ModelBasedFaultTolerantControlBased onSubspace

Methods ofWaterNetworkCanal Systems

Antonio Miguel Vicente Figueira

Department of Mechanical Engineering - IDMEC, Instituto Superior Tecnico, Technical University of Lisbon (TULisbon) Av.Rovisco Pais, 1049-001 Lisboa, Portugal

Abstract

A fault tolerant control for a water canal network is presented in this article. The starting point is the simulation of atheoretical model of the canal water system property of NuHCC. The resulting data is used to build Composite Local LinearModels (CLLM) for subspace methods. Three types of single faults are induced within the system which are the gate fault,the off take valve fault and the sensor fault. The CLLM models are created containing the fault consequence on the systemoutput. With a different input and using at the same time the theoretical model and the CLLM models created, the outputsare related to produce four different residues. The evaluation of the residues by combining the output of several CLLM modelsis made in order to determine when and where a fault is present. This information is used to switch to the most accuratemodel that is included in the model predictive control. This selection is necessary since each MPC assumes different CLLMmodels with different parameters values. The fault tolerant control allows the system to maintain a normal behaviour and soa desired water delivery along the canal even with a fault mechanism present.

Key words: CLLM, Identification, Residue, Decision Model, Fault Implementation, Fault Tolerant Control.

1 Introduction

Irrigation networks of open-water channels are usedthroughout the world to support agricultural activity.On a global scale irrigation accounts for approximately70% of all fresh water usage according with [3]. The dis-tribution of fresh water for irrigation is often achievedvia an extensive civil infrastructure of reservoirs andopen-water channels. From the situation of water levelon August of 2012 in portuguese dumps, the economicneed to control this process is of huge importance andhas resulted in the development of numerous automaticcontrol methods with widely varying conceptual andpractical approaches. According to [9] persistent prob-lems encountered in most canal control methods arethe result of inherent characteristics of the canal flowprocess which are not easy to control with conventionaltechniques. The modernization of irrigation system op-erations was designed by [2]and includes the additionof water level sensors and gates actuators as well asa supervision system. Identification and control cametogether with [6] where an optimal LQR controller wasapplied to a system model.

? This paper was not presented at any IFAC meeting.

Fig. 1. Evora canal system

The canals should be managed in such a way as to createflexible conformities on everything that depends on it.This requires water ready for use at any time and fore-seeable situation. To this end, an elaborate system ofinterconnections between various canal systems shouldbe done in order to have optimal management of water.Since the development is supported by the efficiency thatcomes in the transport of water, minimal losses need tobe guaranteed through canal control. The efficiency of a

Preprint 16 May 2013

canal can be described as the ability to maintain a cer-tain water level. The difficulties in predicting the mo-ments of demand, as well as the amount to use at anygiven time has proved to be one of the main challengesfrom the control perspective.

Since the start of research in this field different types ofcontrollers have been implemented in canal systems [7].But some of these, only had a good performance in sit-uations where the conditions were limited in scope be-cause there is continuous increase in canal systems com-plexity, hence the need to include all physical and oper-ational constraints at the control action. The controllerthat seems to follow all these requirements is the ModelBased Predictive Controller (MPC), given the fact thatit has these requirements and includes them in optimiz-ing the computation of control actions. The basic anal-ysis of the MPC is to calculate the control actions basedon the forced response (unknown) and estimate the fu-ture behavior based on free responses (known) of thesystem. Weight penalties are used for deviations of theset points and the system states are based on the pre-diction horizon to calculate optimal control actions.

For canal systems [13] explains why simple linear inter-nal model is not a good solution for MPC. This is due tolong waves caused by the movement of gates and gatesthemselves producing different types of deviation unpre-dictabilities of water levels in different locations of thegate. Using nonlinear models might guarantee the con-trol of these dynamics preventing resonant waves.

The contribution of this dissertation work is the creationof a single fault tolerant controller. The controller usesthe model predictive control (MPC) technique with aCLLM model. The MPC input signals are not used atthe same time to control the system, the selection be-tween signals is made by a switch. The switch is managedby a decision variable, which is created with a decisionmodel. The decision model uses the residues generatedby the system to make that decision. The residues aregenerated by several CLLM models, which describe thesystem in four different situations. Three of that situa-tions are single faults induced environments and one is anormal system functioning, without fault induced. Theimplementation of such methodology using only CLLMmodels in the controller is the main contribution of thiswork.

2 Data-driven tools used to build nonlinearmodels

Given the difficult analysis of nonlinear systems, oftenthe solution to confront nonlinear systems consists in lin-earizing them thus avoiding nonlinear aspects. This pro-cedure may not be the best to completely describe thesystem dynamics especially for cases where the dynamicbehaviour is typical nonlinear. In the case of the canal

system, the modeling shows dispersive effects due to non-linear propagation of long waves. In the case of ModelBased predictive control, the performance depends a loton this effect [13]. We therefore have a technique of modelnonlinear plant, which is a nonlinear models that makeassumptions about the dynamics of the nonlinear plant.In the nonlinear identification is implemented with com-posite local linear models.

2.1 Composite local linear models

Having developed the approach of Operating Regime [8],with the goal of setting a broader approach to this, Com-posite Local Linear Model (CLLM) were introduced by[11] to model nonlinear systems. The methodology be-hind CLLM are nonlinear discrete-time state-space para-metric equations, which approximate the behaviour ofcomplex dynamic nonlinear systems. It seeks a very par-ticular way to calculate the matrices Ai, Bi, Oi and Cas well as the centres ci and wi for a finite number ofinputs uk and outputs yk. In a particular way based onthe optimization of the cost function JN (θ, c, w), whichis derived from the minimization of the error vector EN .The local model can be initialized based on techniquesof Subspace Identification for CLLM, as proposed in [1].The method of minimization performance is greatly in-fluenced by the initial value minimization Output Er-ror (OE) and this affects in a very comprehensive waythe CLLM. To solve this inconvenience we can estimatethe initial parameters, using independent LTI systems.The performance of CLLM in describing the system dy-namics, is far better when compared with LTI system,according with [9]. Subspace Identification proposed by[12] can be used for the case of class algorithms Multi-variable Output Error State Space (MOESP). Accordingto [11], the initial values for radial basis function are dis-tributed uniformly over the operating range. CLLM hasbeen defined by [11] as:

x(k + 1) =

s∑i=1

pi (φ(k)) (Aix(k) +Biu(k) +Oi) (1)

y(k) = Cx(k) + v(k) (2)

where s is the number of local models,Oi is the offsets onthe state, x(k) ∈ Rn is the state vector, u(k) ∈ Rm is theinput, y(k) ∈ R` is the output, v(k) ∈ R` is a white-noisesequence and pi (φ(k)) ∈ Rs are parameterized usingnormalized radial basis functions, which determines thecombination of local models to use.

pi (φk) =ri (φ(k); ci;wi)∑sj=1 rj (φ(k); ci;wi)

(3)

2

with the following radial basis functions:

ri = exp(− (φ(k)− ci)T diag (wi)

2(φ(k)− ci)

)(4)

where ci is the center and wi the width of the ı-th radialbasis function, φ(k) ∈ Rq is the scheduling vector, de-fined in an operating point of the system and dependson the input and the system state, as shown:

φ(k) = ψ (x(k), u(k)) (5)

with ψ : Rn×m → Rq, which is nonlinear and known.The goal is to determine, from a finite number of mea-surements of the input u(k) and output y(k), the matri-ces Ai, Bi, Oi, C and the centers ci and widths wi thatdescribe the radial basis functions.

2.2 Nonlinear model based predictive controller

When problems of control are faced in the time do-main, Nonlinear Model Based Predictive Controller(NMBPC), schematized in figure 2, has been used as ageneral methodology its effective resolution, given thefact that it deals with problems of optimal and multi-variable control, nonlinear processes and problems withvarious constraints. Regarding to the use of nonlinearinternal models in NMBPC, the optimization algorithmimplemented was the Discrete Search Techniques, no-tably B&B with the control structure shown in figure 2.

GeneratorReference

CostFunction

NMBPC

CLLM

B & B CANAL PLANT

Fig. 2. Implemented control scheme.

2.2.1 Branch and bound

- The problem is divided into smaller sub problems,where the possibilities with no feasible solution are elim-inated. The algorithm takes place in two processes:

(1) Branch, the not eliminated solution is partitionedinto parts that converge increasingly to the optimalsolution.

(2) Bound, the upper and lower bounds are set so asto lead to the optimal value of objective function.

PresentPast Future

Fig. 3. MPC principle.

2.3 Fault Tolerant Control

A system can handle with a fault presence if the faultintensity does not deviate the system output from theexpected. Once the fault consequence changes the nor-mal system behaviour a different controller must be ac-tivated in order to maintain the system output as muchcloser as the system can be from the input reference. Sowith this approach a fault tolerant control can be builtusing a model predictive control with composite linearlocal models. The fault tolerant control can be describedas a set of controllers, with several models containingsome previous defined faults and a model without faultsinduced. In this way and evaluating at each sample timethe system output to determine the system state it ispossible to activate the respective model that better con-trol the system. Knowing that, the fault tolerant controlprocedure can be like: diagnose of the fault followed bya controller selection. This is achieved using a supervi-sion system based on residues. In figure 4 the variablef is the decision variable, yref is the reference that thecontroller must follow, u is the system input; y is thesystem output and J and dare system perturbations andfault switch, respectively.

3

Diagnosis

Controller

Selection

Controller

Plant

Supervision

Execution

Level

LevelJ d

f

Fig. 4. Fault Tolerant Control Scheme.

3 System Identification

3.1 Canal Description

The water canal real system, figure 1, is property fromNucleo de Hidraulica e Controlo de Canais da Uni-versidade de Evora, or NuHCC, and is integrated inAQUANET project, which aims to develop a decentral-ized and reconfigurable control for water delivery mul-tipurpose canal systems, reference PTDC/EEA-CRO/ 102102/2008, financed by Fundacao para a Ciencia eTecnologia.

Table 1NuHCC System Nomenclature Used

Parameter Definition

Q0 Water Canal Flow Intake

U* Gate Position

M* Water Level Sensor at Pool Upstream

C* Water Level Sensor at Pool Centre

J* Water Level Sensor at Pool Downstream

D* Off Take Flow

The notation referring to some important canal featuresis summarized in the table 1 and showed in figure 5. Theasterisk near each letter means that can be referred toany pool number.

Pool 3

M3 C3 J3

U3

Q0

D3

Reservoir

Pool 1Pool 2

Pool 4

Fig. 5. Schematic drawing of the NHCC.

The entire water network is supplied by a reservoir,which flows to the first pool at a specific rate given bythe input flow signal, and then from the first pool tothe second one, and so on until it reaches the end of thelast pool. Represented in figure 5, the parameters fromthe entire canal network are the following: trapezoidalcross section with 0.15 [m] wide crawler, backrest slopeof 1:0.15 [V:H], height of 0.90 [m], total length of 141 [m],average longitudinal slope of 0.0015. The orifice gate inthe first three sections has lengths of 35 [m] and the lastovershot gate has 36 [m]. The project has a nominal flowrate of 0.09 [m3/s]. With 300 [mm] diameter, one elec-tric meter by regulation, operates a MONOVAR valvewhich allows the admission of water into the reservoir.

3.2 Input Signal

The identification input was designed randomly focusingon the excitation of the maximum number of modes ofthe water canal network. It is composed with severalsteps with random duration and amplitude. Being thetop line the gate position, the centre line the nominalflow which enters the water canal flow and the third linethe off take flow.

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Simulation Time

Input Flow

Gate Position

OffTake Flow

Fig. 6. System Input Signal.

4

3.3 Fault Implementation

The fault implementation follows the idea of a fault beinga non-desirable system behaviour, which in most casescan only be detected by evaluating the output. A faultis an intrinsic system malfunction that cannot be rec-ognized from the outside unless the output is evaluated.In order to get several CLLM models which describe theinfluence on the output response when a fault is presentwithin the system, a specific input is changed during acertain time. In case the fault variable takes the valueone it means that the fault is active, the signal where thevalue for input is multiplied by the fault signal goes zero,and in case the value for the switch takes the value zero,the input value keeps unchanged as it passes through thefault block, which means the fault is deactivated. All thefaults are single faults.

Output

INESC

Q0

U3

D3

J3Fault #1

Fault #2

Fault #3

Signal Generator

Gate

Off Take

Sensor

Q0

Q0

U3

U3

D3

D3

Fig. 7. System Implementation.

In figure 7 can be understood as the nominal flow, Q0,the centre output is the gate position, U3, and the loweroutput is the offtake flow, D3. The block in line withthe signal generator through gate position, contains thegate fault implementation, and is denominated as faultnumber X or FaultX.

3.4 CLLM Identification

In the fault identification model the fault behaviour mustbe learned and the system dynamics neglected as muchas possible. If the system dynamics is indeed the princi-pal learning of the CLLM model, during simulation somefalse alarms of fault presence can easily occur, as themodel is induced in error when a specific input combina-tion is present. To avoid that, the user must guaranteethat the linear approximation to the fault behaviour ismade in the simulation and certify that in the test valida-tion some characteristic behaviour occurs near the faultactivation. Parameter tuning procedure can be summa-rized as an iterative process with sequential change ofparameter values with consequently linear model VAFand MSE evaluation. The fault is present between 630and 680 seconds of simulation.

0 100 200 300 400 500 600 700 8000.4

0.42

0.44

0.46

0.48

0.5

0.52

Simulation Time [s]

Dow

nstre

am L

evel

[m]

Canal Water Level

Model Water Level

Fig. 8. CLLM Model Performance on Gate Fault Case.

This methodology of tune model parameters havingsome external information is made by [14]. The num-ber of iterations performed by the function needs to betuned after some results. It should be taken into accountthat the values presented next are the result of the VAFmaximization and MSE minimization.

Table 2Model Performance Values on Gate Fault Case

Linear Model Train Model Test Model

N◦ Iterations 100

VAF 98.1 99.3 96.8

MSE - 4.15x10−6 5.31x10−5

4 System Control

4.1 Residues Generation

Residue is an arithmetic difference between two quan-tifiable signals. In the residue generation the theoreticalmodel and all the CLLM models have the same inputsignal and each one produces a different output. Thoseoutputs are the source of the residues, being the arith-metic difference between the theoretical model and eachof the four CLLM models the so called residues.This ap-proach is schematized in figure 9. the residue analysisis to know in each time step which model best fits thereal system behaviour. The difference between the modeloutput, which best fits in that time the system outputmust be close to zero in that moment. During a simu-lation the residue is always being calculated among thefive models generating four quantities.

5

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Saida Teste

Simulation Time

Reference

No Fault Output

Gate Fault Output

Off Take Fault Output

Sensor Fault Output

Fig. 10. Residues Generated for Three Faults in the SameSimulation.

Constant Input

[Q0 U3 D3]

INESC

Theoretical

Model

CLLM Model

Without Fault

Gate Fault

Off Take Fault

Sensor Fault

Residue #1

Residue #2

Residue #3

Residue #4CLLM Model

CLLM Model

CLLM Model

ANN

or

CLLM

0 = Without Fault

−1 = Gate Fault

0.5 = Off Take Fault

1 = Sensor Fault

Fig. 9. Residue Generation Approach and Decision Model.

The residues generation is presented in figure . In thissimulation a gate fault is activated at the 1000 secondsuntil 1200 seconds, the off take fault is present during100 seconds starting at the 2000 seconds and the sensorfault begins at the 3000 seconds along 100 seconds. Thesimulation total length is also 4000 seconds.

4.2 Decision Model

A decision model is quite important in order to builda robust tolerant control system. To build this kind ofmodel it is necessary to define an input and an output forthe training process. The idea is to use the four residuesas the input of the system and the decision variable asthe output of the system.

Table 3Decision Variable Value

Scenario Decision Variable Value

Without Fault Case 0

Gate Fault Case -1

Off Take Fault Case 0.5

Sensor Fault Case 1

The output or decision variable defined in table 3, hasthe same dimension as the simulation time vector, wherea time step is one second with a variable decision valuefor that instant and so the dimension is equal to thetotal simulation time. The decision variable can take thevalues shown in table 3. Having as reference the previousvalues, the vector is filled manually, where for instantswith no fault active takes the value 0 and for the timewhere the fault is active takes the respective fault value,-1,0:5 or 1 for the gate fault, off take fault or sensor fault,respectively.

0 20 40 60 80 100 120 140 160 180−1.5

−1

−0.5

0

0.5

1

1.5

2

Simulation Time

Decision Variable

Decision Model

Fig. 11. CLLM Performance.

4.3 Fault Tolerant Control

It is visible that the control action also have a oscilla-tory behaviour, which is smooth by the action of thefuzzy filter. The intensity of the wave is directly relatedwith the CLLM model built. Which sometimes is not thebest one for control purposes but is the best between agood performance in identification, residues generationand control purposes. With the validation of the MPCcontrol actions, they can be included in the fault toler-ant controller. The tricky part of the tolerant control iswhere the switch change among MPC, this change canbe seen from the model point of view as a step inducedin the input or in other words, a step in the water level,which in reality, if too high, means a lot of water instan-taneously induced in the system. Sometimes this stepcan be large enough to unstable the model and the con-trol action cannot be performed.

6

0 500 1000 1500 2000 2500 3000 3500 4000 45000.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

Simulation Time

Canal Water Level

Reference

Fig. 12. System Implementation.

In table 5.21 are described the faults presence within thesystem to be controlled.

Table 4Faults Location for Fault Tolerant Control

Gate Fault Offtake Fault Sensor Fault

1000 to 1110 (s) 760 to 910 (s) 1620 to 1870 (s)

2790 to 2900 (s) 2530 to 2700 (s) 3050 to 3190 (s)

For each MPC containing faulty models can be seen thatthe reference is never fitted, except when the fault isactive, or in the intervals described previously. So in thetotal control action the reference is every time followed,with fault or not presence. It can be highlighted the firstsensor fault case, where after 30 seconds of fault presencethe controller turns the signal close to reference again,like moments before when no fault was present withinthe system.

5 Conclusions

5.1 Fault Tolerant Control

The main conclusion of this work is that CLLM mod-els can be used in fault tolerant control applicationsbased on model predictive control. Is validated the use ofCLLM in systems identification, being accurate modelsin data-driven mechanisms. The adopted methodologyof considering a switch mechanism to detect which faultis active at a given period of time, was successfully veri-fied. The switch mechanism is based in the residue evalu-ation by a CLLM, with great results in the three studiedsituations of single faults. The fault tolerant control wasachieved to single faults cases with success, even with aswitch mechanism between the several model predictivecontrollers.

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