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ARTICLE IN PRESSG ModelJPC-2364; No. of Pages 13

Journal of Process Control xxx (2019) xxx–xxx

Contents lists available at ScienceDirect

Journal of Process Control

j ourna l ho me pa ge: www.elsev ier .com/ locate / jprocont

feedback charge strategy for Li-ion battery cells based on Referenceovernor�

affaele Romagnoli ∗, Luis D. Couto, Alejandro Goldar, Michel Kinnaert, Emanuele Garoneniversité Libre de Bruxelles, B-1050 Bruxelles, Belgium

r t i c l e i n f o

rticle history:eceived 30 September 2017eceived in revised form6 November 2018ccepted 30 November 2018vailable online xxx

a b s t r a c t

In this paper, we propose a solution for the control of the state-of-charge of a lithium-ion (Li-ion) batterycell. The aim is to design a control scheme able to charge the battery in a fast way while ensuring that thesafety constraints of the electrochemical model (ECheM) of the battery are not violated. The proposedsolution is based on a novel computationally efficient formulation of the Reference Governor (RG) able todeal with the union of linear constraints and which has been designed on the basis of a reduced ECheM,

eywords:onstrained controli-ion batterylectrochemical modeleference Governor

namely the equivalent-hydraulic model (EHM). Using suitable safety margins, the satisfaction of theconstraints on the EHM ensures the satisfaction of the constraints on the full-order ECheM. Simulationresults obtained on an accurate simulator show that the proposed method, if compared with classicalcharge strategies, is able to considerably improve the charge performance while ensuring the satisfactionof the constraints. These findings are validated experimentally on a Li-ion graphite/LCO battery.

© 2018 Published by Elsevier Ltd.

. Introduction

Batteries are ubiquitous energy storage devices that are morend more present in our daily life, ranging from portable electron-cs devices to electric vehicles. Due to their importance, in the lastew decades batteries have been the subject of a constant researchffort aimed at enhancing their performance in terms of storageapacity, time of charge and lifetime. At the current stage batter-es based on lithium-ion (Li-ion) constitute the most performingechnology among the commercially available ones. Most of theesearch and advancements concerning batteries in general, andi-ion batteries in particular, focuses on material science and elec-rochemical problems. However, motivated by the desire to exploithe full potential of these new technologies and, at the same time,

Please cite this article in press as: R. Romagnoli, et al., A feedback charJ. Process Control (2019), https://doi.org/10.1016/j.jprocont.2018.11.0

y the increasing need to ensure the safety of the new generation ofatteries, in the last few years the research on battery-managementystems (BMSs) has attracted considerable attention.

� This work is performed in the framework of the BATWAL project (Convention318146, PE Plan Marshall 2.vert) financed by the Walloon region (Belgium). Thisesearch has been funded by the Mandats d’Impulsion Scientific “Optimization-freeontrol of Nonlinear Systems subject to Constraints” of the Fonds de la Recherchecientifique (FNRS), Ref. F452617F and by Fonds pour la Formation á la Rechercheans l’Industrie et dans l’Agriculture (FRIA).∗ Corresponding author.

E-mail addresses: [email protected] (R. Romagnoli), [email protected]. Couto), [email protected] (A. Goldar),

[email protected] (M. Kinnaert), [email protected] (E. Garone).

ttps://doi.org/10.1016/j.jprocont.2018.11.008959-1524/© 2018 Published by Elsevier Ltd.

A BMS is a device in charge of performing estimation and controlof the battery operations. Its main goal is to provide information onthe state-of-charge (SOC) and health of the battery, and to managethe charge/discharge operations in an efficient and safe way. Mostof the existing BMSs are based on equivalent-circuit models of thebatteries. Although quite simple, this kind of models are accurateonly in the range of soft operating conditions and are unable togive a physical insight of what happens inside the battery. To over-come these limitations, several electrochemical models (ECheMs)[1] have been proposed in the literature. The main advantage ofECheMs is that their states can be directly related to the processesthat affect the battery performance and lifetime [1]. Early resultsshow that the development of BMSs based on ECheMs has thepotential to considerably improve the battery performance, whileat the same time enhancing its safety [2].

In this paper we will concentrate on the problem of man-aging the charge operations of a Li-ion battery. The mostwidely used battery charging strategy is probably the constant-current/constant-voltage (CCCV) [3]. This strategy consists of twoparts. In the first part of the charge, the strategy imposes on the bat-tery a constant current (CC) until the voltage reaches a pre-definedthreshold. After this threshold, the strategy imposes a constantvoltage (CV) that is kept until the current drops to a given lowerthreshold. Clearly this kind of strategy is an ad hoc solution where

ge strategy for Li-ion battery cells based on Reference Governor,08

the parameters (e.g. current rate, and voltage and current thresh-olds) are specified a priori based on heuristic tuning [4]. In orderto ensure a safe behavior of the battery, injected current, as well

https://doi.org/10.1016/j.jprocont.2018.11.008


http://www.sciencedirect.com/science/journal/09591524

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ARTICLEJPC-2364; No. of Pages 13

R. Romagnoli et al. / Journal of

s current and voltage thresholds, can be rather conservative [1],hile at the same time not giving any assurance that, at some

pecific time, some important electrochemical constraints are notiolated.

To address these issues, in the last few years a few alterna-ive strategies have been proposed which make use of constrainedontrol algorithms based on ECheMs. In spite of the added com-lexity, these solutions are capable of speeding up battery charginghile ensuring a higher level of safety. One of the first contributions

owards the development of constraints-aware BMSs is [5], wherehe optimal current profile for safe fast charging was computedsing the highly accurate Doyle-Fuller-Newman (DFN) ECheM [6]f the battery. While quite interesting from the theoretical view-oint, the huge computational complexity to compute this solutionrevents its use within a model predictive control (MPC) frame-ork and can only be used in open loop, making it unable to copeith modeling uncertainties and/or different initial conditions. Theain source of complexity for this scheme comes from the DFNodel, which consists of coupled nonlinear partial differential and

lgebraic equations that should be solved at each time instant [7]. Inrder to reduce the complexity, an obvious choice is to use reduced-rder ECheMs. Several reduced models [8–10] and their use withinPC schemes [11,12] have been presented in the literature. While

ery promising from the performance perspective, the main limit ofhese approaches is that their computational complexity, although

uch reduced with respect to schemes based on the DFN, is stillignificant and cannot be easily implemented online on low-costevices.

To further reduce the computational complexity, some papersave suggested the use of Reference Governor (RG) ideas. RGethods (see e.g. [13,14]) are low-computational cost constrained

ontrol schemes based on the idea of satisfying the constraints byuitably manipulating the reference of a pre-stabilized system. Onef the first papers suggesting the use of RG ideas to manage con-traints satisfaction in batteries is [15]. In this paper, the authorsake use of a reduced-order linear ECheM and they limit the rate

f variation of the injected current by pre-filtering it with a linearlter. The paper focuses on only one constraint (lithium concentra-

ion) and was tested for mild SOC (i.e. 50%). In [16] two RG-basedchemes are proposed. The first applies a nonlinear RG [14] basedn the DFN model, which cannot be easily implemented in real-ime for a sufficiently long prediction horizon. The second is a RGelying on a reduced model based on a linearization of the DFN,hich however in the proposed formulation was not able to ensure

he constraints when applied within a realistic simulation.In this paper we propose a computationally-efficient con-

rol scheme which makes use of a simple reduced model (thequivalent-hydraulic model, EHM) to ensure the battery safety con-traints during the fast charge of Li-ion batteries. The proposedcheme is based on the Scalar Reference Governor approach [13],hich has been modified to deal with the specific nonconvex con-

traints that arise in the control of batteries. Although based on aeduced model, simulations demonstrate the capability of the pro-osed scheme to enforce the constraints when tested on a realisticimulator (e.g. based on the DFN model). The effectiveness of theresented approach is validated experimentally.

A preliminary version of the scheme proposed in this paperas presented at the 2017 IFAC World Congress [17]. The main

mprovements with respect to such preliminary paper are:

i) a computationally-efficient RG strategy specifically designed


for the management of nonconvex constraints is proposed;ii) two different cell chemistries are considered in the simula-

tion and the results are compared with the conventional CCCVcharging strategy;

PRESSss Control xxx (2019) xxx–xxx

iii) an experimental validation of the proposed approach is car-ried out using a 160 mAh Turnigy Nanotech lithium-ion battery.The obtained results are compared with a conventional CCCVcharging strategy.

The paper is structured as follows. Both the complete ECheMused for simulation (DFN) and for the control (EHM) are describedin Section 2. The proposed control scheme is then presented inSection 3. The effectiveness of the proposed approach is demon-strated in simulation in Section 4 using two different Li-ion batterychemistries. In Section 5 the results of the experimental validationare reported. Some conclusions and a discussion on future workconclude the paper.

2. Li-ion battery cell models

The goal of this paper is to develop a real-time control schemefor charging Li-ion batteries which is able to speed up the batterycharging process while ensuring safe operations (which ensuresthat the battery will not age prematurely [1]).

As a first step, in the next two subsections, two ECheMs of thebattery are presented, together with a characterization of the con-straints that ensure the safe behavior of the battery. The first modelis the Doyle–Fuller–Newman model which will be used to havea realistic simulation of the battery dynamics. The second modelis a reduced model introduced in [18] known as the equivalent-hydraulic model (EHM), which will be used to design the proposedcontrol law. Then, the control law will be validated in simulationon the full-order DFN model. The results will be finally validatedexperimentally on a commercial battery.

2.1. DFN model

The electrochemical processes that take place within a Li-ionbattery cell can be mathematically described by the DFN model[6,19,20]. Such model is physics-based. It has been widely studiedand proven to be accurate [1]. The DFN model considers the bat-tery as composed of three domains, each consisting of two phases.The three domains are: (i) the positive electrode (+); (ii) the sepa-rator (s); and (iii) the negative electrode (−). The two phases are:(i) the porous solid phase (s); and (ii) the electrolyte solution phase(e) (see Fig. 1). The symbols in parenthesis are used as super/sub-script to denote the domain/phase of a model state variable. TheDFN approximates the electrodes as a collection of spheres. TheDFN model involves five variables, namely: the solid and electrolytephase concentration (cs and ce, respectively), the solid and elec-trolyte phase electric potential (ϕs and ϕe, respectively), and thepore-wall molar flux (jn). All these variables depend on the space(r- and x-axis along the radius of the electrodes spherical particlesand the thickness of the battery cell, respectively) and the time (t),and are depicted in Fig. 1.

The main partial differential and algebraic equations describ-ing the system dynamics are shown in Table 1 (see [1,6] for athorough description of model equations, boundary conditions andparameters). During charging, a positive overpotential is applied tothe battery cell, which promotes the deintercalation (oxidation)of lithium from the positive electrode solid phase. The resultinglithium-ions (Li+) travel throughout the electrolyte solution phaseup to the negative electrode solid phase, where they intercalate(reduction). Both oxidation/reduction reactions together constitute


the main reaction in a Li-ion battery cell. The ionic travel of lithium-ions from positive to negative electrode simultaneously involves anelectric current that leaves the system from the positive electrodeand reaches the negative electrode. The opposite mechanism cor-


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Fig. 1. Schematic representation of the charging process of a Li-ion battery cell. Atthe top, the three battery cell domains: negative electrode, separator and positiveelectrode; in the middle part and at the bottom, the two phases: the solution phaseatT

ro

rpmdlcbrttb

wtlat(bw

Table 1Main DFN model equationsa.

Physical process Equation

Solid-phase

Conservation of Li+ ∂c±s

∂t= 1r2

∂

∂r

(D±s r

2 ∂c±s

∂r

)(1)

Conservation of charge∂

2ϕ±s

∂x2= a±

s F

�±s

j±n (2)

Solution-phase

Conservation of Li+ ∂c±e

∂t= ∂

∂x

D±e,eff

ε±e

∂c±e

∂x+ a±

s (1 − t0c )ε±e

j±n (3)

Conservation ofchargeb

∂2ϕ±e

∂x2= − a±

s F

�±e,eff

j±n + �±D

∂2c±e

∂x2(4)

Electrode kinetics

Intercalation reactionc j±n = 2Fi±0 sinh

(˛0F

RT�±s

)(5)

where �±s = ϕ±

s − ϕe − U± − R±fj±n (6)

a Spatial and temporal dependency of state variables not explicitly indicated herefor compactness.

b Linearized equation [21].c Assuming ˛0 = 0.5.

nd the solid phase, respectively. The main state variables and parameters describinghe dynamics in each domain and phase are represented in correspondence withable 1.

esponds to the discharge process. The stored energy is a functionf the potential difference between both electrodes.

Besides the main reaction, Li-ion batteries are prone to sideeactions that may lead to capacity/power fade and can even com-romise the safe operation of the battery [1,22]. Degradationsight have different origins, which could be correlated and are

ependent on the specific chemistry of the battery [22]. Neverthe-ess, once identified, such mechanisms can be modeled as operatingonstraints. The most relevant sources of degradation that haveeen identified for a variety of electrode chemistries include sideeactions at the solid-electrolyte electrodes interfaces, and elec-rode phase changes [23]. Interestingly enough, the occurrence ofhese phenomena strongly depends on the state of operation of theattery and can be avoided by ensuring the following constraints:

here cs and css are the average and surface lithium concentra-ion, respectively, whose normalization by the maximum allowedithium concentration cs,max are lower and upper bounded by rcsnd rcs , respectively, and �sr and Usr are the overpotential and


he equilibrium potential of the side reactions, respectively. Eq.7a) and (7b) hinders the situation where Li is extracted/depositedeyond the maximum concentration allowed by the electrode,hich can cause phase transformations, active material dissolu-

(7)

Fig. 2. Equivalence between (a) single particle and (b) hydraulic model.

tion and oxygen loss in positive electrodes, and lithium dendriteformation in negative electrodes. Eq. (7c) and (7d) prevents sidereactions from taking place, which consume cyclable lithium andreduce the cell capacity (i.e. capacity fade) [24]. Such conditionsoccur first at the negative electrode/separator and the positiveelectrode/separator interfaces, respectively. Eq. (7e) precludeselectrolyte concentration ce depletion, which takes place first atthe current collector/negative electrode interface.

2.2. Reduced-order model

A common simplification of the DFN model is to abstract each

electrode as a single spherical particle (the so-called single-particlemodel framework) [1], as the one shown in Fig. 2(a). On the basisof this abstraction several models have been derived in the litera-


ture. One of the most interesting ones is the equivalent-hydraulicmodel (EHM) proposed in [18]. This model, depicted in Fig. 2(b),allows one to interpret the transportation phenomena within oneelectrode as the mass flow between two tanks with levels q1 and q2,


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Fig. 3. Proposed RG feedback control scheme. The closed-loop system consists of astate feedback controller (K and Kc) with an integrator (Ac , Bc , xc). I(t) is the currentgenerated by the state feedback controller and the block Cd is used to select the“measured” SOC(t) in the vector x(t). The outer loop implements the RG scheme



hich represent the two relevant electrochemical states associatedo the solid-phase diffusion process, namely the state-of-chargeSOC) and the critical-surface-concentration (CSC) [25]. As such,OC and CSC represent the states of each electrode. Since the bat-ery is composed of two electrodes, we denote with SOC+ and CSC+

he states of the positive electrode and with SOC and CSC the statesf the negative one, where the super-script “−” has been droppedor notational convenience.

The main advantage of the EHM with respect to other reducedodels (e.g. electrical equivalent models) is that in the EHM the

tates have a clear physical meaning that can be defined on the basisf the DFN model variables as SOC = cs/cs,max and CSC = css/cs,max.ydraulic parameters g and translate in the lithium diffusion

ontext into the lithium exchange rate between the bulk and theurface of the electrode, and the ratio of bulk to total electrodeolume, respectively. The input u(t) is proportional to the appliedurrent density I(t) as u(t) = �I(t), where � is a constant [25]. Notehat the EHM main approximation is to consider a uniform reactionate along the x-axis, which makes jn spatially independent andonsequently all the other distributed electrochemical variableslso become exclusively time-dependent.

The model can be further reduced by exploiting the fact thathe dynamics of the positive electrode are much faster than thenes of the negative one. As such, it is reasonable to assumehat the diffusion within the positive electrode is instantaneousCSC+(t) = SOC+(t)) and it is linked to the state of the negative elec-rode through the algebraic equation SOC+(t) = − SOC(t) + �, with

and � as positive constants [25]. More details on the EHM can beound in [25].

At this point the dynamics of the battery can be summarized byhe state of the negative electrode

(t) �[

SOC(t)

CSC(t)

](8)

hich evolves as

˙ (t) = Ax(t) + BI(t) (9)

here

=[

0 0g

ˇ(1 − ˇ)− g

ˇ(1 − ˇ)

], B = �

[ −1−1

1 − ˇ

](10)

nd I(t) is the input current. The resulting EHM is a simple linearime-invariant second-order model. Yet, this model is still able toeproduce the main dynamic behavior of an ECheM with a typicallymall model mismatch when compared with the DFN model [25].he state equation (9) is complemented by the output equationepresenting the voltage difference between the battery electrodesiven by

V(t) = �+s (SOC(t) + �, I(t)) − �−

s (CSC(t), I(t))

+U+ (SOC(t) + �) − U− (CSC(t)) − Rf I(t)(11)

here �±s ( · ) and U±(·) are the surface overpotentials and the equi-


ibrium potentials, respectively, which correspond to nonlinearunctions of the input and state variables, and Rf is a film resis-ance. For an analytic description of these functions please refer to25].

that uses the current state of the system (Ad , Bd , x) and the desired reference r(t) togenerate the admissible reference v(t).

Since the variables of the EHM have an explicit link with the onesof the DFN, it is possible to map constraints (7) into the constraintsfor the EHM as

(12)

where rSOC, rCSC are the upper bounds of the SOC and CSC,respectively, and U±

sr are scalars. This constraints mapping can beexplained as follows. Since SOC = cs/cs,max and CSC = css/cs,max, con-straints (7a) of the negative electrode can be rewritten as (12a) and(12b). The left-hand side of these inequalities is added to accountfor the fact that SOC and CSC are necessarily positive. Since thedynamics of the positive electrode are negligible, constraints (7b)of the positive electrode can be satisfied by controlling the nega-tive ones, and therefore they are not necessary. Constraints (12c)and (12d) are obtained by substituting ϕ±

s − ϕe of (6) in (7c) and(7d), under the assumption of uniform reaction rate. Notice thatelectrolyte constraint (7e) requires the extension of the EHM toaccount for electrolyte dynamics. However, such constraint is notviolated under the considered conditions. Extensive simulationshave shown that the satisfaction of constraints (12) implies thesatisfaction of constraints (7) for given values of rSOC, rCSC, and U±

sr .This result is not surprising from the physical viewpoint since theEHM is derived by suitably neglecting the spatial distribution of theDFN variables.

3. Proposed solution

In this paper, the problem of controlling the battery charge issolved making use of the Reference Governor (RG) philosophy.The main idea is to first stabilize the system with an aggressivecontroller, and then to add a RG to ensure the satisfaction of the con-straints [13]. More precisely, we will first design a linear quadraticregulator (LQR). Then, an RG scheme will be designed to providethe reference to the LQR so that the overall system does not vio-late constraints (12). The control scheme for the proposed overallsolution is depicted in Fig. 3.

It is worth remarking that since constraints (12) are noncon-


vex, it is not possible to use existing efficient formulations of theReference Governor (e.g. the Scalar Reference Governor for linearsystems [26]). To overcome this problem, in this paper the non-convex constraints of (12) are approximated as the union of linear


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onstraints and a RG strategy able to deal with this particular classf constraints is proposed.

The proposed strategy will be designed on the basis of a sampledersion of the EHM (9) and (10) considering a sampling time ofs = 1 s. The obtained system is

(k + 1) = Adx(k) + BdI(k) (13)

here Ad = eATs and Bd =∫ Ts

0eABd.

.1. Feedback control

Since the dynamics of the EHM are linear, a control law ensuringhe asymptotic stability of the system can be designed in manyifferent ways [27]. In this paper we use a simple LQR with integralction. The considered reference input r(k) is the desired level ofOC that has to be reached. An integral action is introduced usinghe dynamics

c(k + 1) = Acxc(k) + Bc(r(k) − Cdx(k)) (14)

here Ac = 1 and Bc = 1 represent the dynamics of the integrator andd = [ 1 0 0 ]. The current I(t) is provided by the state feedbackFig. 3) given by

(k) = −[K −Kc

][ x(k)

xc(k)

], (15)

here the gains K and Kc are computed by solving a LQR problem27] over the extended system resulting from (13) and (14), i.e.

x(k + 1)

xc(k + 1)

]=[

Ad 0

−BcCd Ac

][x(k)

xc(k)

]+[Bd

0

]I(k) +

[0

Bc

]r(k).

(16)

et us define xcl = [x, xc]T as the closed-loop states and writehe LQ optimization problem as minI J = minI 1

2

∑∞k=0xT

cl(k)Qxcl(k) +

T(k)RI(k). The weighting matrices Q ≥ 0 and R > 0 are selected so aso obtain an aggressive control of the battery in absence of potentialonstraint violations. Substituting (15) into (16), the closed-loopystem is

cl(k + 1) = Aclxcl(k) + Bclr(k) (17)

here

cl =[Ad − BdK BdKc

−BcCd Ac

], Bcl =

[0

Bc

]. (18)

.2. Constraints reformulation

In order to build a Reference Governor, the first step is to rewritehe constraints (12) derived from the constraints (7) with respecto the state of the closed-loop system (17). Constraints (12a) and12b) can be rewritten as

Tj xcl(k) ≤ sj, j = 1, . . ., 4 (19)

here S1 = −[ 1 0 0 ]T, S2 = − S1, S3 = −[ 0 1 0 ]T, S4 = − S3, and1 = 0, s2 = rSOC, s3 = 0, and s4 = rCSC.

Using the static characteristic of the battery cell (e.g. see Fig. 4),he remaining constraints (12c) and (12d) are accounted for by

apping the DFN constraints (7c)–(7e) on the plane (CSC − I) asossibly nonconvex constraints I ≥ f5(I, CSC), see e.g. the red area

n Fig. 4. These nonconvex constraints can be typically inner-pproximated as the union of linear constraints. For instance, in


ig. 4, the nonconvex constraints are approximated as the union ofwo linear constraints

2

i=1

(STj,ixcl(k) ≤ sj,i

), j = 5 (20)

where the green and red areas are the admissible and non-admissible regions,respectively. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

where STj,i

� aj,i[ 0 1 0 ] − [K − Kc] and the scalars sj,i = − bj,i forj = 5, i = {1, 2} and thus the overall constrained region can be definedas⎧⎨⎩STj

xcl(k) ≤ sj, j = 1, . . ., 4∨2i=1

(STj,i

xcl(k) ≤ sj,i

), j = 5

∀k > 0. (21)

Compactly we can describe the admissible region of the batteryas the intersection of the union of constraints represented by thelogical operators AND and OR as∧cc

j=1

∨nc,j

i=1

(STj,ixcl(k) ≤ sj,i

), ∀k ≥ 0, (22)

where nc,j = 1 in the case of constraints that are not approximatedby the union of linear constraints. In the case of (21) we have nc,j = 1and Sj,1 = Sj for j = 1, . . ., 4, whereas nc,5 = 2.

3.3. The Reference Governor

In this paper, in order to ensure the constraints (22) we proposethe use of a modified Scalar Reference Governor (SRG) able to dealin an efficient way with such type of constraints. Since constraints(22) are defined by the logical OR operator, they are referred to asOR-constraints. For the reader’s convenience the main ideas andnotation of the classical Reference Governors are first reported inthe next subsection. The proposed version of the Reference Gover-nor able to deal with OR constraints is detailed in the subsequentsubsection.

3.3.1. The “classical” Scalar Reference GovernorA Reference Governor [28] is a predictive control law that

enforces constraint satisfaction by suitably manipulating the refer-ence of a pre-stabilized system. The most used Reference Governorwas introduced in [29] for discrete-time linear systems

xcl(k + 1) = Aclxcl(k) + Bclv(k) (23)


where v(k) is the applied reference. The system (23) is subject to cc

linear constraints in the form

Sxcl(k) ≤ s (24)


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cc . The basic idea of all Reference Gov-rnors is the following: given a desired reference r(k), generate onhe basis of the current state xcl(k) an applied reference v(k) thatpproximates r(k) and is so that, if v(k) was to be kept constantrom now to infinity, the constraints are satisfied. In other wordshe choice of v(k) ensures

xcl(�|xcl(k), v(k)) ≤ s, l = 0, . . ., ∞, (25)

here xcl(�|xcl, v) = A�cl

xcl + (I − Acl)−1(I − A�

cl)Bclv represents the �

tep ahead prediction given the initial state xcl and applying theonstant reference v. The set of all the initial states xcl and appliedeferences v for which (25) is satisfied is denoted as the maxi-

al output admissible set O∞. In general, the computation of O∞nvolves an infinite number of constraints. However, if the con-traints (24) define a compact set, for any arbitrarily small ε > 0t is possible to compute a finite integer �* (see [30]) such that thenitely-computable set O∞ is an inner approximation of O∞, where

˜ ∞ ={

(xcl, v)∣∣Hxxcl + Hvv ≤ h

}(26)

here

xi �

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s

s(1 − �)

⎤⎥⎥⎥⎥⎥⎦ . (27)

ll the matrices and the vectors defined above have the same num-er of rows Nc � cc × (� * + 2). Note that O∞ can be made arbitrarilylose to O∞ by reducing ε.

The Scalar Reference Governor [13] computes the applied refer-nce v(k) to be applied as

(k) = v(k − 1) + �(k)(r(k) − v(k − 1)) (28)

here the scalar �(k) ∈ [0, 1] indicates how much the previouslypplied command v(k − 1) can move towards r(k) while ensuringhat (x(k), v(k)) ∈ O∞. The scalar �(k) can be computed as the solu-ion of the optimization problem given by

(k) = max� ∈ [0,1]

� (29)

subject to

Hxxcl(k) + Hv(v(k − 1) + �(r(k) − v(k − 1))) ≤ h.(30)

nder the assumption that at time zero (xcl(0), v(0)) ∈ O∞ thischeme ensures recursive feasibility and, for a constant reference(k) = r, finite time convergence of v(k) to the best steady-state fea-ible approximation of r along the line segment connecting v(0)nd r [29]. As shown in [28], the optimization problem (29) and30) can be computed very efficiently by inspecting one by one theines of (30). By denoting as HT

x,i, HT

v,i, hi the ith line of Hx, Hv, and h,espectively, the ith constraint can be written as(HT

v,i(r(k) − v(k − 1)))

≤ hi − HTx,ixcl(k) − HT

v,iv(k − 1), (31)

here � is the only unknown. Since thanks to recursive feasibility(k − 1) is always admissible, the right hand side of (31) is alwaysositive. At this point two cases have to be considered:

if(HT

v,i(r(k) − v(k − 1)))> 0, any positive � such that


� ≤ �∗i �

hi − HTx,i

xcl(k) − HT,v,iv(k − 1)

HTv,i(r(k) − v(k − 1))


is a feasible solution;• if

(HT

v,i(r(k) − v(k − 1)))

≤ 0, any positive � ∈ [0, 1] is a feasible

solution. Accordingly, for this i-th constraint we can define �∗i

� 1.

At this point the optimal solution of (29) and (30) is

�(k) = min

{min

i=1,...,Nc�∗i , 1

}. (32)

3.3.2. Reference Governor with OR conditionsIn this paper we propose a new computationally efficient Refer-

ence Governor inspired by the above described SRG which is able tomanage a nonconvex set that can be described by OR-constraints.In particular we will concentrate on constraints in the form∧cc

j=1

∨nc,j

i=1STj,ixcl(k) ≤ sj,i. (33)

Again, the idea is to select at each time instant a reference v(k) suchthat, if v(k) were to be held constant from k onward, the futuretrajectory of the state would never violate the constraints, i.e.∧cc

j=1

∨nc,j

i=1STj,ixcl(�|xcl(k), v(k)) ≤ sj,i, l = 0, . . ., ∞, (34)

where xcl(�|xcl, v) is defined as in the previous sub-section. In thiscase, due to the nature of constraints (33), it is not possible to usethe same results as [30] to compute an inner approximation of O∞.The next lemma shows how to compute an arbitrarily tight finitely-computable approximation O∞.

Lemma 3.1. Consider the linear system (23), where Acl ∈ Rn×n is

Schur, and a set of constraints (33). If constraints (33) define a compactset in R

n, then for any arbitrarily small ε > 0, it is possible to computea finite integer �* such that

O∞ ={

(xcl, v)|∧�∗

�=0

∧cc

j=1

∨nc,j

i=1HTx,�,j,ixcl + HT

v,�,j,iv ≤ hj,i

}∩ Oε(35

is an inner approximation of O∞, where HTx,�,j,i

= STj,iA�cl

, HTv,�,j,i =

STj,i

(I − Acl)−1(I − A�

cl)Bcl , hj,i = sj,i and

Oε ={

(xcl, v)|∧cc

j=1

∨nc,j

i=1HT

v,j,iv ≤ (1 − ε)hj,i,}

(36)

with HTv,j,i = ST

j,i(I − Acl)

−1Bcl .

Proof. See Appendix A. �

Using the same ideas as the standard SRG, the reference to beapplied at time k is

v(k) = v(k − 1) + �(k)(r(k) − v(k − 1)) (37)

where �(k) is the solution of the optimization problem

�(k) = max� ∈ [0,1]

� (38)

subject to∧�∗+1�=0

∧ccj=1

∨nc,ji=1(˛�

j,i� ≤ ˇ�

j,i)

(39)

where ˛�j,i

� HTv,�,j,i(r(k) − v(k − 1)) and ˇ�

j,i� hj,i − HT

x,�,j,ixcl(k) −

HTv,�,j,iv(k − 1) for �=0, . . ., � * and ˛�

∗+1j,i

� HTv,j,i(r(k) − v(k − 1)) and

ˇ�∗+1j,i

� hj,i − HTv,j,iv(k − 1).

At this point, using the same arguments as for the standard RG


it is possible to prove the result given next.

Theorem 3.2. Consider a linear system (23) subject to constraints(33) under the usual RG assumptions that Acl ∈ R

n×n is Schur and thatat time k = 0 the applied reference v(0) is such that (x(0), v(0)) ∈ O∞.


IN PRESSG ModelJ

Process Control xxx (2019) xxx–xxx 7

Ta

•

•

P

c

˛

F

Ai∨I

(

w

�

�

unaT�

Tc

•

•

d

a

c(if(

Table 2Values for the DFN model constraints (7) associated to degradation mechanisms.

Parameter Graphite LCO LMO

rcsa 1.0 0.5 0.2

Usr (V) 0.0 4.3 4.2



hen if the applied reference is (37) where �(k) is computed using (38)nd (39):

at each time step k, the optimization problem (38) and (39) alwaysadmits at least one solution and the constraints (33) are satisfied;if r(k) is kept constant over time, i.e. r(k) = r and if (0, v(0) + �(r −v(0))) ∈ Oε, ∀� ∈ [0, 1] then v(k) will converge to r.

roof. See [31]. �

To build an efficient solution of the above optimization problem,onsider the generic constraint with indices �, j, i

�j,i� ≤ ˇ�j,i. (40)

our cases are possible:

C1 – if ˛�j,i, ˇ�j,i> 0, then any positive � such that � ≤ min{��

j,i, 1}

where ��j,i

� ˇ�j,i/˛�j,i

is an admissible solution;

C2 – if ˛�j,i

≤ 0, ˇ�j,i> 0, then any � ≤ ��

j,i� 1 is an admissible solu-

tion;C3 – if ˛�

j,i< 0, ˇ�

j,i< 0, then any positive value � such that � ≥

��j,i

� ˇ�j,i/˛�j,i

is an admissible solution if ��j,i

≤ 1;C4 – there is no solution for � in all other cases.

t this point it is convenient to analyze the OR operation for fixedndices �, j, i.e.nc,j

i=1(˛�j,i� ≤ ˇ�j,i). (41)

t is easy to see that the admissible region will assume the form

� ≤ ��j ) ∨ (� ≥ ��j )

here

�j = max

i=1,...,nc,j��j,i (42)

¯ �j = mini=1,...,nc,j

��j,i (43)

sing the convention that if ��j,i

or ��j,i

are not defined, they areot considered. Note that, because of recursive feasibility, � = 0 islways an admissible solution for the optimization problem (38).his implies that for at least one i the case C1 or C2 apply. As such�j

is always defined. Vice versa, ��j

might or might not be defined.hanks to this analysis we can conclude that for a given set of ORonstraints with indices �, j we have 2 possible cases:

the admissible region it defines is equivalent to a single inequalityconstraint on �

� ≤ ��j ; (44)

the admissible region it defines is equivalent to the OR of twoinequalities on �

� ≤ ��j ∨ � ≥ ��j ; (45)

epending on the fact that at least one ��j,i

is defined or not. Note

lso that in the special case ��j

≤ ��j, it is convenient to rewrite the

ondition as � ≤ ��j

= 1. We will denote as 1 the set of all indices� , j) such that the case (44) happens, and 2 the set of the remain-ng indices falling in case (45). Constraint (39) can be rewritten as


ollows:∧(�,j) ∈ 1

(� ≤ ��j ))

∧(∧

(�,j) ∈ 2

(� ≤ ��j ) ∨ (� ≥ ��j ))

rce (mol m−3) 1.0 1.0 1.0

a rcs for graphite, rcs for LCO, LMO.

which can be further simplified as

(� ≤ � 1) ∧(∧

(�,j) ∈ 2

(� ≤ ��j ) ∨ (� ≥ ��j ))

(46)

where � 1� min

(�,j) ∈ 1

��j. At this point the maximum �(k) satisfying

(46) can be computed according to the following simple algorithm:

1. IF ��j

≤ � 1for any (� , j) ∈ 2

• �(k) = � 1;

• STOP.2. ELSE, for any (� , j) ∈ 2 such that ��

j> � 1

• Set � 1= min{� 1

, ��j}

• Update 2 = 2 \ {(� , j)}3. GO TO 1.

Note that this algorithm solves a maximization problem for thescalar � in the presence of nonconvex constraints without usingany optimization solver. The numerical operations involved do notrequire any special software nor particularly performing hardwarefor its real-time implementation (e.g. see Section 5).

4. Numerical results

To assess the effectiveness of the proposed RG-based approach,we tested it in simulation on two different battery cells. The twocells have different chemistries, namely lithium cobalt oxide (LCO)and lithium manganese oxide (LMO). The behavior of the batteriesis simulated using the DFN model on a simulator that imple-ments the solid-phase diffusion equations through a third-orderPadé approximation whereas all the other partial differential andalgebraic equations (see Table 1) are discretized with the centraldifference method. The simulator uses the cell parameters providedin [32–34].

Each battery cell is subject to constraints (7). The parameters ofthe constraints associated to each cell are different and are reportedin Table 2. Trespassing a constraint induces the occurrence of a spe-cific degradation mechanism. In particular, if inequalities (7a) and(7b) are violated, the graphite electrode deposits lithium [35], theLCO electrode is transformed towards inactive phases and oxygenis released [23], and the LMO electrode becomes structurally unsta-ble and more vulnerable to degradation [36]. If constraints (7c) and(7d) are violated, lithium plating side reaction takes place in thegraphite electrode [1] and both LCO and LMO positive electrodesundergo solvent oxidation [35,37], which is critical for the LMOelectrode since it involves acid generation, active material dissolu-tion [38], and manganese deposition in the graphite electrode [39].Finally, if (7e) is violated, the internal battery cell impedance risesdue to electrolyte depletion.

As detailed in the theory, for the closed-loop system (17) and(18), constraints (7) are mapped into constraints in the form (22).The specific vectors and scalars of (22) for the two specific batteriesare reported in Table 3. Note that the number of constraints differs


for the two batteries, as the nonconvex regions that are approxi-mated (see red areas in Fig. 5) are quite different for the two batterycells. Note also that a certain margin (orange areas) between thetheoretical nonlinear constraint and the linear approximations is



8 R. Romagnoli et al. / Journal of Process Control xxx (2019) xxx–xxx

Fig. 5. DFN model critical conditions for �+sr at the positive electrode/separator interface (electrode dependent), �−

sr = 0.0 V at the negative electrode/separator interface andce = 1 mol m−3 at the current collector/negative electrode interface (common for both batteries).

Fig. 6. Current and voltage profiles from the DFN simulator usi

Table 3Values for the constraints (22).

Graphite/LCO Graphite/LMO

s1,1 0 s2,1 0.712 s1,1 0 s2,1 0.622s3,1 0 s4,1 0.712 s3,1 0 s4,1 0.622s5,1 −9.33 s5,2 −2.59 s5,1 −17.28 s5,2 −5.76– – – – s6,1 −33.6 s6,2 −5.76

cEdscu

d�Tp

mate corresponds to the red solid curve. Note that for high current

a5,1 22.47 a5,2 0 a5,1 24 a5,2 7.2– – – – a6,1 96 a6,2 7.2

onsidered to take into account the model mismatch between theHM and the DFN model. Extensive numerical simulations haveemonstrated that for the reported values, the satisfaction of con-traints (22) in (17) and (18) always ensures the satisfaction ofonstraints (7) on the DFN model, even in presence of a boundedncertainty on the state reconstruction.

For what concerns the control, the reduced-order EHM−3 −1


escribed by (9)–(11) is used where g = 2.60 · 10 s , = 8.81 · 10−6 C−1 and = 0.70 for both battery chemistries.hese parameters are directly derived from the electrochemicalarameters of the DFN model. Note that the models for both

ng CCCV and the proposed RG method for LCO and LMO.

chemistries have the same parameter set because they sharethe same negative electrode. Graphite is indeed the most com-monly used electrode for lithium-ion batteries, which makes theproposed EHM compatible with a wide variety of chemistries.Moreover, its slower dynamics and higher predisposition to sufferdegradation when compared to positive electrodes motivates itscareful monitoring. The batteries are pre-stabilized with respectto the state-of-charge through an optimal LQR state feedbackwith integral action, as detailed in Eqs. (14) and (15). The LQR isdesigned by selecting suitable Q and R values, which yields thematrices K = [−2.8137 − 0.0066] · 104 and Kc = −2.7833 · 103. Thestate is estimated using the extended Kalman filter (EKF) detailedin [25]. The control sampling time is 1 s.

The results of the proposed control scheme in terms of inputand state trajectories are shown in Fig. 5(a) and (b) for LCO andLMO, respectively. The actual state obtained from the DFN modelis represented by the cyan solid curve, whereas the EKF state esti-


in both chemistries the actual state drifts from the estimated one,starting in the green region and staying in the orange safety areafor a certain time. Such difference is due to model mismatch and




ulator

dttr

n

Fig. 7. Profiles of the constrained variables from the DFN sim

ecreases with the charging current. This aspect is expected since


he EHM was developed under the assumptions of constant elec-rolyte and thermal dynamics, which are harder to meet for currentates higher than 1C.1 The safety margin on the constraints has

1 C-rate: normalization of the battery current [A] with respect to the batteryominal capacity [Ah].

using CCCV and the proposed RG method for LCO and LMO.

been tuned to ensure that, despite this mismatch, it never leads toconstraint violations.

To evaluate the improvements of the proposed RG-basedapproach with respect to currently used commercial techniques,


we compared it with the conventional CCCV charging strategy. TheCCCV strategy was set with a 2C charging current for the CC modeuntil the voltage reaches the end of charge voltage (4.06 V and3.96 V for LCO and LMO, respectively), time instant at which the



10 R. Romagnoli et al. / Journal of Process Control xxx (2019) xxx–xxx

Fig. 8. Custom-built experimental setup, where (1) Li-ion battery, (2) cooling fan, (3) current and voltage sensor, (4) acquisition board, (5) battery surface temperature sensor,(6) environment temperature sensor, (7) power supply and (8) security relay.

F tor intb

Cdtaart2cdtv

sFpttt4

ig. 9. In (a), DFN critical conditions for �sr = 0.0 V at the negative electrode/separaattery.

V mode is adopted. Fig. 6(a) and (b) shows current and voltageata, respectively, for LCO and LMO provided by both the CCCV andhe proposed RG charging strategies. The CCCV profile introduces

constant amount of energy during the CC mode (30 min for 2C),fter which the CV mode takes over and gradually reduces the cur-ent rate to zero. Meanwhile, the proposed RG is able to increasehe input current at the beginning and to keep it higher than e.g.C for most part of the charging process. Such strategy allows toharge the battery much faster and, more importantly, it avoids toegrade the battery by ensuring the constraints. Regarding voltage,he proposed RG enables safe incursions beyond the upper cut-offoltage specified for the CCCV strategy.

The temporal evolution of the constrained variables in (7) arehown in Fig. 7 for both tested strategies. The SOC is depicted inig. 7(a) for LCO and LMO subject to both the CCCV and the pro-osed RG charging strategy. For LCO, the proposed RG takes 35 min


o achieve 99.5% SOC, while the CCCV strategy requires more thanwice that time (90 min) for the same SOC level. Such chargingime difference becomes even greater for the LMO cell, needing8 min for the proposed RG to reach 99.5% SOC level whereas CCCV

erface. In (b), current profile using CCCV and the proposed RG method for Turnigy

strategy only gets up to 87% after 120 min of charging. This is dueto the different diffusion time constants �+ of each positive elec-trode, namely �+

LCO = 7.23 s and �+LMO = 722.50 s. A similar trend is

observable also in the CSC evolution, see Fig. 7(b), but with fasterdynamics. For what concerns the SOC+ (see Fig. 7(c)) it is interest-ing to note that the satisfaction of the constraints in the negativeelectrode ensures the satisfaction of constraints in the positive one(rSOC+ ≥ 0.5 for LCO and rSOC+ ≥ 0.2 for LMO) as we translated thepositive electrode constraints into the negative electrode exploitingmaterial balance. Finally, the electrolyte concentration ce as wellas the positive and negative side reaction overpotentials �±

sr aredepicted in Fig. 7(d)–(f) for LCO and LMO under both CCCV and theproposed RG charging strategy. Note that, compared to the CCCV,the proposed RG pushes more the performance of both batteriesgoing closer to the constraints (namely ce ≥ 1.0 mol m−3, �−

sr > 0.0 Vand �+

sr < 0.0 V), especially at the beginning of the charging pro-


cess.Based on the above results, it seems that LCO and LMO batteries

can be charged more than twice as fast by using the proposed RGwhen compared with conventional charging strategies.




Fig. 10. Voltage and SOC profiles using CCCV and the proposed RG method for Turnigy battery.

Table 4EHM parameters for the Turnigy battery.

Parameter Value Units Parameter Value Units

0.7000 [–] g 4.2653 · 10−2 [s−1]�+ 3.7532 · 10−3 [A−1] �− 2.3904 · 10−3 [A−1] 7.9886 · 10−1 [–] � 1.0011 [–]� 5.4581 · 10−6 [C−1] Rf 8.4674 · 10−4 [�]

Table 5Values for the constraints (22) in the implementation.

s1,1 0 s2,1 0.62s3,1 0 s4,1 0.62

5

sc1vcTsR

(ctti

tUrsoimamet

Table 6Experimental comparison of the proposed RG against commercial CCCV chargingstrategies.

Charging time [min] Average computational time [s]

−4

s5,1 −5 a5,1 4s6,1 −3 a6,1 0

. Experimental results

In order to test the effectiveness of the proposed chargingtrategy, we have performed an experimental validation on aommercial lithium-ion battery. The tests were carried out on a60 mAh Turnigy Nanotech graphite/LCO battery of 3.7 V nominaloltage. A custom-built experimental setup consisting of commer-ially available off-the-shelf components was used for these tests.he test-bench is shown in Fig. 8. To enable fast prototyping, theetup is connected to a dual-core computer of 2.5 GHz and 2 GbAM with Matlab

®2012a, which runs the charge algorithm.

The parameters of the discrete-time version of the EHM9)–(11) have been identified experimentally using a series of CCharge/discharge steps, measuring the voltage response, and fit-ing the model using nonlinear least-squares technique similar tohe one used in [40]. The identified model parameters are reportedn Table 4.

The battery constraints are the same ones used in the simula-ion section for the graphite/LCO (Table 2), namely rcs = 1.0 andsr = 0.0 V for graphite and rcs = 0.5 and Usr = 4.3 V for LCO, while

ce = 1.0 mol m−3. The vectors and scalars associated to the repre-entation (22) of the constraints are reported in Table 5. The batteryperating region is depicted in Fig. 9(a). Similarly to what was done

n the previous section, the nonconvex dangerous region, the safetyargin, and the admissible region are represented in red, orange,


nd green, respectively. The solution of the LQR problem yields theatrices K = [−10.9434 − 3.7979] · 104 and Kc = −2.1053 · 104. The

xtended Kalman filter presented in [25] is used for state estima-ion.

Proposed RG 12.60 9.0287 · 102C CCCV 32.30 1.3269 · 10−4

The current profile and voltage response of the battery cell underboth the proposed RG and the 2C CCCV charging strategies areshown in Figs. 9(b) and 10(a), respectively. Charging time and aver-age computational time for both considered charging strategies areshown in Table 6. The time required for the proposed RG strategyto charge the battery up to 97.5% SOC is less than two times the oneof the 2C CCCV (see Fig. 10(b)), whose SOC was Coulomb-counted.The proposed RG-based technique requires a higher computationaltime with respect to the classical CCCV to obtain the charging cur-rent. Yet the charging time is only 7 times the one of CCCV. Theexperimental results confirm those ones obtained in simulation, i.e.the proposed RG is a control strategy able to charge the battery ina fast and safe way without significantly increasing computationalcost.

6. Conclusion

In this paper, we have proposed a low-computational costapproach able to carry out the fast charge of lithium-ion batteriescells while satisfying constraints linked to the cell degradation. Theproposed approach is based on the Reference Governor philosophy.First, the charge process is controlled with a stabilizing feedbackcontrol law with high gains. Then, a Reference Governor is added tosatisfy the constraints, i.e. to keep the battery behavior outside theregions where the side reactions responsible for the degradation ofthe battery occur. Since the admissible region of this control prob-lem is nonconvex, the standard Scalar Reference Governor cannotbe applied as it is. For this reason, in this paper we propose a newcomputationally-efficient reference governor scheme that is able towork with nonconvex constraints defined as the union of polyhedrawithout making use of optimization tools. The proposed approachhas been simulated on two different battery cells that differ onchemistry. Numerical results show an important improvement in


performance (time of charge) with respect to the classical CCCVcharge, while ensuring that the prescribed degradation constraintsare not violated. Experimental results on a commercial battery fur-


ING ModelJ

1 Proce

tt

A

taf

A

gx

V

e

V

w

X

Ai

Ai

•

•

TL(cniVq

�

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


2 R. Romagnoli et al. / Journal of

her confirm these findings. Future work will focus on experimentalests on different battery chemistries and long-term cycling.

cknowledgments

The authors would like to thank Prof. Scott Moura for providinghem with the battery simulation software that he developed. Were also very grateful to Eng. Laurent Catoire and Eng. Serge Torfsor building the battery experimental setup.

ppendix A.

The proof is inspired by [14]. Since the matrix Acl is Schur,iven a constant reference v and the corresponding steady statev = (I − Acl)

−1Bclv, a Lyapunov function in the form

(xcl, xv) = (xcl − xv)TP(xcl − xv), P > 0,

xists such that

(xcl(k + 1), xv) − V(xcl(k), xv) ≤ −(xcl(k) − xv)TQ (xcl(k) − xv). (47)

ith Q > 0. According to (33), the set of admissible states is

={

xcl ∈ Rn|∧cc

j=1

∨nc,j

i=1STj,ixcl ≤ sj,i

}. (48)

ccording to the definition of Oε, the set of admissible equilibria xvs

Xv ={

xv ∈ Rn∣∣∃v ∈ R

m : xv = (I − Acl)−1Bclv and∧cc

j=1

∨nc,j

i=1STj,ixv ≤ (1 − ε)sj,i

}. (49)

t this point since: (i) X and Xv are compact sets; and (ii) since Xvs contained in the interior of X, the following two quantities exist:

The maximum possible value of the Lyapunov function for anyfeasible x ∈ X and any feasible xv ∈ Xv

Vmax � maxxcl ∈ Xxv ∈ X

V(xcl, xv).

The minimum value that the Lyapunov function must have sothat at least one state of the associated Lyapunov level set is ableto violate the constraints if xv ∈ Xv, i.e.

Vmin � minxcl /∈ Xxv ∈ Xv

V(xcl, xv).

he important consideration here is that, if at a given time k: (i) theypunov function V(xcl(k), xv) is less than Vmin; (ii) xv ∈ Xv; andiii) if v is kept constant from k onward, then, because of invariance,onstraints are always satisfied in the future and then there is noeed to check the constraints. At this point to conclude the proof

t is enough to notice that because of (47) whenever V(xcl, xv) >

min, the one step variation always decreases of at least a nonzerouantity �Vmin defined as{


Vmin �mine ∈ RneTQes.t.eTPe ≥ Vmin

.[

[


As a consequence, for any xcl ∈ X and xv ∈ Xv, after at most a finitenumber of predictions �* there is no need to check the constraints.The integer �* can be defined as

�∗ �[Vmax − Vmin

�Vmin

].

Note that (35) and (36) ensure that xcl(k) ∈ X and xv ∈ X.

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