Lasso-MPC – Predictive Control with ℓ1-Regularised Least Squares

209 Pages·2016·6.713 MB·English

by  Marco Gallieri (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Marco Gallieri

ℓ -MPC—
asso
Predictive Control
with ℓ -Regularised
1
Least Squares
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Marco Gallieri
‘ —
-MPC Predictive
asso
‘
Control with -Regularised
1
Least Squares
Doctoral Thesis accepted by
the University of Cambridge, UK
123
Author Supervisor
Dr. MarcoGallieri Prof. Jan M.Maciejowski
Analytics University of Cambridge
McLaren AppliedTechnologies Cambridge
Woking UK
UK
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-3-319-27961-9 ISBN978-3-319-27963-3 (eBook)
DOI 10.1007/978-3-319-27963-3
LibraryofCongressControlNumber:2016934186
©SpringerInternationalPublishingSwitzerland2016
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To Mariya
’
Supervisor s Foreword
It frequently happens that systems have some redundancy of actuators. A given
behaviourofthesystemmightbeobtainedbyusingvariousmanipulatedvariables,
orvariouscombinationsofthem.Aircraftusuallyhaveconsiderableredundancyof
actuators, although there is a preferred and most efficient way of using them to
perform any specific manoeuvre. For example, increasing the thrust of low-slung
engines exerts a pitching torque on the airframe, as well as having the more
intendedeffectofacceleratingtheaircraft.Iftheconventionalwayofchangingthe
pitch angle is not available—due to a failure, for instance, then the engine thrust
could in principle be used for that purpose, and has been used in this way in a
number of well-documented incidents. More routinely, spoilers (air brakes) are
deployed asymmetrically on some aircraft to enhance the roll torque when flying
slowly(usuallywhenlanding)sincetheaileronshavealimitedeffectatlowspeeds.
Therollangleofashipcanbereducedbyusingrollstabilisers,whicharesmallfins
protruding from the sides of the hull, whose angle relative to the water can be
adjusted.Buttherudderofashipalsoexertsarolltorqueonthehull—oftenrather
largerthanthatfromthestabilisers,sinceithasalargersurfaceareaandislocated
lower,hencefartherfromtherollaxis.Soinheavyseas,ifthestabilisersarealready
doingalltheycan,itmaybepossibletousetherudderinordertofurtherreducethe
ship’sroll.Doingsowouldincreasethedragandthusreducethespeedoftheship,
of course, so one would not want to overdo this. In process control there may be
alternativewaysofprovidingheattoaprocess,someofthemmoreexpensivethan
others. For example, using steam which has already been used by another process
may be much more economical than using electrical heating.
When controlling systems which have such actuator redundancy, it is desirable
tousethe“standard”actuatormostofthetime,buttohavethepossibility ofusing
others when an unusually large disturbance occurs, or if the operation of the
standardoneisimpededforsomereason.MarcoGallieri’sthesisisconcernedwith
thisproblem,inthecontextofModelPredictiveControl(MPC).WhenusingMPC
with a quadratic cost function, one can try to obtain this effect by penalising some
actuator movements much more heavily than others. But this typically leads to
vii
viii Supervisor’sForeword
those actuators performing very small movements, rather than not moving at all—
although effects such as friction may result in them not moving in practice. The
solution adopted in Marco’s thesis is to augment the usual quadratic cost function
with an “‘ ” term, namely one penalising the absolute magnitude of an actuator
1
move.Thisisinspiredbytheuseofsuchtermsinthe“LASSO”algorithmusedin
signalprocessingtoobtainsparsemodels,whichaccountsforthename“‘ MPC”.
asso
In the case of control, it allows solutions to exist in which some actuators do not
normallymove.Italsoallowssolutionsinwhichnoneoftheactuatorsmovewhen
onlysmallcorrectionsarerequired,thusgiving“sparseintime”controlsignals.This
behaviour resembles that of “statistical process control”, in which control adjust-
ments are made only rarely, because usually they are very expensive—requiring
manualintervention ortemporary shutdown, for example.
Inthethesis,Marcofirstreviewstheuseof“‘ ”normsinsignalprocessing.The
1
key point is that this penalises moves away from zero significantly, whereas a
quadraticpenaltydoessomuchmoremildly(becauseofthezeroslopeatzero).On
the other hand, of the possible norms that do this, “‘ ” is the first one that is also
1
convex, so its use allows the MPC problem to remain a convex optimisation
problem. This is a most desirable property for a real-time algorithm. Marco then
develops MPC algorithms using the “‘ ” norm to penalise actuator moves (or
1
actuatorlevels,dependingonwhatone istrying toachieve).He developsmethods
of ensuring closed-loop stability and robustness, based on concepts that have
become standard in MPC—mainly, specifying a terminal constraint set with suit-
able invariance properties—but which require some modification in this context.
One chapter is devoted to examining how one can choose theweights in the “‘
asso
MPC” cost function so as to achieve the desired behaviour of the actuators, par-
ticularly when one wants to have “prioritised” and “auxiliary” actuators. Another
chapterillustratestheuseof“‘ MPC”ononeoftheexamplesmentionedearlier,
asso
namely a ship with roll stabilsers and rudder, both being used to reduce the roll
angle in heavy seas.
The topic of “‘ MPC” is far from exhausted. Tuning, namely choosing the
asso
weight(s) on the “‘ ” term in such a way to obtain the desired behaviour, is still
1
difficult,despitetheprogressmadeinMarco’sthesis.Developmentofsomefurther,
easier, guidelines for tuning would make this approach more accessible to the
practitioner. Also finding theoretical connections to “statistical process control”,
which was mentioned above, would be of interest. A possible approach might be
via Stokey’s work on “Economics of Inaction”.1
Cambridge, UK Prof. Jan M. Maciejowski
October 2015
1Nancy L. Stokey, The Economics of Inaction: Stochastic Control Models with Fixed Costs,
PrincetonUniversityPress,2009.
Abstract
High-performance control of large-scale dynamic systems, such as distribution
networks or redundantlyactuatedmarine/air vehicles,requires control policies that
arebothleanandrobust.Whereascontrolresearchisquitematureonthelatter,the
former has been mainly considered in terms of input energy by means of least
squaresapproaches.Thesestrategiesaresometimesnoteconomicforsystemswith
alargenumberofinputsasallofthecontrolchannelsareusedformostofthetime.
The use of hybrid controls can lead to improvement at the expense of increased
computation.
Thisthesispresents‘ -MPC,aModelPredictiveControlstrategythataimsto
asso
produce control inputs that are sparse inthe actuators domain. This isobtained by
means of a particular cost function borrowed from the compressed sensing litera-
ture. As a result, we obtained an MPC strategy that inherits some of the desirable
proprieties of least squares optimal control such as the smoothness of input/output
signalsaswellastheabilitytohandleuncertainty.Inaddition,‘ -MPCallowsthe
asso
designer to exploit further the system’s actuators by producing control sequences
with sparse magnitudes or time increments. This is obtained by adding an ‘
1
regularisation penalty to a standard quadratic cost, resulting in a soft thresholding.
The computational complexity of ‘ -MPC is comparable to that of a standard
asso
MPC.
Sufficient conditions are obtained for closed-loop asymptotic stability of a class
of nonlinear systems under ‘ -MPC. This is done by means of a set-theoretic
asso
controlframeworkasanextension ofstandard MPC theory.Asetofproceduresis
reviewed to perform the required offline computation.
A strategy is proposed to compute the regularisation penalty. This allows the
designer to specify a set of preferred actuators to be used for most of the time as
well as aneighbourhood ofthecontrolerror origin where theremaininginputsare
instandby.Thisconceptisdemonstratedthroughacoupleofillustrativeexamples.
Thefirstoneisthecontrolofanabstractnetworkwithprimaryandsecondarylinks.
Thesecondoneistherollmotioncontrolofalinearisedaircraftmodel.Inthiscase,
the ailerons and the rudders are used for all of the time while the spoilers are used
only when the control error is large.
ix
x Abstract
A contribution is made to the fields of tracking and robust MPC. A novel
strategyisproposedforlinearsystemstoprovideanupperboundonthemagnitude
of the additive uncertainty for which ‘ -MPC is robustly feasible and locally
asso
input-to-statestable.Thestrategyisbasedonset-theoreticcontrolmethodsandsoft
constraints.Theuseofthelatterisalmostinevitablewhendealingwithuncertainty.
In some cases the proposed approach can be much less conservative than the
standard robust MPC.
Theproblemofvesselrollreductioninoceanwavesbymeansofrudderandfins
is used as the main demonstrator for ‘ -MPC. The objective is to reduce the
asso
rudder activity by preferring the fins for the reduction of the roll variance.
Simulationresultsonadetailedvesselmodelillustratehowtheproposed‘ -MPC
asso
can outperform more common MPC strategies in terms of smaller roll variance as
well as significant reduction of the rudder activity.
Cambridge, UK Marco Gallieri
July 2014