Second-Order Sliding Mode Algorithmsfor the Reconstruction of Leaks

Marco Tulio Angulo and Cristina Verde

Abstract—A framework for leak reconstruction in pipelinesusing second-order sliding mode is presented. Two cases areconsidered. In the case of a single leak, the necessary andsufficient condition that allows estimating the position of theleak is determined and discussed. Under such a condition,an algorithm that determines the position and flow of theleak in finite-time is introduced. In the case of two leakswith known positions, a finite-time estimation of the leaksflow is obtained. Experimental results for this second case arepresented, together with a comparison of the performance offirst and second-order sliding mode algorithms.

Index Terms—Sliding mode observers, leak detection.

I. INTRODUCTION

The reliability of pipeline networks is an important andconstant concern in fluid transportation systems. Althoughpipelines are protected against damage, pressure surges in thenetwork often induce leaks and line breaks. Algorithms forthe reconstruction of leaks in pipelines allow the reductionof the number of sensors needed to detect them. Suchalgorithms have two main objectives: estimating the flow andthe position of the leak [1], [2], [3]. These two objectivescan be cast as a Fault Detection and Isolation (FDI) problemin the pipeline system.

FDI for dynamic systems is commonly addressed using atwo-step methodology [4]. In the first step, residual genera-tors are designed using observers or parity equations, whichsupply real-time information of the difference between mea-surements of the plant and signals evaluated from analyticalmodels. In the second step, faults are identified by evaluatingsuch residuals.

Experience has shown that solving the FDI problem inthe pipeline system is particularly difficult for the case ofsmall leaks or multiple leaks in different locations. Forinstance, when only pressures and flows at the extremes ofthe pipe are known, the location of multiple leaks cannot bedetermined if the fluid is in steady-state [5], [6]. Hence, aspecific signal in the valves is required to excite the fluidpermitting the isolation of the leak [7]. Recently, in [8],a high-gain extended Kalman Filter has been proposed todetect both the out-flows of the leaks and their locations. Inthis last reference, the location of the leaks are consideredas additional states in an augmented orthogonal collocation

M. T. Angulo is with Division de Posgrado e Investigacion, Facultad deIngenierıa, UAQ, Mexico, email: darkbyte@gmail.com. Cristina Verde iswith Instituto de Ingenierıa, UNAM, Mexico, email: verde@unam.mx.

model for the fluid. However, special excitation conditionsare required for the convergence of the estimation. In theseprevious works, the necessary conditions for determining thelocations of the leaks are not obtained.

In the last decade, First-order Sliding Mode Observers(FSMOs) have been successfully applied to solve FDIproblems [9]. In [10], they are applied to reconstruct theflow of multiple leaks by successively estimating internalstates. The main characteristic of FSMOs is the presence ofdiscontinuous injections, allowing their convergence despitea mismatch between the actual system and the model used toconstruct the observer. Such a mismatch can be quantified byfiltering the discontinuous injections of the FSMO to obtainthe equivalent output injection error [11]. This informationcan then be used to reconstruct faults in the system [12].Nevertheless, the filtering process is inexact, and selectingthe filter’s time constant in the presence of measurementnoise is difficult [11].

In this paper, our contribution is a framework for leakreconstruction using a minimal order model for the leakstogether with Higher-Order Sliding Mode (HOSM) estima-tors. In particular, we use the Generalized Super-Twisting(GST) algorithm [13] and the HOSM differentiator [14].With respect to first-order sliding modes, HOSM algorithmsoffer improved precision in the presence of noise [14] anddo not require filtration.

The proposed framework considers two cases. Firstly,the case of one leak is considered, and the necessary andsufficient conditions for the identification of its positionand out flow are determined. Under such conditions, aGST-based algorithm that estimates them in finite-time isproposed. In the second case, two leaks with known positionsare considered, and a finite-time estimation of their flowis obtained using HOSM differentiators. In this last case,experimental results in a water pipeline approximately 200meters long are also provided, together with a comparisonof the performance of first- and second-order sliding modealgorithms.

The remainder of the paper is organized as follows.Section II considers the problem of detection and isolationof a single leak. The problem when two leaks are present isanalyzed in Section III. The experimental results for the two-leaks scenario are discussed in Section IV, together with acomparison of the performance of first- and second-ordersliding mode algorithms. Finally, Section V offers someconclusions and directions of future work.

II. DETECTION AND ISOLATION OF A SINGLE LEAK

The simplest and minimal-order discretized model for apipeline of length L with a leak at position z1 has twosections and can be written as

Q1 = −gAz1

(H2 − u1)− µQ1|Q1|,

H2 = − b2

gAz1(Q2 −Q1 − w), (1)

Q2 = − gA

L− z1(u2 −H2)− µQ2|Q2|,

where g = 9.81, A, L and b are known parameters char-acterizing the geometry of the pipe [6]. The parameterµ = f/(2DA) > 0 is also assumed known, with D thediameter of the pipe and f its Darcy-Weisbach frictioncoefficient. The input and output flows Q1 and Q2 aremeasured, as well as the pressures u1 and u2; the pressuressatisfy u1 ≥ u2 > 0. The term w is associated with the leakcaudal at position z1 and pressure H2. The objective is thereconstruction of both the position z1 and the caudal w.

A. Identifiability of the position of the leak

The first objective is to recover z1; therefore, one first askswhen recovery is possible. To answer this question, note thatthe time derivative of (Q2 −Q1)z1 does not depend on theunmeasured state H2. In fact, the following expression canbe derived:

Q2 = Γz1 + v0, (2)

where v0 = gAL−1(u1 − u2) − µQ2|Q2|, and Γ can bewritten as

Γ = (Q2 − Q1)/L− v1

with v1 = (µ/L)(Q1|Q1| − Q2|Q2|). The value of v0 andv1 can be directly obtained using measured information ofthe variables of the system.

Equation (2) has the typical form for the problem of iden-tification of parameters. Therefore, a necessary and sufficientcondition for the identifiability of the leak’s position z1 isthat the regressor Γ(t) is not identically zero.

By using (2) together with (1), deriving the necessaryand sufficient conditions that allow the identification of theparameter z1 is possible. From (2), the condition Γ = 0 isequivalent to

Q2 + µQ2|Q2| =gA

L(u1 − u2),

and by (1) this is also equivalent to

− gA

L− z1(u2 −H2) =

gA

L(u1 − u2).

In addition, after some algebraic manipulations, this lastequation is finally equivalent to the condition

u1 −H2

z1=u1 − u2L

. (3)

Thus, the absence of the condition given by (3) is anecessary and sufficient condition for estimating the positionof the leak z1. In particular, (3) is trivially satisfied whenthe initial conditions of the system are zero together withthe input and output pressures. A more interesting remarkis that, if the initial conditions of the flows Q1 and Q2 areidentical (for instance zero), then condition (3) can be onlysatisfied in the absence of leaks.

For proving this last claim, assume that (3) holds. Thenthe equations for the flows in (1) read as

Qi =gA

L(u1 − u2)− µQi|Qi|, Qi(0) = Qi0,

for i = 1, 2. Since

Q2 − Q1 = −µ (Q2|Q2| −Q1|Q1|) ,

and µ > 0, the difference Q2−Q1 is independent of the inputand satisfies Q2(t)−Q1(t)→ 0 as t→∞. Hence, if Q10 =Q20, Q1(t) = Q2(t) for all t ≥ 0. This means the input andoutput flows are identical, so there is no leak flow. In thecase Q10 6= Q20, we still have Q2(t)→ Q1(t), which meansthat the leak flow asymptotically disappears. This conditioncannot physically occur and, hence, the position of a singleleak is always identifiable in practice provided that there isan input pressure.

B. Estimating the position of the leak

The value of the regressor Γ cannot be directly obtainedusing measured information, since it also depends on thederivative of Q21 = Q2 −Q1.

For the estimation of the regressor Γ, let us assume that|Γ(t)| ≤ c, ∀t ≥ 0 with c ≥ 0 a known constant. Under thisassumption, the following observer based on the generalizedsuper-twisting algorithm [13] can be used

˙Q21 = −l1φ1

(Q21 −Q21

L

)+ Γ + v1

˙Γ = −l2φ2

(Q21 −Q21

L

), (4)

where l1 and l2 are positive gains of the observer and thefunctions φi, for i = 1 and 2, are given by

φ1(θ) =µ1|θ|12 sign(θ) + µ2θ,

φ2(θ) =1

2µ21 sign(θ) +

3

2µ1µ2|θ|

12 sign(θ) + µ2

2θ,(5)

with µ1, µ2 ≥ 0. The parameters µi balance the linear andnonlinear terms of the algorithm. In particular, when µ1 > 0,the observer (4) contains discontinuous terms in its right-hand side.

The discontinuities in the observer (4) allow its exact andfinite-time convergence despite nonvanishing disturbances.It is straightforward to verify that the dynamics of theobservation errors Q21 = Q21 − Q21 and Γ = Γ − Γ takethe same form as in [13]. Then, under the conditions

l1 > 2µ−11

√c and l2 > 2µ−2

1 c,

0 0.5 1 1.5 2 2.5−2.5

−2

−1.5

−1

−0.5

0

time [s]

Γ

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

time [s]

z1

0 2 4 6 8 10−0.2

−0.1

0

0.1

time [s]

w

Fig. 1. Simulation results for the reconstruction of the flow and position for a single leak with uncertainties and measurement noise. The (solid) blue linedenotes the estimation obtained knowing the exact value of the parameters of the system. The (dashed) green and brown lines show the estimation witha deviation of ±10% in the friction coefficient µ, respectively. The (pointed) gray line corresponds to the true value of each variable. Left: estimation ofthe regressor Γ(t). Middle: estimation of the position of the leak z1. Right: estimation of the flow of the leak w(t).

the identity Γ(t) = Γ(t) will be established in finite-time and kept afterwards. When the observer (4) containsdiscontinuous terms, i.e. µ1 > 0, its convergence requiresonly a finite gain.

Once the regressor has been estimated, using any param-eter estimation method is possible to recover the positionz1 by eq. (2). In this paper, a recently reported parameterestimation method that also contains GST terms is used[15]. In contrast to most parameter estimation algorithms,this new method allows to obtain a finite-time estimation ofthe parameters. Moreover, it has been shown to be robustagainst uncertainties in the model and noises.

For the reconstruction of leaks, the algorithm takes theform

˙Q2 = −k1φ1(Q2 −Q2) + Γz1 + v0˙z1 = −k2Γφ2(Q2 −Q2), (6)

where z1 denotes the estimated value of z1. The parametersk1 and k2 are positive gains of the algorithm, and thefunctions φi, for i = 1 and 2, take the same form asin (5). This means that the algorithm (6) also contains itsparticular set of values µi. As proven in [15], for any set ofgains, µ1 > 0 and Γ(t) of persistent excitation, the identityz1(t) = z1 will be established after a finite-time transient.Recall also that Γ(t) will be identical to Γ(t) after a finite-time.

C. Reconstruction of the out flow

Once z1 has been estimated, the second objective is thereconstruction of the out flow w leaving through the leak.After some algebraic manipulations using (1), the followingalgebraic equation for w is obtained

w =gAz1b2

(u1 −

2µ

gA|Q1|Q1 − Q1z1

)+Q2 −Q1

From here, the main idea is to estimate the right-handside of the equation above to reconstruct the leaks flow.Implementing this idea requires derivatives of measuredvariables of the system, together with the value of z1whichhas been estimated using the algorithms (4) and (6) given inSection II-B.

For estimating the required derivatives, employing theHOSM differentiator [14] is proposed. This differentiatoroffers a finite-time exact estimate of the required derivativesand the best accuracy order in the presence of measurementnoises. The second-order HOSM differentiator for estimatingthe required derivatives of Q1 takes the form

ζ0 = ν0 = −3β1/3|ζ0 −Q1|2/3 sign(ζ0 −Q1) + ζ1,

ζ1 = ν1 = −1.5β1/2|ζ1 − ν0|1/2 sign(ζ1 − v0) + ζ2,

ζ2 = −1.1β sign(ζ2 − ν1),

where β > 0 is the gain of the differentiator. Assuming that|Q(3)

1 (t)| ≤ β, ∀t ≥ 0, the identities ζ1 = Q1, ζ2 = Q1 willbe established and retained afterwards. Similarly, the first-order HOSM differentiator for estimating the derivative ofu1 has the form

ξ0 = ν1 = −1.5β1/2|ξ0 − u1|1/2 sign(ξ0 − u1) + ξ1,

ξ1 = −1.1β sign(ξ1 − ν1),

where β > 0 is the gain of the differentiator. Assuming that|u1(t)| ≤ β, ∀t ≥ 0 the identify ξ1 = u1 will satisfy after afinite-time.

With both differentiators and the estimated position of theleak obtained by (6), the following estimator is proposed

w =gAz1b2

(ξ1 −

2µ

gA|Q1|ζ1 − ζ2z1

)+Q2 −Q1, (7)

which depends on measured information of the system andthe variables of the HOSM differentiators. Thus, providedthat the conditions described above hold, a finite-time esti-mation of the flow of the leak will be obtained.

D. Simulation results

Consider the parameters of model (1) given in TableI, together with the pressures at the extremes given byu1 = 1000 and u2 = 30. A band-limited white noise withpower of 9×10−5 was included in the measurement of bothpressures, generating average deviations of 0.2 units. Theoutflow is simulated using w = 0.0009

√H2.

The parameters for observer (4) were chosen as µ1 =0.1, µ2 = 20, l1 = 7, l2 = 12.25. The parameters for theestimation algorithm (6) were chosen as µ1 = 0.5, µ2 =

1, k1 = 5, k2 = 6.25. The gains of the HOSM differentiatorswere chosen as β = 10 for the second-order differentiatorand β = 100 for the first-order differentiator.

Parameter z1 L A µ b gValue 5 10 0.0081 17.057 15.4080 9.81

TABLE IPIPELINE PARAMETERS CONSIDERED IN SECTION II

To validate the proposed framework, three simulationscenarios were considered: the first when the estimationalgorithms know the exact values of the parameters of thesystem and second and third when a ±10% of uncertaintyin the parameter µ is considered, which occurs in practice.

The results by simulation are shown in Fig. 1. When theparameters of the system are exactly known, the positionand leak flow are correctly estimated despite the presence ofnoise. With a ±10% uncertainty in the parameter µ associ-ated with the friction, there is less than 10% error in the leakposition, and the leak flow is still correctly reconstructed.This result is similar to previously tested FDI algorithms[16] where the leak flow is correctly reconstructed, but itsposition is very sensitive to µ.

III. RECONSTRUCTION OF TWO OUTFLOWS

In this section the objective is the reconstruction of theflows leaving through two leaks. In contrast to the previoussection, the position of the leaks are assumed known.

If two leaks occur, the spatial variable z in the modelneeds to be discretized at least in three sections; see Fig. 2.As a result, a minimal model for a pipeline of length L withleaks at positions z1 and z1 + z2 is given by

Q1 =gA

z1(u1 −H2)− µQ1|Q1|,

Q2 =gA

z2(H2 −H3)− µQ2|Q2|,

Q3 =gA

L− z1 − z2(H3 − u2)− µQ3|Q3|, (8)

H2 =b2

gAz1(Q1 −Q2 − w1),

H3 =b2

gA(L− z1 − z2)(Q2 −Q3 − w2),

where Qi and Hi, i = 1, 2, 3, are flows and pressures heads,respectively. The variables w1 and w2 are associated withthe outflows at positions z1 and z1 + z2, respectively.

z1 z2

L

Q1 Q2 Q3

u1 H2 H3 u2

Fig. 2. Variables definition for the case of two leaks

The flows Q1, Q3 and pressures u1, u2 are assumed tobe measured. Thus, the objective is to estimate the flows w1

and w2 using such measurements and their derivatives.For estimating the required derivatives, a Generalized

Super-Twisting (GST) differentiator [13] is proposed insteadof the first-order HOSM differentiator described in Subsec-tion II-C. For estimating the first derivative of a scalar signalσ(t), the GST differentiator takes the form

ξ0 = −k1φ1(ξ0 − σ) + ξ1, ξ1 = −k2φ2(ξ0 − σ), (9)

where k1, k2 are the gains of the differentiator and thefunctions φi, i = 1 and 2, are again given as in (5). Assumingthat |σ(t)| ≤ β, the identity ξ1 = σ will be established aftera finite-time transient if the gains are selected as in [13]

k1 > 2µ−11

√β and k2 > 2µ−2

1 β.

Compared to the first-order HOSM differentiator, the GSTdifferentiator also contains linear terms. These terms provideadditional flexibility to obtain a faster response withoutdeteriorating its precision in the presence of noise.

In what follows, the equations that contain derivatives ofthe measured variables are written down. These derivativeswill be estimated using the GST differentiator (9) to recoverthe flows.

A. Recovering the outflow

Directly from model (8), one obtains the following rela-tions:

H2 = u1 − z1gA (Q1 + µQ1|Q1|),

H3 = L−z1−z2gA (Q3 + µQ3|Q3|) + u2, (10)

which depend only on measured data and its derivatives.Recall that the derivatives appearing above will be obtainedfrom the GST differentiator. From there, one can take onemore derivative to compute

w1 + w2 = Q3 −Q1 −gA

b2[z1H2 − (L− z1 − z2)H3],

which is the caudal of the equivalent leak. The variablesabove constitute the strongly observable part of the system[17]: the part that can be written as a function of measureddata and its derivatives. To distinguish the flows, however,additional information is required.

Such additional information can be obtained from the stateQ2, that is not observable but only detectable. Hence, toreconstruct it, the only possibility is to use an observer thatis only a copy of the system:

˙Q2 =

gA

z2(H2 −H3)− µQ2|Q2|, (11)

where the variable H2 − H3 is an input to this observergiven by the expression (10). Since the friction coefficient µis strictly positive, its error Q2−Q2 is asymptotically stable;nevertheless, its velocity of convergence cannot be adjustedand is determined solely by the friction of the system itself.

From the estimation of Q2, the flow of the first leak isestimated by

w1 = Q1 − Q2 −gAz1b2

H2,

and one can use this estimator on the equivalent leak toobtain w2.

IV. EXPERIMENTAL RESULTS

In this Section, the experimental results for the recon-struction of the leaks using the algorithm given above arepresented.

As discussed before, recall that the proposed frameworkuses the GST differentiator (9) to estimate the requiredderivatives. Here, we also compare the performance of theframework when the GST differentiator is replaced with adifferentiator that uses a first-order sliding mode algorithm toestimate the derivative. This section starts by describing theconstruction of such a differentiator. Afterwards, the waterpipeline plant used for the experimental setup is described,and the experimental results are presented and discussed.

A. Differentiation using first-order sliding modes

The differentiator using first-order sliding mode is basedon the concept of equivalent injection or equivalent control,as appears in the classical reference [11].

For estimating the first derivative of a signal σ(t) witha continuous and bounded derivative, such a differentiatortakes the following form:

˙σ = −β sign(σ − σ). (12)

Assuming |σ(t)| < β,∀t ≥ 0, it is clear that the identityσ(t) = σ(t) will be satisfy after a finite time. Once thatσ = σ, this implies that

z = σ, z = −β sign(σ − σ).

From these relations one identifies a contradiction; z is adiscontinuous signal, and it is simultaneously written asa continuous signal σ. This apparent contradiction can becancelled out by using the equivalent control concept [11].This implies that the average of the discontinuous signal zcoincides with σ. The average zeq of the signal z can beobtained by using the filter

τ zeq = −zeq + z.

It is shown also in [11] that zeq → σ in the limit whenτ → 0 and h/τ → 0, where h is the sampling step which isused to numerically solve the differential equation above.

Note that compared to the GST (9) and first-order HOSMdifferentiators, the differentiator above requires additionalfiltration to obtain the estimation of the derivative which isexact only in a limit.

0 2 4 6 8 10 12 14 16 18 200

0.005

0.01

0.015

0.02

Input Q1 and output Q

3 flows

0 2 4 6 8 10 12 14 16 18 20−5

0

5

10

15

20

Input u1 and output u

2 pressures

normalized time

Fig. 4. Normalized experimental data for the case of two leaks. Top: input(solid) and output (dashed) flows. Bottom: input (solid) and output (dashed)pressures.

B. Description of the water pipeline and results

The experimental results were obtained using a pilot waterpipeline [18]. It consists of a pipeline of galvanized iron witha diameter D = 0.1016 [m], 200.165 [m] of longitude and acoefficient f = 0.0281. The velocity of the pressure wave bis 1248 [m/s]. Two servo-valves located at 11.5 [m] and 49.8[m] are used to generate the leaks. Under these conditions,the parameters of model (7) take the values shown in TableII, together with µ = f/(2DA) = 17.057.

gAz1

gAz2

gAL−z1−z2

b2

gAz1

b2

gA(L−z1−z2)

0.006916 0.002076 0.000529 541233 137859

TABLE IIPARAMETERS OF THE SYSTEM IN SECTION IV

The data was acquired with a sampling time of 1 s duringan interval of 1618 s. For simplicity in the presentation ofthe variables, the time scale was normalized to 20 units,multiplying the time by T = 0.0124. Fig. 4 shows theevolutions of the normalized pressures and flows at thepipeline extremes.

Figs. 5 and 3 show the estimations with both differentia-tors: the GST differentiator (9) and the differentiator (12).The outflows w1, w2, the intermediate flow Q2 and pressuresH2 and H3 at the leaks’ positions are presented. From Fig.3, one can observe that the estimated pressures H2 and H3

using the GST differentiator (9) have fewer high-frequenciesthan the ones obtained using (12). Nevertheless, this im-provement is not reflected in the estimation of leaks’ flows inFig. 5. The reason for this effect is that the estimation of Q2

by the observer (11) acts as a filter that eliminates the highfrequency oscillations of the differentiator (12). Note that inpractice, the most important variable to be quantified is thelost volume per time unit, i.e., V (t) =

∫ t

0(w1(s)+w2(s))ds.

For this last variable, both differentiation algorithms give asimilar performance.

0 2 4 6 8 10 12 14 16 18 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Q2

time [s]

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

normalized time

H2

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

14

16

18

normalized time

H3

Fig. 3. Estimated flow Q2 and pressures H2, H3 for two leaks using experimental data. Blue: estimation using the GST differentiator. Gray: estimationusing the differentiator with first-order sliding mode.

0 2 4 6 8 10 12 14 16 18 20−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

w1

0 2 4 6 8 10 12 14 16 18 20−1

0

1

2

3

4

5

6

7

8

9

x 10−3

normalized time

w2

Fig. 5. Flows Reconstructions using experimental data. Top: w1(t), bot-tom: w2(t). Dashed-black: signal of aperture for the servo valve (0 meansclosed). Blue: estimation using the GST differentiator. Gray: estimationusing the differentiator with first-order sliding mode.

V. CONCLUSIONS AND FUTURE WORK

A framework for the reconstruction of one and two leaksin pipelines using second-order sliding modes was presented.In the case of one leak, the proposed framework provides afinite-time estimation of the location and flow of the leakunder necessary and sufficient conditions. These conditionswere discussed from theoretical and practical points of view.

In the case of two leaks, the framework requires to knowthe position of the leaks and provides a finite-time estimationof the flow of each leak. In this case, the algorithms weretested using experimental data and the performance of first-and second-order sliding mode algorithms was compared.With respect to the final objective of estimating the outflow,both kinds of algorithms demonstrated an equally feasibleperformance.

Future work includes studying the necessary and sufficientconditions for estimating the position of two leaks.

REFERENCES

[1] S. Scott and M. Barrufet, “Worldwide assessment of industry leakdetection capabilities for single and multiphase pipelines,” Departmentof Petroleum Eng, Texas A&M University, Internal Report 18133,August 2003.

[2] P. Lee, J. Vitkovsky, M. Lambert, A. Simpson, and J. Liggett, “Leaklocation using the pattern of the frequency response diagram inpipelines: a numerical study,” Journal of Sound and Vibration, vol.284, pp. 1051–1075, 2005.

[3] M. Ferrante and B. Brunone, “Pipe system diagnosis and leak detec-tion by unsteady-state tests: 1. harmonic analysis,” Advances in Waterresources, vol. 26, pp. 95–105, 2003.

[4] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosisand Fault Tolerant Control. Berlin: 2nd. Ed. Springer, 2006.

[5] L. Billman and R. Isermann, “Leak detection methods for pipelines,”Automatica, vol. 23, no. 3, pp. 381–385, 1987.

[6] C. Verde, N. Visairo, and S. Gentil, “Two leaks isolation in a pipelineby transient response,” Applied Water Resources,, vol. 30, pp. 1711–1721, 2007.

[7] A. F. Colombo, P. Lee, and B. W. Karney, “A selective literaturereview of transient-based leak detection methods,” Journal of Hydro-Environmental Research, no. 2, pp. 212–227, 2009.

[8] L. Torres, G. Besancon, and D. Georges, “Multi-leak estimator forpipelines based on an orthogonal collocation model,” in 48th IEEEConference on Decision and Control, Shanghai, China,, 2009.

[9] C. Edwards, S. K. Spurgeon, and R. J. Patton, “Sliding mode observersfor fault detection and isolation,” Automatica, vol. 36, pp. 541–553,2000.

[10] M. Negrete and C. Verde, “Multi-leak reconstruction in pipelines bysliding mode observers,” in 8th IFAC Symposium SAFEPROCESS.IFAC, August 2012, pp. 934–939.

[11] V. I. Utkin, Sliding Modes in Control and Optimization. SlidingModes in Control and Optimization: Springer, 1992.

[12] C. Edwards, H. Alwi, and C. P. Tan, “Sliding mode methods for faultdetection and fault tolerant control,” in Conference on Control andFault Tolerant Systems Nice, France, October 6-8, 2010.

[13] J. A. Moreno, “Lyapunov approach for analysis and design of second-order sliding mode algorithms,” in Sliding Modes after the first decadeof the 21st Century, L. Fridman, J. Moreno, and R. Iriarte, Eds.Springer-Verlag, 2011, pp. 113–150.

[14] A. Levant, “High-order sliding modes: differentiation and outputfeedback control,” International Journal of Control, vol. 76, no. 9-10, pp. 1924–041, Nov. 2003.

[15] J. Moreno and E. Guzman, “A new recursive finite-time convergentparameter estimation algorithm,” in 18th IFAC World Congress Milano(Italy), 2011, pp. 3439 –3444.

[16] L. Jimenez and C. Verde, “Multi-fault discrimination with fault modeland periodic,” in 8th IFAC Symposium SAFEPROCESS. IFAC, August2012, pp. 49–54.

[17] M. Angulo, L. Fridman, C. Moog, and J. Moreno, “Output feedbackdesign for exact state stability of flat nonlinear systems,” in 2010 11thInternational Workshop on Variable Structure Systems (VSS), 2010,pp. 32–38.

[18] C. Verde, L. Molina, and R. Carrera, “Practical issues of leaksdiagnosis in pipelines,” in 18th IFAC World Congress, August 2011,pp. 12 337 –12 342.