-
Research ArticleSensor Fault Diagnosis and Fault-Tolerant Control forNon-Gaussian Stochastic Distribution Systems
HaoWang and Lina Yao
School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China
Correspondence should be addressed to Lina Yao; michelle [email protected]
Received 3 December 2018; Accepted 23 January 2019; Published 11 February 2019
Academic Editor: Xiangyu Meng
Copyright Β© 2019 Hao Wang and Lina Yao. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
A sensor fault diagnosis method based on learning observer is proposed for non-Gaussian stochastic distribution control (SDC)systems. First, the system is modeled, and the linear B-spline is used to approximate the probability density function (PDF) of thesystem output. Then a new state variable is introduced, and the original system is transformed to an augmentation system. Theobserver is designed for the augmented system to estimate the fault. The observer gain and unknown parameters can be obtainedby solving the linearmatrix inequality (LMI).The fault influence can be compensated by the fault estimation information to achievefault-tolerant control. Slidingmode control is used tomake the PDF of the systemoutput to track the desired distribution.MATLABis used to verify the fault diagnosis and fault-tolerant control results.
1. Introduction
With the rapid development of modern science and tech-nology, control systems have become more complex, large-scale, and intelligent, which puts more demands on thecontrol of engineering control systems [1, 2]. The safety andreliability of the systemmust be ensured during the operationof the system. Otherwise, huge personal and property losseswill be caused. Luckily, fault diagnosis and fault-tolerantcontrol play an important role in detecting and avoiding suchaccidents. It is essential to carry out the research of controllingstochastic distribution systems in the field of modern controltheory, which is widely used in controlling of the ore particledistribution in the grinding process, controlling of pulpuniformity and controlling of particle uniformity in thepapermaking process, and the polymer in the process of thechemical reaction as well as the flame distribution controlduring the combustion process of the boiler.
In reality, there are many production sites that are out ofreach and are accompanied by high temperature, high pres-sure, and toxic environments. However, the fault is inevitable.If it fails to be dealt with in time, it will waste precious timeand may lead to extremely serious consequences. Therefore,it is of great significance to carry out research on fault
diagnosis and fault-tolerant control of stochastic distributionsystems to improve its reliability and safety and avoid lossof personnel and property. Stochastic distribution controlhas been regarded an important topic in the control field inrecent decades. Inmost control projects, practical systems aresubjected to stochastic input. These inputs may be derivedfrom noise, stochastic disturbances, or stochastic parameterchanges. Professor Hong Wang put forward to the idea ofstochastic distribution control [3]. Different from the modelin the existing control systems, the overall shape of theprobability density function of the output of the stochasticsystem is considered. The goal of the controller design isto select a good rigid control input so that the probabilitydensity function shape of the system output can track thegiven distribution.
At present, there are many research results for faultdiagnosis and fault-tolerant control of actuator fault in non-Gaussian stochastic distribution systems. For fault diagnosis,fault diagnosis observers or filters based methods are usu-ally used. In literatures [4β6], the adaptive fault diagnosisobserver is used to diagnose the actuators fault, and the faultamplitude is accurately estimated. In literatures [7, 8], faultreconstruction based on learning observer is recorded. Inliteratures [9β11], sensor fault diagnosis and output feedback
HindawiMathematical Problems in EngineeringVolume 2019, Article ID 5839576, 8 pageshttps://doi.org/10.1155/2019/5839576
http://orcid.org/0000-0003-3819-5643https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/5839576
-
2 Mathematical Problems in Engineering
control are described. In literature [12], the filter basedmethod has been shown to be effective method for fault diag-nosis. For fault-tolerant control of non-Gaussian stochasticdistribution systems, two situations are considered: (1) thedesired PDF is known; (2) the desired PDF is not knownin advance. An adaptive PI tracking fault-tolerant controlleris designed to track the desired PDF in the literature [13].In literatures [14, 15], the minimum entropy fault-tolerantcontrol algorithm is constructed based on the unknown PDF,and the performance index function is designed to findthe control input to make the performance index functionbe minimized. In literature [16], collaborative system faultdiagnosis and model prediction fault-tolerant control forthe stochastic distribution system are described. In literature[17], actuator fault diagnosis and PID tracking fault-tolerantcontrol for n subsystems collaboration are introduced.
The research of sensor fault diagnosis and fault-tolerantcontrol of non-Gaussian stochastic distribution system israrely documented; however, the sensor fault is inevitable.Thus, it is very meaningful for the work to be carried outin this paper. In this paper, the learning observer is used todiagnose the fault of the non-Gaussian stochastic distributionsystem, and the fault is compensated using the fault estima-tion information.The sliding mode control algorithm is usedto make the system output PDF to track the expected PDF.
2. Model Description
The system output probability density function (PDF)πΎ(π¦, π’(π‘)) is approximated by linear B-spline function.π1(π¦), π2(π¦), β β β , ππ(π¦) are basis functions defined in advanceon the interval [π, π]. π1, π2, β β β , ππ are correspondingweights associated with the number of basis functions.πΎ(π¦, π’(π‘)) can be expressed as
πΎ (π¦, π’ (π‘)) = πβπ=1
ππ (π’ (π‘)) ππ (π¦) (1)Since the integral of πΎ(π¦, π’(π‘)) on [π, π] is equal to 1, the
following equation holds:
π1π1 + π2π2 + β β β + ππππ = 1 (2)where ππ = β«ππ ππ(π¦)ππ¦, π = 1, 2, β β β , π. Therefore, only π β 1weights are independent of each other, and the linear B-splinemodel is specifically given as follows:
πΎ (π¦, π’ (π‘)) = πΆ (π¦)π (π‘) + π (π¦) + π0 (3)where πΆ(π¦) = [π1(π¦) β ππ(π¦)π1/ππ, π2(π¦) βππ(π¦)π2/ππ, β β β , ππβ1(π¦) β ππ(π¦)ππβ1/ππ], π(π¦) = ππ(π¦)/ππ βπ 1Γ1, π(π‘) = [π1, π2, β β β ππβ1]π β π (πβ1)Γ1. π0 is the PDFapproximation error that can be ignored. The non-Gaussianstochastic distribution system model can be expressed as
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘)π (π‘) = π·π₯ (π‘) + πΊπ (π‘)
πΎ (π¦, π’ (π‘)) = πΆ (π¦)π (π‘) + π (π¦)(4)
where π₯(π‘) β π π is the state vector, π’(π‘) β π π is the controlinput vector, π(π‘) β π π, π(π‘) β π π is the fault vector and theweight vector, respectively, πΊ β π πΓπ is a full rank matrix.(π΄,π·) is observable; π΄, π΅,π·, πΊ are known matrices withappropriate dimension. A new state variable is introduced forfault diagnosis [18].
βΜ (π‘) = βπ΄ π β (π‘) + π΄ π π (π‘) (5)where βπ΄ π is a Hurwitz matrix, β(π‘) β π π. Combined with(4) and (5), the augmented systemmodel can be expressed asfollows:
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘) + πΊπ (π‘)π (π‘) = π·π₯ (π‘)
πΎ (π¦, π’ (π‘)) = πΆ (π¦)π (π‘) + π (π¦)(6)
where π₯(π‘) = [ π₯(π‘)β(π‘) ] π΄ = [ π΄ 0π΄π π· βπ΄π ] π΅ = [ π΅0 ] πΊ =[ 0π΄π πΊ ] π· = [0 πΌπ]
3. Fault Diagnosis
In order to estimate the size of the fault, the fault diagnosisobserver is designed as follows:
ΜΜπ₯ (π‘) = π΄οΏ½ΜοΏ½ (π‘) + π΅π’ (π‘) + πΊπ (π‘) + πΏπ (π‘)οΏ½ΜοΏ½ (π‘) = π·οΏ½ΜοΏ½ (π‘)
οΏ½ΜοΏ½ (π¦, π’ (π‘)) = πΆ (π¦) οΏ½ΜοΏ½ (π‘) + π (π¦)π (π‘) = πΎ1π (π‘ β π) + πΎ2π (π‘ β π)ΜΜπ (π‘) = ππ (π‘)
(7)
where οΏ½ΜοΏ½(π‘) β π π+π is the estimation of state, οΏ½ΜοΏ½(π‘) β π π isthe estimation of weight vector, π(π‘) β π π is a state variable,π(π‘) is the estimation of π(π‘), and π(π‘) is the residual can beexpressed as
π (π‘) = β«πππ (π¦) [πΎ (π¦, π’ (π‘)) β οΏ½ΜοΏ½ (π¦, π’ (π‘))] ππ¦
= β«πππ (π¦)πΆ (π¦) [π· (π₯ (π‘) β οΏ½ΜοΏ½ (π‘) ] ππ¦
= βπ·ππ₯ (π‘)(8)
β = β«πππ(π¦)πΆ(π¦)ππ¦, π(π¦) = π¦, (π΄,π·) is observable, and it
is easy to know that (π΄,π·) is observable. The parameter πis defined as the learning interval, which can be taken as aninteger multiple of the sampling period or sampling period[19]. ππ₯(π‘) represents the state observation error. πΏ, πΎ1, πΎ2,and π are gain matrices with appropriate dimension thatneed to be determined.
-
Mathematical Problems in Engineering 3
Lemma 1. For two matrices π and π with approximatedimension, the following inequality holds [20]:
2ππ β π β€ ππ β π + ππ β π (9)Assumption 2. βπ(π‘)β β€ πΎπ, where πΎπ is a given positiveconstant.
ππ₯ (π‘) = π₯ (π‘) β οΏ½ΜοΏ½ (π‘) (10)The observation error dynamic system obtained by (6), (7)and (8) is formulated as follows:
Μππ₯ (π‘) = οΏ½ΜοΏ½ (π‘) β ΜΜπ₯ (π‘)= (π΄ β πΏβπ·) ππ₯ (π‘) + πΊπ (π‘) β πΊπ (π‘)
(11)
Theorem 3. It is supposed that Assumption 2 holds. If thereexist positive-definite symmetric matrices π1 β π (π+π)Γ(π+π),π 1 β π (π+π)Γ(π+π), π1 β π (π+π)Γ(π+π), and matrices πΎ1 β π πΓπ,πΎ2 β π πΓπ, π β π π+π, πΏ β π π+π, the following inequalities andequation hold:
π1π΄ + π΄ππ1 β πβπ· β (βπ·)π ππ + π 1+ π1πΊπΊππ1 = βπ1
(12)
0 < (6 + 3π) (πΎ2βπ·)π (πΎ2βπ·) β€ π 1 (13)0 < (6 + 3π)πΎ1ππΎ1 β€ πΌ (14)
where π is a given positive constant. οΏ½e observer gain can beobtained from πΏ = π1β1π. Equation (12) can be converted intothe following LMI:
[[(π΄ β πΏβπ·)π π1 + π1 (π΄ β πΏβπ·) + π 1 + π1 π1πΊ
πΊππ1 βπΎ1πΌ]]
< 0(15)
where πΎ1 is a small positive number. Lemma 1 can ensure thatthe following inequalities hold:
ππ (π‘) π (π‘)β€ 3ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)
+ 3πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)(16)
2 ππ₯ (π‘) π1πΊ βπ (π‘)ββ€ πππ₯ (π‘) π1πΊπΊππ1ππ₯ (π‘) + ππ (π‘) π (π‘)
(17)
In order to prove the stability of the system (11), thefollowing Lyapunov function is selected as
π1 (π‘) = ππ₯π (π‘) π1ππ₯ (π‘) + β«π‘π‘βπ
ππ₯π (π ) π 1ππ₯ (π ) ππ + β«π‘π‘βπ
ππ (π ) π (π ) ππ (18)
The derivative can be obtained as follows:
οΏ½ΜοΏ½1 (π‘) = ππ₯π (π‘) [π1 (π΄ β πΏβπ·) + (π΄ β πΏβπ·)π π1]β ππ₯ (π‘) + 2ππ₯π (π‘) π1πΊπ (π‘) β 2ππ₯π (π‘)
β π1πΊπ (π‘) + ππ₯π (π‘) π 1ππ₯ (π‘) β ππ₯π (π‘ β π)β π 1ππ₯ (π‘ β π) + ππ (π‘) π (π‘) β ππ (π‘ β π)β π (π‘ β π)
(19)
From (14) and (15), the following inequality can be obtained:
οΏ½ΜοΏ½1 (π‘) β€ ππ₯π (π‘) [β +π 1] ππ₯ (π‘) + 2πΎπ π1πΊ ππ₯ (π‘)+ πππ₯ (π‘) π1πΊπΊππ1ππ₯ (π‘) + πππ (π‘) π (π‘) + 2ππ (π‘)β π (π‘) β πππ (π‘) π (π‘) β πππ₯ (π‘ β π) π 1ππ₯ (π‘ β π)β ππ (π‘ β π) π (π‘ β π) β€ ππ₯π (π‘)β [β +π 1 + π1πΊπΊππ1] ππ₯ (π‘) + 2πΎπ π1πΊ ππ₯ (π‘)β ππ (π‘ β π) π (π‘ β π) + (6 + 3π)ππ (π‘ β π)β πΎ1ππΎ1π (π‘ β π) + (6 + 3π) πππ₯ (π‘ β π) (πΎ2βπ·)πβ (πΎ2βπ·) ππ₯ (π‘ β π) β πππ₯ (π‘ β π) π 1ππ₯ (π‘ β π)β πππ (π‘) π (π‘) = ππ₯π (π‘) [β +π 1 + π1πΊπΊππ1]β ππ₯ (π‘) + 2πΎπ π1πΊ ππ₯ (π‘) β πππ (π‘) π (π‘)+ ππ (π‘ β π) ((6 + 3π)πΎ1ππΎ1 β πΌ)π (π‘ β π)+ πππ₯ (π‘ β π)β ((6 + 3π) (πΎ2βπ·)π (πΎ2βπ·) β π 1) ππ₯ (π‘ β π)
(20)
For (20), when inequalities (12), (13), and (14) are satisfied, thefollowing inequality can be further obtained:
οΏ½ΜοΏ½1 (π‘) β€ βπmin (π1) ππ₯ (π‘)2 + 2πΎπ ππΊ ππ₯ (π‘)β πππ (π‘) π (π‘) (21)
where β = π1(π΄ β πΏβπ·) + (π΄ β πΏβπ·)ππ1. The proof iscompleted.
After diagnosing the system sensor fault, the originalsystem is rebuilt to observe the original system PDF afterthe fault occurs and the subsequent fault-tolerant controlis prepared. The original system state, weights, and outputPDF and their observations can be obtained by the followingcoordinate transformation.
π₯ (π‘) = [πΌπ 0] οΏ½ΜοΏ½ (π‘)οΏ½ΜοΏ½ (π‘) = π·π₯ (π‘) + πΊπ (π‘)
πΎ (π¦, π’ (π‘)) = πΆ (π¦) οΏ½ΜοΏ½ (π‘) + π (π¦)(22)
-
4 Mathematical Problems in Engineering
Inequality (16) can be proved as follows:
ππ (π‘) π (π‘)= ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)
+ ππ (π‘ β π)πΎ1ππΎ2βπ·ππ₯ (π‘ β π)+ πππ₯ (π‘ β π) (πΎ2βπ·)ππΎ1π (π‘ β π)+ πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)
(23)
It can be formulated that
2ππ (π‘) π (π‘)β€ 2ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)
+ ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)+ πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)+ ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)+ πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)+ ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)+ 2πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)+ πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)+ ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)+ πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)
= 6ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)+ 6πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)
(24)
It can be further obtained that
ππ (π‘) π (π‘)β€ 3ππ (π‘ β π)πΎ1ππΎ1π (π‘ β π)
+ 3πππ₯ (π‘ β π) (πΎ2βπ·)π (πΎ2βπ·) ππ₯ (π‘ β π)(25)
4. Fault-Tolerant Control
The basic idea of fault-tolerant control (FTC) is to compen-sate the influence of fault on the system performance. Sensorfaultmay exist inmany forms, such as constant, time-varying,or even unbounded. When the system sensor fault occurs, itusually does not work properly. A simple fault compensationscheme is designed in this paper. π(π‘) is the weight vector ofthe system output, which needs to be compensated in orderto ensure the performance of the system after the fault occurs.
The fault estimationπ(π‘) can be obtained by the observer.Thecompensated weight ππ(π‘) can be expressed as follows:
ππ (π‘) = π (π‘) β πΊπ (π‘) (26)Using a compensated sensor output ππ(π‘), control algo-
rithms are given to ensure that whether fault occurs or not,sensor fault compensation can be performed using fault-tolerant operation. To make sure that the probability densityfunction (PDF) of the systemcan still track a given probabilitydensity function after a fault occurs, a fault-tolerant controllerneeds to be designed. A new weight error vector is defined asππ(π‘) = ππ(π‘) β ππ, where ππ is the weight of the expectedoutput. The following weight error dynamic system can beobtained.
Μππ (π‘) = οΏ½ΜοΏ½π (π‘) β οΏ½ΜοΏ½π = π·οΏ½ΜοΏ½ (π‘) + πΊ Μππ (π‘)= π· [π΄π₯ (π‘) + π΅π’ (π‘)] + πΊ Μππ (π‘)= π·π΄π₯ (π‘) + π·π΅π’ (π‘) + πΊ Μππ (π‘)= π·π΄π·β1π·π₯ (π‘) + π·π΄π·β1πΊππ (π‘) β π·π΄π·β1ππ
β π·π΄π·β1πΊππ (π‘) + π·π΄π·β1ππ + πΊ Μππ (π‘)+ π·π΅π’ (π‘)
= π΄1ππ (π‘) β π΄1πΊππ (π‘) + π΄1ππ + πΊ Μππ (π‘)+ ππ’ (π‘)
(27)
where π΄1 = π·π΄π·β1 andπ = π·π΅.With the sliding mode control method, two necessary
conditions should be satisfied: the accessibility of the systemstate and the asymptotic stability of the slidingmode dynamicprocess. The sliding mode control law is designed so that thestate trajectory at any moment can reach the sliding surfaceduring a limited time. The switching function is designed asfollows:
π (π‘) = π»ππ (π‘) β β«π‘0π»(π΄1 + ππΎ3) ππ (π) ππ (28)
where the matrix πΎ3 is selected such that π΄1 + ππΎ3 is aHurwitz matrix. The matrix π» is chosen to make π»π be anonsingular matrix. After the state trajectory of the weighterror dynamic system reaches the sliding surface, π(π‘) = 0 andΜπ(π‘) = 0 should be satisfied simultaneously. The equivalentcontrol law is shown as follows:
π’ππ = (π»π)β1π»[ππΎ3ππ (π‘) β π΄1ππ] (29)Equation (29) is substituted into (27), and (27) can be furtherobtained as
Μππ (π‘) = [π΄1 + π(π»π)β1π»ππΎ3] ππ (π‘)+ [πΌ β π (π»π)β1π»]π΄1ππ β π΄1πΊππ (π‘)+ πΊ Μππ (π‘)
(30)
-
Mathematical Problems in Engineering 5
Theorem 4. If both positive-definite symmetric matrices π2and π2 exist, the following inequality holds:
π2 (π΄1 + ππΎ3) + (π΄1 + ππΎ3)π π2 + π2 β€ 0 (31)οΏ½en the sliding mode dynamic system is stable. οΏ½e followingLyapunov function is selected as
π2 (π‘) = πππ (π‘) π2ππ (π‘) (32)οΏ½e first-order derivative of π2(π‘) is formulated as
οΏ½ΜοΏ½2 (π‘) = πππ (π‘) [π2 (π΄1 + ππΎ3) + (π΄1 + ππΎ3)π π2]β ππ (π‘) + 2πππ (π‘) π2 ([πΌ β π (π»π)β1π»]π΄1ππ+ πΊ Μππ (π‘) β π΄1πΊππ (π‘)) β€ βπmin (π2) ππ (π‘)2+ 2 ππ (π‘) π2 (πΌ β π (π»π)β1π» π΄1ππβ π΄1πΊ ππ (π‘) + βπΊβ Μππ (π‘)) = βπ1 ππ (π‘)2+ 2π2 ππ (π‘)
(33)
where π1 = πmin(π2) and π2 = βπ2β(βπΌ βπ(π»π)β1π»ββπ΄1ππβ β βπ΄1πΊββππ(π‘)β + βπΊββ Μππ(π‘)β).Using (33), the following inequality can be obtained as
οΏ½ΜοΏ½2 (π‘) β€ βπ1 (ππ (π‘) β π2π1)2 + π22π1 (34)
when βππ(π‘)β β₯ 2π2/π1, οΏ½ΜοΏ½2 β€ 0, and the sliding mode dynamicsystem is stable. In order to ensure that the state trajectorystarting at any position can reach the sliding surface during alimited time, the following sliding mode control law is designed:
π’π (π‘) = {{{βπΌ π (π‘)βπ (π‘)β2 π (π‘) ΜΈ= 00 π (π‘) = 0 (35)
where πΌ > 0. οΏ½e sliding mode control law can ensure that thestate trajectory reaches the sliding surface during a limited time,π(π‘) = 0.
οΏ½e following Lyapunov function is chosen as
π3 (π‘) = 12ππ (π‘) (π»π)β1 π (π‘) (36)οΏ½ΜοΏ½3 (π‘) = ππ (π‘) (π»π)β1 Μπ (π‘) = ππ (π‘) (π»π)β1 [π» Μππ (π‘)
β π» (π΄ + ππΎ3) ππ (π‘)] = ππ (π‘) (π»π)β1β π» [π΄1ππ (π‘) β π΄1πΊππ (π‘) + π΄1ππ + πΊ Μππ (π‘)+ ππ’ (π‘)] β ππ (π‘) (π»π)β1π»(π΄ + π΅πΎ3) ππ (π‘)= ππ (π‘) (π»π)β1π»ππ’π (π‘) + ππ (π‘) (π»π)β1β π» [πΊ Μππ (π‘) β π΄1πΊππ (π‘))] = βπΌππ (π‘) π (π‘)βπ (π‘)β2 + π3= βπΌ + π3 < 0
(37)
where πΌ is selected as πΌ > π3, π3 =βππ(π‘)ββ(π»π)β1ββπ»β[βπΊββ Μππ(π‘)β β βπ΄1πΊββππ(π‘)β]. οΏ½estate trajectory reaches the sliding surface during a limitedtime. οΏ½e available fault-tolerant control law is shown asfollows:
π’ (π‘) = π’ππ (π‘) + π’π (π‘)= (π»π)β1π»[ππΎ3 (π (π‘) β πΊπ (π‘) β ππ) β π΄1ππ]
β πΌ π (π‘)βπ (π‘)β2(38)
5. A Simulation Example
In order to verify the effectiveness of the algorithm, theproposed method is applied to the process of molecularweight distribution (MWD) dynamic modeling and control,and a continuous stirring reactor (CSTR) is considered as anexample.The closed-loop control diagram of the polymeriza-tion process is shown in Figure 1. The specific mathematicalmodel is shown as follows:
ΜπΌ (π‘) = πΌ0 β πΌ (π‘)π β πΎππΌ (π‘) + πΎπΌπ’ (π‘)οΏ½ΜοΏ½ (π‘) = π0 β π(π‘)π β 2πΎππΌ (π‘) + πΎππ’ (π‘)
β (πΎπ + πΎπ‘ππ)π (π‘) π π(39)
where π = π/πΉ is the average residence time of reactants(π ), πΌ0 is the initial concentration of initiator (πππ β ππβ1); πΌis the initiator concentration (πππ β ππβ1 ); π0 is the initialconcentration of monomer (πππ β ππβ1 ); π is the monomerconcentration (πππ β ππβ1 );πΎπ, πΎπ, πΎπ‘ππ are the reaction rateconstants; πΎπΌ, πΎπ are constants associated with the controlinput; π π(π = 1, 2, . . . , π) are the free radical. The abovespecific physicalmeaning can be referred to the literature [21].
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘)π (π‘) = π·π₯ (π‘) + πΊπ (π‘)
πΎ (π¦, π’ (π‘)) = πΆ (π¦)π (π‘) + π (π¦)(40)
For this stochastic distribution system, the output proba-bility density function (PDF) can be approximated by a linearB-spline basis function of the form:
π1 (π¦) = 12 (π¦ β 2)2 πΌ1 + (βπ¦2 + 7π¦ β 11.5) πΌ2+ 12 (π¦ β 5)2 πΌ3
-
6 Mathematical Problems in Engineering
Monomer
Monomerinitiator
MWDcontroller
MWDmodeling
Polymer
heatedoil
CSTR
C
Fi
Fm
Figure 1: Schematic diagram of a continuous stirred reactor.
π2 (π¦) = 12 (π¦ β 3)2 πΌ2 + (βπ¦2 + 9π¦ β 19.5) πΌ3+ 12 (π¦ β 6)2 πΌ4
π3 (π¦) = 12 (π¦ β 4)2 πΌ3 + (βπ¦2 + 11π¦ β 29.5) πΌ4+ 12 (π¦ β 7)2 πΌ5
(41)
πΌπ (π¦) = {{{1 π¦ β [π + 1, π + 2]0 ππ‘βπππ€ππ π, (π = 1, 2, 3, 4, 5) (42)
The system parameter matrices and vectors are given asfollows:
π΄ = [ 0 1β2 β2]
π΅ = [11]
π· = [ 1 00.5 1]
πΊ = [01]
π΄ π = [1 00 1]
π΄ =[[[[[[
0 1 0 0β2 β2 0 01 0 β1 00.5 1 0 β1
]]]]]]
π΅ =[[[[[[
1100
]]]]]]
πΊ =[[[[[[
0001
]]]]]]
π· = [0 0 1 00 0 0 1](43)
matrices πΏ, πΎπ(π = 1, 2, 3), π and matrix π» are selected asfollows: πΏ = [ β0.5059β0.05990.9326
β2.9999
], πΎ1 = 0.057, πΎ2 = β0.5, πΎ3 =[ 12.1635 0.0389 ], π = 22.5, π» = [ 0.4 β0.1 ], β =[ β2 β1 ], and π = 0.01. It is assumed that the fault isconstructed as follows:
π (π‘) ={{{{{{{{{
0 0 β€ π‘ < 40.5 4 β€ π‘ < 401.6 β πβ0.16(π‘β40) π‘ β₯ 40
(44)
-
Mathematical Problems in Engineering 7
faultfault estimation
β1
β0.5
0
0.5
1
1.5
2fa
ult a
nd it
s esti
mat
ion
10 20 30 40 50 60 70 800t (s)
Figure 2: Fault and fault estimation.
t (s)y
the desired PDF
0.5
0.4
0.3
0.2
0.1
08
6
4
2 020
4060
80
(y
,u(t)
)
Figure 3: The system expectation output PDF.
the initial PDF
t (s)y
8
6
4
2 020
4060
80
(y
,u(t)
)
1.5
1
0.5
0
β0.5
β1
Figure 4: The system output PDF without fault-tolerant control.
t (s)y
System of three-dimensional graph (PDF)
0.8
0.6
0.4
0.2
0
β0.28
6
4
2 020
4060
80
(y
,u(t)
)
Figure 5: The system output PDF with fault-tolerant control.
the desired PDFthe final PDF
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
(y
,u(t)
)
2.5 3 3.5 4 4.5 5 5.5 6 6.5 72y
Figure 6: The system final and desired PDF.
From Figure 2, it can be seen that the fault occurs at t= 4s, and the fault estimation can track the change of faultafter short transition. The desired output PDF can be seenin Figure 3. Figure 4 shows that the system actual outputPDF cannot track the desired PDF without FTC when thefault occurs. The system postfault output PDF with FTC cantrack the desired PDF, which is shown in Figures 5 and 6, thevalidity of the FTC algorithm is verified.
6. Conclusions
In this paper, the problem of sensor fault diagnosis and fault-tolerant control for non-Gaussian stochastic distributionsystems is studied. The learning observer is used to diagnosethe sensor fault. The fault is compensated by the fault esti-mation information, and the sliding mode control algorithm
-
8 Mathematical Problems in Engineering
is utilized to make the PDF of the system output to trackthe desired distribution. The Lyapunov stability theorem isapplied to prove the stability of the dynamic system of theobservation error and the whole control process. Finally, themathematical model is built by the actual industrial controlprocess and the computer simulation further verifies theeffectiveness of the algorithm.
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Theauthorswould like to thank the financial support receivedfrom Chinese NSFC Grant 61374128, State Key Laboratoryof Synthetical Automation for Process Industries, the Sci-ence and Technology Innovation Talents 14HASTIT040 inColleges and Universities in Henan Province, China, Excel-lent Young Scientist Development Foundation 1421319086 ofZhengzhou University, China, and the Science and Technol-ogy Innovation Team in Colleges and Universities in HenanProvince 17IRTSTHN013.
References
[1] X. Meng and T. Chen, βEvent detection and control co-designof sampled-data systems,β International Journal of Control, vol.87, no. 4, pp. 777β786, 2014.
[2] X. Meng and T. Chen, βEvent triggered robust filter design fordiscrete-time systems,β IET Control οΏ½eory & Applications, vol.8, no. 2, pp. 104β113, 2014.
[3] H. Wang, Bounded Dynamic Stochastic Systems: Modeling andControl, Springer, London, UK, 2000.
[4] L. N. Yao and B. Peng, βFault diagnosis and fault tolerant con-trol for the non-Gaussian time-delayed stochastic distributioncontrol system,β Journal of οΏ½e Franklin Institute, vol. 351, no. 3,pp. 1577β1595, 2014.
[5] L. N. Yao, J. F. Qin, H. Wang, and B. Jiang, βDesign of new faultdiagnosis and fault tolerant control scheme for non-Gaussiansingular stochastic distribution systems,β Automatica, vol. 48,no. 9, pp. 2305β2313, 2012.
[6] L. Yao and L. Feng, βFault diagnosis and fault tolerant trackingcontrol for the non-Gaussian singular time-delayed stochasticdistribution system with PDF approximation error,β Interna-tional Journal of Control Automation and Systems, vol. 14, no.2, pp. 435β442, 2014.
[7] Q. X. Jia, W. Chen, Y. C. Zhang, and X. Q. Chen, βRobustfault reconstruction via learning observers in linear parameter-varying systems subject to loss of actuator effectiveness,β IETControl οΏ½eory & Applications, vol. 8, no. 1, pp. 42β50, 2014.
[8] Q. Jia, W. Chen, Y. Zhang, and H. Li, βIntegrated design offault reconstruction and fault-tolerant control against actuatorfaults using learning observers,β International Journal of SystemsScience, vol. 47, no. 16, pp. 3749β3761, 2016.
[9] Y. Tan, D. Du, and S. Fei, βCo-design of event generator andquantized fault detection for time-delayed networked systemswith sensor saturations,β Journal of οΏ½e Franklin Institute, vol.354, no. 15, pp. 6914β6937, 2017.
[10] M. Liu and P. Shi, βSensor fault estimation and tolerant controlfor ItoΜ stochastic systems with a descriptor sliding modeapproach,β Automatica, vol. 49, no. 5, pp. 1242β1250, 2013.
[11] D. Du and V. Cocquempot, βFault diagnosis and fault tolerantcontrol for discrete-time linear systemswith sensor fault,β IFAC-PapersOnLine, vol. 50, no. 1, pp. 15754β15759, 2017.
[12] S. Xu, βRobust Hβ filtering for a class of discrete-time uncer-tain nonlinear systems with state delay,β IEEE Transactions onCircuits and Systems I: Fundamental οΏ½eory and Applications,vol. 49, no. 12, pp. 1853β1859, 2003.
[13] J. Zhou, G. Li, and H.Wang, βRobust tracking controller designfor non-Gaussian singular uncertainty stochastic distributionsystems,β Automatica, vol. 50, no. 4, pp. 1296β1303, 2014.
[14] H. Jin, Y. Guan, and L. Yao, βMinimum entropy active faulttolerant control of the non-Gaussian stochastic distributionsystem subjected to mean constraint,β Entropy, vol. 19, no. 5, p.218, 2017.
[15] D. Du, B. Jiang, and P. Shi, βSensor fault estimation andcompensation for time-delay switched systems,β InternationalJournal of Systems Science, vol. 43, no. 4, pp. 629β640, 2012.
[16] Y. Ren, A. Wang, and H. Wang, βFault diagnosis and tolerantcontrol for discrete stochastic distribution collaborative controlsystems,β IEEE Transactions on Systems, Man, and Cybernetics:Systems, vol. 45, no. 3, pp. 462β471, 2015.
[17] Y. Ren, Y. Fang, A. Wang, H. Zhang, and H. Wang, βCollabora-tive operational fault tolerant control for stochastic distributioncontrol system,β Automatica, vol. 9, no. 8, pp. 141β149, 2018.
[18] C. Edwards and C. P. Tan, βA comparison of sliding mode andunknown input observers for fault reconstruction,β EuropeanJournal of Control, vol. 12, no. 3, pp. 245β260, 2006.
[19] Q. Jia, W. Chen, Y. Zhang, and H. Li, βFault reconstruction forcontinuous-time systems via learning observers,β Asian Journalof Control, vol. 18, no. 1, pp. 1β13, 2016.
[20] T. Youssef,M. Chadli, H. R. Karimi, and R.Wang, βActuator andsensor faults estimation based onproportional integral observerfor T-S fuzzy model,β Journal of the Franklin Institute, vol. 354,no. 6, pp. 2524β2542, 2017.
[21] L. Yao,W. Cao, and H.Wang, βMinimum entropy fault-tolerantcontrol of the non-gaussian stochastic distribution system,βin Proceedings of the International Conference on InformationEngineering andApplications (IEA) 2012, Springer, London, UK,2013.
-
Hindawiwww.hindawi.com Volume 2018
MathematicsJournal of
Hindawiwww.hindawi.com Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwww.hindawi.com Volume 2018
Probability and StatisticsHindawiwww.hindawi.com Volume 2018
Journal of
Hindawiwww.hindawi.com Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwww.hindawi.com Volume 2018
OptimizationJournal of
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
Engineering Mathematics
International Journal of
Hindawiwww.hindawi.com Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwww.hindawi.com Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwww.hindawi.com Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwww.hindawi.com Volume 2018
Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com
The Scientific World Journal
Volume 2018
Hindawiwww.hindawi.com Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com
DiοΏ½erential EquationsInternational Journal of
Volume 2018
Hindawiwww.hindawi.com Volume 2018
Decision SciencesAdvances in
Hindawiwww.hindawi.com Volume 2018
AnalysisInternational Journal of
Hindawiwww.hindawi.com Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwww.hindawi.com
https://www.hindawi.com/journals/jmath/https://www.hindawi.com/journals/mpe/https://www.hindawi.com/journals/jam/https://www.hindawi.com/journals/jps/https://www.hindawi.com/journals/amp/https://www.hindawi.com/journals/jca/https://www.hindawi.com/journals/jopti/https://www.hindawi.com/journals/ijem/https://www.hindawi.com/journals/aor/https://www.hindawi.com/journals/jfs/https://www.hindawi.com/journals/aaa/https://www.hindawi.com/journals/ijmms/https://www.hindawi.com/journals/tswj/https://www.hindawi.com/journals/ana/https://www.hindawi.com/journals/ddns/https://www.hindawi.com/journals/ijde/https://www.hindawi.com/journals/ads/https://www.hindawi.com/journals/ijanal/https://www.hindawi.com/journals/ijsa/https://www.hindawi.com/https://www.hindawi.com/