Condition Monitoring of Electrolytic Capacitors via ESR Estimation

with Recursive Least Squares and Sliding Mode Techniques

J. M. Andrade

University of Derby, College of Engineering and Technology,

Department of Electronics, Computing and Mathematics, Derby, DE22 3AW, England, U.K.

Keywords:

Condition Monitoring, Capacitor Diagnosis, Capacitor Ageing, Equivalent Series Resistance (ESR), Sliding

Mode Differentiator, Robust Exact Differentiation, Recursive Least Squares (RLS).

Abstract:

A new on-line electrolytic capacitor condition monitoring approach based on sliding mode concepts and the

recursive least squares (RLS) with constant forgetting factor algorithm is proposed in this paper. This scheme

involves robust exact differentiation which outperforms the classical differentiator based on linear approxima-

tions, when the system is affected by noise. The condition monitoring approach proposed in this paper allows

for on-line estimation of the ESR which is considered to be one of the best indicators of capacitor degradation.

Computer simulation results, considering a DC- DC buck converter, provide evidence of the effectiveness of

the capacitor condition monitoring scheme proposed in this paper.

1 INTRODUCTION

Capacitor failure may be classiﬁed into (i) ca ta -

strophic failure when a capacitor has completely lost

its functionality, for instance, short a nd open circuit

failure modes, and (ii) degrad ation failure when a gra-

dual deterioration of a capacitor occ urs. The cau-

ses of the short and open circuit failure s are over-

voltage, mechanical stress, excessive ripple current,

reverse voltage, weak point of electrolytic paper, de-

fective oxide layer, and insufﬁcient connection of tab

and terminal part, where as degradation or wear-out

failures are ma inly caused b y charging and dischar-

ging cycles, high operating tem perature, overvoltage

stress and excessive ripp le current (Nichicon Corpo-

ration, nd). Capacitor degradation is also known as

capacitor ageing.

Electrolytic capacitor degradation results in a gra-

dual increase in the equivalent series resistance (ESR)

(i.e. the sum of the resistances associated with the

aluminium oxid e, electrolyte, paper spacers and elec-

trodes) and also a decre ase in capacitance over time.

An indication of the co ndition of a capacitor may

be obtained by monitoring th ese two electrical pro-

perties (Ab dennadher et al., 2008). The degradation

of an electrolytic capacitor is primarily attributed to

electrolyte evaporation, le akage current, and internal

pressure increa se (Alwitt and Hills, 1965) (Wang and

Blaabjerg, 2014) (Hewitt et al., 2016) . The end-of-

life (EoL) of an alu minium electrolytic capacitor is

reached whe n its c a pacitance chan ges by 20% and/or

its ESR doubles (Hewitt et al., 2016) (Soliman et al.,

2016).

A review of the condition monitoring of capaci-

tors in power converters that summarized different

methods for determining the ESR a nd/or capacitance

has been presented in (Soliman et al., 2016), see re-

ferences therein for details of the different methods.

The Kalman ﬁlter paradigm (Kalman, 1 960) has been

applied to the problem of on- line estimation of ESR

and/or capacitance , e.g. (Abdennadher et al., 2009)

(Celaya et al., 2 011). In (Celaya et al., 2011) besides

using a Kalman ﬁlter, an empirical degradation model

was used fo r prediction of the remaining usef ul life of

electrolytic capacitors. Recursive least squares (RLS)

and least mean squares algorithms were consid e red in

(Buiatti et al., 20 07a) and (Buiatti et al., 2007b) re-

spectively. Anothe r condition mo nitoring system ba-

sed on RLS that a lso incorporated capacitor ageing

models that de scribe the evolution of ESR and capa-

citance with respec t to temperature, was proposed in

(Abdennadher et al. , 2008) and (Abdennadher et al.,

2009). Application of digital signal processing con-

cepts to the problem of ESR and capacitance estima-

tion are reported in (Ama ral et al., 2007) and (Imam

et al., 2005). A wide range of experimental methods

have also been developed, e.g. (Kulkarni et al., 2010)

(Kulkarni et al., 2012) (Hewitt et al., 2016) (Amaral

474

Andrade, J.

Condition Monitoring of Electrolytic Capacitors via ESR Estimation with Recursive Least Squares and Sliding Mode Techniques.

DOI: 10.5220/0006887704740481

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 474-481

ISBN: 978-989-758-321-6

et al., 200 7).

The incr ease of ESR is considered by many re se-

archers, e.g. (Wang et al., 2012), the be st ind ic a tor of

capacitor degradation. In this regard, an ESR estima-

tion scheme based on sliding mode concepts and the

RLS algo rithm is proposed in this paper. The sliding

mode theory (Shtessel et al., 2014) has extensively

been applied to control and estimation/o bservation

problems, and still is a theory under development al-

beit the slidin g mode ideas began in the late 1950s

in the Soviet Union. Af te r a relatively extensive lite-

rature review, it became noticeab le that sliding mod e

concepts a re yet to be applied to the capacitor con-

dition monitoring p roblem. In this paper, a sliding

mode differentiator, which is robust with re spect to

input noise and exact in their absence (Levant, 1998),

is integrated into an ESR estimation scheme for capa-

citor condition monitor ing that a lso involves the RLS

algorithm with forgetting factor.

The main contributions of this paper are: (i) some

level on novelty since, to the best knowledge of the

author, sliding mode ideas have not been applied to

the problem of estimating ESR for capacitor condition

monitoring before, (ii) simplicity in terms of its form

and de sig n, and relative straightforward implementa-

tion at no extra c ost since the existing microcontroller,

DSP or FPGA devices are already used in many po-

wer electronic system s such as power converters, and

(iii) on-line nature which allows for real-time capaci-

tor conditio n monitoring.

2 PROBLEM FORMULATION

The most common type of aluminium electrolytic ca-

pacitor consists of two alum inium foils which form

the ano de and apparent cathode, paper separators, a

liquid electrolyte which is the real cathode, an alum i-

nium o xide layer (Al

) which is the dielectric and

of sufﬁcient thickness to w ithstand the rated voltage

of the capacitor, two terminals, and an aluminiu m can

(Alwitt and Hills, 196 5).

Figure 1: St r ucture of an electrolytic capacitor element.

The electrolytic capacitor elemen t depicted in Fi-

gure 1 may be modelled as the equivalent electric ci-

rcuit shown in Figure 2. The ideal capacitan ce is de-

noted by C

. The resistance R

represents the in su-

lation resistance. The resistance R

correspo nds to the

series resistance of foils, paper and terminals. The in-

ductance of win dings and connections is represented

by L.

Figure 2: Equivalent electric circuit of an electrolytic capa-

citor.

The impedance of the equivalent circuit, shown in

Figure 2, takes the following form

= R

ESR

+ j



ωL

ESL

−

ωC



[Ω] (1)

where the equivalent series inductance (ESL) is given

by L

ESL

= L [H], the ESR is given by

ESR

= R

1 + ω

[Ω] (2)

and the equivalent capacitance is given by

= C

1 +

[F] (3)

Furthermore, the magnitude of the imped a nce and the

resonant frequ e ncy are given b y

| =

ESR



ωL

ESL

−

ωC



[Ω] (4)

and

2π

√

ESL

[Hz] (5)

The impedance characteristics of an electrolytic ca-

pacitor is shown in Figure 3. For frequen cies f < f

impedance is dominated by the ca pacitive r eactance

ωC

, whereas for f > f

the impedance is domi-

nated by th e indu ctive reactance X

= ωL

ESL

. Moreo-

ver for f

< f < f

the impedanc e is mainly resistive

and hence dominated by R

ESR

Without lo ss of generality, it is assumed that the

power electronics system, on which the condition mo-

nitoring approach proposed in this paper will be app-

lied, is a DC-DC Buck converter. Consequently, since

DC-DC power converters typically operate at a lo-

wer frequency band compared with the resonant fre-

quency de ﬁned in (5) and the equivalent series induc-

tance L

ESL

is relatively small, e. g. in the order of nH,

Condition Monitoring of Electrolytic Capacitors via ESR Estimation with Recursive Least Squares and Sliding Mode Techniques

475

Figure 3: Frequency characteristics of |Z

ESL may be neglected. Nevertheless, the switching

frequency of the power co nverter is high enough, i.e.

typically in th e middle frequency range, for the im-

pedance Z

to be dominated by R

ESR

. Note that the

switching frequency that will be considered in Section

(4) is 10 kHz which means that the impedance Z

dominated by R

ESR

(see Figure 3). Sin c e this will be

the case considered in this paper, the equivalent cir-

cuit shown in Figure 2 c a n be simpliﬁed as shown in

Figure 4.

Figure 4: Simpliﬁed equivalent circuit of an electrolytic ca-

pacitor.

A widely app lied criterion states that the EoL of

electrolytic capacitors is reached when its capacitance

changes by 20% and/o r its ESR doubles (Hewitt et al.,

2016) (Soliman et al., 20 16). Other criteria are also

available. For instance, the military standard MIL-C-

62F has been used in (Celaya et al., 2 011) to deﬁne

that the EoF of an electrolytic capacitor is reached if

its ESR increases by 28 0-300% or the capacitance de-

creases by 20% with respect to the respective pristine

condition vales.

From the simpliﬁed equivalent circuit shown in

Figure 4, it follows that the difference of potential be-

tween anode and cathode te rminals is given by

(t) = R

ESR

(t)i

(t) +

(τ)dτ (6)

which, in turn, yields the following differential equa-

tion

(t)

ESR

(t)

+ R

ESR

(t)



(t) −i

(0)



(7)

By deﬁning

(t) =

(t)

−



(t) −i

(0)



(8)

(t) =

(t)

(9)

and

ϑ =

ESR

(t)

ESR

(t)

(10)

the differential equation given in (7) can be written as

a generic linear mode l (Ljung, 1999) (Me ndel, 2008),

which is linear in the unknown vector of parameters

ϑ, and takes the following form

(t) = ϕ

(t)ϑ + η

(t) (11)

where η

(t) ∈ℜ denotes the measurement noise.

Since the inc rease of ESR is the best indicator of

capacitor degradation (Wang et al., 2012), the pro-

blem to be addressed in this paper is the on-line es-

timation of the equivalent series resistance R

ESR

(t)

for condition monitoring of electrolytic capacitors

through the gene ric linear m odel given in (11). This

problem will be tackled within the context of a DC-

DC c onverter as in (Buiatti et al., 2007a) (Soliman

et al., 201 6).

3 PROPOSED SOLUTION

The measurement y

(t) and the observation vector

ϕ(t) of the generic linear model (11) involve the

potential difference of the anode-ca thode termina ls

(t) and the capacitor current i

(t), which are me-

asurable analogue signals, and their ﬁrst der ivatives

(t)

and

(t)

respectively. In this paper, these de-

rivatives will be obtaine d by using sliding mode dif-

ferentiators (Levant, 1998). The voltage v

(t) and

current i

(t) may be affected by noise and henc e the

classical differentiator based on linear approx imati-

ons will not produce satisfactory derivatives as will

be demonstrated in Section 4.

Although this paper is not conc erned with the phy-

sical imp le mentation of the proposed solution, two

conceptual designs are discussed in the sequel. The

scalar and vecto r signals y

(t) and ϕ(t) are obtained

using (8) and (9), which may be realised using opera-

tional ampliﬁers, and then these signals are sampled

at a frequency f

(in the order of tens or hundre ds of

kHz) prior to be processed by an RLS-based e stima -

tor implemented on a microcontroller, DSP or FPGA.

A b lock diagram of the condition m onitoring scheme

proposed in this paper is shown in Figure 5.

The proposed solution may also be im plemented

entirely on a digital programmable device. In this

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

476

Figure 5: Block diagram of the proposed ESR estimation

scheme for electrolytic capacitor condition monitoring.

case, the electrolytic capacitor voltage v

(t) and cur-

rent i

(t) would be sampled and a digital version

of the sliding m ode differentiators could be adopted.

Then, the representation of the signals y

[k] an d ϕ[k]

may be realised by software an d, of course, the RLS

algorithm is also implemented on software.

In what follows, the theory associated w ith the ro-

bust exact differentiator and the RLS estimation algo-

rithm is summarised.

3.1 Sliding Mode Differentiator

The sliding mode robust exact differentiator (Levant,

1998) corresponds to a second-order sliding mode (2-

sliding mode) technique. In order to introduce the

sliding mode differentiator used in the ESR estima-

tion scheme, some deﬁn itions and main theor etical re-

sults a re provided for the sake of mathematical rigour

and completeness. Consider the space of measura-

ble fun ctions (or signals) bounded on an interval [a,b]

denoted b y M

[a,b]

and to which the continuous-time

input signal s

(t) belongs to. It is also assumed that

(t)k = sup|s

(t)|.

A ﬁrst-order differentiator D is said to be an ex-

act differentiator if its output signal, i.e. ˙s

(t) =

(t)

matches the time derivative of the input signal f (t).

Note that the orde r of the differentiator is the order

of the derivative produced. T he robustness of the dif-

ferentiator is an important feature sin ce in real-world

applications the input signal may be corr upted with

relatively small high -freque ncy noise, whic h a lways

exists and may have a large derivative. In this regard,

if the output signal of a differentiator D tends uni-

formly to D s

(t) as the input sig nal tends unifor mly

to s

(t), then the differentiator D is said to be a ro-

bust differentiator. More over, if a differentiator D has

both the exactness and robustness properties deﬁned

previously, then it is said to be a correct differentiator.

For a practical real-time differentiator, as the ones

required by the electrolytic capacitor condition mo -

nitoring for generating

(t)

and

(t)

, it is assu-

med that the input signa l s

(t) is a me asurable locally

bounded functio n deﬁned on the inte rval [0,∞) and

consists o f a base signal involving a de rivative with a

Lipschitz constant L > 0 and noise (Levant, 1998).

Deﬁne the auxiliary differential equation

˙x(t) = y(t) (12)

and the following ﬁrst-order real-time robust exact

differentiator, whic h guarantees that x(t) → s

(t), i.e.

x(t) −s

(t) = 0 (Levant, 1993) (Levant, 1998):

y(t) = z(t) −λ|x(t) −s

(t)|

1/2

sgn(x(t) −s

(t)) (13)

˙z(t) = −κsgn(x(t) −s

(t)) (14)

where the signal y(t) is the output of the differentiator,

and scalars λ, κ ∈ℜ

are selected by the designer for

the convergence of y(t) to ˙s

(t). Sufﬁcient conditions

will be provided below.

The solution of equations (12)-(14) has to be in-

terpreted in the sense of Filip pov’s theory (Filippov,

1988). Following the description presented in (Le-

vant, 1998), deﬁne

Φ(κ,λ, L ) = |Ψ(t

∗

)| (15)

where



Σ(t),Ψ(t)



is the solution of the system

Σ = −|Σ|

1/2

+ Ψ (16)

Ψ =

(

−

(κ−L ) −|Σ|

1/2

+ Ψ > 0

−

(κ+ L ) −|Σ|

1/2

+ Ψ ≤0

(17)

Σ(0) = 0 (18)

Ψ(0) = 1 (19)

where κ > L , λ 6= 0 and t

∗

= inf{t| t > 0, Σ(t) =

0, Ψ(t) < 0} with t

∗

< ∞. Note tha t the function

Φ(κ,λ, L ) has to be d etermined throu gh computer si-

mulations. The convergence criter ion of y(t) to

f (t)

is stated in the following theorem.

Theorem 3.1 (Levant, 1998): Let κ > L > 0, λ > 0

and Φ(κ,λ,L ) < 1. Then, provided f (t) has a deri-

vative with Lipschitz constant L ( f ∈W (L ,2) where

W (L ,2) is the set of all input signals such tha t their

ﬁrst derivatives have a Lip schitz constant L > 0), the

equality y(t) =

f (t) holds identically after a ﬁnite-

time transient p rocess. 

In (Levant, 19 98), the f ollowing sufﬁcient conditions

for the convergence of y(t) to

f (t) are given

κ > L , (20)

≥ 4L

κ+ L

κ−L

. (21)

Condition Monitoring of Electrolytic Capacitors via ESR Estimation with Recursive Least Squares and Sliding Mode Techniques

477

Now consider the following assumptions: κ > L >

0, λ > 0, and Φ < 1.

Theorem 3.2 (Levant, 1998): Let s

(t) = s

(t)+η(t)

be the input signal, where s

(t) is a differentiable

base signal, s

(t) h as a derivative with Lipschitz con-

stant L > 0, and η(t) is the noise satisfying |η(t)| ≤

ε. Then, there exists a constant b > 0 depend e nt on

(κ − L )/λ

and (κ + L )/λ

such that after ﬁnite

time the inequality

|y(t) − ˙s

(t)| < λbε

1/2

(22)

is satisﬁed.

If κ and λ are chosen such that κ = ζ

L and λ =

1/2

respectively, then th e in equality

|y(t) − ˙s

(t)| <

1/2

(23)

holds for some

b(ζ

,ζ

) > 0. 

A discrete-time version of the sliding mode d if-

ferentiator can be obtained by applying the one-step

Euler method to (12)-(14) (Livne and Levant, 2014):

x[k + 1] = x[k] + T

z[k] −λ

|e[k]|

1/2

sgn(e[k]) (24)

z[k + 1] = z[k] −κ

sgn(e[k]) (25)

where

e[k] = x[k] −s

[k] (26)

and T

is the sample time. The positive scalars λ

and κ

are the design p a rameters of th e sliding mode

differentiator.

3.2 Recursive Least Squares Estimator

Since the p arameter of interest for electrolytic capaci-

tor condition monitorin g is R

ESR

(t), which is a slo-

wly time-varying para meter, RLS with exponential

forgetting is integrated into the condition mon itoring

scheme proposed in this pap e r. The calculations in-

volved in this RLS method are given in the following

theorem.

Theorem 3.3 (Recursive Least Square s with Expo-

nential Forgetting): Given the estimation linear mo-

del

ˆy

[k] = ϕ

[k]

ϑ[k] + η

[k] (27)

where ˆy

∈ ℜ is the model output, ϕ ∈ ℜ

is the re-

gressor vector which is a kn own deterministic vector

(the components of ϕ are said to b e the regression va-

riables), and

ϑ ∈ℜ

is the parameter vector estimate.

Provided that

ϑ[0] =

and P[0] = P

, the least

squares estimate of the vector of unkn own parameters

ϑ satisﬁes the following recursive equations:

ϑ[k] =

ϑ[k −1]+K [k]



[k] −ϕ

[k]

ϑ[k −1]



(28)

where

K [k] = P [k]ϕ[k] (29)

= P[k −1]ϕ[k]



RLS

+ ϕ

[k]P [k −1]ϕ[k]



−1

(30)

and

P[k] =

RLS



I −K [k]ϕ

[k]



P[k −1] (31)

and minimises the cost function

ϑ,τ) =

∑

k=1

τ−k

RLS



[k] −ϕ

[k]

ϑ[k]



(32)

where the forgetting factor λ

RLS

is such that 0 <

RLS

< 1. 

The initial value P (0) is chosen as

P(0) = γ

RLS

I (33)

where γ

RLS

∈ ℜ is a large number.

The gain vector K [k] in Theorem (3.3 ) provides

a p roportional correction to the difference between

the ac tual measurement y

[k] and the estimate out-

put ˆy

[k] based on the previous parameter estimate

ϑ[k − 1]. Moreover, cost function (32) involves a

time-varying weighting of the squared of this diffe-

rence, i.e . e

= (y

[k] −ϕ

[k]

ϑ[k −1])

, through the

factor λ

τ−k

RLS

. Thus, it is possible to give more empha-

sis to recently observed measurements ra ther than to

older data.

Note that λ

RLS

= 1 corresponds to the fundamental

RLS a lgorithm in which all data are weighted equally

and the algorithm has an inﬁnite memory length (Zhu-

ang, 1998). The ch oice of λ

RLS

is a trade-off between

tracking and noise sensitivity ( Ljung, 1999). That is,

less we ight to older me a surements, i.e. λ

RLS

small,

means that older data is forgo tten faster. However,

the algorithm is more sensitive to noise. Usually,

the forgetting factor λ

RLS

is c hosen from the interval

[0.95 , 0.99]. In (Isermann and M¨unchhof, 2011), it

is suggested that λ

RLS

has to be selected as follows:

(i) a small λ

RLS

if the rate of c hange of the parameter

is large and only small noise is allowed; (ii) a large

RLS

if th e rate of change of the parameter is small

and noise can be larger.

4 DESIGN AND SIMULATION

In this section, a DC-DC buck converter (see Figure

6) with nominal parameter values given in Tab le 1 is

considered. For this pu rpose, a model of th e entire sy-

stem (i.e. DC-DC converter and condition monitor ing

scheme) was implemented on MATLAB/ Simulink.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

478

Figure 6: DC-DC buck converter schematic.

Table 1: Nominal parameter values for the DC-DC Buck

Converter.

L r

C ESR R

DS V

[V] [H] [µΩ] [mF] mΩ Ω [kHz] [%] [V]

200 0.21 68.7 1.041 2 5 1.584 10 50 100

Sliding mode differentiators for estimation of

(t)

and

(t)

were designed and their gains are

shown in Table 2.

Table 2: Sliding mode differentiator gains.

Differentiator 1 Differentiator 2

λ 420 ×10

8.40 ×10

κ 4.7250 ×10

2.2 ×10

The RLS with forgetting factor algorithm, presen-

ted in Theorem 3 .3, was implemented as a MATLAB

function. The following parameter values w e re used

on the computer simulations: λ

RLS

= 0.99, ϑ(0) =

[0 0]

and F (0) = 100I. A large constant forget-

ting factor has been selected sinc e ESR is a slowly

time-varying parameter. Furthermore, this large λ

RLS

allows for a reduced estimation no ise. The sample

time considered for the RLS estimator was 100 kHz,

i.e. 10 f

. Regarding, the MATLAB/Simulink model

conﬁguration parameters, they were set a s follows:

ode1 (Euler) solver with ﬁxed-step size (fu ndamental

sample time) of 1 ×10

−9

seconds.

An experime nt involving a simulated change in

the ESR value (see, for example, the black colour

curve in Figure 12) was carried out. Note that in or-

der to emulate the electroly tic capacitor d egra dation,

the E SR value was gradually increased. Moreover,

the capacitor voltage and curr e nt measurements were

corrupted by noise. In this respect, two different uni-

formly d istributed random signals were injected to the

capacitor voltage and current measurements, i.e. η

amplitude = ±1 ×10

−3

[V] and seed = 2, and η

amplitude = ± 5 ×10

−3

[A] and seed = 4 . The sample

time of 1 ×10

−6

seconds was used for both signals.

The results obta ined with the pr oposed scheme are

compare d against the following c la ssical ﬁrst-order li-

near appr oximation differentiator (LAD):

G(s) =

100 ×10

s + 1

(34)

This differentiator was used alongside the very same

RLS with forgetting algorithm used for the sliding

mode based scheme. In this paper, this approach will

be called the classical scheme or method.

A detail of the perf ormance of the DC-DC buck

converter is shown in Figure 7.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-3

100

150

out

(t) [V]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-3

(t) [A]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-3

(t) [A]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

time [s]

-3

-0.5

0.5

1.5

Figure 7: DC-DC buck converter dynamic response (zoom).

The ﬁrst derivatives

(t)

and

(t)

, obtained

with the sliding m ode differentiato rs, are shown in Fi-

gures 8 and 9 respectively. On the other hand, the

same derivatives but produced by the classic LAD

are depicted in Figures 10 and 11. The better per-

formance of the sliding mode differentiator is evident

from these graphs. This will become even more evi-

dent wh e n assessing the performance of b oth condi-

tion monitoring schemes later on.

The time evolution of the true and estimated ESR,

using the proposed scheme and the classical met-

hod, are shown in Figure 12 and 13. Furthermore,

the corresponding errors e

ESR

SMS

= |R

ESR

−

ESR

SMS

and e

ESR

= |R

ESR

−

ESR

| are provided in Figure

14. The per formance of the classical method (CM)

is indeed unaccep ta ble because it cannot estimate the

ESR in the case of noisy measurements and hence

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

-1

d v

/dt [V/s]

First derivative of v

(t)

0 0.5 1 1.5 2 2.5 3 3.5 4

time [s]

-3

-1

d v

/dt [V/s]

First derivative of v

(t) (zoom)

Figure 8: First derivative of the capacitor voltage, i.e.

(t)

, obtained with the sliding mode differentiator.

Condition Monitoring of Electrolytic Capacitors via ESR Estimation with Recursive Least Squares and Sliding Mode Techniques

479

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

-1

d i

/dt [A/s]

First derivative of i

(t)

1 1.5 2 2.5 3 3.5 4

time [s]

-4

-1

d i

/dt [A/s]

First derivative of i

(t) (zoom)

Figure 9: First derivative of the capacitor current, i.e.

(t)

obtained with the sliding mode differentiator.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

-1

d v

/dt [V/s]

First derivative of v

(t)

0 0.5 1 1.5 2 2.5 3 3.5 4

time [s]

-3

-1

d v

/dt [V/s]

First derivative of v

(t) (zoom)

Figure 10: First derivative of the capacitor voltage, i.e.

(t)

, obtained with the classical ﬁrst order LAD.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

-1

d i

/dt [A/s]

First derivative of i

(t)

1 1.5 2 2.5 3 3.5 4

time [s]

-4

-1

d i

/dt [A/s]

First derivative of i

(t) (zoom)

Figure 11: First derivative of the capacitor current, i.e.

(t)

, obtained with the classical ﬁrst order LAD.

has a much bigger estimation error than the propo-

sed scheme. The sliding mode-ba sed scheme (SMS)

outperforms the classical approach.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

ESR [ ]

Equivalent Series Resistance (ESR)

Figure 12: True and estimated ESR obtained with the pro-

posed scheme.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

-50

ESR [ ]

Equivalent Series Resistance (ESR)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

-1

-0.5

0.5

ESR [ ]

Equivalent Series Resistance (ESR)

Figure 13: True and estimated ESR obtained with the clas-

sic method.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

Abs(error) [ ]

-3

Estimation error (Proposed Scheme)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

time [s]

Abs(error) [ ]

Estimation error (Classical Method)

Figure 14: Estimation errors.

5 CONCLUSIONS

The sliding mod e RLS based scheme for ele ctrolytic

capacitor con dition monitoring proposed in this pa-

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

480

per is capable of estimating the equivalent series re-

sistance (ESR) on-line desp ite noisy me asurements.

A constant forgetting factor has been used in the RLS

algorithm since the ESR is a slowly time-varying pa-

rameter. Robust slidin g mode differentiation ha s been

satisfactorily applied for calculating signals required

by the RLS algorithm.

The new co ndition monitoring approach outp er-

forms an equivalent scheme based on linear approx-

imation differentiators and the RLS with forgetting

factor algorithm. The scheme is relatively simple in

its form and design. A detailed design example has

illustrated the simplicity of th e metho d. Moreover,

computer simulation results have demonstrated the ef-

fectiveness of the new capa citor condition monitoring

scheme in which th e degradation of an electrolytic ca-

pacitor on a DC-DC buck converter has been conside-

red as proof of co ncept.

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