Abstract

An approach to fault estimation systems design, adjusted for linear continuous-time systems, is proposed in the paper. Based on LMI approach, the method exploits the state-space observer principle in an adaptive scheme intended for single actuator faults. A simulation example, subject to different type of failures, demonstrates the effectiveness of the proposed form of the fault estimation technique

Liščinský, Pavol

Serbák, Vladimír

Advances in Electrical and Electronic Engineering

Vladimir Serbak

Pavol Liscinsky

Crossref

Adaptive Observer Based Actuator Faults Estimation

DSpace at VSB Technical University of Ostrava

CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHAdaptive Observer Based Actuator FaultsEstimationVladimir SERBAK, Pavol LISCINSKYDepartment of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics,Technical University of Kosice, Letna 9, 042 00 Kosice, Slovakiavladimir.serbak@tuke.sk, pavol.liscinsky@tuke.skDOI: 10.15598/aeee.v13i1.1226Abstract. An approach to fault estimation systems de-sign, adjusted for linear continuous-time systems, isproposed in the paper. Based on LMI approach, themethod exploits the state-space observer principle in anadaptive scheme intended for single actuator faults. Asimulation example, subject to different type of failures,demonstrates the effectiveness of the proposed form ofthe fault estimation technique.KeywordsAdaptive state observer, fault estimationschemes, state-space system description.1. IntroductionOperating conditions in modern engineering systemsare still exposed to possibility of system failure. Anyfailure of sensors, actuators or other system com-ponents can drastically change the system behavior.Fault tolerant control (FTC) heavily relies on fault de-tection, identification and isolation schemes (FDI) al-lows a strategy to improve reliability of the system.Estimation of an occurred actuator fault can be uti-lized in the processes of FDI. The final results from FDIshould provide sufficient information for the secondpart of the remedial action, performed in the systemafter the fault occurrence, namely the control recon-figuration that is reconfiguring or adapting the nomi-nal control to compensate undesired effects caused bya fault. The used principles of fault estimation coverschemes based on the reduced order observers [4], adap-tive observers [1], [9], the unknown input observers [3]and sliding-mode observers [7]. This approach utilizingtwo step remedial process is typical for active FTC.Other option is to utilize passive FTC, but this how-ever does not offer possibility to find solution that isoptimal after the fault occurrence and is beneficial onlyin case of a priori considered faults.An approach described in this paper utilizes theadaptive state observer and is usually denoted as thefast adaptive fault estimation [2], [8].2. Problem FormulationA linear dynamic multi-input, multi-output (MIMO)system in presence of an unknown fault can be de-scribed by the state-space equations in the followingform:q˙(t) = A~q(t) +B~u(t) +E~f(t), (1)~y(t) = C~q(t), (2)where A ∈ <n×n, E ∈ <n×s, B ∈ <n×r and C ∈ <p×nare real matrices. To estimate faults, the followingadaptive state estimator is proposed in the generalform:q˙e(t) = A~ee(t)+B~u(t)+E~fe(t)+J(~y(t)−~ye(t)), (3)~ye(t) = C~qe(t), (4)where J ∈ <n×p and ~fe(t) is an estimate of thefault ~f(t). The above used vectors ~q(t), ~qe(t) ∈ <n,~u(t) ∈ <r and ~y(t), ~ye(t) ∈ <p are vectors of the state,estimated state, input, output and estimated outputvariables, respectively.The task is to design the matrix J in such a way thatthe observer system matrixAe = A−JC be stable andthe estimated fault ~fe(t) approximates time propertiesof ~f(t). Considering single actuator faults, the matrixE takes form of the corresponding column of the inputmatrix B, i.e. E = Bi for i ∈ 〈1, 2, ...r〉. Moreover, itis required that rank(CE) is equal to rank(E).c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 48CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHThe state observer Eq. (3) and Eq. (4), is connectedwith the fault estimation updating law of the form [6],[8]:f˙e(t) = GHT~ey(t), (5)where H ∈ <p×s is the gain matrix and the matrixG = GT > 0, G ∈ <s×s is a learning weight matrix tobe set interactively.The design of the observer matrix parameters has toensure asymptotic convergence of the estimation errorsin Eq. (6) to zero values.~ef (t) = ~f(t)− ~fe(t),~ey(t) = ~y(t)− ~ye(t).(6)Assumption 1 The couple (A,B) is controllable andthe couple (A,C) is observable.Assumption 2 The unknown fault vector, changingunexpectedly when a fault occurs, is differentiable andbounded, i.e., |~f(t)| < ~f , ~f is known, and the value of~f(t) is set to zero until a fault occurs.Moreover, Ass. 2 implies that the derivative ~ef (t)with respect to time can be considered as follows:~ef (t) = ~f0 − ~fe(t) ⇒ e˙f (t) = −f˙e(t). (7)From the point of the paper main results, the follow-ing lemma is needed to indicate the reasons for substan-tial modification of the actuator fault estimator designcondition.Lemma 1 [2] The estimator system matrix is sta-ble if there exist symmetric positive definite matrixP ∈ <n×n and matrices H ∈ <p×s, Y ∈ <n×psuch that:P = PT > 0, (8)PA+ATP−YC−CTYT < 0, (9)PE = CTH. (10)When the above conditions hold, the observer gainmatrix is given by:J = P−1Y (11)and the adaptive fault estimation algorithm is:f˙e(t) = GHTC~eq(t), (12)where~eq(t) = ~q(t)− ~qe(t). (13)Since single actuator faults act on the systemthrough different input vectors (columns of the matrixB), it is possible to avoid design of different estimatorswith the tuning matrix parameter G > 0 and formu-late the task of the estimator design through the set ofmatrix equalities:PEi = CTH. (14)Therefore solutions of Eq. (8), Eq. (9) and Eq. (14)are very conservative since the Lyapunov matrix P ver-ifies the fault observer stability not only forA, but alsowith respect to all polytops defined by Ei. To suppressthis disadvantage it is proposed to decouple the Lya-punov matrix from all system parameters A and Eiand so to obtain less conservative the enhanced designconditions. Another reason to apply this principle is ifEq. (10) is close to singular or singular.3. Enhanced DesignConditionsIn case of the enhanced design conditions the followingtheorem holds. It is evident that the theorem can besimply modified to respect Eq. (14).Theorem 1 The fault estimator is stable if for givenpositive tuning parameter δ ∈ < there exist a sym-metric positive definite matrix P ∈ <n×n, matricesQ ∈ <n×n, H ∈ <p×s, Y ∈ <n×p, identity matrixIs ∈ <s×s and a positive scalar γ ∈ < such that:P = PT > 0, γ > 0, (15) P•11 ∗ ∗P•21 −δ(Q+QT) ∗0 δETQ −γIs < 0, (16)whereP•11= QTA+ATQ−YC−CTYT +CTC,P•21 = P−Q+ δQTA− δYC(17)andCTH = QE. (18)When the above conditions hold, the estimator gainmatrix is given by:J = P−1Y (19)and the adaptive fault estimation algorithm is:f˙e(t) = GHTC~eq(t). (20)Here and hereafter ∗ denotes the symmetric item ina symmetric matrix.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 49CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHProof: From system model Eq. (1), Eq. (2) and esti-mator model Eq. (3), Eq. (4) in this case can be seenthat the time derivative of ~eq(t) is:e˙q(t) = (A−JC)~eq(t)+E~ef (t), (21)and, moreover, can be expressed in the following form:(A−JC)~eq(t)+E~ef (t)− e˙q(t) = 0. (22)The Lyapunov function is considered as follows:v(~eq(t)) = ~eTq (t)P~eq(t)+~eTf (t)G−1~ef(t)++t∫0(~eTy (r)~ey(r)−γ~eTf (r)~ef (r))dr,(23)where P = PT > 0, G = GT > 0 and γ is square ofthe H∞ norm of the transfer matrix function between~ef and ~eq. Then the time derivative of v(~eq(t)) is:v˙(~eq(t)) = ~eTq (t)P~eq(t) + ~eTq (t)Pe˙q(t)++e˙Tf (t)G−1~ef (t) + ~eTf (t)G−1e˙f (t)++~eTy (t)~ey(t)− γ~eTf (t)~ef (t).(24)If it is assumed, that the following statements Eq. (7)andf˙e(t) = GHT~ey(t) = GHTC~eq(t), (25)hold, then the substitution of Eq. (25) into Eq. (24)leads tov˙(~eq(t)) = e˙Tq (t)P~eq(t) + ~eTq (t)Pe˙q(t)−−~eTq (t)CTHGG−1~ef (t)−−~eTf (t)G−1GHTC~eq(t)++~eTy (t)~ey(t)− γ~eTf (t)~ef (t),(26)that can be modified into the following form:v˙(~eq(t)) = e˙Tq (t)P~eq(t) + ~eTq (t)Pe˙q(t)−−~eTq (t)CTH~ef (t)− ~eTf (t)HTC~eq(t)++~eTy (t)~ey(t)− γ~eTf (t)~ef (t).(27)It is possible to define the condition based onEq. (22) as:(~eTq (t)ST1 + e˙Tq (t)ST2 )××((A−JC)~eq(t)+E~ef (t)− e˙q(t)) = 0(28)and inserting into Eq. (27), the following expression isobtained:v˙(~eq(t)) = e˙Tq (t)P~eq(t) + ~eTq (t)Pe˙q(t)−−~eTq (t)CTH~ef (t)− ~eTf (t)HTC~eq(t)++(~eTq (t)ST1 + e˙Tq (t)ST2 )((A−JC)~eq(t)−−e˙q(t)) + ((A−JC)~eq(t)− e˙q(t))T(S1~eq(t) + S2e˙q(t)) + (~eTq (t)ST1++e˙Tq (t)ST2 )E~ef (t) + ~eTf (t)ET(S1~eq(t)++S2e˙q(t)) + ~eTy (t)~ey(t)− γ~eTf (t)~ef (t).(29)If the following condition is defined:0 = ~eTf (t)[ETS1−HTC]~eq(t)−−~eTq (t)[ST1E−CTH]~ef (t), (30)then the equalityST1E−CTH = 0 (31)impliesCTH = ST1E (32)and this allows to transcribe Eq. (29) as:v˙(~eq(t)) = e˙Tq (t)P~eq(t) + ~eTq (t)Pe˙q(t)++(~eTq (t)ST1 + e˙Tq (t)ST2 )((A−JC)~eq(t)−−e˙q(t)) + (~eTq (t)(A−JC)T − e˙Tq (t))(S1~eq(t) + S2e˙q(t)) + e˙Tq (t)ST2E~ef (t)++~eTf (t)ETS2e˙q(t) + ~eTy (t)~ey(t)−−γ~eTf (t)~ef (t)(33)and with the notation:~e •Tq (t) =[~eTq (t) e˙Tq (t) ~eTf (t)], (34)it can be written as:v˙(~eq(t)) = ~e•Tq (t)P•~e •q (t) < 0, (35)whereP• = P•11 ∗ ∗P•21 −ST2 − S2 ∗0 ETS2 −γIs < 0 (36)andP•11= ST1 (A−JC) +CTC+ (A−JC)TS1,P•21 = P+ ST2 (A− JC)− S1.(37)Inserting into Eq. (32), Eq. (36) and Eq. (37) thefollowing substitutions:S1 = Q, S2 = δS1, ST1 J = QTJ = Y (38)then Eq. (32), Eq. (36) and Eq. (37) implies Eq. (16),Eq. (17) and Eq. (18). This concludes the proof.Note, Eq. (17) and Eq. (18) implies that the Lya-punov matrix P is decoupled from all system parame-ters.4. Illustrative ExampleTo illustrate the proposed method, the system whosedynamics is described by the equations Eq. (1) andEq. (2) is considered with the matrix parameters:A =0.1650 −0.0767 0.5173 2.06380.2151 0.1866 −0.3077 −1.5260−0.3851 0.4033 0.2887 −0.3833−2.0829 1.5170 0.3093 0.1109 ,c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 50CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHCT=2 21 22 12 2 , B = E =−0.9042 0−1.6493 0.43610 1.85290 −0.0355 .For simulation purposes only, the equilibrium of thesystem was stabilized by the state feedback controller:~u(t) = −K~q(t),where the gain matrix K was designed using the pole-placement method as follows:K=[−3.1315 1.6633 3.5641 11.3920−4.2235 3.1471 1.0377 0.9104].Solving Eq. (8), Eq. (9) and Eq. (10) with respectto the LMI matrix variables P, H, and Y using Self-Dual-Minimization (SeDuMi) package [5] for Matlab,the estimator parameter design problem was solved asfeasible and the LMI matrix variables were:P =0.1770 0.2107 0.0321 0.04570.2107 0.6888 0.0678 0.20600.0321 0.0678 0.1586 −0.09550.0457 0.2060 −0.0955 0.6740 ,Y=−0.3691 0.3090−0.7451 −0.12670.0251 0.16470.2357 −0.8632,H=[ 0.4026 −0.1207−0.2489 0.2774],so that the observer gain matrix were given asJ =−1.2534 2.8937−1.1442 −0.68721.5003 −0.01820.9967 −1.2693 .It can verify that the eigenvalue spectrum ρ(Ae) ofthe estimator system matrix Ae is stable, where:ρ(Ae) ={−2.1745 − 2.6067 − 3.3377 ± 0.1566 i}.Setting the tuning parameter G as:G =[1.7 11 5.5],the observer fault response is given in Fig. 2.This figure presents the fault signal, as well as its es-timation, reflecting a single actuator fault in the secondactuator. Fault starts at the time instant t = 30 s andis applied for 40 s. The tuning parameter was experi-mentally set considering the maximal value of the faultsignal amplitude. Then, at the time instant t = 100 s,the first actuator fault is applied for 40 s.From the simulation results in Fig. 2 it can be ob-served that the differences between the signals reflect-ing a single actuator fault and the observer approxi-mate ones tends to zero. Moreover, the states of theFig. 1: The actuator faults and its estimation.system converge to the equilibrium when the actuatorfault disappeared, via the used controller.Applying the same toolbox to solve LMIs Eq. (15),Eq. (16), Eq. (17) and Eq. (18) conditioned byδ = 0.001, the resulting set of the matrix variableswas as follows:P =23.2065 −5.2282 6.5739 5.9199−5.2282 16.1633 −5.5997 4.23216.5739 −5.5997 11.1507 4.67935.9199 4.2321 4.6793 24.1004 ,Q =23.2346 −5.2370 6.5874 5.8961−5.2420 16.1435 −5.6122 4.24796.5646 −5.6171 11.1679 4.65165.9477 4.2403 4.7055 24.0935 ,Y=−4.9513 3.7898−6.9495 11.805210.1032 −1.616911.5372 −3.4027, H=[ 9.5143 13.2220−15.7000 −8.3657]and the estimator gain matrix J, obtained usingEq. (19), was:J =−0.7142 0.4462−0.5347 1.14820.8107 0.39020.5906 −0.5282 ,which gives the following stable eigenvalue spectrum:ρ(Ae) ={−0.4091±3.2455 i− 0.8964±1.0815 i}.Finally, setting the tuning parameters G as followsG =[0.6 0.10.1 0.6],the observed actuator fault estimation on the outputof the fault observer is given in Fig. 2.This figure expresses the fault signal and its estima-tion, indicating a single actuator fault in the secondactuator, starting at the time instant t = 30 s and ap-plied for 40 s, as well as a single actuator fault in thefirst actuator occurring at the time instant t = 100 sand lasts for 40 s. The tuning parameters G and δc© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 51CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHFig. 2: The actuator faults and its estimation using the en-hanced design condition.were set experimentally considering the maximal valueof fault signal amplitude. It is obvious that the ob-tained fault estimation dynamics is slightly faster thencan be obtained using the standard design criteria.Comparing the both methods, it is evident that theFrobenius norms of the observer gain matrices J, G,implying from the enhanced design criterion, are sub-stantial less than the Frobenius norms of J, G de-signed using the standard approach. This explains thehigher sensitivity and faster dynamics of the proposedmethod.Of course, the both fault observers which parameterswere obtained using the solutions of the LMI prob-lems specified by Lem. 1 and Thm. 1 can success-fully provide for the observer steady-state propertiesand asymptotic dynamics.5. ConclusionFault estimation for linear continuous-time systemsbased on the standard design conditions provides usefuland in the process easily implementable fault detection,isolation and identification. The proposed approachto fault estimation for linear continuous-time systems,utilizing the enhanced design conditions, allows evenbetter results, when the occurred actuator faults areestimated more precise then in previous case as it canseen in the simulation results.Tuning parameters G and δ were set experimentally,incorrect values of the parameters would potentially re-sult in unstable or noisy response of the fault estima-tion signal. It would be beneficial in the perspective todetermine values of the parameters G and δ using anyanalytical or numerical solution.AcknowledgmentThe work presented in the paper was supported byVEGA, the Grant Agency of the Ministry of Educationand the Academy of Science of Slovak Republic, underGrant No. 1/0348/14. This support is very gratefullyacknowledged.References[1] DING, S. X. and P. M. Frank. An AdaptiveObserver-Based Fault Detection Scheme for Non-linear Dynamics Systems. In: Preprints of the12th IFAC World Congres. Sydney: IEEE, 1993,pp. 307–310. ISBN 0-8493-0053-3.[2] FILASOVA, A., D. GONTKOVIC and D.KROKAVEC. Observer-Based Fault Estimationfor Linear Systems with Distributed Time-Delay.Archives of Control Sciences. 2013, vol. 23, no. 2,pp. 169–186. ISSN 1230-2384. DOI: 10.2478/acsc-2013-0010.[3] KROKAVEC, D., A. FILASOVA and V.HLADKY. Takagi-Sugeno Model-Based State Es-timation for One Class of Nonlinear Systems withUnknown Disturbances. In: Proceedings of the13th Internatonal Carpathian Control ConferenceICCC’12. Podbanske: IEEE, 2012, pp. 398–402.ISBN 978-1-4577-1866-3. DOI: 10.1109/Carpathi-anCC.2012.6228675.[4] KROKAVEC, D. and A. FILASOVA. Actua-tor Faults Reconstruction Using Reduced-OrderFuzzy Observer Structures. In: Proceedings ofthe 12th EUCA European Control ConferenceECC’13. Zurich: IEEE, 2013, pp. 4299–4304.ISBN 978-3-9524173-4-8.[5] PEAUCELLE, D., D. HENRION, Y. LABIT andK. TAITZ. User’s Guide for SeDuMi Interface1.04. Toulouse: LAAS-CNRS, 2002.[6] WANG, H. and S. DALEY. Actuator fault di-agnosis: an adaptive observer-based technique.IEEE Transactions on Automatic Control. 1996,vol. 41, iss. 7, pp. 1073–1078. ISSN 0018-9286.DOI: 10.1109/9.508919.[7] YAN, X.-G., C. EDWARDS and V. COC-QUEMPOT. Nonlinear robust fault recon-struction and estimation using a slidingmode observer. Automatica. 2007, vol. 43,iss. 9, pp. 1605–1614. ISSN 0005-1095.DOI: 10.1016/j.automatica.2007.02.008.[8] ZHANG, K., B. JIANG and V. COCQUEM-POT. Adaptive Observer-Based Fast Fault Es-timation. International Journal of Control, Au-tomation and Systems. 2008, vol. 6, no. 3, pp. 320–326. ISSN 1598-6446.[9] ZHANG K. and J. BIN. Analysis and design ofadaptive fault estimation for time-varying delayc© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 52CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHsystems. In: 27th Chinese Control Conference.Kunming: IEEE, 2008, pp. 38–42. ISBN 978-7-900719-70-6. DOI: 10.1109/CHICC.2008.4605097.About AuthorsVladimir SERBAK was born in Sobrance, Slovakia.He graduated in Cybernetics in 2011 from the Facultyof Electrical Engineering and Informatics, TechnicalUniversity of Kosice (FEI TU Kosice), Slovakia. Since2011 he is a Ph.D. student with the Department ofCybernetics and Artificial Intelligence, FEI TU Kosice.His scientific research is focusing on fault tolerantcontrol systems and control reconfiguration.Pavol LISCINSKY was born in Kosice, Slo-vakia. He graduated in Cybernetics in Kosice fromthe Faculty of Electrical Engineering and Informatics,Technical University of Kosice, Slovakia. Since 2012he is a Ph.D. student with the Department of Cyber-netics and Artificial Intelligence, FEI TU Kosice. Hisscientific research is focusing on fault tolerant controlsystems and fault detection and isolation (FDI).c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 53

Adaptive observer based actuator faults estimation

An approach to fault estimation systems design, adjusted for linear continuous-time systems, is proposed in the paper. Based on LMI approach, the method exploits the state-space observer principle in an adaptive scheme intended for single actuator faults. A simulation example, subject to different type of failures, demonstrates the effectiveness of the proposed form of the fault estimation technique

Directory of Open Access Journals

CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHAdaptive Observer Based Actuator FaultsEstimationVladimir SERBAK, Pavol LISCINSKYDepartment of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics,Technical University of Kosice, Letna 9, 042 00 Kosice, Slovakiavladimir.serbak@tuke.sk, pavol.liscinsky@tuke.skDOI: 10.15598/aeee.v13i1.1226Abstract. An approach to fault estimation systems de-sign, adjusted for linear continuous-time systems, isproposed in the paper. Based on LMI approach, themethod exploits the state-space observer principle in anadaptive scheme intended for single actuator faults. Asimulation example, subject to different type of failures,demonstrates the effectiveness of the proposed form ofthe fault estimation technique.KeywordsAdaptive state observer, fault estimationschemes, state-space system description.1. IntroductionOperating conditions in modern engineering systemsare still exposed to possibility of system failure. Anyfailure of sensors, actuators or other system com-ponents can drastically change the system behavior.Fault tolerant control (FTC) heavily relies on fault de-tection, identification and isolation schemes (FDI) al-lows a strategy to improve reliability of the system.Estimation of an occurred actuator fault can be uti-lized in the processes of FDI. The final results from FDIshould provide sufficient information for the secondpart of the remedial action, performed in the systemafter the fault occurrence, namely the control recon-figuration that is reconfiguring or adapting the nomi-nal control to compensate undesired effects caused bya fault. The used principles of fault estimation coverschemes based on the reduced order observers [4], adap-tive observers [1], [9], the unknown input observers [3]and sliding-mode observers [7]. This approach utilizingtwo step remedial process is typical for active FTC.Other option is to utilize passive FTC, but this how-ever does not offer possibility to find solution that isoptimal after the fault occurrence and is beneficial onlyin case of a priori considered faults.An approach described in this paper utilizes theadaptive state observer and is usually denoted as thefast adaptive fault estimation [2], [8].2. Problem FormulationA linear dynamic multi-input, multi-output (MIMO)system in presence of an unknown fault can be de-scribed by the state-space equations in the followingform:q̇(t) = A~q(t) +B~u(t) +E~f(t), (1)~y(t) = C~q(t), (2)where A ∈ <n×n, E ∈ <n×s, B ∈ <n×r and C ∈ <p×nare real matrices. To estimate faults, the followingadaptive state estimator is proposed in the generalform:q̇e(t) = A~ee(t)+B~u(t)+E~fe(t)+J(~y(t)−~ye(t)), (3)~ye(t) = C~qe(t), (4)where J ∈ <n×p and ~fe(t) is an estimate of thefault ~f(t). The above used vectors ~q(t), ~qe(t) ∈ <n,~u(t) ∈ <r and ~y(t), ~ye(t) ∈ <p are vectors of the state,estimated state, input, output and estimated outputvariables, respectively.The task is to design the matrix J in such a way thatthe observer system matrix Ae = A−JC be stable andthe estimated fault ~fe(t) approximates time propertiesof ~f(t). Considering single actuator faults, the matrixE takes form of the corresponding column of the inputmatrix B, i.e. E = Bi for i ∈ 〈1, 2, ...r〉. Moreover, itis required that rank(CE) is equal to rank(E).c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 48CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHThe state observer Eq. (3) and Eq. (4), is connectedwith the fault estimation updating law of the form [6],[8]:ḟe(t) = GHT~ey(t), (5)where H ∈ <p×s is the gain matrix and the matrixG = GT > 0, G ∈ <s×s is a learning weight matrix tobe set interactively.The design of the observer matrix parameters has toensure asymptotic convergence of the estimation errorsin Eq. (6) to zero values.~ef (t) = ~f(t)− ~fe(t),~ey(t) = ~y(t)− ~ye(t).(6)Assumption 1 The couple (A,B) is controllable andthe couple (A,C) is observable.Assumption 2 The unknown fault vector, changingunexpectedly when a fault occurs, is differentiable andbounded, i.e., |~f(t)| < ~f , ~f is known, and the value of~f(t) is set to zero until a fault occurs.Moreover, Ass. 2 implies that the derivative ~ef (t)with respect to time can be considered as follows:~ef (t) = ~f0 − ~fe(t) ⇒ ėf (t) = −ḟe(t). (7)From the point of the paper main results, the follow-ing lemma is needed to indicate the reasons for substan-tial modification of the actuator fault estimator designcondition.Lemma 1 [2] The estimator system matrix is sta-ble if there exist symmetric positive definite matrixP ∈ <n×n and matrices H ∈ <p×s, Y ∈ <n×psuch that:P = PT > 0, (8)PA+ATP−YC−CTYT < 0, (9)PE = CTH. (10)When the above conditions hold, the observer gainmatrix is given by:J = P−1Y (11)and the adaptive fault estimation algorithm is:ḟe(t) = GHTC~eq(t), (12)where~eq(t) = ~q(t)− ~qe(t). (13)Since single actuator faults act on the systemthrough different input vectors (columns of the matrixB), it is possible to avoid design of different estimatorswith the tuning matrix parameter G > 0 and formu-late the task of the estimator design through the set ofmatrix equalities:PEi = CTH. (14)Therefore solutions of Eq. (8), Eq. (9) and Eq. (14)are very conservative since the Lyapunov matrix P ver-ifies the fault observer stability not only for A, but alsowith respect to all polytops defined by Ei. To suppressthis disadvantage it is proposed to decouple the Lya-punov matrix from all system parameters A and Eiand so to obtain less conservative the enhanced designconditions. Another reason to apply this principle is ifEq. (10) is close to singular or singular.3. Enhanced DesignConditionsIn case of the enhanced design conditions the followingtheorem holds. It is evident that the theorem can besimply modified to respect Eq. (14).Theorem 1 The fault estimator is stable if for givenpositive tuning parameter δ ∈ < there exist a sym-metric positive definite matrix P ∈ <n×n, matricesQ ∈ <n×n, H ∈ <p×s, Y ∈ <n×p, identity matrixIs ∈ <s×s and a positive scalar γ ∈ < such that:P = PT > 0, γ > 0, (15) P•11 ∗ ∗P•21 −δ(Q+QT) ∗0 δETQ −γIs < 0, (16)whereP•11= QTA+ATQ−YC−CTYT +CTC,P•21 = P−Q+ δQTA− δYC(17)andCTH = QE. (18)When the above conditions hold, the estimator gainmatrix is given by:J = P−1Y (19)and the adaptive fault estimation algorithm is:ḟe(t) = GHTC~eq(t). (20)Here and hereafter ∗ denotes the symmetric item ina symmetric matrix.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 49CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHProof: From system model Eq. (1), Eq. (2) and esti-mator model Eq. (3), Eq. (4) in this case can be seenthat the time derivative of ~eq(t) is:ėq(t) = (A−JC)~eq(t)+E~ef (t), (21)and, moreover, can be expressed in the following form:(A−JC)~eq(t)+E~ef (t)− ėq(t) = 0. (22)The Lyapunov function is considered as follows:v(~eq(t)) = ~eTq (t)P~eq(t)+~eTf (t)G−1~ef(t)++t∫0(~eTy (r)~ey(r)−γ~eTf (r)~ef (r))dr,(23)where P = PT > 0, G = GT > 0 and γ is square ofthe H∞ norm of the transfer matrix function between~ef and ~eq. Then the time derivative of v(~eq(t)) is:v̇(~eq(t)) = ~eTq (t)P~eq(t) + ~eTq (t)Pėq(t)++ėTf (t)G−1~ef (t) + ~eTf (t)G−1ėf (t)++~eTy (t)~ey(t)− γ~eTf (t)~ef (t).(24)If it is assumed, that the following statements Eq. (7)andḟe(t) = GHT~ey(t) = GHTC~eq(t), (25)hold, then the substitution of Eq. (25) into Eq. (24)leads tov̇(~eq(t)) = ėTq (t)P~eq(t) + ~eTq (t)Pėq(t)−−~eTq (t)CTHGG−1~ef (t)−−~eTf (t)G−1GHTC~eq(t)++~eTy (t)~ey(t)− γ~eTf (t)~ef (t),(26)that can be modified into the following form:v̇(~eq(t)) = ėTq (t)P~eq(t) + ~eTq (t)Pėq(t)−−~eTq (t)CTH~ef (t)− ~eTf (t)HTC~eq(t)++~eTy (t)~ey(t)− γ~eTf (t)~ef (t).(27)It is possible to define the condition based onEq. (22) as:(~eTq (t)ST1 + ėTq (t)ST2 )××((A−JC)~eq(t)+E~ef (t)− ėq(t)) = 0(28)and inserting into Eq. (27), the following expression isobtained:v̇(~eq(t)) = ėTq (t)P~eq(t) + ~eTq (t)Pėq(t)−−~eTq (t)CTH~ef (t)− ~eTf (t)HTC~eq(t)++(~eTq (t)ST1 + ėTq (t)ST2 )((A−JC)~eq(t)−−ėq(t)) + ((A−JC)~eq(t)− ėq(t))T(S1~eq(t) + S2ėq(t)) + (~eTq (t)ST1 ++ėTq (t)ST2 )E~ef (t) + ~eTf (t)ET(S1~eq(t)++S2ėq(t)) + ~eTy (t)~ey(t)− γ~eTf (t)~ef (t).(29)If the following condition is defined:0 = ~eTf (t)[ETS1−HTC]~eq(t)−−~eTq (t)[ST1E−CTH]~ef (t),(30)then the equalityST1 E−CTH = 0 (31)impliesCTH = ST1 E (32)and this allows to transcribe Eq. (29) as:v̇(~eq(t)) = ėTq (t)P~eq(t) + ~eTq (t)Pėq(t)++(~eTq (t)ST1 + ėTq (t)ST2 )((A−JC)~eq(t)−−ėq(t)) + (~eTq (t)(A−JC)T − ėTq (t))(S1~eq(t) + S2ėq(t)) + ėTq (t)ST2 E~ef (t)++~eTf (t)ETS2ėq(t) + ~eTy (t)~ey(t)−−γ~eTf (t)~ef (t)(33)and with the notation:~e •Tq (t) =[~eTq (t) ėTq (t) ~eTf (t)], (34)it can be written as:v̇(~eq(t)) = ~e•Tq (t)P•~e •q (t) < 0, (35)whereP• = P•11 ∗ ∗P•21 −ST2 − S2 ∗0 ETS2 −γIs < 0 (36)andP•11= ST1 (A−JC) +CTC+ (A−JC)TS1,P•21 = P+ ST2 (A− JC)− S1.(37)Inserting into Eq. (32), Eq. (36) and Eq. (37) thefollowing substitutions:S1 = Q, S2 = δS1, ST1 J = QTJ = Y (38)then Eq. (32), Eq. (36) and Eq. (37) implies Eq. (16),Eq. (17) and Eq. (18). This concludes the proof.Note, Eq. (17) and Eq. (18) implies that the Lya-punov matrix P is decoupled from all system parame-ters.4. Illustrative ExampleTo illustrate the proposed method, the system whosedynamics is described by the equations Eq. (1) andEq. (2) is considered with the matrix parameters:A =0.1650 −0.0767 0.5173 2.06380.2151 0.1866 −0.3077 −1.5260−0.3851 0.4033 0.2887 −0.3833−2.0829 1.5170 0.3093 0.1109 ,c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 50CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHCT=2 21 22 12 2 , B = E =−0.9042 0−1.6493 0.43610 1.85290 −0.0355 .For simulation purposes only, the equilibrium of thesystem was stabilized by the state feedback controller:~u(t) = −K~q(t),where the gain matrix K was designed using the pole-placement method as follows:K=[−3.1315 1.6633 3.5641 11.3920−4.2235 3.1471 1.0377 0.9104].Solving Eq. (8), Eq. (9) and Eq. (10) with respectto the LMI matrix variables P, H, and Y using Self-Dual-Minimization (SeDuMi) package [5] for Matlab,the estimator parameter design problem was solved asfeasible and the LMI matrix variables were:P =0.1770 0.2107 0.0321 0.04570.2107 0.6888 0.0678 0.20600.0321 0.0678 0.1586 −0.09550.0457 0.2060 −0.0955 0.6740 ,Y=−0.3691 0.3090−0.7451 −0.12670.0251 0.16470.2357 −0.8632,H=[ 0.4026 −0.1207−0.2489 0.2774],so that the observer gain matrix were given asJ =−1.2534 2.8937−1.1442 −0.68721.5003 −0.01820.9967 −1.2693 .It can verify that the eigenvalue spectrum ρ(Ae) ofthe estimator system matrix Ae is stable, where:ρ(Ae) ={−2.1745 − 2.6067 − 3.3377 ± 0.1566 i}.Setting the tuning parameter G as:G =[1.7 11 5.5],the observer fault response is given in Fig. 2.This figure presents the fault signal, as well as its es-timation, reflecting a single actuator fault in the secondactuator. Fault starts at the time instant t = 30 s andis applied for 40 s. The tuning parameter was experi-mentally set considering the maximal value of the faultsignal amplitude. Then, at the time instant t = 100 s,the first actuator fault is applied for 40 s.From the simulation results in Fig. 2 it can be ob-served that the differences between the signals reflect-ing a single actuator fault and the observer approxi-mate ones tends to zero. Moreover, the states of theFig. 1: The actuator faults and its estimation.system converge to the equilibrium when the actuatorfault disappeared, via the used controller.Applying the same toolbox to solve LMIs Eq. (15),Eq. (16), Eq. (17) and Eq. (18) conditioned byδ = 0.001, the resulting set of the matrix variableswas as follows:P =23.2065 −5.2282 6.5739 5.9199−5.2282 16.1633 −5.5997 4.23216.5739 −5.5997 11.1507 4.67935.9199 4.2321 4.6793 24.1004 ,Q =23.2346 −5.2370 6.5874 5.8961−5.2420 16.1435 −5.6122 4.24796.5646 −5.6171 11.1679 4.65165.9477 4.2403 4.7055 24.0935 ,Y=−4.9513 3.7898−6.9495 11.805210.1032 −1.616911.5372 −3.4027, H=[ 9.5143 13.2220−15.7000 −8.3657]and the estimator gain matrix J, obtained usingEq. (19), was:J =−0.7142 0.4462−0.5347 1.14820.8107 0.39020.5906 −0.5282 ,which gives the following stable eigenvalue spectrum:ρ(Ae) ={−0.4091±3.2455 i− 0.8964±1.0815 i}.Finally, setting the tuning parameters G as followsG =[0.6 0.10.1 0.6],the observed actuator fault estimation on the outputof the fault observer is given in Fig. 2.This figure expresses the fault signal and its estima-tion, indicating a single actuator fault in the secondactuator, starting at the time instant t = 30 s and ap-plied for 40 s, as well as a single actuator fault in thefirst actuator occurring at the time instant t = 100 sand lasts for 40 s. The tuning parameters G and δc© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 51CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHFig. 2: The actuator faults and its estimation using the en-hanced design condition.were set experimentally considering the maximal valueof fault signal amplitude. It is obvious that the ob-tained fault estimation dynamics is slightly faster thencan be obtained using the standard design criteria.Comparing the both methods, it is evident that theFrobenius norms of the observer gain matrices J, G,implying from the enhanced design criterion, are sub-stantial less than the Frobenius norms of J, G de-signed using the standard approach. This explains thehigher sensitivity and faster dynamics of the proposedmethod.Of course, the both fault observers which parameterswere obtained using the solutions of the LMI prob-lems specified by Lem. 1 and Thm. 1 can success-fully provide for the observer steady-state propertiesand asymptotic dynamics.5. ConclusionFault estimation for linear continuous-time systemsbased on the standard design conditions provides usefuland in the process easily implementable fault detection,isolation and identification. The proposed approachto fault estimation for linear continuous-time systems,utilizing the enhanced design conditions, allows evenbetter results, when the occurred actuator faults areestimated more precise then in previous case as it canseen in the simulation results.Tuning parameters G and δ were set experimentally,incorrect values of the parameters would potentially re-sult in unstable or noisy response of the fault estima-tion signal. It would be beneficial in the perspective todetermine values of the parameters G and δ using anyanalytical or numerical solution.AcknowledgmentThe work presented in the paper was supported byVEGA, the Grant Agency of the Ministry of Educationand the Academy of Science of Slovak Republic, underGrant No. 1/0348/14. This support is very gratefullyacknowledged.References[1] DING, S. X. and P. M. Frank. An AdaptiveObserver-Based Fault Detection Scheme for Non-linear Dynamics Systems. In: Preprints of the12th IFAC World Congres. Sydney: IEEE, 1993,pp. 307–310. ISBN 0-8493-0053-3.[2] FILASOVA, A., D. GONTKOVIC and D.KROKAVEC. Observer-Based Fault Estimationfor Linear Systems with Distributed Time-Delay.Archives of Control Sciences. 2013, vol. 23, no. 2,pp. 169–186. ISSN 1230-2384. DOI: 10.2478/acsc-2013-0010.[3] KROKAVEC, D., A. FILASOVA and V.HLADKY. Takagi-Sugeno Model-Based State Es-timation for One Class of Nonlinear Systems withUnknown Disturbances. In: Proceedings of the13th Internatonal Carpathian Control ConferenceICCC’12. Podbanske: IEEE, 2012, pp. 398–402.ISBN 978-1-4577-1866-3. DOI: 10.1109/Carpathi-anCC.2012.6228675.[4] KROKAVEC, D. and A. FILASOVA. Actua-tor Faults Reconstruction Using Reduced-OrderFuzzy Observer Structures. In: Proceedings ofthe 12th EUCA European Control ConferenceECC’13. Zurich: IEEE, 2013, pp. 4299–4304.ISBN 978-3-9524173-4-8.[5] PEAUCELLE, D., D. HENRION, Y. LABIT andK. TAITZ. User’s Guide for SeDuMi Interface1.04. Toulouse: LAAS-CNRS, 2002.[6] WANG, H. and S. DALEY. Actuator fault di-agnosis: an adaptive observer-based technique.IEEE Transactions on Automatic Control. 1996,vol. 41, iss. 7, pp. 1073–1078. ISSN 0018-9286.DOI: 10.1109/9.508919.[7] YAN, X.-G., C. EDWARDS and V. COC-QUEMPOT. Nonlinear robust fault recon-struction and estimation using a slidingmode observer. Automatica. 2007, vol. 43,iss. 9, pp. 1605–1614. ISSN 0005-1095.DOI: 10.1016/j.automatica.2007.02.008.[8] ZHANG, K., B. JIANG and V. COCQUEM-POT. Adaptive Observer-Based Fast Fault Es-timation. International Journal of Control, Au-tomation and Systems. 2008, vol. 6, no. 3, pp. 320–326. ISSN 1598-6446.[9] ZHANG K. and J. BIN. Analysis and design ofadaptive fault estimation for time-varying delayc© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 52CONTROL ENGINEERING VOLUME: 13 | NUMBER: 1 | 2015 | MARCHsystems. In: 27th Chinese Control Conference.Kunming: IEEE, 2008, pp. 38–42. ISBN 978-7-900719-70-6. DOI: 10.1109/CHICC.2008.4605097.About AuthorsVladimir SERBAK was born in Sobrance, Slovakia.He graduated in Cybernetics in 2011 from the Facultyof Electrical Engineering and Informatics, TechnicalUniversity of Kosice (FEI TU Kosice), Slovakia. Since2011 he is a Ph.D. student with the Department ofCybernetics and Artificial Intelligence, FEI TU Kosice.His scientific research is focusing on fault tolerantcontrol systems and control reconfiguration.Pavol LISCINSKY was born in Kosice, Slo-vakia. He graduated in Cybernetics in Kosice fromthe Faculty of Electrical Engineering and Informatics,Technical University of Kosice, Slovakia. Since 2012he is a Ph.D. student with the Department of Cyber-netics and Artificial Intelligence, FEI TU Kosice. Hisscientific research is focusing on fault tolerant controlsystems and fault detection and isolation (FDI).c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 53

http://dspace.vsb.cz/bitstream/10084/111829/1/1226-7206-6-PB.pdf

Adaptive observer based actuator faults estimation

Abstract

Similar works

Full text

Available Versions

Crossref

DSpace at VSB Technical University of Ostrava

Directory of Open Access Journals