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Tuning of multivariable model predictive controllersthrough expert bandit feedback

Alex. S. Ira, Chris Manzie, Iman Shames, Robert Chin, Dragan Nesic, Hayato Nakada, Takeshi Sano

Abstract—For certain industrial control applications an ex-plicit function capturing the nontrivial trade-off between com-peting objectives in closed loop performance is not available.In such scenarios it is common practice to use the humaninnate ability to implicitly learn such a relationship and manuallytune the corresponding controller to achieve the desirable closedloop performance. This approach has its deficiencies because ofindividual variations due to experience levels and preferencesin the absence of an explicit calibration metric. Moreover, asthe complexity of the underlying system and/or the controllerincrease, in the effort to achieve better performance, so does thetuning time and the associated tuning cost. To reduce the overalltuning cost, a tuning framework is proposed herein, whereby asupervised machine learning is used to extract the human-learnedcost function and an optimization algorithm that can efficientlydeal with a large number of variables, is used for optimizing theextracted cost function. Given the interest in the implementationacross many industrial domains and the associated high degreeof freedom present in the corresponding tuning process, a ModelPredictive Controller applied to air path control in a diesel engineis tuned for the purpose of demonstrating the potential of theframework.

I. INTRODUCTION

From a high level perspective, controller tuning involvesmodifying the gains to improve the closed loop performanceof a system. For controllers with low degrees of freedom, suchas PI loops that have been commonplace in industrial applica-tions, this approach has typically involved online experimentsand a human expert with sufficient experience to evaluateand then improve the quality of the response. However, withincreasing degrees of freedom in modern controllers and thecomplexity in many systems, faster and more efficient methodsof controller tuning are being sought to reduce the time burden.To utilize a digital twin of the system in an initial offlinecalibration, two linked problems can be addressed. The firstof these relates to interpreting the goal of the tuning fromthe perspective of human experts. This is often not explicitlyknown, yet is instrumental in forming an optimization problemthat can be tackled using an offline approach. The secondproblem involves posing an efficient solution to the resulting

Alex. S. Ira, Chris Manzie, Iman Shames, Robert Chin, Dragan Nesic,are with the Department of Electrical and Electronic Engineering atthe University of Melbourne, Parkville, Australia. Robert Chin is alsoaffiliated with the University of Birmingham, United Kingdom. E-mails:{alex.ira, manziec, ishames, dnesic}@unimelb.edu.au,chinr@student.unimelb.edu.au

Hayato Nakada and Takeshi Sano are with Advanced Unit ManagementSystem Development Division, Toyota Motor Corporation, Higashi-Fuji Tech-nical Center, 1200, Mishuku, Susono-city, Shizuoka, 410-1193. E-mails:{hayato_nakada, takeshi_sano_aa}@mail.toyota.co.jp

optimization problem, which may involve high numbers ofparameters, constraints and have non-unique solutions.

Because of the existence of human experts who, for a givendata point (e.g., a pair of closed loop output trajectories) canprovide a label (e.g., a quantitative or qualitative evaluation),supervised machine learning (SML) is propsoed to addressthe first problem. Using human labeled data to learn a rankfunction (Xia, 2019) has been extensively used in addressingthe information retrieval (IR) problem. Some applications aresearch engines, review and recommendation sites and servicessuch as Netflix watch recommendation and Amazon buyrecommendations. Inverse Reinforcement Learning (IRL) (Nget al., 2000), can be thought of as an approach of extractinga (human) reward function for sequence of actions. Sincethe labelling consists of only one action, IRL is conceptuallyequivalent to SML.

Although similar challenges exist for any modern controlarchitecture, in this work the focus is on the tuning of modelpredictive control (MPC) controllers. The reason for thischoice is the growing prevalence of MPC in many industrialdomains (Samad, 2017), along with the challenges it presentsfor traditional calibrators by having a large number of variablesthat are not directly linked to the closed loop system response.

Early approaches for the calibration of MPCs includedtuning rules-of-thumb and general guidelines documentedin (Garriga and Soroush, 2010; Morari and Lee, 1999; Qin andBadgwell, 2003; Rani and Unbehauen, 1997). However, thesefocused on only a few parameters of the MPC. Replacement ofthe existing controllers with MPC counterparts was consideredin (Maciejowski, 2007). There, the guidelines on how toreverse-engineer the existing controller for the MPC designare provided. This served as a useful method for a rapidintroduction of MPC, although its performance was limitedto that achievable with the original controller.

More recently, metaheuristic optimizers, such as particleswarms in (Junior et al., 2014), genetic algorithms in (Van derLee et al., 2008) and a gradient-descent algorithm in (Buninet al., 2012) have been considered. There are also metaheuris-tic off-the-shelf methods of goal attainment in (Exadaktylosand Taylor, 2010) and (Vega et al., 2008), and analytical ap-proaches (employing appropriate simplifications) in (Bagheriand Khaki-Sedigh, 2014; Shah and Engell, 2010; Shridharand Cooper, 1998). The potential shortcoming of the aboveapproaches is they are not necessarily tailored for the optimiza-tion problem resulting from the proposed tuning framework.Thus, they may result in slower convergence properties orlower performance when included in an offline optimizationroutine involving a high-fidelity plant model. This becomes

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particularly problematic as the controller complexity increases,and to date we are unaware of any of the approaches beingadopted in an industrial context.

To address the potential shortcomings of existing MPCtuning algorithms not exploiting the full potential of themethodology and the issues with scalability for more genericapproaches, in this paper we propose a framework to effec-tively tune advanced multivariable controllers such as MPC.In a similar vein to (Nguyen et al., 2017), a simulated humanresponse to a closed loop output is generated, although hereinformed initially by real experts. The simulated responsescan then be used within bespoke optimisers to efficientlytune the controller gains and deliver near-optimal closed loopresponses. To demonstrate the proposed approach, the appli-cation considered here is a diesel engine air path control. Theobjective is to ensure multiple outputs track separate referencetrajectories while satisfying the corresponding constraints. Theexperimental results are promising and provide incentive forfurther research and development.

The reminder of the paper is organized as follows. Section IIpresents the two main challenges associated with the proposedtuning framework. In Section III, the proposed framework isdemonstrated by tuning an MPC controller which is used ina diesel engine air path control. The conclusions are providedin Section IV.

II. PROBLEM FORMULATION

Many industrial control applications achieve the desiredclosed loop performance via a tuning process. This involvesmany choices and tasks but essentially it is an optimizationproblem, solved by a human, see Fig. 1. This problem can be

Controller

Plant

Human

Controller parameter <latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Output of interest Y()<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Fig. 1. Controller tuning framework with a human in the loop. Dashed linesindicate a different time scale to a closed loop time scale.

abstracted as the following optimization problem

minimize Ch(Y(κ))

subject to κ ∈ K, (1)

where κ is a controller parameter, K is the set of admissiblecontrol parameters, Y(κ) is a closed loop output of interestwith respect to a controller with the parameter κ, and Ch(·) isan approximation of a cost function C(·).

The cost function C(·) captures a cost-benefit relationshipbetween multiple, often competing, objectives. Unfortunately,more often than not, it is not available in an explicit form.Therefore, a common approach is to approximate it by exploit-ing the innate human ability to recognize and learn patterns,resulting in Ch(·).

Even though humans are good at obtaining Ch(·), they arenot necessarily efficient at solving (1), particularly when there

is not an explicit relationship between the tuning variables andCh(·). A more efficient alternative might be to use a machine.

One approach to accomplish this is to first extract thehuman learned cost function Ch(·), resulting in Ch(·). Giventhe availability of a human who can label1 the correspondingclosed loop performance, SML may be used (Abu-Mostafaet al., 2012).

Once Ch(·) is obtained, the tuning problem (1) is approxi-mated by

minimize Ch(F(Y(κ)))

subject to κ ∈ K,(2)

where F(·) is a feature extractor function which transformsan output of interest (time signal) into a vector consisting ofrelevant features, see Fig 2.

Controller

Plant

Optimisation algorithm

Y()<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

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Featureextraction

SMLF(Y())

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Ch(F(Y()))<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Fig. 2. Controller tuning framework with a machine in the loop. Dashed linesindicate a different time scale to a closed loop time scale.

There are two major steps in the transition from a tuningframework with a human-in-the-loop (as in Fig. 1) to theproposed automated tuning framework (as in Fig. 2): (i)estimation of a human learned cost function, Ch(·), and (ii)selection of an optimization algorithm for solving (2).

A. Estimation of a human learned cost function

In this section SML is proposed to estimate the implicitcost function of an experienced calibrator using an appropri-ate regressor, for which there exist already well establishedtools (Abu-Mostafa et al., 2012).

One potentially time consuming aspect of the estimationof a human cost function is to collect the corresponding dataused for training, development (validation) and testing. In thiswork the data set is defined as D := {(Y(i)(κ),L(i)) =

(Y,L)(i)|i ∈ {1, . . . , N}, N ∈ N,Y : I → RpI ,L ∈ R≥0},where superscript (i) refers to the i-th data point, N is thenumber of data points, I is the relevant time interval ofthe output of interest, p ∈ N and Li is the correspondingcontinuous label (or grading) assigned by the human for thegiven Yi .

One of the challenges in obtaining D is related to the costassociated with labelling. Namely, the high cost may lead tosmall N which further may result in a poor estimate of thehuman learned cost function. In some cases, one might beable to increase N by using data synthesis together with activelearning (Settles, 2012).

Another challenge is related to the extraction of the featuresof the output of interest Y as it is not always clear whichfeatures to use. Depending on the control application and the

1Quantitative and/or qualitative evaluation.

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experience of the human expert, s/he may be able to providean insight into which ones to use. Otherwise, one might use anappropriate statistical approach, such as principal componentanalysis, for selecting features.

Let F : RpI → Rl, l ∈ N be a feature extractor functionresulting from human expert providing an insight or someappropriate statistical analysis. Then, the raw data set D canbe transformed into

DF := {(F(Y),L)(i)|i ∈ {1, . . . , N}} (3)

which is used for training, development and testing.Given a regressor the challenge is to obtain the correspond-

ing data set DF , which is dealt with on a case by case basis,as will be illustrated in Section III.

B. Gradient-free optimization

The final step in the proposed tuning framework of Fig. 2is to select or develop a bespoke optimization algorithm.

As mentioned in Section I, one objective of the proposedframework is to efficiently deal with a large number ofcontroller parameters (e.g., optimization variables). To thatend, optimization approaches requiring no gradients or higherorder derivatives, such as in (Kolda et al., 2003; Torczon,1997), are required.

Further, in some cases (as demonstrated in the case studybelow) an approach which can efficiently accommodate con-straints is necessary. This may require enforcement of certainrestrictions on a problem so it becomes amenable to a certainfamily of algorithms.

The following case study is used to illustrate one possibleimplementation of the high level framework and demonstratethe potential benefits of this approach relative to the existingtuning methodologies.

III. CASE STUDY

In what follows, the proposed tuning framework is appliedto the problem of air path control in a diesel engine usingMPC. This is a particularly relevant example, as the potentialfor advanced control implementations in the diesel air path toimprove efficiency have been widely reported in the researchliterature, yet the industry uptake of these controllers hasbeen slow due to the time and cost of using existing tuningmethods on the new controllers. This is coupled with the issueof the non-explicit relationship between tuning parameters inalgorithms like MPC and their time domain outputs, alongwith the limited exposure of many engine calibrators to thesecontrol methods.

A. Engine air path model

An illustration of a simplified diesel engine air path is givenin Fig. 3. Modern (automotive) diesel engines improve the fueleconomy by using the variable geometry turbine (VGT), whilethey decrease the nitrogen oxide (NOx) emissions by usingthe exhaust gas recirculation (EGR) (Huang et al., 2016). Thedynamics of the air path is manipulated via throttle, EGR valveand VGT vane. In particular, the compressor, followed by the

intercooler, increases the density of the fresh air entering theair path. Then, the EGR system is used to cool and recirculatepart of the burnt gas in the exhaust manifold. Finally, thepart of the burnt gas is used to drive VGT which spins thecompressor.

Throttle

Intercooler

EGR cooler

EGR valve

Compressor

Air

Air-flow sensor

Variable Geometry Turbine (VGT)

Exhaust manifold

Intake manifold

Fuel rail and injectors

Fig. 3. Illustration of a diesel engine air path.

A diesel engine air path is typically modeled as acontinuous-time system2. Although slight variations on themodel structure exist in (Broomhead et al., 2015; Wahlstromand Eriksson, 2010, 2011), they can all be captured in thefollowing general form

x = f(x, u, θ), (4a)y = h(x, u, θ), (4b)

where x ∈ X ⊆ Rn is the state, u ∈ U ⊆ Rm is the input,θ ∈ Θ ⊆ Rb is the parameter, y ∈ Y ⊆ Rp is the output,mappings f : X × U ×Θ→ X and h : X × U ×Θ→ Y arenonlinear, and n,m, b, p ∈ N.

A four-state model is adopted here (Huang et al., 2016;Sankar et al., 2019). The model consists of

x = (pin, pex,Wcomp, FEGR), (5a)θ = (w,wfuel), (5b)u = (uthr, uEGR, uV GT ), (5c)y = (pin, FEGR), (5d)

where pin is the pressure in the intake manifold; pex is thepressure in the exhaust manifold; Wcomp is the compressormass flow rate; FEGR is EGR rate; w is the engine speed;wfuel is the volumetric fuel-rate; uthr is the throttle position;uEGR is the EGR valve position; uV GT is the setting of VGT.Note that full state feedback is available either via direct sensormeasurements or it can be reliably estimated within the enginecontrol unit (ECU). It is possible to incorporate more statesin order to improve the accuracy of the model predictions butthe cost for doing that is the increased online computationalload and the requirement for a state estimation.

One approach to handling the nonlinearities in the dieselair path is to use a switched linear-time-invariant (LTI) modelto represent the engine. In such an approach, the engineoperating range Θ is divided into Nr subregions, with one

2Note that this is a shorthand notation, for relevant details see AppendixA.

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possible division of 12 regions illustrated in Fig. 4, whereθ]l ∈ Θ] := {θ ∈ Θ|∀w ∈ {wL, wM , wH}, ∀wfuel ∈{wLfuel, wML

fuel, wMHfuel, w

Hfuel}}, ∀l ∈ {1, 2, . . . , 12}.

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Fuel

rate

wf

uel[m

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wHfu

el

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fuel

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fuel

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Fig. 4. Illustration of a parameter space Θ divided into 12 regions withrespect to to 12 evenly spaced operating points.

Each region is approximated with a linear model aroundthe corresponding operating point. In particular, considerθ]l = θ] ∈ Θ], l ∈ {1, 2, . . . , Nr}. Then, the correspondingdiscrete-time linear approximation is denoted as

x(k + 1) = Alx(k) +Blu(k), (6a)y(k) = Clx(k) +Dlu(k), (6b)

where x, u and y are the corresponding perturbed quantities(see (13)) and Al, Bl, Cl and Dl are the corresponding matri-ces for the active region; see Appendix A for the details.

Diesel engine control is challenging due to the presence ofmutliple competing objectives and constraints. The overarch-ing problem requires attaining good drivability and fuel econ-omy whilst ensuring that the legislated emissions constraint ismet (along with other constraints on states and inputs). Thisrequires close tracking of boost pressure and EGR.

For a given engine operating condition θ, the correspondingreference r(θ) ∈ R2 (see (12)) corresponds to the “optimal”driver demand responsiveness and fuel efficiency, and the satis-faction of the emission regulations. These optimal trajectoriesare often unable to be tracked exactly, and the calibrationproblem involves a tradeoff when attempting to track bothsets of references. An experienced calibrator is well-versed inmanaging this tradeoff at different engine operating points.

B. Control architecture

In the switched-LTI approach, there are Nr MPC controllerswith the active controller dependent on the value of θ. TheMPC cost function is quadratic and it is denoted as

J(x(0),{u}

)= ||x(N)||P +

N−1∑i=0

(||x(i)||Q+ ||u(i)||R

), (7)

where the tuning parameters of interest consist of P ∈ R4×4,Q ∈ R4×4 and R ∈ R3×3, while ||v||M := v>Mv, M ∈

{P,Q,R}. Note that although the calibration problem con-siders the closed loop response of only two states on a highfidelity model, the MPC (for stability reasons) must containa positive definite Q and so considers four states in an openloop optimization on a reduced order model.

The corresponding MPC optimization problem is defined as

minimize J(x(0), {u}

)subject to

(6a),

x(0) = x(kTs)− x?r(θ]),x ∈ X ,u ∈ U ,

(8)

where x(kTs) is the sampled state of system (4) at time t =kTs, x?r(θ

]) is “optimal” steady state value with respect toa given output reference r(θ]), and X and U represent therelevant state and input constraints, respectively. The solutionof the optimization problem (8) is a sequence of optimallypredicted control values over the horizon N , that is

{u∗} := {u∗(0), u∗(1), . . . , u∗(N − 1)}. (9)

Only the first element of the sequence is applied to the engineat each time step.

For more details on the MPC problem formulation referto Appendix A and Appendix B, and for details on theparameters used in the case study problem here (includinga discussion on state and input constraints and predictionand control horizon lengths) interested readers are referred to(Shekhar et al., 2017).

C. Automated tuning framework

As discussed above, the first step towards the transitionfrom a tuning framework with a human in the loop to anautomated tuning framework is to identify the human learnedcost function Ch. For the purposes of this paper, the followingassumption is made.

Assumption 1 (Human learned cost function). The humanlearned cost function Ch is invariant to θ ∈ Θ. �

This assumption is limiting in the context of industrialapplication, in that different priorities may arise at differentengine operating points. For instance, the calibrator maypreference greater torque in certain subregions, however thisis dealt with simply by increasing the training dataset.

1) Data collection and training stage: The output trajectoryof interest, Y(κ), is (6b) over the time interval I and in thiscase study consists of the pressure in the intake manifold pin(or boost pressure) and the EGR rate FEGR.

The human expert uses plots like in Fig. 5 to evaluatethe behavior of the corresponding outputs of interest, whichcan be considered to represent the performance of the chosencontroller parameters. In order to extract the correspondingcost function the first step is necessary to generate a setof output trajectory pairs and query for its labels. This stepcan be achieved either by using closed loop responses, or byartificially generating trajectories. The advantage of the latter

5

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r1(✓])

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r2(✓])

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Fig. 5. A typical pair of normalized boost pressure (y1) and normalised EGRrate (y2) responses with respect to the step reference r1(θ]) and r2(θ]),respectively; T is the duration of a reference signal.

approach is the trajectories can explore a more specific partof the operational space at significantly lower computationaleffort, but not all trajectories may be achievable. This is notnecessarily a significant issue as the key information beingextracted is the labels of the trajectory pairs.

For the specific case study used here, the undershoot,overshoot, settling time and steady state error, for both boostpressure and EGR rate represent possible features of interestand are used to populate the Yi data. Each collected stepresponse is a finite array of numerical values. Extracting thefeatures of interest (undershoot, overshoot, settling time andsteady state error) is done with a bespoke algorithm whichoutputs these values from the collected step response usingthe standard definitions (e.g. the step response has settledto within 1% of the steady state value). To allow for noisymeasurements, a small tolerance band is applied to all thefeatures (e.g. noise on the step response causing a temporaryexcursion from the 1% boundary does not invalidate thecalculation). The data set, D in (3), can then be populatedfrom a number of trajectories with the extracted features asdetailed above and their associated grading by the calibrator.

For this specific implementation of the framework, a mul-tilayer feedforward neural network is used to approximatethe relationship between the features and the labels providedby the human expert, Ch. The developed network has twohidden layers, with 24 and 8 nodes in the first and secondlayers respectively. This choice of regressor typically requiresat least 10 times as many data pairs as there are weightsin the network (Abu-Mostafa et al., 2012), and so relevanttrajectory generation is crucial in order to not overburdenthe human expert. Active learning approaches (Settles, 2012)may be useful in maximising the new information content inthe trajectories presented to the human for labelling, and canbe tailored to only present features in a relevant region ofoperation.

To avoid issues with the optimisation problem later, thefunction Ch should have good coverage of the entire featureset, however this is at odds with attempts to minimise thetrajectories considered by the human to those that are likelyto be acceptable. One method of dealing with this issue isto synthesise labels for features that are clearly unacceptableand use this additional data to supplement the rated responses.We found that modelling a human expert who monotonicallypenalizes the trajectories as the features become more unde-sirable as a way of synthesising the required data (e.g. as the

overshoot increases, the synthesised label decreases).2) Tuning stage: As discussed earlier, the engine operating

range Θ is divided into Nr regions, each approximated witha separate linear model (6) and each governed with thecorresponding MPC controller (7) and (8).

For notational simplicity, in the subsequent analysis wereplace θ]l with θ] and intend that the relevant index l isalways considered. Further, by Assumption 1, the same Chis used in all Nr optimization problems. The tuning problemthen amounts to:

minimize Ch(fy(Pθ] , Qθ] , Rθ]))

subject to

Pθ] = P>θ] , Pθ] � 0,

Qθ] = Q>θ] , Qθ] � 0,

Rθ] = R>θ] , Rθ] � 0,

(10)

where symmetric constraints3 lower the number of parametersfrom 46 to 26, while definiteness constraints are due to stabil-ity reasons, (Rawlings and Mayne, 2009). The tuning stage forregion l ∈ {1, 2, . . . , Nr} is performed in a simulation settingusing the digital twin as illustrated in Fig. 6.

MPC

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Optimiser(gradient-

free)

“Optimal”steady states

(2)

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r1(✓])

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r2(✓])

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NNfy

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DA

Fig. 6. Block diagram illustration of tuning MPC controller with respect toregion l ∈ {1, 2, . . . , 12}. DA stands for digital-to-analog.

The chosen optimizer utilizes an algorithm for randomsearch for nonsmooth and stochastic optimization, (Nesterovand Spokoiny, 2011, Section 4). This algorithm considers con-strained optimization problems, i.e., it does not need gradientor any higher derivative information. The former is importantbecause of constraints in (10) while the latter is important forpractical purposes. The tuning parameters are the elementsof P ∈ R4×4, Q ∈ R4×4 and R ∈ R3×3 matrices. Findingthe gradient or any other higher derivative information wouldresult in a considerable computational cost. Details regardingthe optimization algorithm used to solve (10) are provided inAppendix C.

3In addition to lowering the number of parameters, they are imposed dueto implementation reasons, (Ira et al., 2018).

6

The performance of the tuned switched-MPC architectureis then ready to be experimentally tested over the extra-urbandriving cycle (EUDC).

Remark 1. We note that although the implemented controlleris MPC, other advanced controllers with tuning gains replac-ing (Pθ] , Qθ] , Rθ]) can be substituted into (10). However,the constraints on the optimisation variables must be alteredfrom requiring positive semi-definite matrices as in (10) toappropriate conditions to guarantee stability under the chosencontroller architecture.

D. Experimental results

The experimental testing of the tuning efficacy of theproposed framework was performed at Toyota’s Higashi-Fuji Technical Center in Susono, Japan. The test bench isequipped with a diesel engine and a transient dynamometer.A dSPACE DS1006 real-time processor board is used toimplement the switched MPC architecture. A block diagram ofthe experimental configuration is shown in Fig. 7. The ECUlogs sensor data from the engine and transmits the currentstate information to the controller. Note that ECU directlycontrols all engine subsystems. However, its commands forthe three actuators, i.e., throttle, EGR valve and VGT, canbe overridden with the MPC commands by toggling theswitch (from the ControlDesk interface), see Fig. 7. Finally,depending on the current engine speed and fueling rate, i.e.,θ, the corresponding MPC controller is selected based on theswitched MPC architecture.

MPC

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“Optimal”steady states

12

Diesel engine

Transient dynamometer

Dynamometer controller

Engine speed setpoint

ECU

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Fig. 7. Experimental setup.

For real time implementation of the controller on thedSPACE platform, it is desirable to choose a quadratic program(QP) solver that runs on embedded hardware. To this end, first,the MPC is formulated using the condensed formulation (Jerezet al., 2011) (which reduces the number of decision variablesin the QP) while the choice of the solver is QPKWIK algo-rithm(Schmid and Biegler, 1994). This solver is implemented

in MATLAB’s MPC toolbox and it is used for both tuning (insimulation time) and testing (in real time).

The tracking performance of the tuned switched MPCarchitecture is tested over the EUDC. The results are displayedin Fig. 8.

Fig. 8. Tuned switched MPC architecture performance on EUDC.

The performance in Fig. 8 is promising but there aretracking errors on both trajectories. In what follows, twopossible error origins are presented and potential solutions arediscussed.

The first possible origin is insufficient fidelity of the plantmodel (4). Since the proposed tuning framework is imple-mented in the offline setting of Fig. 6, and so the correspondingresults depend on the accuracy of the approximation (4). Twopossible solutions are either to i) use a higher-fidelity modelwhich would increase the computational cost; or ii) implementthe tuning structure online which would amount to replacingmodel (4) in Fig. 6 with a real engine. Indeed, the latter mayappear appealing from an accuracy perspective but would incuran implementation cost as the time per optimisation step isincreased and requires human intervention. One might addressthis through mixing the approaches by initialising the onlinetuning with an offline optimised solution to reduce the numberof optimisation iterations requiring the plant in-the-loop. Thishas no strict guarantees on improvements but practically maybe beneficial.

The second potential origin for tracking errors is the amountof data collected or the number of features considered toapproximate the human cost function. Increasing the amountof data collected (as would be required through removingAssumption 1 for example) would address this problem butalso comes at a cost.

Both of these issues are of course fundamental to datascience and therefore not unexpected.

IV. CONCLUSIONS

A tuning framework, using machine learning to extract thehuman learned cost function, is implemented and experimen-tally tested. The results are promising but there are also areasfor improvement. The issues related to the approximation of

7

the human learned cost function can be addressed by collectingmore data. However, this increases the overall cost. The issuesrelated to the plant model can be addressed by using a higher-fidelity models or implementing the tuning framework in anonline setting.

Even though better results might necessitate more datawhich would lead to the implementation cost increase, itis important to note that extracting the human learned costfunction might be one-off investment. Namely, once the costfunction is extracted, the tuning reduces to solving an offlineoptimization problem that may be applicable across multipleengine models.

Finally, although the case studies presented here focuson MPC-based architectures, the high level framework isequally deployable to other existing and proposed algorithmicsolutions to closed loop control. We also note that othercombinations of optimiser and machine learning tools mayprovide superior performance than that depicted in the casestudy, although the same core elements as the algorithms usedherein should be retained.

APPENDIX

A. Approximation of system dynamics

Consider the following continuous-time nonlinear system

dx(t)

dt= f(x(t), u(t), θ(t)), (11a)

y(t) = h(x(t), u(t), θ(t)), (11b)

where x ∈ X ⊆ Rn is the state, u ∈ U ⊆ Rm is the input,θ ∈ Θ ⊆ Rb is the parameter and y ∈ Y ⊆ Rp is the outputof the system (11), t ∈ R≥0 ⊂ T is the continuous-time,n = 4,m = 3, b = 2 and p = 2 (see (5)).

Now, consider a scenario where system dynamics (4) is suchthat practically it is expensive to consider it for each θ ∈ Θ.In such scenarios, one can resort to a discretization of theparameter space Θ, and an approximation of (4) with respectto the resulting discrete values.

Discretized version of the parameter space Θ ⊆ R2 isdefined as Θ] := {θ = (θ1, θ2) ∈ Θ|∀θ1 ∈ Θ1, ∀θ2 ∈ Θ2},where θ1 = w, θ2 = wfuel, Θ1 = {wL, wM , wH} andΘ2 = {wLfuel, wML

fuel, wMHfuel, w

Hfuel}.

Now, consider a θ] ∈ Θ] and system (11), and let a givenreference value be denoted with r(θ]) ∈ Y , while the corre-sponding state and input with xr(θ]) and ur(θ]), respectively.Determining the pair (xr(θ

]), ur(θ])) consists of solving the

following system of equations subject to operational and inputconstraints:

0 = f(xr(θ]), ur(θ

]), θ]), (12a)

r(θ]) = h(xr(θ]), ur(θ

]), θ]). (12b)

yields a feedfoward mapping of input-reference pairs.u?r(θ

]) and x?r(θ]).

To improve the transient response, in this case study a MPCcontroller is used in conjunction with the feedforward control.At each time instant, MPC solves an optimization problem, itis desirable to reduce the complexity of the considered plant

dynamics (if possible) and to transform it from continuous-time to discrete-time. To that end, consider again a θ] ∈ Θ],system (11), and let r(θ]) ∈ Y be the given reference value.Next, define the following perturbed quantities

x(t, θ]) := x(t)− x?r(θ]), (13a)

u(t, θ]) := u(t)− u?r(θ]), (13b)

y(t, θ]) := y(t)− r(θ]). (13c)

(Note that in the sequel, when appropriate, for notationalconvenience, time t and parameter θ] are dropped out in (13).)After linearisation and discretisation of (11), one arrives at:

x(k + 1) = Ax(k) +Bu(k), (14a)y(k) = Cx(k) +Du(k). (14b)

B. MPC cost functionAt each time instant k, the MPC controller samples the state

of the system (11). This sampled state xk is used to initializethe plant model (14). The cost of using a sequence of controlvalues {u} := {u(k), u(k+1), . . . , u(k+N−1)} on the plantmodel (14), over a horizon N ∈ N, is defined as

J(x(0), {u}

):= ||x(N)||P +

N−1∑i=0

(||x(i)||Q + ||u(i)||R

),

where matrix P ∈ Rn×n, Q ∈ Rn×n and R ∈ Rm×m,||v||M := v>Mv, M ∈ {P,Q,R} and v is the relevant vector.

C. Optimization algorithmTo obtain a desirable behaviour of the closed loop system

using the controller of the preceding section, one needs to tuneparameters P, Q and R. In this section we define the conceptof optimisation using a random oracle as defined below forthe MPC tuning problem:

Definition 1 (Random Oracle). Let MPj ,M

Qj and MR

j berandom symmetric matrices of appropriate dimensions anddefinitenesses. Then,

gµ(Pj) :=((Ch(fy(Pj + µMP

j , Qj + µMQj , Rj + µMR

j ))

− Ch(fy(Pj , Qj , Rj)))MPj

)/µ

is a random oracle for Pj , Qj , and Rj . �

Recall that due to the stability requirements, matrices P,Qand R need to have a particular structure. In order to generate arandom matrix with such a structure, a random unitary matrix,constructed using a technique described in (Ozols, 2009), isused. The algorithm based on (Nesterov and Spokoiny, 2011,Section 4), applied to the present problem, consists of thefollowing steps,

Choose: P0, Q0, R0,

While: j ≤ N,N ∈ N0,

Compute: Pj+1 ← πP (Pj − hjB−1gµ(Pj)),

Qj+1 ← πQ(Qj − hjB−1gµ(Qj)),

Rj+1 ← πR(Rj − hjB−1gµ(Rj)),

j ← j + 1,

8

where hj > 0, while πP and πQ are projections on a positivesemidefinite cone and πR is a projection on a positive definitecone4.

Note that the approximated cost function Ch might consistof many local extrema. Thus, the performance of the usedalgorithm depends on the initial point (P0, Q0, R0). There-fore, a generation of a batch of initial points is performedand a promising candidate is chosen. The promising candi-date is defined as the initial point (P0, Q0, R0), such thatCh(fy(P0, Q0, R0)) achieves the smallest value with respectto the corresponding batch. In this work, µ is chosen to bevery small and hj is defined according to (42) in (Nesterovand Spokoiny, 2011). Specifically µ = 10−9 and hj =10−6/

√j + 1, j ∈ [1, 2. . . . , 50]. are used for all controllers

in 12 regions introduced in Fig. 4.

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