Fault diagnosis for the vertical three-tank system via high-ordersliding-mode observation

N. Orani, A. Pisano and E. Usai.

Abstract— This paper presents an approach to the faultdiagnosis and disturbance observation for the hydraulic verticalthree-tank system. An observer is designed which contains acorrective term based on a second-order sliding mode controlalgorithm featuring global convergence properties. Thanks tothe proposed global observer, the reconstruction of certainfaults and/or disturbances can be performed directly from thecontinuous observer output injection signal, without requiringany filtration. The effectiveness of the proposed method isshown by means of a simulation example and is then validatedby performing real experiments.

I. INTRODUCTION

Recent years witness a large interest from the researchcommunity towards the theory and applications of faultdetection and isolation (FDI) techniques, among which themodel-based approaches [1]-[6] play an important role. Determination of the kind, size, location and time of occurrenceof a fault is the main goal of FDI. Generally speaking, theterm “fault” is referred to any disturbances or failures act-ing/occurring in the plant, actuator or sensor functional units.In particular, faultdetectionreveals the mere occurrence ofa fault, fault isolation determines the location and/or thetype of the fault, and faultidentificationalso specifies themagnitude of the fault. The informations provided by thediagnostic system can be useful for the development of faultaccommodation strategies.

The main problem of model based FDI is that there isalways a mismatch between the actual process behaviourand its mathematical model even under “nominal” (i.e.,unperturbed and not faulty) conditions. Such a mismatchcan act as source of false and/or missed alarms. Rejectingthe effect of modeling uncertainties, external disturbances,and measurement noises, is therefore the key-issue for thepractical success of model based FDI [5].

Sliding mode based observation techniques offer goodpotential in the so-called field of “robust” FDI due to theirremarkable properties of robustness against uncertainties anddisturbances [19].

In opposition to earlier works, where in presence of faultsthe sliding motion was destroyed [7], more recently [13],[10] it was exploited a new idea where the sliding motionis wanted to be preserved also in the presence of faultsand disturbances (both treated as unknown input terms).In this way, faults and disturbances acting on the systemcan be recovered by appropriate processing of the so called

The authors are with the Department of Electrical and Electronic Engi-neering (DIEE), University of Cagliari,P.zza d’Armi, I-09123, Cagliari, Italy.emails: {n.orani,pisano,eusai}@diee.unica.it Corre-sponding author: A. Pisano.

”output error injection”. This processing normally includesthe low-pass filtration of the discontinuous observer inputsignal. From the continuous output of the filter, explicitinformations on the location and magnitude of, both, faultsand disturbances can be extracted.

In [14] an approach to achieve the reconstruction of faultsand disturbances is presented for a class of nonlinear uncer-tain systems. Based on a kind of the previously mentionedlow pass filtering, the method is arbitrarily “precise” but notexact.

In this paper we address the problem of detecting andexactly identifying certain faults and disturbances acting ona vertical three tank system. The laboratory size setup thatwe used for the experimental verifications is shown in thenext Figure 1.

Fig. 1. The experimental setup

Several authors dealt with the problem of fault detectionand isolation for multi-tank systems, or similar hydraulicprocesses, using linear [22]-[24] and nonlinear [25]-[29]mathematical models. In the present work the faults areassociated to a malfunctioning in the control valves, andconsidered disturbance is a possible uncertain outflow fromthe lower tank. We exploit a robust model-based techniquewhere a second order sliding mode observer (SMO) isconsidered instead of a conventional first-order SMO like in[14]. Basically we consider, as compared to [14], a differentform for the observer output injection signal that permits the

reconstruction of the fault and disturbance termswithoutrequiring any filtration . This means that the reconstructionis theoretically exact and is achieved in finite time.

The plan of the paper is as follows: Section II presentsthe three-tank system model and some general preliminaryconsiderations. Section III describes the observer design,Sections IV and V provide some simulation and experimentalresults respectively. The final Section VI gives some conclud-ing remarks.

II. SYSTEM DESCRIPTION AND PRELIMINARIES

As one of popular experimental systems in control labora-tories, the three-tank water process is regarded as a valuablesetup for investigating, theoretically and experimentally, non-linear multivariable feedback control as well as fault diagno-sis schemes [22], [25]. The multi-tank system that we shallconsider [30] is composed of three tanks of different shapeslocated vertically, a fourth reflux tank placed under the lowertank, a variable-velocity pump that supplies the upper tank,and three electrical servo valvesRV1,RV2,RV3 that determinethe outflow from each tank. A thorough description of theexperimental setup can found in the manufacturer’s website[30]. The three tanks are equipped with piezo-resistive pres-sure transducersPZ1,PZ2,PZ3 which permit to measure thewater levelsH1 , H2, H3.

As depicted in figure 2, we select, as the modifiable controlinputs, the water inflow to the Tank 1 (provided by thevariable-speed DC pump) and the the opening of the valvesRV1 andRV2. The flow through the valveRV3 is regarded asa disturbance term.

3 – TANKSYSTEM

DC pump

RV1

RV2

U(t)

U1(t)

U2(t)

H1(t)

H2(t)

H3(t)

PZ1

PZ2

PZ3

Inputs Outputs

RV3Disturbance

ψ

Fig. 2. System inputs, outputs and disturbances

The input variablesU(t),U1(t),U2(t) (whose exact mean-ing shall be explained later on) control the DC pump andthe servo valvesRV1 and RV2, respectively. The opening ofthe third servo valveRV3 is adjusted in open-loop in orderto model a sudden demandψ of water from the downstreamwater distribution network. This term, to be treated as anexternal disturbance, will be reconstructed by the proposed

observer. Concerning the FDI aspects, it is also our aim todetect and reconstruct additive fault signals, calledf1 andf2, involving the regulation servo valvesRV1 andRV2.

The flow balance equations lead to the following simplemathematical model

V1 = q−C1√

H1 (1)

V2 = C1√

H1−C2√

H2 (2)

V3 = C2√

H2−C3

√

H3 (3)

whereV1,V2,V3 represent the actual volume of water con-tained in the three tanks,q = q(t) is the water inflow intothe upper tank (provided by the DC pump),H1,H2,H3 arethe nonnegative water levels, andC1,C2,C3 are adjustablecoefficients representing the resistance of the valves openingorifices.

The time derivative of the actual volumes of water dependon the time derivative of the water level inside the tankaccording to the simple relationships

Vi = βi(Hi)Hi , i = 1,2,3 (4)

where the nonnegative functionsβi(·), i = 1,2,3 representthe cross sectional area of thei-th tank at the level heightHi .

Taking into account the shapes of the three tanks, thecross-sectional areas are described the following analyticformulas, wherea,b,c,w,R,HMAX are constant parameterswhose values are reported in the Table 1 (see Section IV)

β1(H1) = aw (5)

β2(H2) = cw+bw(H2/Hmax) (6)

β3(H3) = w√

R2− (R−H3)2 (7)

Clearly, all the above functions are strictly positive. Ityields the simple model

H1 = 1β1(H1)

[

q−C1√

H1]

(8)

H2 = 1β2(H2)

[

C1√

H1−C2√

H2]

(9)

H3 = 1β3(H3)

[

C2√

H2−C3√

H3]

(10)

In order to account for unexpected phenomena that mightinvalidate the Bernoulli law (which is valid for a laminar flowonly) a more general representation is used, that is reportedas follows, where the flow across the valves takes a differentexpression

H1 = (q−C1H1α1)β−1

1 (H1) (11)

H2 = (C1H1α1 −C2H2

α2)β−12 (H2) (12)

H3 = (C2H2α2 −C3H3

α3)β−13 (H3) (13)

The nonnegative real coefficientsα1,α2,α3, whose“nominal” value is 0.5, will be identifiedin the experimentalpart of the present work.

The coefficientsC1, C2, C3 actually depend on the valvesopening. With the fully closed valve one has thatCi = 0,while with the fully open valveCi = C∗

i . We represent themby the notationCi = Ci(t) = C∗

i Ui(t), whereUi(t) ∈ [0,1]

is time-varying and represent the actual valve opening. Weassume thatU1(t), U2(t) andq= q(t) are user-modifiable andtherefore known. We also assume thatC3H3

α3 is an unknownterm to be treated as an external disturbance, and we denotethe overall outflow from the lower tank 3 in compact formas

ψ = C3H3α3 (14)

We also denote the adjustable water inflow generated bythe DC pump as

q = C∗U(t), U(t) ∈ [0,1] (15)

It results the following model

H1 =(C∗U(t)−C1

∗U1(t)H1α1)

β1(H1)(16)

H2 =(C1

∗U1(t)H1α1 −C2

∗U2(t)H2α2)

β2(H2)(17)

H3 =(C2

∗U2(t)H2α2 −ψ(t))

β3(H3)(18)

We assume that actuator faults (such as permanent orintermittent biases, or gain degradation) and componentfaults (such as leakage in the tanks and clogs in the pipes)can occur in the valvesRV1 andRV2.

Assuming that an appropriate closed-loop control systemhas been designed, capable of guaranteeing that the tanksnever become empties and never exceeded a prescribedmaximum level, in such a way that 0< Hi < Himax, fori = 1,2,3, our aim is to realize an observer-based FDI schemeguaranteeing the precise reconstruction of the (possibly si-multaneous) faultsf1(t) and f2(t) that can be occur in controlvalvesRV1 andRV2, and of the disturbance termψ as well.It worth noting that the knowledge of the flow disturbanceψ would be very useful for level feedback control purpose.

The actuator fault signals can be modeled by additiveterms “corrupting” the adjustable valve driving signalsU1(t)andU2(t). Thus, by including such possible fault signals inthe model one can rewrite system (16)-(18) in the generalform

x = G(x,u)+D(x)f(x,u,t)+E(x)Ψ(x,t) (19)

where x = [H1,H2,H3]T ∈ Ξ ⊂ ℜ3, u = [U,U1,U2]

T ∈ ℜ3,andG(x,u) = G∗(x)u with

G∗(x) =

C∗β1(H1) −C1

∗H1α1

β1(H1)0

0 C1∗H1

α1

β2(H2) −C2∗H2

α2

β2(H2)

0 0 C2∗H2

α2

β3(H3)

(20)

D(x) =

−C1∗H1

α1

β1(H1) 0C1

∗H1α1

β2(H2) −C2∗H2

α2

β2(H2)

0 C2∗H2

α2

β3(H3)

(21)

E(x) =

00

− 1β3(H3)

, f(x,u,t) =

[

f1f2

]

, Ψ(x,t) = ψ(t)

(22)

The nonlinear vector fieldsG∗(x), D(x), E(x) are assumedto be known, while vectorsf(x,u,t) and Ψ(x,t) representthe considered actuators faults and the external disturbance,respectively.

The measurements actually represent the complete statevector. Nevertheless, the design of an appropriate observeris still necessary to identify and reconstruct the faults anddisturbance signals. An important Assumption, which isimplicitly fulfilled in the considered example, is that thedimension of the state should be not less than the overalldimension of vector[fT(x,u,t),ΨT(x,t)]T .

Assumption 1 Matrix

M(x) = [ D(x) E(x) ] (23)

is full column-rank.

Such a constraint has been imposed by many authors inorder to reconstruct both the faults and disturbances signalvectors [13], [5]. For the considered example, assumption 1implies that the following condition holds

rank([D(x) E(x) ]) = 3 (24)

which means that the matrixM(x) is globally nonsingular.

III. OBSERVER DESIGN

Our aim is to design a dynamical observer allowing toreconstruct, exactly and in finite time, the actuator vectorfaults f(x,u,t) and the disturbance termΨ(x,t). Some as-sumptions are met about the uncertain terms and the faultactuator signals affecting the considered system.

Assumption 2Let known functionsρ(t),ξ (t),ρd(t),ξd(t)exist, possibly depending on the system input and/or statevariables, such that

‖f(x,u)‖ ≤ ρ(t), ‖Ψ(x,t)‖ ≤ ξ (t) (25)

‖f(x,u)‖ ≤ ρd(t), ‖Ψ(x,t)‖ ≤ ξd(t) (26)

Assumption 3Let known functionsζ (t),ζd(t),η(t),ηd(t)exist, possibly depending on the system input and/or statevariables, such that

‖E(x)‖ ≤ ζ (t), ‖D(x)‖ ≤ η(t) (27)

‖E(x)‖ ≤ ζd(t), ‖D(x)‖ ≤ ηd(t) (28)

The proposed observer takes the following form:

˙x = G∗(x)u+w (29)

wherex is the estimated vector state andw is an appropriatecontrol law to be designed. It is worth noting that in (29)we are usingG∗(x), instead ofG∗(x), although the fullstate vector is actually directly available for measurement.This choice is made in order to attenuate the effects of themeasurement noise in the piezo-resistive transducers (seealso Remark 5).

Let e = x− x be the state observation error, then, from(19) and (29), the observer error dynamics is

e = Gd +D(x)f(x,u,t)+E(x)Ψ(x,t)−w (30)

whereGd denotes

Gd = (G∗(x)−G∗(x))u (31)

A. Fault/disturbance reconstruction

Assuming that it can be found an observer control inputw(t) guaranteeing that vectorse and e both tend to zero infinite time (as shown in Subsection III.B), the aim is nowto demonstrate that the simultaneous exact reconstructionofsignal faults and disturbance term is possible. Assume thatthere isT∗ such that

e= e= 0, t ≥ T∗ (32)

From (30) and (32), the following relationship holds atany t ≥ T∗

D(x)f(x,u,t)+E(x)Ψ(x,t)−w = 0 (33)

It yields

w = [D(x) E(x)]

[

f(x,u,t)Ψ(x,t)

]

≡ M(x)

[

f(x,u,t)Ψ(x,t)

]

(34)

In the three tank system, matrixM(x) takes the form

M(x) =

−C1∗H1

α1

β1(H1) 0 0C1

∗H1α1

β2(H2) −C2∗H2

α2

β2(H2)0

0 C2∗H2

α2

β3(H3) −1/β3(H3)

(35)

and it is easy to show that matrixM(x) is nonsingularunder the conditionsHi > 0 (i = 1,2,3).

Thus, by considering the condition[

f(x,u,t)Ψ(x,t)

]

= M−1(x)w (36)

it can be formulated the following explicit reconstructionformulas for the signal faults and disturbance.

f1(t) = − w1β1

(C1∗H1

α1)(37)

f2(t) = − (w1β1 +w2β2)

(C2∗H2

α2)(38)

ψ(t) = −w3β3 +C2∗H2

α2 f2 (39)

Remark 3 If Im(E(x))⋂

Im(D(x)) 6= {0} the exact recon-struction of the faults is no longer possible since matrixM−1(x) does not exists. By(33) it can be derived thefollowing approximate expression for the reconstructed faultssignals

f(t) = D+(x)w−D+(x)E(x)Ψ(x,t) (40)

where D+(x) is the left pseudo-inverse ofD(x). Theapproximate fault reconstruction formula (40) gives rise toan estimation error that can be overestimated as follows [14]‖f(t)− f(x,u,t)‖ ≤ ‖D+(x)E(x)‖ξ (t)

B. Design of the observer input

It has to to designed an observer control vectorw capableof guaranteeing the required global and finite-time conver-gence to zero ofe and e. In [14] a solution was suggestedbased on standard first-order sliding mode control technique[19]. Such an approach opens the way to achieve the faultreconstruction via using the equivalent control principle(i.e.,via low pass filtering) and it is therefore anapproximatemethod [19], [13]-[14].

Here we propose a different approach based on second-order sliding modes enabling us to reconstruct the faults andthe disturbanceswithout any filtration , therefore leading toan exact solution.

Consider the second time derivative ofe

e= Gd + EΨ+EΨ+ Df +Df− w (41)

where, for simplicity, the arguments of the vector functionsare neglected. Such system can be rewritten in compact formas follows:

e= ϕ(e,x,u, u,w,t)− w (42)

with implicit definition of the “drift term” ϕ(·). Equation(42) definesp= n decoupled single input subsystems havingthe following form

{

γ1i = γ2i , i = 1,2, ..,nγ2i = ϕ i(·,t)+vi

(43)

where γ1i and ϕ i are the i-th entry of vectore and ϕ ,respectively, andvi = −wi .

The problem is to find a set ofdiscontinuous controlinputs vi stabilizing the uncertain SISO systems (43) infinite time. To solve this problem the second-order slidingmode control approach appears to be particularly appropriatesince the single input systems (43) have relative degreetwo [18] . The boundedness properties of the drift termϕ i

play a crucial role and critically affect the solution to thecontrol problem. In the standard literature on 2-SMC it wasoften assumed the existence of aconstantupperboundΦ∗

i ,known a-priori, such that|ϕi | ≤ Φ∗

i [18] . In more recentworks the problem was solved considering linear growth ofthe drift term upperbound with respect toγ1i and γ2i , i.e.|ϕi | ≤ Φ∗

i + Ki(|γ1i |+ |γ2i |), whereKi is a positive constant[17]. In the actual case, none of the above relationships canbe assumed to hold.

Here we refer to a recently proposed ”Global” version[20], [21] of the suboptimal algorithm which can work underthe more general assumption

|ϕi(·,t)| ≤ Φi(t) (44)

where Φi(t) is a function (possibly depending on states,inputs and/or explicitly on time) given in real time. Theproposed solution is summarized in the following Theorem1.

Theorem 1: Consider system(43), whose uncertain dy-namics satisfies(44). Apply the control law

vi(t)=

− [Φi(t)+ χ ]sign(γ1i(t)− γ1i(0)) 0≤ t ≤ tMi j

− [Φi(t)+ χ ]sign(γ1i(tMi j )) tMi j < t ≤ tci j

[Φi(t)+ Π + χ ]sign(γ1i(tMi j )) tci < t ≤ tMi, j+1(45)

where Π ≥ 13η2, χ and η are positive arbitrary constants

tMi j ( j = 1,2, . . .) is the sequence of time instants at whichγ2i(t) = 0, and tci j is the first time instant subsequent tMi j atwhich one of the following relationships is verified

γ1i(tci j ) = 12γ1(tMi j )

γ2i(tci j ) = η√

|γ1i(tMi j )|(46)

whereγ2i(t), which represents an instantaneous upperboundof |γ2i(t)|, is defined as

γ2i(tMi j ) = 0 j = 1,2, . . .

γ2i(t) =

2(Φi(t)+ χ) tMi j ≤ t ≤ tci j

0 tci j < t < tMi, j+1

(47)

Then, the global finite-time attainment of conditionsγ1i = γ2i = 0 (i=1,2,...,n) is provided.

Proof of Theorem 1 See [20], [21].

Remark 4: The proposed algorithm requires the sequenceof the values ofγ1i at the time instants at whichγ2i is zero(i.e the the singular values ofγ1i). The corresponding timeinstants can be detected, with an arbitrarily-small delay,byexisting peak-detector devices. In previous works [18], thesub-optimal algorithm has been shown to be robust againstthe approximate detection of the singular values, and thesame considerations still apply to the actual case.

C. Computation of bounds to the uncertainties

In order to implement the observer input design algorithmdescribed in the Theorem 1, the bounding functionsΦi(t) in(44) must be available, whereϕi(t) (i = 1,2,3) is the i-thentry of vector

ϕ(·,t) = Gd + EΨ+EΨ+ Df +Df (48)

Define

Φ(t) = col(Φ1(t),Φ2(t), ...,Φp(t)) (49)

Let us derive an expression for the norm of the upper-bounding vectorΦ(t). An upperbound of‖Gd‖ is derivedbelow

Gd =

(

∂G∗

∂xx− ∂G∗

∂ x˙x)

u+(G∗(x)−G∗(x))u (50)

By performing some manipulations it yields the followinginequality

‖Gd‖ ≤ LG∗‖e‖‖u‖+LG∗‖e‖‖u‖ (51)

whereL∗G is a Lipschitz constant of functionG∗(x):

LG∗ = maxx∈Ξ

∥

∥

∥

∥

∂G∗

∂x

∥

∥

∥

∥

(52)

Note thatLG∗ can be a function of the state, instead ofbeing a constant, without any loss of validity.

From(25) and(30) an upperbound for‖e‖ can be derivedas follows

‖e‖ ≤ LG∗‖e‖‖u‖+‖E‖ξ +‖D‖ρ +‖w‖ (53)

By (48), and considering the restrictions(25)-(28) to-gether with(51)-(53), it can be found a scalar functionΦ∗(t)such that

Φi(t) ≤ Φ∗(t) ∀i (54)

FunctionΦ∗(t) has the following structure

Φ∗(t) = Γ1(t)‖e‖+ Γ2(t)ξ + Γ3(t)ρ(t)+

+ζ (t)ξd(t)+ η(t)ρd(t)+‖w‖ (55)

where the positive functionsΓi(t) (i = 1,2,3) are defined as

Γ1(t) = LG∗2 +LG∗‖u‖ (56)

Γ2(t) = ζ (t)+ ζd(t) (57)

Γ3(t) = η(t)+ ηd(t) (58)

The upperbound defined in (55)-(58) is available in realtime and can be used in all the control laws in Theorem 1 (inother words, the conservative approximationΦi(t) = Φ∗(t)could be made, for the simplicity sake, when implementingthe proposed observer).

D. Overall observer

The overall observer design os summarized by the follow-ing Theorem 2.

Theorem 2. Consider the three tank system (19)-(22),satisfying Assumptions 1-3 with the state vectorx availablefor measurement. Implement the observer (29), with theobserver inputw(t) = [w1(t),w2(t),w3(t)] expressed as

wi = −vi , i = 1,2,3 (59)

where signalsvi(t) are computed as specified in the Theorem1 with the uncertainty upperboundΦi(t) ≡ Φ∗(t) given in(55), (58). Then, after a finite transient time the formulas(37)-(39) allow for the exact reconstruction of the faults andof the disturbance.

Proof of Theorem 2. The proof follows from the abovepresented developments and from the proof of Theorem 1.

Remark 5: The studied three tank system has the peculiar-ity that the state is fully available. Then, a different formforthe observer could be implemented instead of (29), namely

˙x = G∗(x)u+w (60)

The difference is that the actualmeasuredstate is put directlyin the right-hand side of (60). As a result, the formulation ofΦ∗(t) given in (55)-(58) could be much easier because thevector functionGd in (31) would become identically zero.As a consequenceΦ∗(t) simplifies as follows

Φ∗(t) = Γ2(t)ξ + Γ3(t)ρ(t)+ ζ (t)ξd(t)+ η(t)ρd(t)+‖w‖ (61)

The “price to pay” for reducing the uncertainties is theincrease of the sensitivity against the measurement noise.The proposed observer (29) performed better than the alter-native formulation (60), both in simulations (including themeasurement noises) and real experiments.

IV. SIMULATION RESULTS

The effectiveness of the suggested fault reconstructionscheme is studied preliminarily by making some simulativeanalysis. In the next section, experimental result will bediscussed. In all tests (of both simulative and experimentalnature) it has been implemented an appropriate PI-basedcontrol system capable of guaranteeing a stable closed-loopbehaviour. The task for the control system is to keep constantthe water levels in the tanks.

The simulations have been carried out using model (16)-(18) with parameter values that have been evaluated througha procedure of system identification. The dimensional param-etersa,w,b,c,R have been directly measured, and an identi-fication procedure has been applied to evaluate appropriatevalues forC∗, C1

∗, C2∗, α1, α2 and α3. A representative

scheme of the adopted identification methodology is depictedin the Figure 3. A model with adjustable parameters wasconstructed, and a performance index weighting the squareddeviations between the data measured during outflow exper-iments (that were carried out for each tank) and the datapredicted by the model, was optimized by means of theFMINS MATLAB Optimization Toolbox function.

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Fig. 3. Principle of parameter identification

The overall parameter values are reported in Table I.

TABLE I

TABLE 1. PARAMETERS OF THETREE TANK SYSTEM

Parameter Value Unit

C∗ 1.296∗10−4

C1∗ 4.702∗10−5

C2∗ 4.49∗10−5

α1 0.28a 0.035 mw 0.035 mα2 0.28b 0.348 mc 0.1 m

Hmax 0.35 mα3 0.29R 0.365 m

A mismatch between the initial values of the actual andobserved state variables has been imposed. The initial con-ditions of the actual levels are set as[H1(0),H2(0),H3(0)] =[1,1.5,1]m, while the initial conditions of the observed onesare [H1(0),H2(0),H3(0)] = [2,1,1.5]m.

Two tests have been performed with sampling timeTs =0.01s. In the first TEST 1 noise-free measurements are used.In the second TEST 2, a band-limited additive white noise istaken into account. Fault and disturbance signals of differentshape and magnitude have been considered. Note that a faultof magnitude of 0.5 represent an error which is the 50% ofthe overall valve opening run.

In the figures 4 and 5 the results of the two simulationtests are shown. A precise reconstruction of the faults anddisturbances is achieved in both tests.

V. EXPERIMENTAL RESULTS

Experimental results using the three-tank laboratory-sizeapparatus by Inteco [30] are presented and commented inthis section. A picture of the experimental setup was shownin the Figure 1. The multi-tank system is interfaced with anexternal PC-based data acquisition and control system. Thedevelopment of, both, the controller and observer systemsis made in the MATLAB/Simulink environment, and theassociated executable code is automatically generated bythe RTW/RTWI rapid prototyping environment. The waterlevel in the tanks are measured with piezo resistive pressuretransducer and acquired by a DAC multipurpose I/O board.There are four control signals sent out from the DAC boardto the multi-tank system: the three valve control signals andthe DC pump control signal.

The sampling time is set to 0.01 s. The closed-loop controlsystem is made up of anti wind-up PI controllers. Theidentification of the plant, sensor and actuator parameters, iscarried out in order to minimize the discrepancies betweenthe real process and its mathematical model. This procedureis indeed very important in model-based FDI.

In order to reproduce faulty conditions in the experimentalsystem, we acted on adjustable manual valvesMV1,MV2that are located besides the electrically controlled servo-valvesRV1 and RV2. Since the outflow rate from tank 1,obtained acting onMV1, is at the same time an inflow for

tank 2, this situation well represent a bias in the regulatingvalvesV1. Similar reasoning can be applied to valveV2.

It has been made a test during which both faultsf1 andf2 have been generated by opening the respective manualvalve. The fault f1 is applied after 20 seconds, the faultf2is applied ten seconds later. An exact measurement of thefault magnitude is not available. However, the shape of thereconstructed fault seems very plausible.

In the figure 6 it is reported the difference between theactual and estimated value ofH1 and H2, and the recon-structed fault signalsf1 and f2. Is is worth noting that bothsignals f1 and f2 start to grow at the correct time instant.There is, however, some small coupling between them, asindicated by the fact that signalf2 also slightly responds tothe occurrence of faultf1 too. This is due to the unavoidablemodeling imperfection that cannot be avoided in any realsystem. The last figure 7 shows the measured and observedwater levels in the tanks.

VI. CONCLUSIONS

A scheme for the simultaneous reconstruction of valveactuator faults and water outflow disturbance in a three-tanksystem has been presented. The proposed scheme, whichmakes use of a global version of a second order slidingmode controller, permits the direct reconstruction of thefaults and disturbance from thecontinuousobserver outputinjection signal, thereby avoiding the delays and intrinsicerrors associated to any filtration process. Simulations andexperimental result made on a laboratory size setup confirmthe effectiveness of the proposed scheme.

It would be interesting to implement “enhanced” feed-back controllers compensating the reconstructed disturbance.Also, it could be also possible to design an accommodationscheme that make use of the proposed FDI observer. Thesetasks will be pursued in next research activities.

VII. ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial supportfrom the FP7 European Research Project ”PRODI - Powerplants Robustification by fault Diagnosis and Isolation tech-niques”, grant n. 224233.

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[30] http://www.inteco.com.pl/index.php

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