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Flotation Process Fault Diagnosis Via Flotation Process Diagnosis Via FlotationStructural Process Fault Fault Diagnosis Via Analysis FlotationStructural Process Fault Diagnosis Via Analysis Structural Analysis Structural Analysis ∗ C. G. P´ erez-Zu˜ niga ∗,∗∗ ∗,∗∗ J. Sotomayor-Moriano ∗

C. G. P´ erez-Zu˜ ∗∗niga ∗,∗∗ J. Sotomayor-Moriano ∗∗ E. Chanthery e-Massuy` e M. Soto ∗∗∗ C. G. P´ erez-Zu˜ igaTrav´ J. Sotomayor-Moriano ∗∗nL. ∗,∗∗ e-Massuy` E. Chanthery ess ∗∗ ∗∗nL. ∗∗ M. Soto ∗ C. G. P´ erez-Zu˜ igaTrav´ J. Sotomayor-Moriano E. Chanthery L. Trav´ e -Massuy` e s M. Soto ∗ ∗∗ E. Chanthery L. Pontifical Trav´ e-Massuy` es ∗∗ M. Sotoof Peru, ∗ Engineering Department, Catholic University ∗ ∗ Engineering Department, Pontifical Catholic University of Peru, PUCP (e-mail: [email protected], [email protected], Engineering Department, Pontifical Catholic University of Peru, ∗ PUCP (e-mail: [email protected], [email protected], Engineering Department, Pontifical Catholic University of Peru, [email protected]) PUCP (e-mail: [email protected], [email protected], [email protected])[email protected], ∗∗ PUCP (e-mail: [email protected], ee de [email protected]) ∗∗ LAAS-CNRS, Universit´ de Toulouse, Toulouse, CNRS, CNRS, INSA, INSA, Toulouse, Toulouse, ∗∗ LAAS-CNRS, Universit´ [email protected]) LAAS-CNRS, Universit´ e de Toulouse, CNRS, INSA, Toulouse, France (e-mail: [email protected], [email protected]). ∗∗ France (e-mail: [email protected], [email protected]). LAAS-CNRS, Universit´ e de Toulouse, CNRS, INSA, Toulouse, France (e-mail: [email protected], [email protected]). France (e-mail: [email protected], [email protected]). Abstract: For For the the improvement improvement of of safety safety and and efficiency, efficiency, fault fault diagnosis diagnosis becomes becomes increasingly increasingly Abstract: important in mining industry. The expansion of flotation processes with high-tonnage cooper Abstract: For the improvement of safety and efficiency, fault diagnosis becomes increasingly important inFor mining industry. The expansion ofefficiency, flotation fault processes with becomes high-tonnage cooper Abstract: the improvement of safety and diagnosis increasingly concentrators demands the use of large flotation circuits in which the large amount of important in mining industry. The expansion of flotation processes with high-tonnage cooper concentrators demands the use large flotation circuits in which large amount of important in mining industry. Theof expansion of flotation processes withthe high-tonnage cooper instrumentation and interconnected subsystems (with coupled measured and non-measured concentrators demands the use of large flotation circuits in which the large amount of instrumentation and interconnected subsystems (with coupled measured non-measured concentrators demands the use of large flotation inprocess, which any theand large amount of variables) makes makes this process process complex. Moreover, in aacircuits flotation equipment failure instrumentation and interconnected subsystems (with coupled measured non-measured variables) this complex. Moreover, in flotation process, anyand equipment failure instrumentation and interconnected subsystems (with coupled measured and non-measured can lead to a fault condition, which will affect the operation of this process. This paper proposes variables) makes this process complex. Moreover, in a flotation process, any equipment failure can lead tomakes a faultthis condition, which will Moreover, affect the operation of this process. This paper proposes variables) process complex. in a flotation process, any equipment failure an approach approach for on-line fault which diagnosis useful for large flotation circuit based on distributed can lead to a for fault condition, willuseful affect for the aaoperation of this process. This paper proposes an on-line fault diagnosis large flotation circuit based on aa distributed can lead to a fault condition, which will affect the operation of this process. This paper proposes an approach for on-line fault diagnosis useful for a large flotation circuit based on a distributed architecture. In In this this approach, approach, structural structural analysis analysis is is used used for for the the design design of of the the distributed distributed fault fault architecture. an approach for on-line fault diagnosis useful forimplementation aislarge flotation circuit based on a distributed diagnosis system. Finally, a procedure for the of local diagnosers for on-line architecture. In this approach, structural analysis used for the design of the distributed fault diagnosis system. Finally, a procedure for the implementation of localofdiagnosers for on-line architecture. In this approach, structural analysis is used for design the distributed fault operation system. is presented presented and aillustrated illustrated with an application to the flotation process. diagnosis Finally, procedure for theapplication implementation of local process. diagnosers for on-line operation is and with an to aa flotation diagnosis system. Finally, a procedure for the implementation of local diagnosers for on-line operation is presented and illustrated with an application to a flotation process. © 2019, IFAC (International of with Automatic Control) Hosting by Elsevierprocess. Ltd. All rights reserved. operation is presented andFederation illustrated an application to a flotation Keywords: Fault Fault diagnosis, diagnosis, Flotation Flotation process, process, Distributed Distributed architecture, architecture, Structural Structural analysis analysis Keywords: Keywords: Fault diagnosis, Flotation process, Distributed architecture, Structural analysis Keywords: Fault diagnosis, Flotation process, Distributed architecture, Structural analysis 1. INTRODUCTION INTRODUCTION detection in in flotation flotation process process operation operation that that use use analysis analysis of of 1. detection 1. INTRODUCTION detection in flotation process operationThe thatuse useofanalysis of variables measurement are proposed. Principal variables measurement are proposed. The Principal 1. INTRODUCTION detection in flotation operation thatuse useof of measurement are proposed. use ofanalysis Principal Component Analysis process (PCA) models isThe proposed in (Bergh Nowadays, the the recovery recovery is is one one of of the the most most important important variables Component Analysis (PCA) models is proposed in (Bergh variables measurement are proposed. The use of Principal Nowadays, is proposedfailures in (Bergh and Acosta, Acosta,Analysis 2009) to to(PCA) detectmodels instrumentation on Nowadays, the recovery is one of the most important process in in mining mining industry. Currently, the recovery recovery of Component and 2009) detect instrumentation failures on Component Analysis (PCA) models is proposed in (Bergh process industry. Currently, the of Acosta,column. 2009) toThe detect instrumentation failures on a flotation development of fault diagnosis Nowadays, the recovery isisone of the most important process inin mining industry. Currently, the through recovery of and minerals this industry mainly made the aand flotation column. development of faultfailures diagnosis Acosta, 2009) industry toThe detect on minerals in mining this industry is Currently, mainly made through flotation column. The development of faultbecause diagnosis systems in mining mining is instrumentation very important important an process industry. the recoverythe of asystems minerals in this industry is mainly made through the flotation in processing technique around the the world. in industry is very because an a flotation column. The development of fault diagnosis flotation processing technique around world. systems in mining industry effective diagnosis of faults may have a high economic is very important because an minerals in this industry is mainly made through the flotation processing technique around the world. diagnosis of faults ismay have a highbecause economic systems in mining industry verydiagnosis important an Froth flotation flotation usestechnique the difference difference inthe surface properties effective effective diagnosis of faults fault may have a high economic and safety safety impact. However, in large large flotaflotation processing aroundin world.properties Froth uses the surface and impact. However, fault diagnosis in flotaeffective diagnosis of faults may have a high economic Froth flotationseparate uses the difference in gangue surface and properties to physically minerals from is one and safety impact. However, fault diagnosis in large flotation circuits is a difficult task due not only to the large to physically separate minerals from and is one tion circuits is a difficult task duediagnosis not onlyintolarge the large Froth flotation uses used the difference in gangue surface properties and safety impact. However, fault to physically separate minerals from gangue and is one tion of the the most widely widely methods of ore concentration. circuits is a difficult task due only to the flotalarge amount of instrumentation, instrumentation, but alsonot to its its interconnected of most used methods of ore concentration. amount of but also to interconnected to physically separate minerals from gangue and is one tion circuits is a difficult task due not only to the large of the most widely used methods ore concentration. In order to improve the recovery of valuable minerals, amount of instrumentation, but also to its interconnected subsystems with coupled (measured and non-measured) In order to improve the methods recovery of minerals, subsystems with coupled (measured and non-measured) of the most widelypractice used of valuable orecells. concentration. amount instrumentation, but case, also tothe itsimplementation interconnected In order to improve the recovery of valuable minerals, industrial flotation uses multiple multiple These cells subsystems with coupled and non-measured) variablesofbetween between them. In In(measured this industrial flotation practice uses cells. These cells variables them. this case, the implementation In order to improve the recovery of valuable minerals, subsystems with coupled (measured and implementation non-measured) industrial flotation practice usesamultiple cells. These cells are arranged in series forming bank. A combination of variables between them. In this case, the of a global diagnoser may be an impractical option because are arranged in series forming bank. Acells. combination of of a globalbetween diagnoser may In be this an impractical option because industrial flotation practice usesa multiple These cells variables them. case, the implementation are arranged in series forming acircuit. bank. It banks is referred as flotation is common for A combination of of a global diagnoser may be an impractical option because the amount of needed communication, (Blanke et al., al., banks is referred as flotation It common for amount of needed communication, (Blanke et are arranged in series forming acircuit. bank. A is combination of of of athe global diagnoser may be an impractical option because banks is referred as flotation circuit. It is common for conventional flotation cells to be assembled in a circuit, of the amount of needed communication, (Blanke et al., 2016). Thus the use of centralized architecture for on-line conventional flotation cells to be assembled in a circuit, Thus theofuse of centralized architecture for on-line banks is referred as flotation circuit. Itcells, is common for 2016). of thediagnosis amount needed communication, (Blanke et al., conventional flotation cells be assembled inwhich a circuit, with rougher, rougher, cleaner, andtoscavenger scavenger can 2016). Thus thecan use of very centralized fault be expensive and lack lack robustness architecture for on-line with cleaner, and cells,inwhich can fault diagnosis can be very expensive and robustness conventional flotation cells to be assembled a circuit, 2016). Thus theinterconnected usebe of very centralized architecture for on-line with rougher,in cleaner, and configuration. scavenger cells, which can fault be arranged a designed On the other diagnosis can expensive and lack robustness for large-scale subsystems, (P´ e rez-Zuniga be arranged in cleaner, a designed configuration. On which the other large-scale interconnected subsystems, (P´erez-Zuniga with rougher, and scavenger can for fault diagnosis canpossibility be very expensive and this lack robustness be arranged in decades, a designed hand, in recent recent theconfiguration. expansion cells, of On flotation with the other for large-scale interconnected (P´edifficulty rez-Zuniga et al., al., 2018). One One tosubsystems, overcome is hand, in decades, the expansion of flotation with et 2018). possibility to overcome this difficulty is be arranged in a designed configuration. On the other for large-scale interconnected subsystems, (P´ e rez-Zuniga hand, in recent decades, the expansion of flotation with high-tonnage copper concentrators in Peru, Chile, etc. et al., 2018). One possibility to overcome to employ a distributed diagnosis architecture. this difficulty is high-tonnage copper concentrators in Peru, Chile, with etc. to employ a distributed diagnosis architecture. hand, in recent decades, the been expansion of flotation et al., 2018). One possibility to overcome this difficulty is high-tonnage concentrators in Peru, Chile, (O’Connell et copper al., 2016), has demanding the use useetc. of to employ a distributed diagnosis architecture. (O’Connell et al., 2016), has been demanding the of Recently, a distributed diagnosis framework for for physical physical high-tonnage copper concentrators in Peru, Chile, etc. to employ aa distributed distributed diagnosis diagnosis architecture. (O’Connell et al., 2016),consisting has been of demanding the use of Recently, large flotation circuits a large number framework large flotation circuits consisting of a large number of Recently, a distributed diagnosis framework for physical systems with continuous behavior using structural model (O’Connell et al., 2016), has been demanding the use of large banks,flotation with several several cells consisting each one. one. of a large number of systems circuits with continuousdiagnosis behavior framework using structural model Recently, aproposed distributed for physical banks, with cells each with continuous behavior using structural model has been in (Bregon et al., 2014) and dislarge flotation circuits banks, with several cells consisting each one. of a large number of systems beenwith proposed in (Bregon et using al., 2014) and model aa dissystems continuous behavior structural Flotation equipment requires machine for for mixing mixing and and has has beendiagnosis proposed in (Bregon eta al., 2014) and a that distributed approach with set of diagnosers banks, with several cells each one. Flotation equipment requires aa machine diagnosis approach witheta al., set of diagnosers has been proposed in (Bregon 2014) and a that disFlotation requires a machine mixing and tributed dispersing equipment air throughout throughout the mineral mineral slurry while removforwhile tributed diagnosis approach with a set of diagnosers that are as as local as possible possible was presented in (Khorasgani dispersing air the slurry removare local as was presented in (Khorasgani Flotation equipment requires a machine for mixing and tributed diagnosis approach with a set of diagnosers that dispersing air throughout the mineral slurry while removing the the froth froth product. product. Instrumentation Instrumentation is is also also necessary necessary for for are local Inasdistributed possible was presented in (Khorasgani et al., al.,as2015). 2015). diagnostic architectures, unlike ing Inasdistributed diagnostic architectures, unlike dispersing air throughout theofmineral slurry removare localones, possible was presented in the (Khorasgani ing the froth product. Instrumentation is also while necessary for et a successful successful implementation control strategies. strategies. The ultiet al.,as2015). In distributed diagnostic architectures, unlike centralized it is not mandatory to know model of aing implementation of control The ultiones, it is not mandatory to know the model of the froth product. Instrumentation is also necessary for et 2015). In distributed diagnostic architectures, unlike amate successful of control strategies. The ultiaim of implementation control is to to increase the economic economic efficiency of centralized centralized ones, itDistributed is not mandatory to knowuse thesubsystem model of theal., global system. architectures mate aim of control is increase the efficiency of the global system. Distributed architectures use subsystem athe successful implementation of control strategies. Thethere ulticentralized ones, itDistributed is not to know thesubsystem model of mate aim of by control is totoincrease the economic efficiency of the process seeking optimise performance, and global use models forsystem. diagnosis and mandatory localarchitectures diagnosers (LDs), so they they the process by seeking optimise performance, and there for diagnosis and local diagnosers (LDs), so mate aim ofstrategies control iswhich toto the economic efficiency of models the global system. Distributed architectures use subsystem the process by seeking toincrease optimise performance, and there are several can be adopted to achieve this, formore diagnosis and local (LDs), so would be be appropriate for diagnosers complex systems, systems, (P´eethey rezare several strategies which can be adopted to achieve this, models appropriate for complex (P´ rezthe process byIn seeking to optimise performance, and there models formore diagnosis and local diagnosers (LDs), so are several strategies which (Wills, 2006). the flotation flotation process, any equipment equipment fail- would canprocess, be adopted to achieve this, would be more appropriate for complex systems, (P´ethey rezZuniga et al., 2017), such as the large flotation circuits. (Wills, 2006). In the any failet more al., 2017), such as for the large flotation circuits. are strategies which canprocess, beetc.), adopted to achieve this, would be appropriate systems, (P´erez(Wills, In the flotation any fail- Zuniga ure several (in 2006). valves, sensors, pipelines, canequipment lead to aa fault fault Zuniga et al., 2017), such as thecomplex large flotation circuits. ure (in valves, sensors, pipelines, etc.),any can lead to The aim of this paper is to propose an approach for onon(Wills, 2006). In the flotation process, equipment failZuniga et al., 2017), such as the large flotation circuits. ure (in valves, sensors, condition, which will affect the operation of this process. In pipelines, etc.), can lead to a fault The aim of this paper is to propose an approach for condition, which will affect the operation of this In The aim of this paper is to propose ancircuit approach foron on-a line fault fault diagnosis in flotation process based ure (in valves, sensors, pipelines, etc.), can leadprocess. tofor a fault fault condition, which will affect the operation of this process. In line (Xu et al., 2012; Ming et al., 2015), methodologies diagnosis in flotation process based The fault aim of this paper is to propose ancircuit approach foron on-a (Xu et al., which 2012; Ming et al.,the 2015), methodologies for fault diagnosis in flotation process circuit based on a condition, will affect operation of this process. In line (Xu et al., 2012; Ming et al., 2015), methodologies for fault line fault diagnosis in flotation process circuit based on a (Xu et al., 2012; Ming et al., 2015), methodologies for fault 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2019 IFAC 235 Copyright © under 2019 IFAC 235 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 235 10.1016/j.ifacol.2019.09.191 Copyright © 2019 IFAC 235

235 235 235 235

2019 IFAC MMM 226 Pérez-Zuñiga et al. / IFAC PapersOnLine 52-14 (2019) 225–230 Stellenbosch, South Africa, August 28-30,C.G. 2019

distributed architecture propose in (P´erez-Zuniga et al., 2017). In this approach, structural analysis is used as an efficient tool for the design of fault diagnosis systems for nonlinear processes, (Isermann, 2006). Likewise, in order to optimize the offline design of LDs, Fault-Driven Minimal Structurally Overdetermined (FMSO) sets are calculated and guarantee minimal redundancy of analytical redundancy relations (ARR) generators, (P´erez-Zuniga et al., 2015). At last, a procedure for the residual generation for on-line operation is presented and shown with the flotation process. 2. PROBLEM STATEMENT In a flotation process, the pulp is introduced into the first cell, the froth is collected through launders and the remaining pulp flows to the next cell. The magnitude of the flow depends on the pressure difference between two adjacent cells, the position of the control valves, and the viscosity and density of the pulp. Figure 1 shows the flotation process under study. y1

h1

u1

y3

T-1

h2

u2 f1

f2

T-2

T-3

y4

f4

f3

h3

u3 f5

f6

T-4

y5

T-5 f8

f7

h4

u4 f9

f10

T-6

T-7

u5

f12 f11

f13

f14

T-8

3. BACKGROUND THEORY In this section, we summarize some important concepts presented in previous works related to the generation of diagnostic tests using structural analysis. Structural analysis allows to obtain structural models that are very useful for the design of Model Based Diagnosis (MBD) systems. The main assumption is that each system component is described by one or several constraints; thereby, violation of at least one constraint indicates that the system component is faulty. The structural model of the system Σ(z, x, f), also denoted with some abuse of notation by Σ(z, x, f) or Σ in the following, can be obtained abstracting the functional equations. It retains a representation of which variables are involved in the equations. This abstraction leads to a bipartite graph G(Σ ∪ X ∪ Z, A), or equivalently to G(Σ ∪ X, A), where A ⊆ A and A is a set of edges such that a(i, j) ∈ A iff variable xi is involved in equation ej . The structural model Σ(z, x, f) for this system is composed of 41 equations e1 to e41 relating the known variables Z = {u1 , u2 , ..., u5 , y1 , y2 , ..., y5 , qin , qout }, the unknown variables X = {x˙ 1 , x1 , x˙ 2 , x2 , x˙ 3 , x3 , ..., x˙ 8 , x8 , q0 , q1 , q2 , ..., q8 } and the set of sensors, actuators and process faults F = {f1 , f2 , f3 , f4 , f5 , ..., f16 }.

y2 qin

related to the tailings. There are a set of 5 measurements y1 to y5 and a set of 5 control valves u1 to u5 .

qout f16

3.1 Analytical Redundancy Relations

f15

Fig. 1. Diagram of the flotation process under study. Due to the physical characteristics of the flotation process, and considering the disturbances caused by the composition of the minerals and the constant and arduous work of the system, these systems usually have a limited efficiency, which is evidenced by faults in sensors, actuators and the system such as leaks in tanks and pipes, (Jamsa et al., 2003). For the application of the structural analysis approach, let the system description consist of a set of n equations involving a set of variables partitioned into a set Z of nZ known (or measured) variables and a set X of nX unknown (or unmeasured) variables. We refer to the vector of known variables as z and the vector of unknown variables as x. The system may be impacted by the presence of nf faults that appear as parameters in the equations. The set of faults is denoted by F and we refer to the vector of faults as f. Definition 1. (System). A system, denoted Σ(z, x, f) or Σ for short, is any set of equations relating z, x and f. The equations ei (z, x) ⊆ Σ(z, x, f), i = 1, . . . , n, are assumed to be differential or algebraic in z and x. The flotation process under study has 5 levels at different altitudes (h1 to h5 ) and is composed of 41 equations (36 for the system and 5 linked to the level control of each stage). Later, we assumed each level with outlet pipe as a subsystem so this system is composed by 5 subsystems. The flow qin refers to the pulp inflow, while the flow qout is 236

Analytical redundancy relations (ARR) are equations that are deduced from an analytical model and only involve measured variables. Definition 2. (ARR for Σ(z, x, f)). Let Σ(z, x, f) be a system. Then, a relation arr(z, z, ˙ z¨, ...) = 0 is an ARR for Σ(z, x, f) if for each z consistent with Σ(z, x, f) the relation is fulfilled. Definition 3. (Residual generator for Σ(z, x, f)). A system taking a subset of the variables z as input, and generating a scalar signal arr as output, is a residual generator for the model Σ(z, x, f) if, for all z consistent with Σ(z, x, f), it holds that lim arr(t) = 0. t→∞

We use the decomposition of Dulmage Mendelshon as a tool to compute redundant sets using structural analysis, (Dulmage and Mendelsohn, 1958). Making use of this permutation, a system model Σ can be divided into three parts: the structurally overdetermined (SO) part Σ+ with more equations than unknown variables; the structurally just determined part Σ0 , and the structurally underdetermined part Σ− with more unknown variables than equations, (?). Definition 4. (Structural redundancy). The structural redundancy ρΣ of a set of equations Σ ⊆ Σ is defined as the difference between the number of equations and the number of unknown variables in Σ . Definition 5. (Fault support). The fault support FΣ of a set of equations Σ ⊆ Σ is defined as the set of faults that are involved in the equations of Σ .

2019 IFAC MMM Pérez-Zuñiga et al. / IFAC PapersOnLine 52-14 (2019) 225–230 Stellenbosch, South Africa, August 28-30,C.G. 2019

227

Definition 6. (PSO and MSO sets). A set of equations Σ is proper structurally overdetermined (PSO) if Σ = Σ+ and minimally structurally overdetermined (MSO) if no proper subset of Σ is overdetermined (Krysander et al. (2010)).

the models of each subsystem to design LDs independently considering minimizing the communication between them until reaching the same diagnosis as with a centralized diagnosis. Let us consider the system Σ and define the following:

Since PSO and MSO sets have more equations than variables, they can be used to generate ARRs and residuals.

A decomposition of the system Σ(z, x, f), into several subsystems Σi (zi , xi , fi ) is defined as a partition of its equations. Let Σ(z, x, f) = {Σ1 (z1 , x1 , f1 ), ..., Σn (zn , xn , fn )} n with Σi (zi , xi , fi ) ⊆ Σ(z, x, f), Σi (zi , xi , fi ) = Σ,

A Fault-Driven Minimal Structurally Overdetermined (FMSO) set can be defined as an MSO set of Σ(z, x, f) whose fault support is not empty. Let us define Zϕ ⊆ Z, Xϕ ⊆ X, and Fϕ ⊆ F as the set of known variables, unknown variables involved in the FMSO set ϕ, and its fault support, respectively. Next, we summarize the definition of FMSO set, Definition 7. (FMSO set). A subset of equations ϕ ⊆ Σ(z, x, f) is an FMSO set of Σ(z, x, f) if (1) Fϕ = ∅ and ρϕ = 1 that means |ϕ| = |Xϕ | + 1, (2) no proper subset of ϕ is overdeterminated. (P´erez-Zuniga et al., 2017) We propose the use of FMSO sets that guarantee to always be impacted to faults contrary to the MSO sets that not may not be impacted by faults. Based on the concept of FMSO set, we summarize the concept of detectable fault, and isolable fault: Definition 8. (Detectable fault). A fault f ∈ F is detectable in the system Σ(z, x, f) if there is an FMSO set ϕ ∈ Φ such that f ∈ Fϕ . Definition 9. (Isolable fault). Given two detectable faults fj and fk of F , j = k, fj is isolable from fk if there exists an FMSO set ϕ ∈ Φ such that fj ∈ Fϕ and fk ∈ Fϕ . Additionally, a Clear Minimal Structurally Overdetermined (CMSO) set is a MSO set of Σ(z, x, f) whose fault support is empty. 3.2 Distribution and Related Notions A distributed diagnosis architecture assumes a decomposition of the process into subsystems, each with its corresponding LD, with similar functions and with possible communication between them. This communication must be properly designed; therefore, the local diagnoses are globally consistent. This architecture is shown in Figure 2.

i=1

Σi (zi , xi , fi ) = ∅ and Σi (zi , xi , fi ) ∩ Σj (zj , xj , fj ) = ∅ if i = j. where zi is the vector of known variables in Σi , xi is the vector of unknown variables in Σi and fi is the vector of faults in Σi . The set of variables and faults of the ith subsystem Σi , denoted as Xi , Zi , and Fi respectively, are defined as the subset of variables of X, Z, and F respectively, that are involved in the subsystem Σi (zi , xi , fi ) also denoted by Σi . For the flotation process, we consider each level as a subsystem, therefore, the first subsystem includes a tank and the outlet pipe, the second to the fourth subsystems, contain 2 tanks, the pipe between them and the outlet pipe and the fifth subsystem includes a tank and the outlet pipe, see Table 1. Table 1. Model decomposition of the flotation process system into subsystems Σi (zi , xi , fi ), i = 1, 2, 3, 4, 5. Σ1 X1 Σ2 X2 Σ3 X3 Σ4 X4 Σ5 X5

= {e1 , e2 , e3 , e4 , e5 , e6 , e7 } = {x˙1 , x1 , q0 , w1 } = {e8 , e9 , e10 , ..., e16 } = {x˙2 , x3 , x˙3 , q2 , w2 } = {e17 , e18 , e19 , ..., e25 } = {x˙4 , x5 , x˙5 , q4 , w3 } = {e26 , e27 , e28 , ..., e34 } = {x˙6 , x7 , x˙7 , q6 , w4 } = {e35 , e36 , e37 , ..., e40 } = {x˙8 , q8 , w5 }

F1 Z1 F2 Z2 F3 Z3 F4 Z4 F5 Z5

= {f1 , f2 } = {u1 , y1 , qin } = {f3 , f4 , f5 , f6 } = {u2 , y2 } = {f7 , f8 , f9 , f10 } = {u3 , y3 } = {f11 , f12 , f13 , f14 } = {u4 , y4 } = {f15 , f16 } = {u5 , y5 }

The set of local variables of the ith subsystem, denoted by Xil , is defined as the subset of variables of Xi that are only involved in the subsystem Σi . Definition 10. (Shared variables). The set of shared variables of the ith subsystem, denoted as Xis , is defined as: Xis

=

n

(Xi ∩ Xj ) = Xi \ Xil

(1)

j=1,j=i

The set of shared variables of the whole system Σ is denoted by X s . Without loss of generality, we consider that all known variables of Zi are local to the subsystem Σi , for i = 1, . . . , n. If the same input was applied to several subsystems, it could be artificially replicated. Fig. 2. Distributed diagnosis architecture

3.3 Distributed FMSO sets

For the flotation process, in this paper we propose the design of the distributed system taking into account only

Definition 11. (Local FMSO set). ϕ is a local FMSO set of Σi (zi , xi , fi ) if ϕ is an MFSO set of Σ(z, x, f) and if

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ϕ ⊆ Σi , Xϕ ⊆ Xi and Zϕ ⊆ Zil . The set of local FMSO sets of Σi is denoted by Φli . The set of all local FMSO sets n Φli . is denoted by Φl = i=1

Definition 12. (Shared FMSO set). ϕ is a shared FMSO set of subsystem Σi (zi , xi , fi ) if ϕ is an FMSO set of ˜ i (˜ ˜i ), where z˜i is the vector of variables in Z˜i = Zi ∪ zi , x ˜i , f Σ s ˜ i = X l , and f ˜i = fi ). ˜i is the vector of variables in X Xi , x i The set of shared FMSO sets for Σi is denoted by Φsi . The n Φsi . set of all shared FMSO sets is denoted by Φs = i=1

From the above definition, a shared FMSO set ϕ for subsystem Σi (zi , xi , fi ) is such that ϕ ⊆ Σi , Xϕ ⊆ Xil , Zϕ ∩ Xis = ∅, and Zϕ ⊆ (Zi ∪ Xis ).

Definitions 11 and 12 can also be applied to CMSO sets to define local CMSO sets Λli and shared CMSO sets Λsi . The set of all shared CMSO sets is denoted by Λs . Definition 13. (Compound FMSO set). A global FMSO set ϕ that includes at least one shared FMSO set ϕ ∈ Φsi is called a compound FMSO set. The set of compound FMSO sets of Σi is denoted by Φci . The set of all compound FMSO n Φci . sets is denoted by Φc = i=1

Definition 14. (Root FMSO set). If a compound FMSO set ϕ ∈ Φc includes a shared FMSO set ϕ ∈ Φs , then ϕ is a root FMSO set of ϕ. Definition 15. (Locally detectable fault). f ∈ Fi is locally detectable in the subsystem Σi (zi , xi , fi ) if there is an FMSO set ϕ ∈ Φli such that f ∈ Fϕ . Definition 16. (Locally isolable fault). Given two locally detectable faults fj and fk of Fi , j = k, fj is locally isolable from fk if there exists an FMSO set ϕ ∈ Φli such that fj ∈ Fϕ and fk ∈ Fϕ .

Some properties required for the generation of compound FMSO sets starting from shared FMSO sets are detailed in P´erez-Zuniga et al. (2017) 4. DISTRIBUTED DIAGNOSIS First, a set of distributed local diagnosers (LD) that together make the entire system completely diagnosable through compound FMSO sets is obtained, then residual generators that make it possible to detect and isolate all system faults are implemented. First, Algorithm 1 for generating local diagnostics off-line is applied and then Algorithm 2 is proposed for on-line residual generation.

Algorithm 1. Offline Generation of LDs. 1: for i=1...n do 2: Φi = ∅; 3: Φli ← Calculate local FMSO sets of Σi ; 4: if there is any fault f ∈ Fi not locally detectable 5: or not locally isolable with the set of local 6: FMSO sets Φli then 7: Φsi ← Calculate shared FMSO sets of Σi ; 8: Λsi ← Calculate shared CMSO sets of Σi ; 9: end if 10: while it exists f ∈ Fi that is not detectable 11: or isolable do 12: Let ϕ∗ ∈ Φsi such that f ∈ Fϕ∗ be the ’best’ 13: (not already selected) shared FMSO set of Φsi ; 14: Label ϕ∗ as root FMSO set: ϕr ← ϕ∗ ; 15: Let Xϕs r be the set of shared variables of ϕr ; 16: Φc∗ i ← Build a ’good’ compound FMSO set 17: including ϕ∗ by always selecting the ’best’ 18: shared FMSO sets to cover newly introduced 19: shared variables; 20: Φi ← Φi ∪ Φc∗ i ; 21: 22: 23: 24: 25: 26:

Φl∗ i ← Find a minimal cardinality set of local FMSO sets achieving the same diagnosability as all local FMSO sets; Φi ← Φi ∪ Φl∗ i ; end while end for

4.2 On-line distributed residual operation of LDs After the off-line design of the LDs performed with algorithm 1, the online operation of the distributed diagnoser relies on the bank of residual generators ARRi selected for each LD LDi , i = 1, . . . , n, fed by measured signals from their corresponding subsystems. As shown in Figure 3, fault isolation is carried out after fault detection using local fault signature matrices according to Definition 17. Definition 17. (FSM of a subsystem). Given a set ARRi composed of nri ARRs and Fi the set of considered nfi faults for the subsystem Σi and consider the function ARRi × Fj,i −→ 0, 1, then the signature of a fault f ∈ Fi is the binary vector F Si (f ) = [τ1 , τ2 , ...τnri ]T where τk = 1 if f is involved in the equations used to form arrk ∈ ARRi , otherwise τk = 0. The signatures of all the faults in Fi together constitute the fault signature matrix (FSM) F SMi for subsystem Σi , i.e. F SMi = [F Si (f1 ), . . . , F Si (fnf )]T . i

5. APPLICATION TO THE FLOTATION PROCESS

4.1 Offline distributed generation of LDs

5.1 Offline distributed generation of LDs

The LD design is done off-line in Algorithm 1. First, local FMSO sets are computed for every subsystem Σi . If there is any fault not locally detectable, then a set of compound FMSO sets is calculated to achieve full diagnosability for all the faults in Fi . The procedure to compute ’good’ compound FMSO sets starting with ϕ∗ as a root FMSO set makes use of an optimization heuristic based on the number of shared variables and on the number of subsystems involved with the aim of minimizing communication between subsystems.

In this section, the construction of the LD for each subsystem is presented in order to diagnose all system faults. Below the steps of the offline design: 1.- The local FMSOs are calculated for each of the subsystems, considering only local information.

238

Φl1 = Φl2 = Φl3 = Φl4 = Φl5 = ∅

(2)

No local FMSOs were found considering only information from each subsystem. The shared FMSOs for each

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Φs1 = {ϕ1 , ϕ2 , ϕ3 } ϕ1 = {e2 , e5 , e6 }, ϕ2 = {e1 , e3 , e4 , e5 } ϕ3 = {e1 , e2 , e3 , e4 , e6 } Φs2 = {ϕ4 , ϕ5 , ϕ6 , ϕ7 , ϕ8 , ϕ9 , ϕ10 , ϕ11 } ϕ4 = {e12 , e14 , e15 }, ϕ5 = {e9 , e11 , e13 , e14 } ϕ6 = {e9 , e11 , e12 , e13 , e14 , e15 }, ϕ7 = {e8 , e10 , e11 , e13 , e14 } ϕ8 = {e8 , e10 , e11 , e12 , e13 , e15 }, ϕ9 = {e8 , e9 , e10 , e14 } ϕ10 = {e8 , e9 , e10 , e12 , e15 }, ϕ11 = {e8 , e9 , e10 , e11 , e13 } Φs3 = {ϕ12 , ϕ13 , ϕ14 , ϕ15 , ϕ16 , ϕ17 , ϕ18 , ϕ19 } ϕ12 = {e21 , e23 , e24 }, ϕ13 = {e18 , e20 , e22 , e23 } ϕ14 = {e18 , e20 , e21 , e22 , e24 }, ϕ15 = {e17 , e19 , e20 , e22 , e23 } ϕ16 = {e17 , e19 , e20 , e21 , e22 , e24 }, ϕ17 = {e17 , e18 , e19 , e23 } ϕ18 = {e17 , e18 , e19 , e21 , e24 }, ϕ19 = {e17 , e18 , e19 , e20 , e22 } Φs4 = {ϕ20 , ϕ21 , ϕ22 , ϕ23 , ϕ24 , ϕ25 , ϕ26 , ϕ27 } ϕ20 = {e30 , e32 , e33 }, ϕ21 = {e27 , e29 , e31 , e32 } ϕ22 = {e27 , e29 , e30 , e31 , e33 }, ϕ23 = {e26 , e28 , e29 , e31 , e32 } ϕ24 = {e26 , e28 , e29 , e30 , e31 , e33 }, ϕ25 = {e26 , e27 , e28 , e32 } ϕ26 = {e26 , e27 , e28 , e30 , e33 }, ϕ27 = {e26 , e27 , e28 , e29 , e31 } Φs5 = {ϕ28 , ϕ29 , ϕ30 } ϕ28 = {e36 , e39 , e40 }, ϕ29 = {e35 , e37 , e40 } ϕ30 = {e35 , e36 , e37 , e39 }

Σ1

Σ2

Σ3

Fig. 3. Scheme of distributed generation of LDs.

Σ4

Algorithm 2. On-line Residual Operation of LDs. 1: for i=1...n do 2: For each LD: 3: Compute ARRs for LDi 4: for j=1...m do 5: For all selected compound FMSO sets: 6: ARRi,j ← Compute analytical residual 7: generators of LDi ; 8: Save the set of known variables of 9: each ARRj,i ; 10: ZLDi ← ZLDi ∪ ZARRj,i ; 11: end for 12: By means of the fault signature matrix (F SMi ) 13: verify the isolability of faults of each subsystem; 14: end for 15: Add the known variables of the vector ZLDi to the

fault diagnosis software for the online calculation of the ARRs of the LDs; 16: Generate a on-line scalar signal arrk from 17: the respective ARRj,i using the signals of ZLDi .

subsystem are then determined by considering the vector of shared variables (X s = {x2 , x4 , x6 , x8 , q1 , q3 , q5 , q7 }) as part of the vector of known variables for each subsystem. 2.- For subsystems σ1 to σ5 , shared FMSO sets are computed, Results are given in Table 3. 3.- For each subsystem, Algorithm 1 chooses from the set of shared FMSO sets, a subset that is labeled as root FMSO set and complete with a shared FMSO set each of its shared variables until get a set of compound FMSO sets that can diagnose all the faults of that subsystem. The set of compound FMSO sets capable of detecting and isolating the faults constitute the LD of the corresponding subsystem. Results are given in Table 3.

229

Σ5

Table 2. Shared FMSO sets of Σ1 to Σ5 .

LD1 LD2

ϕ31 = ϕ32 = ϕ33 = ϕ34 =

LD3

ϕ35 = ϕ36 =

LD4

ϕ37 = ϕ38 =

LD5

ϕ39 = ϕ40 =

{e1 , e2 , e3 , e4 , e5 , e6 , e8 , e9 , e10 , e14 } {e2 , e5 , e6 , e8 , e9 , e10 , ..., e15 , e17 , e18 , e19 , e23 } {e1 , ..., e6 , e9 e11 e13 , e17 , ..., e24 , e26 , e27 , e28 , e32 } {e1 , ..., e6 , e8 , e10 , e11 , e13 , e14 e17 , ..., e22 , e23 , e24 , e26 , e27 , e28 , e32 } {e2 , e5 , e6 , e8 , ..., e15 , e18 , e20 , e22 , e23 , e26 , ..., e33 , e38 } {e2 , e5 , e6 , e8 , ..., e15 , e17 , e19 , e20 , e22 , e23 , e26 , ..., e33 , e38 } {e12 , e14 , e15 , e17 , ..., e24 , e27 , e29 , e31 , e32 , e35 , e37 , e38 , e40 } {e12 , e14 , e15 , e17 , ..., e24 , e26 , e28 , e29 , e31 , e32 , e35 , e37 , e38 , e40 } {e30 , e32 , e33 , e35 , e37 , e38 , e40 } {e36 , e38 , e39 , e40 }

Table 3. Compound FMSO sets of LDs.

arr1 ∈ ARR1,1 arr2 ∈ ARR1,2

Table 4. isolation capability for ARRs for LD1 .

f3 arr3 ∈ ARR2,1 arr4 ∈ ARR2,2

X

239

Faults f4 f5 X X X

f6

Table 5. isolation capability for ARRs for LD2 .

5.2 On-line distributed residual operation of LDs Using Algorithm 2, the ARRs are calculated and the isolation of the 16 faults of this system is verified, as shown in Table 4 to 8. As example, Figure 4 shows the ARRs operating online for subsystem 1, as can be seen in the case of a momentary fault of the tank level sensor 1 (f1 ) from 600 s. up to 650 s., there is a detection of ARR1 and no detection of ARR2 , which demonstrates the isolation of this fault locally.

Faults f1 f2 X X

f7 arr5 ∈ ARR3,1 arr6 ∈ ARR3,2

X

Faults f8 f9 X X X

f10

Table 6. isolation capability for ARRs for LD3 . Finally, Figure 5 shows the human machine interface of the fault diagnosis software running on-line where a fault alarm is shown in valve 2 (f6 ). This software is executed

2019 IFAC MMM 230 Pérez-Zuñiga et al. / IFAC PapersOnLine 52-14 (2019) 225–230 Stellenbosch, South Africa, August 28-30,C.G. 2019

f11 arr7 ∈ ARR4,1 arr8 ∈ ARR4,2

X

Faults f12 f13 X X X

validating that the 16 faults can be detected and isolated locally or at a higher level. Likewise, a procedure for residual generation was presented and it has been tested into a programmable automation controller for on-line operation of fault diagnosis software.

f14

Table 7. isolation capability for ARRs for LD4 .

arr9 ∈ ARR5,1 arr10 ∈ ARR5,2

7. AKNOWLEDGMENTS

Faults f15 f16 X X

This work was funded by Proyecto de Mejoramiento y Ampliaci´ on de los Servicios del Sistema Nacional de Ciencia Tecnolog´ıa e Innovaci´ on Tecnol´ ogica 8682-PE, Banco Mundial, CONCYTEC and FONDECYT through grant N48-2018-FONDECYT-BM-IADT-MU.

Table 8. isolation capability for ARRs for LD5 . ARR1

100

REFERENCES

80 60 40 20 0 0

100

200

300

400

500

600

700

800

900

1000

600

700

800

900

1000

ARR2

100 80 60 40 20 0 0

100

200

300

400

500 time(s)

Fig. 4. LD for subsystem 1 in a programmable automation controller (PAC) that receives the signals from the sensors and generates the control signals.

Fig. 5. Fault diagnosis software In fact, the proposed approach is applicable to fault diagnosis of a large floating circuit, decomposing the latter into subsystems. Here, as shown above, each subsystem in the distributed architecture will have its own LD. 6. CONCLUSION An approach for on-line fault diagnosis in a flotation process was proposed based on a distributed architecture. The application of the approach allows the development of diagnosis systems for large-scale flotation circuits. The fault diagnosis system developed, was tested by simulation 240

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