1

Health-Aware and User-Involved BatteryCharging Management for Electric Vehicles:

Linear Quadratic StrategiesHuazhen Fang and Yebin Wang

Abstract—This work studies control-theory-enabledintelligent charging management for battery systemsin electric vehicles (EVs). Charging is crucial for thebattery performance and life as well as a contributoryfactor to a user’s confidence in or anxiety about EVs.For existing practices and methods, many run with alack of battery health awareness during charging, andnone includes the user needs into the charging loop.To remedy such deficiencies, we propose to performcharging that, for the first time, allows the user to spec-ify charging objectives and accomplish them throughdynamic control, in addition to accommodating thehealth protection needs. A set of charging strategiesare developed using the linear quadratic control theory.Among them, one is based on control with fixed termi-nal charging state, and the other on tracking a referencecharging path. They are computationally competitive,without requiring real-time constrained optimizationneeded in most charging techniques available in theliterature. A simulation-based study demonstrates theireffectiveness and potential. It is anticipated that charg-ing with health awareness and user involvement guar-anteed by the proposed strategies will bring majorimprovements to not only the battery longevity butalso EV user satisfaction.

Index Terms—Intelligent charging, battery manage-ment, fast charging, electric vehicles, linear quadraticcontrol, linear quadratic tracking

I. INTRODUCTION

Holding the promise for reduced fossil fuel useand air pollutant emissions, electrified transporta-tion has been experiencing a surge of interest inrecent years. Over 330,000 plug-in electric vehicles(EVs) are on the road in the United States as ofMay 2015 [1], with strong growth foreseeable in

H. Fang is with Department of Mechanical Engineering,University of Kansas, Lawrence, KS 66045, USA (e-mail:fang@ku.edu).

Y. Wang is with Mitsubishi Electric Research Laboratories,Cambridge, MA 02139, USA (e-mail: yebinwang@ieee.org).

the coming decades. Most EVs rely on battery-based energy storage systems, which are crucial forthe overall EV performance as well as consumeracceptance. An essential problem in the battery useis the charging strategies. Improper charging, e.g.,charging with a high voltage or current density,can induce the rapid buildup of internal stressand resistance, crystallization and other negativeeffects [2]–[5]. The consequence is fast capacity fadeand shortened life cycle, and even safety hazardsin the extreme case, eventually impairing the con-sumer confidence.

Literature review: The popular charging ways, es-pecially for inexpensive lead-acid batteries used forcars and backup power systems, are to apply aconstant voltage or force a constant current flowthrough the battery [6]. Such methods, thougheasy to implement, can lead to serious detrimentaleffects for the battery. One improvement is theconstant-current/constant-voltage charging [6], [7],which is illustrated in Figure 1a. Initially, a tricklecharge (0.1C or even smaller) is used for depletedcells, which produces a rise of the voltage. Then aconstant current between 0.2C and 1C is applied.This stage ends when the voltage increases to adesired level. The mode then switches to constantvoltage, and the current diminishes accordingly tocharge. Yet its application is empirical, with the op-timal determination of the charge regimes remain-ing in question [8]. In recent years, pulse charginghas gained much interest among practitioners. Itscurrent profile is based on pulses, as shown inFigure 1b. Between two consecutive pulses is ashort rest period, which allows the electrochemi-cal reactions to stabilize by equalizing throughoutthe bulk of the electrode before the next charg-ing begins. This brief relaxation can accelerate thecharging process, reduce the gas reaction, inhibitdendrite growth and slow the capacity fade [9]–[11]. Its modified version, burp charging, applies

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(a)

(b)

Fig. 1: (a) Constant-current/constant-voltage charg-ing; (b) pulse charging and burp charging.

a very short negative pulse for discharging duringthe rest period , see Figure 1b, in order to removethe gas bubbles that have built up on the electrodes.

A main issue with the above methods is the lackof an effective feedback-based regulation mecha-nism. With an open-loop architecture, they sim-ply take energy from power supply and put itinto the battery. As a result, both the chargingdynamics and the battery’s internal state are notwell exploited to control the charging process forbetter efficiency and health protection. This moti-vates the deployment of closed-loop model-basedcontrol. Constrained optimal control is appliedin [2], [12], [13], in conjunction with electrochem-ical or equivalent circuit models, to address fastcharging subject to input, state and temperatureconstraints for health. With the ability of dealingwith uncertain parameters, adaptive control is usedfor energy-efficient fast charging in [14]. In [15],the reference governor approaches are investigatedfor charging with state constraints. Leveraging thePontryagin minimum principle, optimal control de-sign of charging/discharging is studied in [16] tomaximize the work that a battery can performover a given duration while maintaining a desiredfinal energy level. However, we observe that theresearch effort for feedback-controlled charging hasremained limited to date. The existing works aremostly concerned with the fast charging scenario

and employ a restricted number of investigationtools, thus leaving much scope for further research.

Research motivation: In this paper, we propose toperform control-based EV charging managementin a health-aware and user-involved way. Since thebattery system is the heart as well as the mostexpensive component of an EV, health protectionduring charging is of remarkable importance toprevent performance and longevity degradation.As such, it has been a major design considera-tion in the controlled charging literature mentionedabove. Furthermore, we put forward that the userinvolvement, entirely out of consideration in theliterature, will bring significant improvements tocharging. Two advantages at least will be createdif the user can give the charging managementsystem some commands or advisement about thecharging objectives based on his/her immediatesituation. The first one will be improved batteryhealth protection against charging-induced harm.Consider two scenarios: 1) after arriving at thework place in the morning, a user leaves the carcharging at the parking point with a forecast inmind that the next drive will be in four hours;2) he/she will have a drive to the airport in onehour, and a half full capacity will be enough. Inboth scenarios, the user needs can be translated intocharging objectives (e.g., charge duration and targetcapacity). The charger then can make wiser, morehealth-oriented charging decisions when aiming tomeet the user specifications with such information,rather than pumping, effectively but detrimentally,the maximum amount of energy into the batterieswithin the minimum duration. Second, a directand positive impact on user satisfaction will resultarguably, because offering a user options to meethis/her varying and immediate charging needs notonly indicates a better service quality, but also en-hances his/her perception of level of involvement.

Statement of contributions: We will build health-aware and user-involved charging strategies viaexploring two problems. The first one is chargingwith fixed terminal charging state. In this case, the userwill give target state-of-charge (SoC) and chargingduration, which will be incorporated as terminalstate constraint. The second problem is tracking-based charging, where the charging is implementedvia tracking a charge trajectory. The trajectory isgenerated on the basis of user-specified objectivesand battery health conditions. The solutions, devel-oped in the framework of linear quadratic optimalcontrol, will be presented as controlled charging

3

(a)

(b)

Fig. 2: (a) the battery RC model; (b) the kineticbattery model.

laws expressed in explicit equations. The proposedmethods differ from those in the literature, e.g., [2],[12]–[16] in either of both of the following twoaspects: 1) from the viewpoint of application, theykeep into account both user specifications and bat-tery health — such a notion is unavailable beforeand will have a potential impact on transform-ing the existing charging management practices;2) technically, they, though based on optimizationof quadratic cost functions, do not require real-time constrained optimization needed in many ex-isting techniques [2], [12], [13], [15] and thus arecomputationally more attractive. In addition, thelinear quadratic control is a fruitful area, so futureexpansion of this work can be aided with manyestablished results and new progresses, e.g., [17]–[20].

Organization: The rest of the paper is organizedas follows. Section II introduces an equivalent cir-cuit model oriented toward describing the batterycharging dynamics. Section III presents the devel-opment of charging strategies. Section III-A studiesthe charging with fixed terminal charging statespecified by the user. In Section III-B, tracking-based charging is investigated. Section IV offers nu-merical results to illustrate the effectiveness of thedesign. Finally, concluding remarks are gathered inSection V.

II. CHARGING MODEL DESCRIPTION

While the energy storage within a battery re-sults from complex electrochemical and physicalprocesses, it has been useful to draw an analogybetween the battery electrical properties and anequivalent circuit which consists of multiple linearpassive elements such as resistors, capacitors, in-ductors and virtual voltage sources. While plentyof equivalent circuit models have been proposed,we focus our attention throughout the paper ona second-order resistance-capacitance (RC) modelshown in Figure 2a.

Developed by Saft Batteries, Inc., this model wasintended for the simulation of battery packs in hy-brid EVs [21], [22]. Identification of its parameters isdiscussed in [23]. The bulk capacitor Cb representsthe battery’s capability to store energy, and thecapacitor Cs accounts for the surface effects, whereCb Cs. Their associated resistances are Rb andRs, respectively, with Rb Rs. Let Qb and Qsbe the charge stored by Cb and Cs, respectively,and define them as the system states. The state-space representation of the model is show in (1). Itcan be verified that this system is controllable andobservable, indicating the feasibility of controlledcharging and state monitoring.

Based on the model, the overall SoC is given by

SoC =Qb −Qb + Qs −Qs

Qb −Qb + Qs −Qs, (2)

where Qj and Qj for j = b, s denote the unusableand the maximum allowed charge held by thecapacitor Cj. When the equilibrium Vb = Vs isreached, the SoC can be simply expressed as thelinear combination of SoCb and SoCs, i.e.,

SoC =Cb

Cb + CsSoCb +

Cs

Cb + CsSoCs. (3)

The RC model can well grasp the “rate capacityeffect”, which means that the total charge absorbedby a battery goes down with the increase in charg-ing current and is often stated as the Peukert’slaw. To see this, consider that a positive current isapplied for charging. Then both Qb and Qs, andtheir voltages, Vb and Vs,will grow. However, Vs,increases at a rate faster than Vb. When the currentI is large, the terminal voltage V, which is largelydependent on the fast increasing Vs, will growquickly as a result. Then V will reach the maximumin a short time. This will have the charging processterminated, though Qb still remains at a low level.

4

[Qb(t)Qs(t)

]=

−1

Cb(Rb + Rs)

1Cs(Rb + Rs)

1Cb(Rb + Rs)

− 1Cs(Rb + Rs)

[Qb(t)Qs(t)

]+

Rs

Rb + RsRb

Rb + Rs

I(t)

V(t) =[

Rs

Cb(Rb + Rs)

RbCs(Rb + Rs)

] [Qb(t)Qs(t)

]+

(Ro +

RbRs

Rb + Rs

)I(t)

. (1)

Fig. 3: The schematic diagram for charging based on linear quadratic control with fixed terminal chargingstate.

In addition, the RC model can also describe anotheressential characteristic, the “recovery effect”. Thatis, when the charging stops, the terminal voltage Vwill decrease, due to the charge transfer from Cs toCb.

To develop a digitally controlled chargingscheme, the model in (1) is discretized with a sam-pling period of ts. The discrete-time model takesthe following standard form:

xk+1 = Axk + Buk

yk = Cxk + Duk. (4)

where x =[Qb Qs

]>, u = I, y = V, and A, B, Cand D can be decided based on the discretizationmethod applied to (1).

The RC model is closely connected with thekinetic battery model (KiBaM) [24], [25], which isused in [16] for charging/discharging analysis. Asis shown in Figure 2b, the charge in the KiBaM isstored in two wells: the available-charge well (la-beled as #1) and bound-charge well (labeled as #2).The amount of charge in each well is determinedby the its height h and cross-sectional area s withz = hs. The current supplies ions to each well witha ratio coefficient c. The bound-charge well, dueto its smaller capacity, may grow faster in heightand deliver ions to the available-charge well with

a coefficient p driven by the pressure difference.Mathematically, it is given by

[z1(t)z2(t)

]=

−ps1

ps2p

s1− p

s2

[z1(t)z2(t)

]+

[c

1− c

]I(t).

(5)By letting s1 = Cb, s2 = Cs, p = 1/(Rb + Rs) andc = Rs/(Rb + Rs), the equivalence between (5) andthe state equation of (1) will be observed.

For health consideration, we need to constrainthe difference between Vb and Vs throughout thecharging process. Note that it is Vs −Vb that drivesthe migration of the charge from Cs to Cb. Itshares great resemblance with the gradient of theconcentration of ions within the electrode. Cre-ated during charging, the concentration gradientinduces the diffusion of ions. However, too largea gradient value will cause internal stress increase,heating, solid-electrolyte interphase (SEI) forma-tion and other negative side effects [26]–[28]. Me-chanical degradation in the electrode and capacityfade will consequently happen. Thus to reduce thebattery health risk, uniformity of the ion concen-tration should be pursued during charging. It isalso noteworthy that such a restriction should beimplemented more strictly as the SoC increases,because the adverse effects of a large concentration

5

difference would be stronger then. The same argu-ment can be extended to the KiBaM-based charging,for which the height difference between h1 and h2should be limited in order to defend the batteryagainst the harm during charging.

Next, we will build the charging strategies onthe basis of the RC model. The development willbe laid out in the framework of linear quadraticcontrol, taking into account both health awarenessand user needs.

III. HEALTH-AWARE AND USER-INVOLVEDCHARGING STRATEGIES

In this section, we will develop two chargingstrategies. For both, the user specifies the desiredcharging duration and target capacity. The firststrategy accomplishes the task via a treatmentbased on linear quadratic control subject to fixedterminal state resulting from the user objective. Inthe second case, charging is managed via trackinga charging trajectory which is produced from theuser objective. A discussion of the strategies willfollow.

A. Charging with Fixed Terminal Charging StateA charging scenario that frequently arises is:

according to the next drive need, a user will informthe charging management system of his/her objec-tive in terms of target SoC and charging duration.This can occur for overnight parking at home anddaytime parking at the workplace, or when a driveto some place will set off in a predictable time. Asafore discussed, the objective offered by the user,if incorporated into the dynamic charging decisionmaking process, would create support for healthprotection more effectively than fast charging. Thismotivates us to propose a control-enabled chargingsystem illustrated in Fig. 3. The charging objectivegiven by the user is taken and and translatedinto the desired terminal charging state. A linearquadratic controller will compute online the charg-ing current to apply so as to achieve the target statewhen the charging ends. Meanwhile, a chargingstate estimator will monitor the battery status usingthe current and voltage measurements, and feedthe information to the controller. In the following,we will present how to realize the above chargingcontrol.

From the perspective of control design, the con-sidered charging task can be formulated as anoptimal control problem, which minimizes a cost

function commensurate with the harm to healthand subject to the user’s goal. With the model in (4),the following linear quadratic control problem willbe of interest:

minu0,u1,··· ,uN−1

12

x>NSNxN

+12

N−1

∑k=0

(x>k G>QkGxk + u>k Ruk

),

subject to xk+1 = Axk + Buk, x0

xN = x.

(6)

where SN ≥ 0, Qk ≥ 0, R > 0 and

G =

[− 1

Cb

1Cs

].

In above, Gxk represents the potential differencebetween Cb and Cs, and the time range N and thefinal state x are generated from the user-specifiedcharging duration and target SoC. Note that, to-gether with (2)-(3), x can be easily determinedfrom the specified SoC value. The quadratic costfunction intends to constrain the potential differ-ence and magnitude of the charging current duringthe charging process. The minimization is subjectto both the state equation and the fixed terminalstate. In the final state, the battery should be atthe equilibrium point with Vb = Vs. The weightcoefficient Qk should be chosen in a way such that itincreases over time, in order to offer stronger healthprotection that is needed as the SoC builds up.

Resolving the problem in (6) will lead to a state-feedback-based charging strategy, which can beexpressed in a closed-form [17]:

Kk = (B>SN B + R)−1B>Sk+1A, (7)

Sk = A>Sk+1(A− BKk) + Qk, (8)

Tk = (A− BKk)>Tk+1, TN = I, (9)

Pk = Pk+1 − T>k+1B(B>Sk+1B + R)−1B>Tk+1,PN = 0, (10)

Kuk =

(B>Sk+1B + R

)−1B>, (11)

uk = −(

Kk − Kuk Tk+1P−1

k T>k)

xk

− Kuk Tk+1P−1

k x. (12)

This procedure comprises offline backward compu-tation of the matrices Kk, Sk, Tk, Pk and Ku

k from theterminal state and online forward computation ofthe control input (i.e., charging current) uk.

The state variable xk is not measurable directly inpractice, so its real-world application necessitates

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the conversion of the above state-feedback-basedstrategy to be output-feedback-based. One straight-forward avenue to achieve this is to replace xk byits prediction xk. This is justifiable by the certaintyequivalence principle, which allows the optimaloutput-feedback control design to be divided intothe separate designs of an optimal state-feedbackcontrol an optimal estimator [29]. The optimal esti-mation can be treated via minimizing

minx0,x1,··· ,xk

12(x0 − x0)

>Σ−10 (x0 − x0)

+12

k−1

∑i=0

w>i W−1wi +12

k

∑i=0

v>i V−1vi, (13)

where Σ0 > 0, W > 0, V > 0, and

wk = xk+1 − Axk − Buk,vk = yk − Cxk − Duk.

The optimal solution resulting from (13), is the one-step-forward Kalman predictor given by

Lk = AΣkC>(CΣkC> + V)−1, (14)xk+1 = Axk + Buk + Lk(yk − Cxk − Duk), (15)

Σk+1 = AΣk A> + W − AΣkC>

· (CΣkC> + V)−1CΣk A>. (16)

Substituting xk with its estimate xk, the optimalcontrol law in (12) will become

uk = −(

Kk − Kuk Tk+1P−1

k T>k)

xk − Kuk Tk+1P−1

k x.(17)

Putting together (7)-(11), (14)-(16) and (17), wewill obtain a complete description of the chargingmethod via linear quadratic control with fixed ter-minal state, which is named LQCwFTS and sum-marized in Table I. The LQCwFTS method performsstate prediction at each time instant, and then feedsthe predicted value, which is a timely update aboutthe battery’s internal state, to generate the controlinput to charge the battery. Much of the compu-tation for LQCwFTS can be performed prior to theimplementation of the control law. The sequences,Kk, Sk, Tk, Pk and Ku

k can be computed offline, andthen Kk, Ku

k Tk+1P−1k T>k and Ku

k Tk+1P−1k are stored

for use when the control is applied. On the sideof the Kalman prediction, offline computation andstorage of Lk can be done. Then the only workto do during charging is to compute the optimalstate prediction and control input by (15) and (17),reducing the computational burden.

B. Charging Based on Tracking

Tracking-control-based charging is another wayto guarantee health awareness and user objectivesatisfaction. A realization is shown in Fig. 4. Whena user specifies the charging objective, a chargingpath can be generated. A charging controller willbe in place to track the path. The path generationwill be conducted with a mix of prior knowledgeof the battery electrochemistries, health awarenessand user needs. It is arguably realistic that anEV manufacturer can embed path generation algo-rithms into BMSs mounted on EVs, from which theuser can select the one that best fits the needs whenhe/she intends to charge the EV. Leaving optimalcharging path generation for our future quest, wenarrow our attention to the focus of path-tracking-based charging control here.

Suppose that the user describes the target SoCand duration for charging, which are translated intothe final state x. Then a reference trajectory rk fork = 0, 1, · · · , N is calculated with rN = x. Notethat the trajectory should constrain the differencebetween Vb and Vs to guarantee health. A linearquadratic state-feedback tracking can be consideredfor charging:

minu0,u1,··· ,uN−1

12(xN − rN)

> SN (xN − rN)

+12

N−1

∑k=0

[(xk − rk)

> Q (xk − rk) + u>k Ruk

],

subject to xk+1 = Axk + Buk, x0

(18)

where SN ≥ 0, Q ≥ 0 and R > 0. Referringto [17], the optimal solution to the above problemis expressed as follows:

Kk = (B>Sk+1B + R)−1B>Sk+1A, (19)

Ksk = (B>Sk+1B + R)−1B>, (20)

Sk = A>Sk+1(A− BKk) + Q, (21)

sk = (A− BKk)>sk+1 + Qrk, sN = SNrN , (22)

uk = −Kkxk + Ksksk+1. (23)

Resembling (7)-(12), the execution of the above pro-cedure is in a backward-forward manner. Specifi-cally, (19)-(22) computed offline and backward priorto charging, and (23) online and forward from themoment when charging begins.

Following lines analogous to the development ofLQCwFTS, the output-feedback tracker for chargingcan be created based on (19)-(23) running with theKalman predictor in (14)-(16). That is, (23) will use

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Offline backward computation (from time N to 0)Kk = (B>SN B + R)−1B>Sk+1ASk = A>Sk+1(A− BKk) + QkTk = (A− BKk)

>Tk+1, TN = IPk = Pk+1 − T>k+1B(B>Sk+1B + R)−1B>Tk+1, PN = 0

Kuk =

(B>Sk+1B + R

)−1 B>

Online forward computation (from time 0 to N)Battery state prediction

Lk = AΣkC>(CΣkC> + V)−1

xk+1 = Axk + Buk + Lk(yk − Cxk − Duk)Σk+1 = AΣk A> + W − AΣkC>(CΣkC> + V)−1CΣk A>

Charging decisionuk = −

(Kk − Ku

k Tk+1P−1k T>k

)xk − Ku

k Tk+1P−1k x

TABLE I: The LQCwFTS charging strategy (Linear Quadratic Control with Fixed Terminal State).

Fig. 4: The schematic diagram for charging based on linear quadratic tracking.

xk rather than xk in practical implementation, i.e.,

uk = −Kk xk + Ksksk+1. (24)

Summarizing (19)-(22), (14)-(16) and (24) willyield the linear quadratic tracking strategy, or LQT,for charging, see Table II. Similar to the aforepro-posed LQCwFTS, the LQT can have much computa-tion completed offline. Then only the Kalman stateprediction and optimal tracking control (24) needto be computed during the actual control run.

The computational cost of LQT can be furtherreduced if we use its steady-state counterpart, mak-ing it more desirable in the charging application.The steady-state tracker is deduced as follows. Itis known that, if (A, B) is stabilizable and (A, Q

12 )

is detectable, Sk, as N − k → ∞, will approach aunique stabilizing solution of the discrete algebraic

Riccati equation (DARE)

S = A>SA− A>SB(B>SB + R)−1B>SA + Q.

Then Kk and Ksk will approach their respective

steady-state values, K and Ks. In a similar way,the Kalman gain Lk will achieve steady state L ask → ∞ given the detectability of (A, C) and sta-bilizability of (A, Q

12 ), which is unique stabilizing

solution to the DARE

Σ = AΣA> − AΣC>(CΣC> + V)−1CΣA> + W.

According to the DARE theory, S and Σ can besolved for analytically. With the steady-state gainsK, Ks and L, the optimal prediction and control forcharging will be

uk = −Kxk + Kssk+1, (25)

8

Offline backward computation (from time N to 0)Kk = (B>Sk+1B + R)−1B>Sk+1AKs

k = (B>Sk+1B + R)−1B>

Sk = A>Sk+1(A− BKk) + Qsk = (A− BKk)

>sk+1 + Qrk, sN = SNrN

Online forward computation (from time 0 to N)Battery state prediction

Lk = AΣkC>(CΣkC> + V)−1

xk+1 = Axk + Buk + Lk(yk − Cxk − Duk)Σk+1 = AΣk A> + W − AΣkC>(CΣkC> + V)−1CΣk A>

Charging decisionuk = −Kk xk + Ks

ksk+1

TABLE II: The LQT charging strategy (Linear Quadratic Tracking).

Offline computation of DAREs and gainsS = A>SA− A>SB(B>SB + R)−1B>SA + QΣ = AΣA> − AΣC>(CΣC> + V)−1CΣA> + WK = (B>SB + R)−1B>SAKs = (B>SB + R)−1B>

L = AΣC>(CΣC> + V)−1

Offline computation of s0 (from time N to 0)s0 = (A− BK)>sk+1 + Qrk, sN = SNrN

Online forward computation (from time 0 to N)Battery state prediction

xk+1 = Axk + Buk + L(yk − Cxk − Duk)Charging decision

sk+1 = (A− BK)−>sk − (A− BK)−>Qrkuk = −Kxk + Kssk+1

TABLE III: The SS-LQT charging strategy (Steady-State Linear Quadratic Tracking).

xk+1 = Axk + Buk + L(yk − Cxk − Duk). (26)

If (A− BK) is invertible, the backward computationof sk can be substituted by the forward computationgoverned by

sk+1 = (A− BK)−>sk − (A− BK)−>Qrk. (27)

Its implementation is initialized by s0 computedoffline by (22). We refer to this suboptimal charg-

ing strategy (25)-(27) as the steady-state LQT, orSS-LQT and outline it in Table III. The SS-LQTstrategy, due to its exceptional simplicity, has morecomputational appeal in terms of time and spacecomplexity.

C. DiscussionThe following remarks summarize our discussion

of the proposed charging strategies.

9

Remark 1: (Soft-constraint-based health awareness).As is seen, the proposed LQCwFTS, LQT andSS-LQT strategies incorporate the health awarenessas part of the cost functions rather than hard con-straints. This soft-constraint-based treatment willbring the primary benefit of computational effi-ciency and convenience. This compares with thetechniques based on real-time constrained opti-mization, which involve time-consuming computa-tion and on occasions, face the issue that no feasiblesolution exists in the constrained region. Usingsoft constraints, however, does not compromisethe effectiveness to protect the battery health. It isnoted that the harm to health is associated witha weighted penalty. When a proper weight Q isselected, minimizing the penalty cost will ensurea sufficient consciousness of the health.

Remark 2: (Robustness of SS-LQT). The SS-LQTstrategy is based on a combination of linearquadratic tracker and a Kalman filter. Such a designmay engender weak robustness in terms of gainand phase margins. To overcome this limitation,the loop transfer recovery can be used to buildrobust control design on the linear quadratic controlstructure [17].

Remark 3: (Continuous-time charging). The pro-posed charging strategies are developed in thecontext of digital control. Their extension tocontinuous-time control can be achieved throughusing the continuous-time in (1) and formulat-ing integral-based cost functions. The solutions arereadily available in the literature, and the interestedreader is referred to [17], [30].

Remark 4: (Generality to other models). The pro-posed development has a potential applicability toother battery models based on equivalent circuitsor electrochemical principles. Let us take the well-known single particle model (SPM) as an example.This model represents each battery electrode as aspherical particle and delineates the migration ofions in and between the particles as a diffusionprocess [31]. The PDE-based SPM can be convertedinto the standard linear state-space form, as shownin [32]. Then following similar lines to this work,linear quadratic problems can be established andsolved for charging tasks, where the differenceof ion concentration gradients are constrained topenalize charging-induced harm. Additionally, if amodel has a nonlinear output equation, the stateestimate can be acquired using a nonlinear Kalmanfilter.

IV. NUMERICAL ILLUSTRATION

In this section, we present two simulation exam-ples to illustrate the performance of the proposedcharging strategies.

Let us consider a lithium-ion battery describedthe RC model in (1) with known parameters. As-sume Cb = 82 kF, Rb = 1.1 mΩ, Cs = 4.074 kF,Rs = 0.4 mΩ, and Ro = 1.2 mΩ [21]. It has anominal capacity of 7 Ah. The model is discretizedby a sampling period of ts = 1 s. The initial SoCis 30%. The user will specify that certain SoC mustbe achieved within certain duration.

Example 1 - Application of LQCwFTS: Suppose thatthe user wants to complete the charging in 2 hours.Meanwhile, he/she specifies the target SoC value.For the simulation purpose, different target SoCvalues, 55%, 65%, 75%, 85% and 95%, are set here.The total number of time instants is N = 7200.We apply the LQCwFTS method to carry out thecharging tasks. For the control run, Qk = 0.1 · (5×107)k/N and R = 0.1. The exponential increase ofQk illustrates increasing emphasis on health owingto the growth of charge stored in the battery.

The computational results are illustrated in Fig-ure 5. It is observed from Figure 5a that the dif-ferent target SoCs are satisfied when the chargingends, meeting the user-specified objectives. The SoCincreases approximately proportionally with timefor the first 1.25 hours. Then the rate slows downgradually to zero as the charging objective is beingapproached. This results from a much larger weightQk in the later stage for health protection. Thecharging current is kept at almost a constant levelinitially during each charging implementation, asillustrated in Figure 5b. For a higher target SoC,the magnitude is larger accordingly. However, thecurrent drops quickly as the SoC grows further.The concerned health indicator, voltage differentbetween Cs and Cb is characterized in Figure 5c.For each case, Vs − Vb remains around a constantvalue in the first hour, despite high-frequency fluc-tuations due to noise. This is because a battery canaccept a higher internal stress at a low SoC level.However, the differences decreases drastically asmore charge is pumped into the battery, in orderto maximize the health of the battery’s internalstructure. For comparison, we enforce a constantcurrent of 2.275 A through the battery for 2 hours toreach 95% SoC. The potential difference, as shownin Figure 5d, will be kept at a fixed level unsur-prisingly, which, however, will cause much more

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Fig. 5: Example 1 - Application of LQCwFTS to charge the battery from 30% to 55%, 65%, 75%, 85%and 95%: (a) the SoC trajectories; (b) the charging current profiles; (c) the potential differences as healthindicator; (d) potential difference due to constant current charging to 95% SoC.

detrimental effects to the battery when SoC grows.

Example 2 - Application of SS-LQT: We considerthe use of SS-LQT for charging in this example,which is an upgraded version of LQT but morecomputationally efficient. The problem setting isthe same as in Example 1 — the tasks of chargingthe battery to from 30% of SoC to 55%, 65%, 75%,85% and 95% in 2 hours. The charging trajectoryis generated based on the task. For simplicity andconvenience, we assume that the desired trajecto-ries for x1 and x2, denoted as rb and rs, is generatedby

rj,k =1− e−kts/τj

1− e−Nts/τj(rj,N − rj,0) + rj,0,

where j = b or s, k = 1, 2, · · · , N − 1 and rj,0 is

the initial charge, rj,N the target charge, and τj thetime coefficient for j = b or s. Note that rj,0 andthat rj,N can be calculated from the initial SoC anduser-specified target SoC. The resultant trajectorieshave a steep increase followed by a gentle slope,which are reasonable in view of health protection.Letting τb = τs = Nts/4, Vs and Vb are forced tobe equal through the charging process. Thus at thetrajectory design stage, we put the minimization ofthe detrimental effects well into consideration.

With the reference trajectories generated, theSS-LQT strategy is applied to charging. The actualSoC increase over time is demonstrated in Fig-ure 6a. All the targets are met. In each case, theSoC grows at a fast rate when the SoC is at a lowlevel but at a slower rate when the SoC becomeshigher. Figure 6b shows the current produced by

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Fig. 6: Example 2 - Application of LQT to charge the battery from 30% to 55%, 65%, 75%, 85% and 95%:(a) the SoC trajectories; (b) the charging current profiles; (c) the potential differences; (d) tracking of x1(i.e., Qb) for 95% target SoC; (e) tracking of x2 (i.e., Qs) for 95% target SoC.

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SS-LQT. The current usually begins with a largemagnitude but decreases quickly. The potential dif-ference, given in Figure 6c, has the similar trend.It is relatively high when the charging starts, andthen reduces fast. The state tracking for the taskof 95% SoC is shown in Figures 6d and 6e. It isobserved that tracking of rb by x1 exhibits high ac-curacy. Tracking of rs by x2, however, is increasinglyaccurate, despite a minor deviation in the first hour.Overall, the closer the target SoC is approached, thesmaller the tracking error becomes.

In the above examples, different charging currentprofiles are generated for the same charging task.While the contributory factors include the selectionof Q and the reference charging path generation,such a difference poses another important question:how to assess and compare the charging strategies?There is no clear-cut answer yet as it involves a mixof battery electrochemisitry, charging performance,computational complexity, economic cost, and evenuser satisfaction. Though beyond scope of this pa-per, evaluation of charging strategies through theo-retical analysis and experimental validation will bepart of our future quest.

V. CONCLUSIONS

Effective battery charging management is vitalfor the development of EVs. In recent years, fastcharging control has attracted some research effort.However, the problem of health-aware and user-involved charging has not been explored in theliterature. In this paper, we propose a set of first-of-its-kind charging strategies, which aim to meetuser-defined charging objectives with awarenessof the hazards to health. They are developed inthe framework of linear quadratic control. Oneof them is built on control with fixed terminalstate, and the other two on tracking a referencecharging trajectory. In addition to the merits ofhealth consciousness and user involvement, theyare more computationally competitive than mostexisting charging techniques requiring online real-time optimization solvers. The usefulness of theproposed strategies is evaluated via a simulationstudy. This work will provide further incentivesfor research on EV charging management and isalso applicable to other battery-powered applica-tions such as consumer electronics devices andrenewable energy systems. Our future research willinclude optimal charging trajectory generation anda multifaceted assessment of charging strategies.

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