disturbance input, engine speed∗, and ambient conditions are measured or known, while theperformance variables, actual torque, emissions, and fuel efficiency are typically not available (exceptduring laboratory testing) The reference set-point, engine torque demand, is available based on theaccelerator pedal position In the next three sections, we briefly review important paradigms of enginecontrol system design: fuel consumption set-point optimization and feedback regulation In each case,

an advanced optimization or control method has been tried In each case, experimental tests were run toconfirm the achieved benefit

2.3.1 Fuel-Consumption Optimization

In today’s engines, several optimization devices are added to improve an attribute such as fuel economy

In most cases, the optimal set-point for each device varies with engine operating conditions and is usuallyfound experimentally Combining these devices has made it increasingly difficult and time consuming

to map and calibrate such engines The complexity increases not linearly, but exponentially with thenumber of devices, that is, degrees of freedom Each additional degree of freedom typically increases thecomplexity in terms of mapping time and size of the calibration tables by a factor between 2 (for twoposition devices) and 3–10 (for continuously variable devices) For a high-degree-of-freedom (HDOF)

Torque Emissions FE

Air flow (MAF) Engine speed A/F Air temperature

.

.

.

.

.

.

Disturbance inputs Performanceoutputs

Engine

Control inputs

Measured outputs

Engine control unit

Torque demand

Throttle Fuel injectors Spark VCT

Engine speed Ambient condition

FIGURE 2.1 The input–output structure of a typical engine control system.

∗ Engine speed can be viewed as a disturbance input in some modes of operation and a system state in others (such as

engine idle).

engine, the conventional process that generates the final calibration has become very time consuming.

For example, in the dual-independent variable cam timing (diVCT) gasoline engine, which we shall use

as the platform for the results in this section, the mapping time could increase by a factor of 30 or moreover the conventional (non-VCT) engine or the process could end up sacrificing the potential benefit

From the product development point of view, either outcome is undesirable

The diVCT engine has the intake and exhaust cam actuators that can be varied independently (Jankovicand Magner, 2002; Leone et al., 1996) A typical VCT hardware is shown in Figure 2.2 The intake valveopening (IVO) and exhaust valve closing (EVC) expressed in degrees after top dead center (ATDC)(TDC is at 360 deg crank inFigure 2.2) are considered the two independent degrees of freedom Theexperimental 3.0L V6 engine under consideration has the range of−30 to 30 deg ATDC for IVO and 0

to 40 deg ATDC for EVC

2.3.1.1 Engine Drive Cycle and Pointwise Optimization

We consider the problem of optimizing vehicle fuel efficiency while assuring that specific emissionsregulations, defined over a drive cycle, are satisfied A drive cycle, in which the vehicle speed and conditionsare specified by regulations, is intended to evaluate vehicle emissions or fuel consumption under a generic

Advance

Exhaust profile

Intake profile Retard

FIGURE 2.2 Valve lift profiles versus crank angle in a VCT engine.

3000 2000 1000

0

200 100

FIGURE 2.3 Vehicle speed, engine speed, and torque, during the first and last 505 s (bags 1 and 3) of the US75 drive cycle.

or particular drive pattern The top plot in Figure 2.3 shows the vehicle speed profile during a part of theUS75 drive cycle Given the transmission shift schedule, the vehicle speed uniquely determines the enginespeed and torque needed to follow the drive trace Hence, from the point of view of engine optimization,engine speed (middle plot) and engine torque (bottom plot) are constrained variables

Due to the presence of three-way catalysts, which become very efficient in removing regulated exhaustgases after light-off (50–100 s from a cold start), this optimization problem basically splits into two disjointproblems:

• Achieve a fast catalyst light-off while managing feedgas (engine-out) emissions

• Optimize fuel economy after catalyst light-off

In this section, we shall only consider the latter problem

Given that the engine speed and torque are constrained by the drive cycle, and the air–fuel ratio iskept close to stoichiometry to assure high efficiency of the catalyst system, the variables one can use tooptimize fuel consumption are IVO, EVC, and the spark timing (spk) Basically, we are looking for thebest combination of these three variables at various speed and torque points of engine operation (see

brake-specific fuel consumption (BSFC) Figure 2.4 shows BSFC versus two optimization variables at a typicalspeed/torque operating point (1500 rpm engine speed, 62 Nm torque) The top plot shows BSFC versusIVO and EVC at the spark timing for best fuel economy (called maximum brake torque (MBT) spark) Thebottom plot shows BSFC versus IVO and spark, at EVC= 30◦ Hence, the dash curves in Figure 2.4a and b

show the same set of mapped points The plots have been obtained by the full-factorial mapping, which isprohibitively time consuming to generate On the other hand, achieving the fuel economy potential of the

0.345 (a)

(b)

0.34 0.335 0.33

0.325 0.32 0.315

0.37 0.36 0.35 0.34

0.33 0.32 0.31 50 45 40 35 30 25

20 –30 –20 –10

0 IVO degrees Spark advance degrees

10 20 30

0 10 20 30

40 30

20 10

0 IVO EVC

–10 –20 –30

FIGURE 2.4 BSFC: (a) BSFC versus IVO and EVC at MBT spark; (b) BSFC versus IVO and spark at EVC = 30 ◦.

2.3.1.2 Extremum Seeking (ES) Methods for Engine Optimization

To operate the engine with best fuel consumption under steady-state conditions only requires the edge of the optimal IVO–EVC pair, and the corresponding MBT spark timing Several methods ofobtaining these optimal combinations are available: ES with sinusoidal perturbation (Ariyur and Krstic,2003), Direct Search methods such as Nealder–Mead (Wright, 1995; Kolda et al., 2003) and the GradientSearch methods (Box and Wilson, 1951; Spall, 1999; Teel, 2000) Such algorithms have already been used

knowl-for engine optimization For example, Draper and Lee (1951) used the sinusoidal perturbation to mize engine fuel consumption by varying air–fuel ratio and spark, while Dorey and Stuart (1994) used agradient search to find the optimal spark setting.

opti-ES is an iterative optimization process performed in real time on a physical system, that is, without

a model being built and calibrated before The function being optimized is the steady-state relationshipbetween the system’s input parameters and its performance output The optimization function, in our case

BSFC, denoted by f (.) is usually called the response map Since f (.) is not known (otherwise, it could be

optimized on a computer), ES controllers rely only on measurements to search for the optimum Startingfrom some initial parameter values, the ES controller iteratively perturbs the parameters, monitors theresponse, and adjusts the parameters toward improved performance This process runs according to somechosen optimization algorithm, usually as long as there is improvement

Several ES algorithms that belong to the Gradient Search category have been developed and tally tested on a diVCT engine (Popovic et al., 2006) They include modified Box–Wilson, simultaneouslyperturbed stochastic approximation (SPSA) (Spall, 1999), and persistently exciting finite differences(PEFD) (Teel, 2000) The last two are closely related in implementation though not in theoretical under-pinnings Therefore, we shall describe how they work on our problem of optimizing the BSFC response

experimen-map f (x) with the optimization parameters vector x = [x1x2x3]T consisting of IVO, EVC, and sparktiming

The SPSA and PEFD algorithms in 3D start with selection of one of the four perturbation directions:

v1= [1, 1, 1], v2= [−1, 1, 1], v3= [1, −1, 1], v4= [1, 1, −1]

SPSA selects the direction randomly and PEFD periodically in the round-robin fashion The current

values of the parameters x(k) are perturbed in the direction of the selected vector vi, by the amount

controlled by the parameterλ, and the measurement of f (xk + λvi) is obtained Because the map is generated by a dynamical system, the response of f to a change in input parameters can be measured only

after a period of time, usually after large transients have died down The measurements are usually noisyand may require filtering or averaging For the BSFC optimization, we have waited for 1 s after the change

in the set-point, and then averaged the measurement over the next 3 s

Next, the current parameter value x(k) is perturbed in the direction of −vi and the measurement

of f (xk − λvi) is taken The wait time and averaging are used again, and the total time of a parameter

update step, dominated by the time needed to take the measurements, is found to be 8 s With both themeasurements in, the parameters are updated in the direction opposite to the directional derivative:

versus time The starting parameter estimate is IVO = 0, EVC = 10, spk = 30.

The noise-like jumps in BSFC are generated by the perturbations introduced by this method whilethe parameter plot only shows the estimates (without perturbation) so that they are smooth After about

20 min, parameters have converged to the (local) minimum in BSFC For comparison, generating thecomplete response surface, shown in Figure 2.5, takes about 15–20 h using the conventional enginemapping method The initial and final points are indicated by arrows on the BSFC response surface (inIVO and EVC) as shown in the lower plot of Figure 2.5 In case it is not obvious, the minimum found

by the algorithm is local The (slightly lower) global minimum is on the other side of the ridge Thus, ingeneral, to find the global minimum, the algorithms will have to be run several times from different initialconditions