Enhanced Interpreted Petri Nets for Industrial Processes

Enhanced Interpreted Petri Nets for Industrial Processes Eid M. Al-Hajri , J.A. Rossiter Department of Automatic Control and Systems Engineering University of Sheffield, Sheffield, S1 3JD UK

Abstract: Hybrid control systems are an exciting field of research. These contain two distinct types of system; one with continuous dynamics and the other with discrete dynamics, that interact with each other. Model-based fault detection methods were proposed to detect and isolate faults as may occur in such systems; however, these methods are computationally demanding. This paper enhances a diagnoser interpreted Petri net (IPN) model to locate and isolate faults in such systems. The enhanced IPN model has two notable features, one it is simpler and the other it is faster. The latter feature stemmed from the fact that the proposed model consists of a single place and the same number of transitions that the system model has. Unlike, the conventional IPN-based diagnoser, the enhanced IPN-based diagnoser deals with complex hybrid control systems with ease. Industrial processes are employed to test the enhanced IPN diagnoser.

Keywords: Discrete event systems, Hybrid systems, Fault detection and isolation, Interpreted Petri Nets, Industrial processes

1. INTRODUCTION The field of Discrete Event Systems (DES) is a relatively new research area that combines different formalisms, methodologies and tools [1-2]. The domains of DES are: manufacturing automation, protocols, robotics, process control, nuclear reactors, fault diagnosis, and other man-made technological systems [3]. Historically, DES was introduced in the early 1980s, in the field of chemical engineering. They quickly gained popularity in modeling and supervision of hybrid systems [4]. A hybrid system comprises continuous levels and discrete event ones and thus may be more complex than the DES. The common tools of discrete event systems are finite state automata [5] and Petri nets [6]. The latter has a good descriptive power compared with the former. Petri nets models are more compact than automata based models for representing DES and hybrid systems. The central component of abnormal event management deals with detection, diagnosis and correction of faults in a process. Many authors addressed the technical challenges in research and development that need to be directed for the successful design and implementation of practical intelligent supervisory control systems for the process industries. They classified the different Fault Detection and Isolation (FDI) methods into three categories [7-9]: Quantitative modelbased methods, Qualitative model –based methods, and Process history based methods. The authors also presented a framework that shows how these different approaches relate to and differ from each other regarding the transformation of

information from the measurement space to the decision space. In fact, these diagnostic methods are computationally demanding. An efficient method for obtaining a diagnoser interpreted Petri net (IPN) model was proposed based on the DES model [10]. This model consists of a single place and the same number of transitions that the system model has. However, this diagnoser has some limitations when it is employed for hybrid systems. The aim of this paper is to enhance the conventional IPN diagnoser to locate and isolate faults in such hybrid systems. The rest of the paper is organized as follows. Section 2 describes the IPN and its diagnosability. The enhanced IPN-based diagnoser is detailed in section 3. Section 4 reports the results obtained from detecting and isolating faults in industrial processes using the proposed diagnoser. Section 5 concludes the topics in this paper by summarizing the contributions made and presenting suggestions for future research. 2. IPN AND ITS DIAGNOSABILITY This section presents the basic concepts and notation of IPN used in this paper. For more details the reader can be directed to read [10-11]. Definition 1: A Petri Net structure G is a bipartite digraph represented by the 4-tuple G=(P,T,I,O) where: P = {p1,p2,...,pn} and T ={t1,t2,...,tm} are finite sets of vertices named places and transitions respectively,

I (O ) : P × T → Z + is a function representing the weighted arcs going from places to transitions (transitions to places); Z + is the set of nonnegative integers. Pictorially, places are represented by circles, transitions are represented by rectangles, and arcs are depicted as arrows. The symbol ∗ t j (t •j ) denotes the set of all places pi such that I ( pi, t j ) ≠ 0

O ( pi, t j ) ≠ 0 .

Analogously,

∗

pi ( pi• )

denotes the set of all transitions tj such that O( pi , t j ) ≠ 0 ( I ( pi , t j ) ≠ 0) . The pre-incidence matrix of G is C − = [cij− ] where C ij− = I ( pi , t j ) ; the post-incidence matrix of G is C + = [cij+ ] , where

C ij+ = O( pi , t j ) the

incidence matrix of G is C = C + − C − . A marking function M : P → Z + represents the number of tokens (depicted as dots) residing inside each place. The marking of a PN is usually expressed as an n-entry vector. An IPN (Q, M 0 ) is an Interpreted Petri Net structure Q=(G,Σ,λ, ϕ ,) with an initial marking M 0 , G is a PN structure, Σ = {α1,α2,...,αr} is the alphabet of input symbols αi.,  λ : T→ ∑ ∪{ε } is a labeling function of transitions with the following constraint: ∀t j , t k ∈ T , j ≠ k , if ∀pi I ( pi , t j ) = I ( pi , t k ) ≠ 0 and both

λ (t j ) ≠ ε , λ (t k ) ≠ ε , and λ (t j ) ≠ λ (t k ); ε  represent a system internal event,

ϕ : R(Q, M 0 ) → ( Z + ) q

is an output

function, that associates to each marking in R(Q, M 0 ) qentry output vector; q is the number of outputs. Fault detection and isolation IPN diagnoser models were proposed in [11]. This model consists of a single place, the same number of the system model’s transitions, and a set of weights. The latter is the contents of the incident matrix Cd that is defined as C d = B T φ T φ C N , where B is nx1 vector with internal elements x which can be computed using the exponential base, b. That is b= 2 MAX [abs(Cij)] + 1 and x=bi, where, i is an integer numbers, 1, 2, ….., for the measurable places, and b=0 for nonmeasurable places. The matrix φ is qxn output matrix where, q is the measurable outputs, and n is the number of places. The matrix CN is nxm matrix that describes the normal process. In this model, the current marking of this place is enough to determine and locate faults occurring within a discrete event system. When ek ≠ 0 , an error is detected, and then a faulty marking was reached. The mechanism used to find out the faulty marking is named fault isolation. The main problem of the use of such model is that its limitation to detect and isolate faults in simple processes with single-input and single-output transitions (diagnosable systems). Let (Q,M0) be an IPN obtained such that it is strongly connected , live, and event detectable then it is diagnosable [11]. Diagnosability, means the diagnoser can be structured. However, event detectable PN-model can be obtained iff the firing of any pair of transitions, Ti and Tj of (Q, Mo), can be distinguished from each others. This is not guarantied for many PN models. For complex non-diagnosable systems these diagnosers may fail

to locate and isolate faults. To overcome this problem, this paper enhances the IPN-based diagnosers as shown in section 4.

3. THE ENHANCED IPN MODEL This section describes the conventional and the proposed enhanced IPN-based diagnosers; algorithm #1. The problem statement is also formulated. A. The Conventional IPN-based Diagnoser; Algorithm #1 [10] Inputs: M k , M kd , e k ; where, M k is the marking vector of the normal process,

M kd

is the marking place of the

diagnoser, and e k is the error between them. Outputs: p(faulty place),

M f (faulty marking)

d

Constants: C is the IPN diagnoser structure incidence matrix described in section 2. i = index of the column of C

d

such that C

d

(1,i)= ek

•

1- ∀p ∈ t i , M k ( P) = 0 •

2- ∀p ∈ t i , M k ( P) = 0 3-

∀p F ∈( • t i ) • • ∩ P F , M k ( p F ) = 1

4- M

f

= Mk

5 - Return (p, M f ) B. The Problem Statement Although, the current marking of this place is enough to determine and locate faults occurring within a discrete event system that has transitions with single-input and singleoutput transition (SISO) as shown Fig. 1 [10], it fails to determine and locate faults occurring within a complex hydride control systems that has multi-input and multi-output (MIMO) transitions as shown in section 4. The reason for this shortcoming is that the firing transitions of most IPNs describing complex systems are usually not detectable. The diagnoser incidence matrix C d that represents the faulty transition should have columns which are not null and also are different from each other. This is the problem that faces the IPN diagnoser when dealing with complex systems containing MIMO transitions. To overcome this problem this paper proposes that the diagnoser should check faulty marking place M k ( p F ) to detect the faults of the undetectable transition systems when no fault is detected by the conventional IPN-based diagnoser.

So, the next subsection enhances the conventional IPN-based diagnoser described in algorithm A.

d

5- If ek ≠ C (the error does not match any value in the incidence matrix of the diagnoser) Then apply the following proposed routine: 5.1- Define the ith position of the faulty place using, M k (i,1) = −μ where μ is the number of moved tokens I

from the faulty marking vector, and the negative sign indicates that the faulty marking vector losses token unlike the normal marking vector. 5.2- Define the jth position of the faulty place using, C ( I , j ) = −μ at the ith position of the faulty place, I. J

5.3-

Detect

faulty place using ⎞ ⎛ ⎟ ⎜ ⎟ ⎜ f p ∈ ⎜ ∑ C ( J1 ) ∪ C ( , J 2 ) ⎟ . Using this equation, the J ⎟ ⎜ i J ⎟ ⎜ ⎠ ⎝ faulty place can be detected with its link with the fault transition.

.,

the

.

5.4- Return (p, M f ) Fig. 1 A typical part of the producer-consumer PN model [10]

else

6-

M f = Mk

7- Return (p, M f ) C. The Enhanced IPN-based Diagnoser Inputs: M k , M kd , e k These variables are described in the conventional diagnosed described in section 3-A. Outputs: p(faulty place), M f (faulty marking) Constants: C d is the IPN diagnoser structure incidence matrix i = index of the column of C d such that C d (1,i)= ek 1- ∀p ∈ • t i , M k ( P) = 0 •

2- ∀p ∈ t i , M k ( P) = 0 F

•

3- ∀p ∈ ( t i )

• •

F

F

∩ P ,Mk (p ) =1

//updating the faulty and normal marking vectors 4- M f (k + 1) = M f (k ) + C [q; v] , M n (k + 1) = M n (k ) + C N [q ] , where C is the incidence matrix of the faulty plant model, and the CN is the incidence matrix of the normal plant model respectively, q is the normal input vector, and v is the faulty input. To compute the weights of the multi-input mutioutput transition, this paper proposes the following subroutine.

In summary, the key modification is step number 5, which enables the enhanced dignoser to detect the faulty place only, unlike the conventional IPN-based diagnose. The efficacy of the proposal will be demonstrated with simulation results described in section 4. 4. NUMERICAL EXAMPLES USING THE ENHANCED IPN DIAGNOSER The common examples of industrial systems are batch processes which are characterized by combination of discrete and continuous dynamics [12]. Their automation and optimization pose difficult issues mainly because it is necessary to operate concurrently with continuous (algebraic or differential equations) and discrete (Petri net) models. Batch plants consist of many transport resources (transporters) like valves and pipes, and processing resources (processors) like mixing tanks, batch reactor vessels, and other container like units [13-15]. These complex structures have many expected faults that should be located and isolated. Many research studies have been conducted for this issue; however, most are computationally demanding. L. Zhenjuan et al. [16] introduced a new method of complex batch process fault diagnoses based on fuzzy Petri nets. In [17], the bond graph approach was employed to develop an integrated chemical reactor model, able to be exploited for

supervision and diagnosis of industrial process. Chetouani presented a FDI strategy for nonlinear dynamic systems [1]; they introduced a methodology of tackling the fault detection and isolation issue by combining a technique based on the residuals signal and a technique using the multiple Kalman filters. The structure and the PN model of the batch process to be used in this paper and its supervisory control have been proposed in [18]; this is summarized in subsection A. Fault detection and isolation of the batch process using the conventional and enhanced IPN-based are discussed in subsection B.

A. The System Description The employed chemical batch process cell is described in this subsection. It consists of three input buffers and two process lines; each contains one reactor vessel as shown in Fig. 2. It represents the final stage of large scale chemical batch process [18].

Fig. 3. Detailed view of batch reactor.

represent the continuous part in this process. The glycol flow is controlled by adjusting the proportional valves Vre and Vrf and the temperature of the reactor contents is measured by the temperature sensor TT. The basic recipe for the reactor is given in Table 1.

Table 1. The basic recipe for the reaction

Fig. 2. Batch process cell with two reactors. A single batch reactor is detailed in Fig. 3. Each reactor has three inlet valves for three incoming substances and the output is produced by the reaction of the three chemicals at a specified temperature. The filling of the reactor is controlled by the three on/off valves (Vra, Vrb, and Vrc) and the discharging is controlled by the on/off valve Vrd. The charging of each of the three chemical substances is sensed by three level switches (Sra, Srb,and Src). The temperature of each reactor is controlled by two closed loop local controllers (one for heating phase and the other for cooling phase) by feeding hot or cold glycol through the reactor jacket, which surrounds the reactor vessel. This control loops

Step 1 2 3 4 5 5 6 7

Actions and Their Associated Places Load chemical A (places pr4, pr5) Load chemical B (places pr6, pr7) Start the stirrer and load chemical C (places pr8, pr9, pr10) Switch on the temperature controller to heat up the mixture to the required reaction temperature (places pr11, pr12) When the set-point temperature is reached start the timer to time the duration of the reaction (place pr13). Cool down the reactor contents (place pr14) Discharge the product from the reactor (place pr15) Wait for the start of new batch (place pr16).

The PN model of the chemical batch process described in subsection 4-A and its supervisory control shown in Fig. 4 was proposed in [18]. The readers who are interested in modeling and supervision of such processes are referred to [18-19].

Similarly, the consuming unit could reach a faulty state from its consuming state. The incidence matrix, Cd = [-1 27 1 9 9 -27 3 -3], has different and non zero values. Simulation results obtained by firing the 2nd and 3rd transitions respectively are: •

•

•

Fig. 4. Unified Plant/Supervisor model

B. Simulation Results Firstly, consider the producer-consumer PN model shown in Fig. 5. The model consists of a producer unit (PU), a consumer unit (CU) and a buffer of 2-slots. The behavior of this system is the following. The producer unit PU creates and delivers products into the free buffer positions. The consumer unit CU retrieves products from the buffer when there is a product stored into a buffer slot. The producer unit PU could reach a faulty state from its producing state.

At firing the 2nd transition, the output of the IPNbased process model, y(k)=30, and the output of the IPN-based diagnoser, yd=30, and the difference between them e(k)=0. The diagnoser indicates no fault happened. At firing the 3rd transition, the output of the IPNbased process model, y(k)=31, and the output of the IPN-based diagnoser, yd=31, and the difference between them, e(k)=0. The diagnoser also indicates no fault happened. At firing the 1st transition and the 9th faulty transition, the error, e(k)=-1. Compared this error with the contents of the incidence matrix Cd defined above, the diagnoser indicates fault happened at event number 1 and the faulty place is its output; p1.

Fig. 5 the producer-consumer PN model

Secondly, applying the same procedure using the conventional IPN-based diagnoser for detecting and isolating the faults on the batch chemical process described in section 4-A, results the following. At firing the 9th transition shown in Fig. 4, the output of the IPN-based process model, y(k)=0, and the output of the IPNbased diagnoser, yd=0, and the difference between them e(k)=0. The diagnoser indicates no fault happened.

At firing the 1st transition, the output of the IPN-based process model, y(k)=4, and the output of the IPN-based diagnoser, yd=4, and the difference between them, e(k)=0. The diagnoser also indicates no fault happened. At firing the 2nd transition and the 19th faulty transition, the error, e(k)=-32. The diagnoser should indicate that the faulty place is Pf34 only, however, its outputs two faulty places , Pf34, and Pf36 respectively.

diagnoser, yd=0, and the difference between them e(k)=0. The diagnoser indicates no fault happened.

This means that, the conventional diagnoser confused between Pf34, and Pf36 as shown in Fig. 6. This can be investigated using the diagnoser equation, ∀p F ∈ ( • t i ) • • ∩ P F , M k ( p F ) = 1 , that includes the places

At firing the 2nd transition and the 19th faulty transition, the error, e(k)=3. Unlike the conventional IPN-based diagnoser, the enhanced diagnoser indicates that the faulty place is p34 shown in Fig. 7. This is due to the virtue of using the effectiveness of the proposed diagnoser to discriminate between the faulty states (places). The current faulty marking in the proposed diagnoser is enough to determine and locate faults occurring within a hybrid system not only that has transitions with SISO but also that has transitions with MIMO.

Pf34 and Pf36 respectively as a subset of the general set that includes the faulty place, PF. This is totally wrong because the transition T21 is not fired as shown in the figure. Although, the current marking of this place is enough to determine and locate faults occurring within a discrete event system that has transitions with single-input and singleoutput (SISO) as shown in the producer-consumer process discussed in the first example, it fails to determine and locate faults occurring within a complex hydride control systems that has transitions multi-input and multi-output transitions (MIMO) depicted in Fig. 6. The problem has been sorted out using the enhanced IPN-based diagnoser proposed in section 3. Its simulation results are discussed in the following subsection.

At firing the 1st transition, the output of the IPN-based process model, y(k)=4, and the output of the IPN-based diagnoser, yd=4, and the difference between them, e(k)=0. The diagnoser also indicates no fault happened.

Fig. 7 A typical part of the Unified Plant/Supervisor model

5. CONCLUSIONS AND FUTURE WORK Fig. 6 A typical part of the Unified Plant/Supervisor model

The proposed IPN-based diagnoser discussed in section 3 has been tested using the batch process described in section 4, at the same conditions of using the conventional IPN-based diagnoser. The PN model of the batch chemical process shown in Fig. 4 is employed. The simulation results obtained are described as follows. At firing the 9th transition, the output of the IPN-based process model, y(k)=0, and the output of the IPN-based

The main problem inherent to fault detection and isolation in batch processes is due to its hybrid nature. State variables like the tank level or the pump speed are continuous, others like the on-off valves, are discrete-state components. Moreover, the whole process behaves as a cycle of discrete events. Model-based fault detection methods were proposed to detect and isolate faults in such systems; however, these models are computationally demanding. This paper enhanced a diagnoser interpreted Petri net (IPN) model to locate and isolate faults in such systems with ease. The enhanced IPN model has two notable features, one it is simpler and the other it is faster. The latter is stemmed from the fact that the proposed model consists of a single place and the same number of transitions that the system model has. Simulation results reflected that the proposed IPN-based diagnoser is

promising for fault detection and isolation for any type of PN-model, unlike the conventional IPN-based diagnoser. Although good results have been obtained using the proposed scheme, further experimentation in a wide range of fault detecting and isolating complex hybrid processes would be worth pursuing to demonstrate conclusively the general validity of the proposed scheme. 6. REFERENCES [1] Y. Chetouani, " Design of a Multi-Model Observer-based Estimator for Fault Detection and Isolation (FDI) Strategy: Application to a chemical reactor”, Brazilian Journal of Chemical Engineering, ISSN 0104-6632, Vol. 25, No. 04, pp. 777 - 788, October - December, 2008. [2] Yu-Chi. Ho, DEDS analyzing complexity and performance in the modern world. IEEE Press ,ISBN 087942-281-5, 1992. [3] D. L., Pepyne, and C. G. Cassandras, (2000) “Optimal control of hybrid systems in manufacturing”, Proceedings of the IEEE, vol. 88, pp. 1108-1123, 2000. [4] M. V. Iordache, and P. J. Antsaklis, "Supervision based on place invariants: A survey", Technical Report of the ISIS Group at the University of Notre Dame, ISIS-2004-003, 2004. [5] W. M. Wonham, “Supervisory control of discrete event systems”, Systems Control Group, Dept. of ECE, University of Toronto, ECE 1636F/1637S, 2007. www.control.utoronto.ca/people/profs/wonham/wonham.htm l. [6] T. Murata, “Petri net: Properties, analysis, and applications”, Proceedings of IEEE, vol. 77, pp. 541-580, 1989. [7] V., Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “A review of process fault detection and diagnosis. Part I: Quantitative model-based methods”, Computers and Chemical Engineering, vol. 27, pp. 293-311, 2003. [8] V., Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “A review of process fault detection and diagnosis. Part II: Qualitative model-based methods”, Computers and Chemical Engineering, vol. 27, pp. 313-326, 2003. [9] V., Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “A review of process fault detection and diagnosis. Part III: Process history based methods”, Computers and Chemical Engineering, vol. 27, pp. 327-346, 2003. [10] J. Arámburo-Lizárraga, A. Ramírez-Treviño, E. LópezMellado and E. Ruiz-Beltrán, " Fault Diagnosis in Discrete Event Systems using Interpreted Petri Nets”, Advances in Robotics, Automation and Control, Book edited by: J. Arámburo and A. Ramírez-Treviño, ISBN 78-953-7619-16-9, pp. 472, October 2008, I-Tech, Vienna, Austria. [11] A. Ramírez-Treviño, I. Rivera-Rangel, and E. López Mellado, “Observability of Discrete Event Systems Modeled by Interpreted Petri Nets”, IEEE Transactions on Robotics and Automation, vol. 19, pp. 557-565, 2003. [12] B. Lennartson, B. Egardt, and M. Tittus, "Hybrid Systems in Process Control", In Proceedings of 33rd

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