toward a systematic design for turbocharged engine control

Chapter 1

Toward a Systematic Design forTurbocharged Engine Control

Greg Stewart, Francesco Borrelli, Jaroslav Pekar, David Germann, DanielPachner, and Dejan Kihas

Abstract The efficient development of high performance control is becomingmore important and more challenging with ever tightening emissions legisla-tion and increasingly complex engines. Many traditional industrial control de-sign techniques have difficulty in addressing multivariable interactions amongsubsystems and are becoming a bottleneck in terms of development time. Inthis article we explore the requirements imposed on control design from a va-riety of sources: the physics of the engine, the embedded software limitations,the existing software hierarchy, and standard industrial control developmentprocesses. Decisions regarding the introduction of any new control paradigmmust consider balancing this diverse set of requirements. In this context wethen provide an overview of our work in developing a systematic approach tothe design of optimal multivariable control for air handling in turbochargedengines.

1.1 Introduction

The goal of this chapter is to present the emerging problems facing develop-ment of control for increasingly complex turbocharged engines and to discusspotential solutions. In turbocharged diesel engines ever-tightening emissionslegislation drives the incorporation of new sensors [27, 33], actuators, andsubsystems such as multistage turbochargers, complex exhaust gas recircu-

Greg Stewart, David Germann, Dejan KihasHoneywell Automation and Control Solutions, 500 Brooksbank Avenue, North Vancouver,BC, Canada, V7J 3S4

Francesco BorrelliDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720

Jaroslav Pekar, Daniel PachnerHoneywell Prague Laboratory, V Parku 2326/18, 148 00 Prague 4, Czech Republic

1

2 Stewart et al.

lation (EGR) topologies [31], and aftertreatment devices (such as selectivecatalytic reduction [26, 32], lean NOx traps, and diesel particulate filters[21]). These changes are all introduced in the context of difficult analysis anddecisions regarding the tradeoff of development and product cost, reliability,fuel economy, drivability, and emissions.

The rising complexity of engines and the demand of tighter performance isincreasing the complexity of the control functionality that is required to man-age the engine. It is widely recognized that the development of the controlis becoming a bottleneck in the development of engines and systematic ap-proaches to developing performant controls would be welcome provided theymitigate these burdens. Typically most engine makers require control designtechniques that provide some combination of improved closed-loop perfor-mance and reduced development effort. The relative importance of each con-sideration depends to a large extent on the business model of the particularcontrol system developer.

In this chapter we will discuss a wide view of the problem of developingsuch a systematic approach with the goal of integration into industrial prac-tice. In so doing we need to consider the interaction of many practical issues;including engine physics and its resulting nonlinearities and multivariableinteractions, the desired closed-loop performance, engine variability due toproduction dispersion and ageing, the restrictions in CPU time and memorydue to the embedded electronic control unit (ECU) platform, the hierarchicalsoftware structure into which the final control function must integrate, theexisting development process for engine control, and the range of personnelwith whom any new process must interact.

The scope of a solution to such a problem is large, but is aided by thesignificant body of previous research - both academic and industrial - inthe domain. The requirements associated with engine modeling and closed-loop control performance are considered in [28, 20, 15, 34, 16] and referencestherein. As will be introduced below, our approach includes a model predic-tive control (MPC) component and previous MPC engine control examplesmay be found in [23, 17].

The interplay of control design and computational restrictions1 (CPU andmemory) were noted in [17] where a fully nonlinear MPC (NMPC) is pro-posed which demonstrates improved performance with respect to linear statefeedback and input-output linearization approaches. As pointed out in [17],the main drawback of most NMPC techniques comes from the fact that theytypically require far more computational power than is available on mod-ern automotive ECUs. To address these issues we have implemented severalpractical simplifications and have separately explored the question of onlineimplementation of MPC in depth. This approach has allowed us to success-fully implement a multivariable MPC controller in a production ECU [29].

1 More detailed discussions of the very important subject of requirements and managementof engine control software design and its innovations may be found in references such as[25, 30].

1 Toward a Systematic Design for Turbocharged Engine Control 3

Selected aspects of the underlying techniques are outlined in Section 1.4 be-low.

Section 1.2 presents the requirements to be met by control design, Section1.3 describes how the proposed modeling and model predictive control tech-niques are suited to address these requirements, Section 1.4 presents somemore detailed results on one of the many technical aspects that are neededto be overcome in order to achieve an industrial quality systematic approachto control design.

1.2 Engine Control Requirements

In this section we present the specific requirements that must be addressedby any engine control design process. These are discussed with reference toa high-level description of a typical engine control design process which hasevolved over many years to address the industrial need of creating controllersfor highly nonlinear engines which are to be hosted on ECUs and deployedacross a fleet of thousands to millions of vehicles that may stay in activeservice as long as 20 years. As will be seen below, the support of all phases ofthis process requires a wide range of skills that include an understanding ofengine physics, performance specifications (including emissions legislation),embedded software issues, and calibration and post-release support. Typicallythese activities involve separate skills and groups within a company and itis important that any proposed technology change anticipates and addressesall of these areas.

While specific engine control design processes will vary site to site, manysimilarities exist and a high level outline of a typical process is illustrated inFigure 1.1 which is similar to those described in references such as [25].

1.2.1 Steady-state engine calibration

Also known as “base mapping” in this pre-control phase an engine is cal-ibrated in order to produce the actuator positions at a coarse grid of en-gine speed and load points such that certain desired steady-state criteria aretraded off. This step is often performed by executing a design of experimentthat sweeps the relevant engine actuators at a selection of speed and loadpoints while recording the engine’s performance in terms of fuel consump-tion and emissions. Upon completion of the experiment the desired enginesteady-state operating points (outputs and actuator positions as a function ofexogenous variables such as speed and injected fuel quantity) are determinedby optimizing with respect to requirements imposed by legislated drive cycle

4 Stewart et al.

Certification and Release

Steady-state engine calibration

Control functional development

shor

t-pa

th

func

tiona

l ite

ratio

n so

ftwar

e ite

ratio

n

long

-pat

h fu

nctio

nal i

tera

tion

Software development

(specification, coding, testing)

Integration (testing and debugging)

Functional testing (simulation, testbench,

vehicle)

Calibration (simulation, testbench,

vehicle)

Fig. 1.1 Simplified illustration of the engine control development process.

limits. This is a heavily experimental stage and requires much insight intothe engines’ behavior.

1.2.2 Control functional development

In this step control engineers decide on the partitioning of the engine func-tionality and, where needed, propose and configure new control functionswith the goal of approximately delivering the steady-state engine calibrationwhile simultaneously providing enough flexibility to meet the certificationand drivability requirements that are evaluated over transient driving condi-tions. This stage is often performed with the aid of rapid prototyping toolsthat support the porting of high-level code such as Matlab or Simulink intoan ECU bypass system. Commercially available examples include [12, 2, 22].


Figure 1.2 is helpful for understanding the environment into which a con-troller must be designed. An engine is a highly nonlinear plant (see for ex-ample [28, 20, 15, 16, 19]) and in most cases requires the development ofnonlinear control strategies. For example, when considering the response ofairside parameters such as compressor flow and boost pressure to the vari-able geometry turbine (VGT) actuator, the control designer must address asteady-state gain that changes sign and the fact that the nonlinearities are afunction of the engine speed and fuel injection quantity and thus may changemore quickly than the dynamics of the air loop which are often dominatedby the turbocharger inertia [20, 29]. In addition one must often consider thefact that multiple engine configurations may be required to be addressed byrelatively minor configuration changes to a single control strategy.

Figure 1.2 illustrates the hierarchical software environment into which thedeveloped controller function must be integrated. An individual subcontrolleris typically configured to be responsible for an engine subsystem and a givensubcontroller may be responsible for fuel injection control, aftertreatmentcontrol, exhaust gas recirculation (EGR) control, or turbocharger (waste-gate or VGT) control. The illustrated higher level function is responsible forengine monitoring and distribution of signals to the subcontrollers which typ-ically includes information about special modes (cold start, regeneration ofaftertreatment, etc.), setpoints, feedforward actuator values, and time vary-ing constraints for actuators and engine states. Actuator constraints typicallyinclude the enforcement of minimum and maximum bounds on actuator po-sitions and a common example of a state constraint is the imposition ofan upper bound on the turbocharger speed in order to prevent damage tothe turbocharger wheels. In our work we are concentrating on developingtechniques that enable the user design multivariable controllers that may besubstituted for one or more subcontrollers, initially focusing on unifying thecontrol of the air induction subsystems and coordinating their interactionwith fuel injection and aftertreatment subcontrollers.

Multivariable interactions among subsystems have typically been neglectedin traditional engine control design, but are rapidly becoming crucial to con-sider as performance requirements become more stringent due to legislationand the emergence of aftertreatment devices that are most effective in certainoperating windows of temperature, flow, and composition of the exhaust gas.The techniques and tools required in the “control functional development”stage are those that enable the developer to systematically create models anddesign controllers for integration into the existing control hierarchy that willprovide acceptable performance for a wide variety of engine configurationsand performance specifications.

6 Stewart et al.

Engine

from sensors to actuators

high level (monitoring and

decision making)

subcontroller 1

sens

or p

roce

ssin

g

y h (t)

actu

ator

pro

cess

ing

subcontroller N

u 1 (t) f 1 (t)

y 1 (t)

y N (t) f N (t) u N (t)

ECU

Fig. 1.2 Illustration of relative position of subcontroller algorithms within the ECU soft-ware hierarchy. The symbols yi(t) and uj(t) represent the ith and jth subsets of sensor andactuator information respectively. The symbols fk(t) denote the information transferredto the kth subcontroller and may include setpoints, feedforward actuator values, and timevarying constraints for actuators and engine states. The techniques discussed in this chap-ter consider controller design for the replacement of one or more subcontrollers with anoptimal multivariable controller.

1.2.3 Functional testing

In this phase the developing control functionality may be progressively testedin simulation, in the engine test cell, in vehicle, and over a range of ambi-ent conditions of temperature and altitude. This step is typically performediteratively with the control functional development and the developer mustevaluate how well the designed control function interacts with the engine andany other existing control loops to deliver the desired performance.

1.2.4 Software development

This phase may be performed in-house by the engine maker or may involvea third party software provider. The software developers receive a functionalspecification from the above step and then specify, code, and test the de-sired function into the embedded software environment which may includethe reformulation of the code to use fixed-point arithmetic. In this phase akey requirement is to respect the memory and processor limitations of the


ECU. Furthermore, as this phase is potentially long and expensive, whereverpossible one prefers to use control algorithm structures with the flexibility toenable control changes without requiring re-entry into the software develop-ment phase.

The duration and expense of this phase has motivated the developmentof autocoding tools and techniques which enable the conversion of high-levellanguages such as Matlab and Simulink into production-quality embeddedcode. Such tools have begun to make inroads into various production appli-cations [30].

1.2.5 Integration

The engine developer then integrates the developed code which may be testedin a simulated engine environment before proceeding to testing and debuggingin the test cell and vehicle.

1.2.6 Calibration

This is a detailed phase in which the numerous free parameters of the en-gine controller are tuned or calibrated to provide the desired steady-stateand transient performance. This phase requires that controller tuning toolsbe efficient, intuitive, and well-integrated into the existing calibration en-vironment. This phase also includes all of the calibration required for thediagnostics portions of the software which are indirectly related to the con-trol functionality.

1.2.7 Certification

Refers to the set of tasks related to preparing, documenting and performingthe certification testing that is relevant to the vehicle class and geographicalregion in which it will be used.

1.2.8 Release and post-release support

Once the software and control have been accepted, then the engine or vehicleis released to market. Subsequent control calibration or other fixes may berequired in the post-release support of the product. Since post-release revision

8 Stewart et al.

to the control is limited and expensive, the controllers designed and calibratedthroughout the above steps must perform well over a fleet of engines eachwith differences in the manufactured components, and must maintain systemperformance over the lifetime as the engine components age and change indifferent ways.

1.2.9 Iteration loops

Figure 1.1 illustrates the above steps along with some standard iterationpaths. The “short-path functional iteration” between functional testing andcontrol functional development has been made relatively rapid by modernrapid prototyping systems. It is these iterations in which the major decisionsregarding overall controller structure are made and thus it is critical thatthe developer be provided tools that direct or otherwise assist control struc-ture decisions to be made systematically. In this stage it is noted that thedesigner must design a controller to work with the engine physics of its des-ignated task, and also to interact appropriately with “neighboring” controlfunctions. In [29], the hierarchical nature of engine control is described andit is pointed out that any developed control function must be integrated intothis hierarchy where the supervisory levels deliver signals including setpoints,actuator feedforward, time-varying constraints for actuators or engine statesto the lower-level components of the control hierarchy.

The illustrated “software iterations” on the lower-right portion of the di-agram indicates any rework that is required for fixing bugs or other errors inthe software coding. The “long-path functional iteration” illustrated on theleft side of the diagram indicates the required iteration path when controllercode is found to be unacceptable at the functional level at a later stage ofdevelopment. Since this path contains the entire software development cycle,it may incur a cost of several months of development time2 using existingdevelopment tools and processes, and is thus considered undesirable in themajority of cases.

The above discussion has presented a high-level description of the enginecontrol development process from which it is clear that several key require-ments to be met by any control technology include:

(i) to maintain performance of the family of highly nonlinear and multivari-able engines that lie within the uncertainty bounds of a production fleet,

(ii) integration into the hierarchical control structure (including time varyingconstraints),

(iii) to fit within the tight processor and memory requirements for implemen-tation in an ECU, and,

2 The time and effort incurred in the “long-path functional iteration” is sometimes reducedby the use of autocoding tools such as [30] in the Software Development phase.


(iv) to provide a systematic, efficient development process to facilitate softwaredevelopment and enable a reduced calibration burden.

1.3 Modeling and Control for Turbocharged Engines

Section 1.1 discussed the trends and pressures facing the development of mod-ern engine control while Section 1.2 overviewed the scope of the requirementsfor any strategy - new or old. In this section we discuss the merits of usingoptimal model predictive control technology to address these issues.

In light of the above requirements, MPC shows several strong advantages:

• MPC is ideally suited for handling the signals from the higher level ofthe hierarchy illustrated in Figure 1.2. Particularly the setpoints and timevarying input and output constraints [29].

• MPC has a general algorithmic structure which can cover many differentcontrol problems (e.g. unconstrained or constrained, single or multivari-able) without requiring software structure changes.

• The tuning of MPC (while not quite as straightforward as is often claimedin MPC literature) can be made intuitive with the appropriate softwaretools to allow practical usage by people with a wide variety of technicalbackgrounds and interests [14].

Figures 1.3 and 1.4 contain representative examples of the on-engine re-sults achieved using model predictive control on two very different enginecontrol problems. Application details may be found in [29]. Figure 1.3 is anillustration of the simultaneous control of measured EGR flow and engine-outNOx concentration in a heavy-duty diesel engine using the EGR valve andvariable cylinder valve as actuators. Such a control configuration could haveapplication in coordinated engine-aftertreatment control. Figure 1.4 repre-sents a more standard control problem where the intake manifold pressureand the compressor flow are simultaneously controlled to respective setpointsusing the VGT vanes and EGR valve while the engine traverses the indicatedtransient in engine speed and injected fuel quantity.

On the other hand, we will have to consider and address the challengesfaced by MPC to achieve the desired reduction in development time andeffort. These include:

• Modeling: one is obliged to develop an efficient and reliable process togenerate the control oriented models that are required by all advancedcontrol techniques

• Computation: MPC - and especially nonlinear MPC - typically require toomuch computing power and memory for implementation on a modern ECU(see for example [17]). This topic is discussed in some detail in Section 1.4below.

10 Stewart et al.

260 280 300 320 340 360 380 40060

80

100

EG

R fl

ow [%

]time [sec]

260 280 300 320 340 360 380 400

40

60

80

100

NO

x [%

]

time [sec]

260 280 300 320 340 360 380 400

70

80

90

100

VV

A [%

]

time [sec]

260 280 300 320 340 360 380 400−10

0

10

20

30

EG

R v

alve

[%]

time [sec]

Fig. 1.3 Simultaneous control of EGR flow and engine-out NOx concentration using theEGR valve and variable cylinder valve actuators in a heavy duty diesel engine. The set-points are indicated by the dashed blue lines and the sensor measurements in red in theupper two subplots. The actuator constraints are illustrated as the broken magenta linesand the actuator positions as the solid green lines in the lower two subplots. The axes havebeen scaled for deidentification considerations.

• Usability: for deployment, advanced control techniques must be developedto the point of industrial quality, such that they do not require such de-tailed knowledge of MPC or modeling that disqualifies all but a smallgroup of specialists for their use.

1.3.1 Modeling

It is almost accepted as a truism that model based control design techniquesare used in order to cut down on the development time. A better phrasing of


0 20 40 60 80 100 120 140 160 180 200

60

80

100

boos

t pre

ssur

e [%

]

0 20 40 60 80 100 120 140 160 180 200

50

100ai

r flo

w [%

]

0 20 40 60 80 100 120 140 160 180 200

50

100

VG

T [%

]

0 20 40 60 80 100 120 140 160 180 2000

50

100

EG

R v

alve

[%]

0 20 40 60 80 100 120 140 160 180 2000

50

100

Fue

l flo

w [%

]

Time [seconds]

0 20 40 60 80 100 120 140 160 180 200

60

80

100

Eng

ine

Spe

ed [%

]

Time [seconds]

Fig. 1.4 Simultaneous control of boost pressure and compressor flow using the VGT andEGR valve actuators in a small diesel engine. The MPC was implemented on a produc-tion ECU (Motorola MPC555). In the upper two subplots the setpoints are representedby the solid blue lines and the measured sensor signals by the dashed green lines. Theactuator constraints are constant at 5% and 95% respectively. The axes have been scaledfor deidentification considerations.

that statement may be that one should take care to ensure that the modelingresults in a reduction of the overall the control development effort. Tradi-tional, non-model-based approaches to engine control design have an ironicadvantage of avoiding a potentially onerous and difficult modeling process.Of course traditional approaches are acknowledged to have serious limita-tions in the complexity of control problem they can address and furthermoreengine models have many benefits beyond their use in control synthesis (seefor example [11]).

12 Stewart et al.

The models we develop in the course of our design are highly nonlinear andthe reliable identification of their free parameters is a serious challenge, thedetailed description of which is outside the scope of this chapter. We presenta high level overview of the key issues here and will publish the technicaldetails elsewhere.

A practical modeling approach needs to be configurable to a wide variety ofengines including single and multistage turbochargers, low and high pressureEGR, various actuators such as valves, throttles, wastegates, VGT vanes,variable valve actuation, and various sensor selections and locations on theengine. With an eye on their intended use, one must implicitly trade off modelcomplexity and precision since low order models are preferable for modelbased control design. Figure 1.5 illustrates two different engine layouts whichhave been built from our library of components. The dynamic response isgoverned by the states associated with the intake and exhaust manifolds andalso the turbocharger speed(s). In more complicated engine layouts, such asthe illustrated multiple turbocharger example, correspondingly more statesare required to include the associated dynamics.

HP turbine (VGT)

LP compressor

LP turbine

intake manifold

exhaust manifold

engine

HP compressor air filter

aftertreatment

EGR cooler

EGR valve

throttle

air filter compressor

VGT turbine

charge air cooler

EGR cooler

EGR valve

intake manifold

exhaust manifold

engine

turbocharger shaft

LP turbocharger

shaft

HP turbocharger

shaft

Fig. 1.5 Example engine layouts for a standard single-stage turbocharged and a seriesturbocharged engine both with high-pressure exhaust gas recirculation (EGR).


Next the configured model must be calibrated to the engine such thatit matches the true nonlinear and dynamic input-output response across alloperating conditions. The model identification is typically required to workwith some mix of steady-state and transient data (see e.g. [16]). To addressthis nonlinear model identification problem we take a two-step approach.First the individual components (listed above and illustrated in Figure 1.5)are fit one by one to component maps (if available) and recorded engine data.In practice, the overall model quality achieved from an assembled collection ofcomponents is typically insufficient for capturing the required input-outputbehavior of the engine and the second step of the model identification isperformed by executing a global nonlinear optimal fitting of the availableparameters. In practice, this step typically results in a dramatic improvementin model accuracy and a representative example is illustrated in Figure 1.6.

On a technical level the model identification is further complicated by thefact that turbocharged engine models have inherent feedback paths via theEGR and the turbocharger shaft. Still more feedback paths are often addedduring standard modeling procedures [20, 16]. Thus the sensitivity and evenstability of the model with respect to its tuning parameters must be treatedwith due care to preserve the desired match to the true engine response.

This modeling process results in a continuous-time nonlinear model inwhich f and h denote the state update and output functions, respectively:

x(t) = f(x, u, v), y(t) = h(x, u, v) (1.1)

where y represent the controlled variables, the array u represents the ac-tuator setpoints to be computed by the controller, and v represents the ex-ogenous inputs to the system. The content of these variables is problem spe-cific and configurable. The controlled variables y may include some subset ofboost pressure, compressor air flow, EGR flow, turbocharger speed, engine-out NOx, exhaust temperature, etc. The actuators u may include variablegeometry turbine (VGT) vanes, variable (cylinder) valve actuators (VVA),EGR valve, intake or exhaust throttles. The exogenous inputs v will typicallyinclude engine speed and fuel injection quantity as a baseline and may be fur-ther refined by coolant temperature, and ambient pressure and temperature,etc.

The following section will use the nonlinear model (1.1) as input to an MPCdesign and will discuss the simplifications required for practical controllersynthesis.

14 Stewart et al.

0.18 0.19 0.2 0.21 0.22 0.23

0.16

0.18

0.2

0.22

0.24

0.26

0.28

data

mod

el

Oxygen Concentration Intake Manifold

305 310 315 320 325 330 335 340260

280

300

320

340

360

380

400

data

mod

el

Temperature Post−Charge Air Cooler

1 1.5 2 2.5 3 3.5 4

x 105

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

data

mod

el

Intake Manifold Pressure

4000 6000 8000 10000 12000 140000.4

0.6

0.8

1

1.2

1.4

1.6x 10

4

data

mod

el

High−Pressure Turbocharger Speed

Fig. 1.6 Representative accuracy plots obtained from modeling a two-stage turbochargedengine with the proposed medium-fidelity (7-11 dynamic states in this example) nonlinearcontrol oriented model. Steady-state data from the full range of engine speeds and loads isrepresented. The blue circles indicate the model results when each component has been fitlocally and the red squares illustrate the improved model results following the developedoptimization-based simultaneous fitting of all model parameters. The straight lines indicateerror bounds of 5%, 10%, and 15% with respect to the measured engine data.

1.4 Model Predictive Control and ComputationalComplexity

1.4.1 Explicit Predictive Control

We consider a piecewise affine (PWA) discrete-time approximation of thesystem dynamics (1.1)


x(k + 1) = Aσx(k) + Bσu(k) + Bvσv(k) + Bw

σ w(k) + fσ

y(k) = Cσx(k) + Cvσv(k) + Cw

σ w(k) + gσ

for[

xuvw

](k) ∈ Cσ

(1.2)

where x(k), u(k), y(k), v(k), w(k) are the state, input, output, measuredand unmeasured disturbances, respectively at time kTs where Ts is the sam-pling time, fσ and gσ are constant vectors. The natural number σ(k) ∈{1, 2, . . . ,M} is the operating point at time kTs and it is a function of in-puts u(k), states x(k) and disturbances v(k). The set {Cσ}M

i=1 is a polyhedralpartition of the state, input and measured disturbance set. System (1.2) issubject to the following time varying constraint on inputs and outputs for allk ≥ 0.

u(k) ∈ U(umin(k), umax(k)), y(k) ∈ Y(ymin(k), ymax(k)) (1.3)

where U(umin(k), umax(k)) and Y(ymin(k), ymax(k)) are polyhedra for allk ≥ 0.

Consider the problem of letting the output of system (1.2) track a givenreference yref,k while satisfying input and output constraints (1.3). Assumethat estimates/measurements of the state x(k) and disturbances v(k) areavailable at the current time k and consider the following cost function

Jk(x, v, w, ∆U, ε) := ‖yHp+k − yref,Hp+k)||P2 ++

∑k+Hp−1t=k ‖(yt − yref,t)‖Q

2 + ‖δut‖R2 + ρε2

(1.4)

where ‖v‖M2 = v′Mv. Then, the finite time optimal control problem is solved

at time k,

16 Stewart et al.

min∆UHc ,ε

Jk(x(k), v(k), w(k),∆UHc , ε) (1.5a)

s.t. xt+1 = Aσxt + Bσut + Bvσvt + Bw

σ wt + fσ

wt+1 = Awσ wt + Bu

σut

yt = Cσxt + Cvσvt + Cw

σ wt + gσ (1.5b)

if[

xuvw

]

k

∈ Cσ (1.5c)

t = k, . . . , k + Hp − 1ut = δut + ut−1

ut ∈ U(umin(k), umax(k)), (1.5d)yt ∈ Y(ymin(k), ymax(k))⊕ ε (1.5e)

t = k, . . . , k + Hc − 1, ε > 0xk+Hp

∈ Xf (1.5f)δut = 0, t = k + Hc, . . . , k + Hp − 1 (1.5g)vt = v(t− 1), t = k + 1, . . . , k + Hp (1.5h)uk−1 = u(k − 1) (1.5i)xk = x(k), wk = w(k), vk = v(k) (1.5j)

where the column vector ∆UHc := [δu′k, . . . , δu′Hc−1]′ and ε are the optimiza-

tion vectors, Hp and Hc denote the output prediction horizon and the controlhorizon and Xf is the terminal region. Note that the subscript notation is usedto distinguish between the variables of the optimization problem (1.4)-(1.5)and the state, input, disturbances and outputs of the system model (1.2).

Let ∆U∗Hc

= {δu∗k, . . . , δu∗k+Hc−1} and ε∗ be the optimal solution of (1.4)-(1.5) at time k. Then, the first sample of U∗

Hc(obtained from ∆U∗

Hcand

u(k − 1)) is applied to the system:

u(k) = u∗k. (1.6)

The optimization (1.4)-(1.5) is repeated at time k + 1, based on the newstate xk+1 = x(k + 1), measured disturbances vk+1 = v(k + 1), additiveunmeasured disturbances wk+1 = w(k + 1), input and output constraints,yielding a moving or receding horizon control strategy. In (1.4) we assumethat Q = Q′ º 0, R = R′ Â 0, P º 0.

In problem (1.4)-(1.5) the following assumptions are used

A1 Hp > Hc and the control signal is assumed constant for all Hc ≤ k ≤ Hp.This allows the reduction of the computational complexity of the MPCscheme.

A2 The exogenous disturbance v is assumed constant over the horizon. If PWAprediction models for v(k) are available they could be included in the MPCformulation (1.4)-(1.5).

A3 The region σ is constant over the horizon (1.5c).


A4 Soft Constraints on outputs, i.e. Y(ymin(k), ymax(k)) ⊕ ε := {y|y + ε ∈Y(ymin(k), ymax(k))}.

Remark 1. Assumption (A3) basically implies that for any given time wesimply implement a linear MPC for one member of the set of linear sys-tems. Ideally the assumption should be removed in order to predict switchesbetween affine dynamics over the horizon Hp. This would improve both per-formance and attractivity region of the closed loop system. Nevertheless, wehave been forced to use assumption (A3) by the current limitations of au-tomotive ECUs. In fact, by removing (A3), problem (1.4)-(1.5) becomes amixed integer quadratic program (MIQP) whose explicit solution [7] requiresmore floating point operations for its evaluations and more memory for itsstorage. ¤

The optimization problem (1.4)-(1.5) can be recast as a quadratic program(QP) .

min∆UHc ,ε

12∆UHc

′Hσ∆UHc+ Hε,σε2 + φ(k)′Fσ∆UHc

subj. to Gσ∆UHc ≤ Wσ + Eσφ(k)(1.7)

where

φ(k) := [x(k) u(k − 1) v(k) w(k) yref,k · · ·· · · yref,k+Hp umin(k) umax(k) ymin(k) ymax(k)]

and φ(k) ∈ Rnp . Problem (1.7) is a multiparametric quadratic program thatcan be solved by using the algorithm presented in [5]. Once the multipara-metric problem (1.7) has been solved, the solution ∆U∗

Hc= ∆U∗

Hc(φ(k)) of

problem (1.4)-(1.5) and therefore u∗(k) = u∗(x(k)) is available explicitly as afunction of the set of parameters φ(k) for all φ(k) ∈ X0. X0 ⊆ Rnp is the setof initial parameters φ(0) for which the optimal control problem (1.4)-(1.5)is feasible. The following result [5] establishes the analytical properties of theoptimal control law and of the value function.

Theorem 1. [5] The control law δu∗(k) = fσ(φ(k)), fσ : Rnp → Rm, ob-tained as a solution of (1.7) is continuous and piecewise affine on polyhedra

fσ(φ) = F iσφ + gi

σ if φ ∈ CRiσ, i = 1, . . . , Nr

σ (1.8)

where the polyhedral sets CRiσ = {φ ∈ Rnp |Hi

σφ ≤ Kiσ}, i = 1, . . . , Nr are a

partition of the feasible polyhedron X0.

As discussed in [5] the implicit form (1.7) and the explicit form (1.8) areequal, and therefore the stability, feasibility, and performance properties areautomatically inherited by the piecewise affine control law (1.8). Clearly, theexplicit form (1.8) has the advantage of being easier to implement, M lookuptables, one for each operating point σ, are uploaded on the ECU and ateach time step k the MPC resorts to selecting the current operating point

18 Stewart et al.

σ, searching for the region CRjσ containing the current vectors of parameters

φ(k) and implementing the corresponding controller F jσφ(k) + gj

σ.The following lists some of the major practical issues which have been

encountered while implementing the proposed MPC on an automotive ECU.In this chapter we will focus on only the final listed issue.

• State estimation. Estimation of the state x(k) and w(k) of system (1.2)is a nontrivial task. We have used a bank of M Kalman filters which runin parallel in order to smoothen the estimation during switch betweendifferent operating points.

• Time-varying actuator and state constraints. The proposed formulationhas been designed to address constraints that vary arbitrarily as a functionof time and thus independently of the state variables or gain schedulingparameters.

• Constraint satisfaction under steady state disturbances. For a range ofsteady-state exogenous disturbance v(k) = v the reference yref might beinfeasible for the given input and output constraints and thus not be track-able. At steady-state, the objective function will be composed of two termswith conflicting objectives: satisfy the constraints (ρε2) and track yref . Inaddition, model uncertainty and high ρ weights (usually used to strictlyenforce soft constraints) can lead to oscillating behavior and poor perfor-mance. A possible solution to this problem based on computing regions ofattraction and switching tuning has been given in [29]

• Limited ECU memory. We had to modify the explicit implementation inorder to be able to run the MPC (1.4)-(1.6) in an industrial ECU (even forshort horizons). We have used the Karush-Kuhn-Tucker conditions whichlead to (1.8) in order to reduce the memory required for storing Fσ andgσ in (1.8). More details can be found in [8].

In the next section we will provide more details on the last steps. The inter-ested reader is refereed to [29] for a thorough treatment of the aforementionedtopics.

1.4.2 On the Complexity of Explicit MPC ControlLaws

The efficient solution of the optimization problem (1.7) depends on the prob-lem properties and on the hardware platform. In [9, 10] we have comparedthe computational time and storage demand associated to (i) active-set QPsfor solving (1.7) and to (ii) the evaluation of the explicit solution (1.8). Inparticular we have shown that there might be alternative ways for solvingthe optimization problem (1.7) which are more efficient than evaluating theexplicit linear MPC (1.8). However, the corresponding state-feedback con-trollers do not have the nice piecewise affine closed-form solution as the ex-


plicit solution presented in [5, 4]. Next we provide a brief overview of themain results and refer the reader to [9, 10] for a more formal and detaileddiscussion.

The evaluation of the explicit solution (1.8) in its simplest form would re-quire: (i) the storage of the list of polyhedral regions and of the correspondingaffine control laws, (ii) a sequential search through the list of polyhedra forthe i-th polyhedron that contains the current state in order to implementthe i-th control law. Since verifying if a point φ belongs to a critical regionmeans to verify primal and dual conditions, then the on-line search for thepolyhedron containing φ can be compared to the main steps of a QP solver.In fact in [9] we have shown that an active set QP solver requires more opera-tions at each iteration than an explicit solver. This is obtained at the price ofincreased memory requirement. In fact, for the evaluation of (1.8) the poly-hedral partition and the gains have to be stored which, in general, largelysurpass the memory required for an active set QP (simply the matrices ofthe QP (1.9)).

The main idea behind alternative approaches can be simply explained asfollows. Rewrite the optimization problem (1.7) for a fixed σ compactly as [5]

minz

12z′Hz

subj. to Gzz ≤ bz(φ)(1.9)

where z is the optimization variable, and bz(φ) is an affine function of φ.Let I be the set of constraint indices. Consider a subset A of the constraintsindex A ⊆ I. Given a matrix M , MA denotes the submatrix of M consistingof the rows indexed by A. Consider the solution z∗(φ) and λ∗(φ) when theset A is active at the optimum [9]:

z∗ = H−1z G′z,A(Gz,AH−1G′z,A)−1bz,A(φ) = TAbA(φ)

λ∗ = −(Gz,AH−1z G′z,A)−1bz,A(φ) = SAbA(φ) (1.10)

The alternative class of algorithms presented in [9, 10], computes and storesthe matrices SA and TA off-line for all optimal sets of active constraints A.Then, the online steps are: (1) compute bz(φ), verify duality conditions bycomputing λ∗ from (1.10), compute the optimizer candidate z∗ by using (1.10)and verify primal feasibility conditions Gzz

∗ ≤ bz(φ).In conclusion, one can identify three classes of algorithms as illustrated

in Figure 1.7: the explicit solvers (upper-left in the figure), the QP solvers(lower-right in the figure) and hybrid solvers which trade-off memory andcomputation in a different way and which, for certain classes of problems,can be more efficient than both QP solvers and explicit solvers [9, 10].

Remark 2. We remark that there exist other very efficient approaches ap-peared in the literature for solving predictive control problems for linear andPWA systems [18, 3, 6, 24, 13]. The comparison of our approach to otherapproaches would be problem dependent and requires the simultaneous anal-

20 Stewart et al.

Mem

ory

Number of Flops per Iteration

Active-Set MPC Solvers

Explicit MPC Solvers

Hybrid MPC Solvers

Fig. 1.7 Comparison of various algorithms for solving QP (1.9).

ysis of several issues such as speed of computation, storage demand and realtime code verifiability. This is an involved study and as such is outside of thescope of this chapter. We refer the reader to [1] for a good review of explicitpredictive control.

1.5 Summary and Conclusions

The demand for systematic and efficient techniques for the development,calibration, and deployment of control algorithms is undisputed in modernindustrial engine design. The key issues are the achieved closed-loop perfor-mance and also the time, effort, and expense that are required to achieveit.

Within the context of a standard industrial engine control developmentprocess we surveyed the set of control requirements at a high level, including:

• to maintain performance of the family of highly nonlinear and multivari-able engines that lie within the uncertainty bounds of a production fleet,

• to integrate into the hierarchical control structure (including time varyingconstraints),

• to fit within the tight processor and memory requirements for implemen-tation in an ECU, and,

• to provide a systematic, efficient development process to facilitate softwaredevelopment and enable a reduced calibration burden.


With respect to these requirements we discussed our recent work in at-tempting to address this set of requirements in the form of a systematicprocess and set of software tools that allow the user to configure a model us-ing components from a library, to automatically and robustly fit this modelto engine and component data, to use this model in the synthesis of both thefeedforward and feedback components of a multivariable control strategy. Weselected model predictive control (MPC) as the underlying technology dueto its ability to address the multivariable interactions among subsystems, itsability to be integrated into the existing control software hierarchy foundin industrial electronic control units (ECUs) - including its straightforwardaccommodation of time varying input and output constraints, and its gen-eral algorithmic structure which can cover many control problems withoutrequiring software structure changes.

We next overviewed some of the issues involved when considering imple-menting MPC for a nonlinear plant within the limited resources of a modernECU, particularly paying attention to balancing the tradeoff between memoryand processor usage. We presented examples of the overviewed control beingused to control the air handling on two very different engine applications.

Acknowledgements

This work would not have been possible without the support of Honeywell.

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toward a systematic design for turbocharged engine control

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