Actuator and sensor fault estimation based on a proportional-integral quasi-LPV observer with inexact scheduling parameters⁎

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Actuator and sensor fault estimation based Actuator and sensor fault estimation based Actuator and sensor fault estimation based Actuator and sensor fault estimation based on a proportional-integral quasi-LPV Actuator and sensor fault estimation based on a proportional-integral quasi-LPV on a proportional-integral quasi-LPV on a proportional-integral quasi-LPV with inexact scheduling onobserver a proportional-integral quasi-LPV observer with inexact scheduling observer with inexact scheduling  observer with inexact scheduling  parameters observer with inexactscheduling parameters parameters parameters  parameters S. G´ omez-Pe˜ nate, ∗∗∗ F.R. L´ opez-Estrada, ∗∗∗ S. G´ o mez-Pe˜ n ate, L´ o pez-Estrada, ∗∗ ∗∗∗,∗∗∗∗ S. G´ o mez-Pe˜ n ate, F.R. L´ o pez-Estrada, ∗ F.R. ∗ G. Valencia-Palomo, D. Rotondo, ∗ F.R. ∗ S. G´ o mez-Pe˜ n ate, L´ o pez-Estrada, ∗∗ ∗∗∗,∗∗∗∗ ∗∗ ∗∗∗,∗∗∗∗ S. G´ o mez-Pe˜ n ate, F.R. L´ o pez-Estrada, G. Valencia-Palomo, D. Rotondo, ∗ ∗∗ D. Rotondo, ∗ ∗ G. Valencia-Palomo, ∗∗∗,∗∗∗∗ S. G´ omez-Pe˜ nate, F.R. L´ opez-Estrada, J. Enr´ ıquez-Z´ a rate. ∗∗ D. ∗∗∗,∗∗∗∗ G. Valencia-Palomo, Rotondo, ∗ ∗ G. Valencia-Palomo, Rotondo, J. Enr´ ıquez-Z´ a rate. ∗∗ D. ∗∗∗,∗∗∗∗ J. Enr´ ıquez-Z´ a rate. ∗ G. Valencia-Palomo, D. Rotondo, J. a J. Enr´ Enr´ıquez-Z´ ıquez-Z´ arate. rate. ∗ ∗ J.deEnr´ ıquez-Z´ arate. ∗ Tecnol´ Tecnol´ o gico Nacional M´ e xico / Instituto o ∗ ∗ Tecnol´ ogico gico Nacional Nacional de de M´ M´eexico xico / / Instituto Instituto Tecnol´ Tecnol´ ogico gico de de Tuxtla Tuxtla o o gico de Tuxtla ∗ Tecnol´ Guti´ e rrez, TURIX-Dynamics Diagnosis and Control Group, ∗ Tecnol´ o gico Nacional de M´ e xico / Instituto Tecnol´ o gico de gico TURIX-Dynamics Nacional de M´exico-- /Diagnosis Instituto and Tecnol´ ogico Group, de Tuxtla Tuxtla Guti´ eeorrez, Control ∗ Tecnol´ Guti´ rrez, TURIX-Dynamics Diagnosis and Control Group, Tecnol´ gico TURIX-Dynamics Nacional dekm M´e1080, xico- /Diagnosis Instituto Tecnol´ oMexico. gico Group, de Tuxtla Carretera Panam. Tuxtla Guti´ eerrez, Guti´ eeorrez, and Control Guti´ rrez, TURIX-Dynamics Diagnosis and Control Group, Carretera Panam. km 1080, Tuxtla Guti´ rrez, Mexico. ∗∗ Carretera Panam. km 1080, Tuxtla Guti´ errez, Mexico. Guti´ errez,ogico TURIX-Dynamics Diagnosis and Control Group, Tecnol´ Nacional M´ ee- xico / Instituto Tecnol´ o gico de Carretera Panam. km 1080, Tuxtla Guti´ Mexico. ∗∗ ∗∗ Tecnol´ Carretera Panam. kmde 1080, Tuxtla Guti´eerrez, rrez, Mexico. o gico Nacional de M´ xico / Instituto Tecnol´ o gico de o gico Nacional de M´ e xico / Instituto Tecnol´ o gico de ∗∗ Tecnol´ Carretera Panam. km 1080, Tuxtla Guti´ e rrez, Mexico. Hermosillo, Av. Tec. y Perif´ e rico Poniente SN, 83170, Mexico. ∗∗ Tecnol´ o gico Nacional de M´ e xico / Instituto Tecnol´ o gico de ogico Nacional de eeM´ exico / Instituto Tecnol´ ogico de Hermosillo, Av. Tec. yy Perif´ rico Poniente SN, 83170, Mexico. ∗∗ Tecnol´ ∗∗∗ Hermosillo, Av. Tec. Perif´ rico Poniente SN, 83170, Mexico. Tecnol´ o gico Nacional de M´ e xico / Instituto Tecnol´ o gico de Research Center for Supervision, Safety and Automatic Control Hermosillo, Av. Tec. y Perif´ e rico Poniente SN, 83170, Mexico. ∗∗∗ ∗∗∗ Hermosillo, Av. Tec. y Perif´ e rico Poniente SN, 83170, Mexico. Research Center for Supervision, Safety and Automatic Control Research Center for Supervision, Safety and Automatic Control ∗∗∗ Hermosillo, Av. Tec. ySupervision, Perif´ erico Poniente SN, 83170,Rambla Mexico. (CS2AC), Universitat Polit` e cnica de Catalunya (UPC), Sant ∗∗∗ Research Center for Safety and Automatic Control Center for Supervision, Safety and(UPC), Automatic Control (CS2AC), Universitat Polit` eecnica de Catalunya Rambla Sant ∗∗∗ Research (CS2AC), Universitat Polit` cnica de Catalunya Rambla Sant Research Center for Supervision, Safety and(UPC), Automatic Control Nebridi, Rambla Sant Nebridi, 10, Terrassa 08222, Spain. (CS2AC), Universitat Polit` e cnica de Catalunya (UPC), Rambla Sant (CS2AC), Universitat Polit` e cnica de Catalunya (UPC), Rambla Sant Nebridi, Rambla Sant Nebridi, 10, Terrassa 08222, Spain. ∗∗∗∗ Nebridi, Rambla Sant Nebridi, 10, Terrassa 08222, Spain. (CS2AC), Universitat Polit` e cnica de Catalunya (UPC), Rambla Santi Institut de Rob` o tica i Inform` a tica Industrial,CSIC-UPC,Llorens Nebridi, Rambla Sant Nebridi, 10, Terrassa 08222, Spain. ∗∗∗∗ ∗∗∗∗ Institut Nebridi, Sant Nebridi, Terrassa 08222, Spain. deRambla Rob` otica tica i Inform` Inform` atica tica10, Industrial,CSIC-UPC,Llorens de Rob` o i a Industrial,CSIC-UPC,Llorens iii ∗∗∗∗ Institut Nebridi, Rambla Sant Nebridi, 10, Terrassa 08222, Spain. Artigas 4–6, Barcelona 08028, Spain. ∗∗∗∗ Institut de Rob` o tica i Inform` a tica Industrial,CSIC-UPC,Llorens otica 4–6, i Inform` atica Industrial,CSIC-UPC,Llorens i Artigas Barcelona 08028, Spain. ∗∗∗∗ Institut de Rob` Artigas 4–6, Barcelona 08028, Spain. Institut de Rob` otica i Inform` atica Industrial,CSIC-UPC,Llorens i Artigas Artigas 4–6, 4–6, Barcelona Barcelona 08028, 08028, Spain. Spain. Artigas 4–6, Barcelona 08028, Spain.

Abstract: Abstract: Abstract: This paper presents aa method for actuator and sensor fault estimation based Abstract: Abstract: This paper presents method for actuator and sensor fault estimation based on on aaa proportionalproportionalThis paper presents a method for actuator and sensor fault estimation based on proportionalAbstract: integral observer (PIO) for a class of nonlinear system described by a polytopic quasi-linear This paper presents a method for actuator and sensor fault estimation based on This paper presents a method actuator and sensor fault estimation based on aa proportionalproportionalintegral observer (PIO) for aa for class of nonlinear system described by aa polytopic quasi-linear integral observer (PIO) for class of nonlinear system described by polytopic quasi-linear This paper presents a method for actuator and sensor fault estimation based onapproach, a proportionalparameter varying (qLPV) mathematical model. Contrarily to the traditional which integral observer (PIO) for a class of nonlinear system described by a polytopic quasi-linear integral observer (PIO) formathematical a class of nonlinear system described by a polytopic quasi-linear parameter varying (qLPV) model. Contrarily to the traditional approach, which parameter varying (qLPV) mathematical model. Contrarily to the traditional approach, which integral observer (PIO) formathematical a class of scheduling nonlinear system described by aproposes polytopic quasi-linear considers measurable or unmeasurable parameters, this work a methodology parameter varying (qLPV) model. Contrarily to the traditional approach, which parameter varying (qLPV) mathematical model. Contrarily to the traditional approach, which considers measurable or unmeasurable scheduling parameters, this work proposes a methodology considers measurable or unmeasurable scheduling parameters, proposes aa methodology parameter varying (qLPV) mathematical model. Contrarily tothis thework traditional approach, which that considers inexact scheduling parameters. This condition is present many physical systems considers measurable or unmeasurable scheduling parameters, this work proposes considers measurable or unmeasurable scheduling parameters, this workin proposes a methodology methodology that considers inexact scheduling parameters. This condition is present in many physical systems that considers inexact scheduling parameters. This condition is present in many physical systems considers measurable or unmeasurable scheduling parameters, this work proposes a methodology where the scheduling parameters can be affected by noise, offsets, calibration errors, and other that considers inexact scheduling This condition is in physical systems that considers inexact parameters scheduling parameters. parameters. Thisby condition is present present in many manyerrors, physical systems where the scheduling can be affected noise, offsets, calibration and other where the scheduling parameters can be affected by noise, offsets, calibration and other that considers inexact scheduling parameters. This condition is present in manyerrors, physical systems factors that have a negative impact on the measurements. A H performance criterion is where the scheduling parameters can be affected by noise, offsets, calibration errors, and ∞ where the scheduling parameters can be affected by noise, offsets, calibration errors,criterion and other other factors that have aa negative impact on the measurements. A H performance is ∞ factors that have negative impact on the measurements. A H performance criterion is ∞ where the scheduling parameters can be affected by noise, offsets, calibration errors, and other considered in the design in order to guarantee robustness against sensor noise, disturbance, factors that have a negative impact on the measurements. A H performance criterion is ∞ factors that have a negative impact on the measurements. A H performance criterion is considered in the design in order to guarantee robustness against sensor noise, disturbance, ∞ considered in the design in order to guarantee robustness against sensor noise, disturbance, factors that have a negative impact on the measurements. A H performance criterion is and inexact scheduling parameters. Then, a set of linear matrix inequalities (LMIs) is derived ∞sensor noise, considered in the design in order to guarantee robustness against disturbance, considered in the design in order to guarantee robustness against sensor noise, disturbance, and inexact scheduling parameters. Then, a set of linear matrix inequalities (LMIs) is derived and inexact scheduling parameters. Then, a set of linear matrix inequalities (LMIs) is derived considered in the design in order to guarantee robustness against sensor noise, disturbance, by the use of a quadratic Lyapunov function. The solution of the LMI guarantees asymptotic and inexact scheduling parameters. Then, aa set of matrix inequalities (LMIs) is and inexact scheduling parameters. Then, setThe of linear linear matrix inequalities (LMIs)asymptotic is derived derived by the use of a quadratic Lyapunov function. solution of the LMI guarantees by the use of a quadratic Lyapunov function. solution of the LMI guarantees asymptotic and inexact scheduling parameters. Then, a setThe of linear matrix inequalities (LMIs) is derived stability of the PIO. Finally, the performance and applicability of the proposed method are by the use of a quadratic Lyapunov function. The solution of the LMI guarantees asymptotic by the use of a quadratic Lyapunov function. The solution of the LMI guarantees asymptotic stability of the PIO. Finally, the performance and applicability of the proposed method are stability of the PIO. Finally, the performance and applicability the proposed method are by the use of a quadratic Lyapunov function. The solution of theof LMI guarantees asymptotic illustrated through a numerical experiment in a nonlinear system. stability of the PIO. Finally, the performance and applicability of the proposed method are stability of the PIO. Finally, the performance and applicability of the proposed method are illustrated through a numerical experiment in a nonlinear system. illustrated through a numerical experiment in a nonlinear system. stability of the PIO. Finally, the performance and applicability of the proposed method are illustrated through a numerical experiment in a nonlinear system. Keywords: qLPV system, Inexact scheduling parameters, PI observer, fault diagnosis. illustrated through a numerical experiment in a nonlinear system. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: qLPV system, Inexact scheduling parameters, PI observer, fault diagnosis. Keywords: qLPV system, Inexact scheduling parameters, PI observer, fault diagnosis. illustrated through a numerical experiment in a nonlinear system. Keywords: Keywords: qLPV qLPV system, system, Inexact Inexact scheduling scheduling parameters, parameters, PI PI observer, observer, fault fault diagnosis. diagnosis. Keywords: qLPV system, Inexact scheduling parameters, PI observer, fault diagnosis. 1. INTRODUCTION It is well known that an effective system 1. INTRODUCTION INTRODUCTION It is is well well known known that that an an effective effective model-based model-based FD FD system 1. It model-based FD system requires a mathematical model that captures the nonlinear 1. INTRODUCTION It is well known that an effective model-based FD system 1. INTRODUCTION It is wellaaknown that an model effective model-based FD system requires mathematical that captures the nonlinear requires mathematical model that captures the nonlinear It is wellaknown that an model effective model-based FD system 1. faults INTRODUCTION dynamics inherent in most of the physical systems. In that requires mathematical that captures the nonlinear Timely diagnosis of is particularly essential in order requires a mathematical model that captures the nonlinear dynamics inherent in most of the physical systems. In that Timely diagnosis diagnosis of of faults faults is is particularly particularly essential essential in in order order requires dynamics in most of the physical systems. In that Timely a inherent mathematical model that captures the nonlinear sense, convex systems such as linear parameter varying dynamics inherent in most of the physical systems. In that to increase the safety and reliability of a system. Fault Timely diagnosis of faults is particularly essential in order dynamics inherent in most of the physical systems. In that sense, convex systems such as linear parameter varying Timely diagnosis of faults is particularly essential in order to increase the safety and reliability of a system. Fault sense, convex systems such as linear parameter varying to increase the safety and reliability of a system. Fault dynamics inherent in most of the physical systems. In that (LPV) and quasi-LPV (qLPV) systems have proved to sense, convex systems such as linear parameter varying diagnosis algorithms and their applications have received Timely diagnosis of faults is particularly essential in order to increase the safety and reliability of a system. Fault sense, convex systems such as linear parameter varying (LPV) and quasi-LPV (qLPV) systems have proved to to increase the safety and reliability of a system. Fault diagnosis algorithms and their applications have received (LPV) and quasi-LPV (qLPV) systems have proved to diagnosis algorithms and their applications have received sense, convex systems such as linear parameter varying represent complex nonlinear systems by a set of linearto increase the safety and reliability of a system. Fault (LPV) and quasi-LPV (qLPV) systems have proved to considerable attentionand have been the the subject subject of inin- represent diagnosis algorithms their applications have received (LPV) and quasi-LPV (qLPV) systems have proved to complex nonlinear systems by a set of lineardiagnosis algorithms and their applications have received considerable attention and have been of represent complex nonlinear systems by a set of linearconsiderable attention and have been the subject of in(LPV) and quasi-LPV (qLPV) systems have proved to time varying models interpolated by weighting functions. represent complex nonlinear systems by a set of lineartensive research during the last decades (Li et al., 2018). diagnosis algorithms and their applications have received considerable attention and have been the subject of inrepresent complex nonlinear systems by a set of lineartime varying models interpolated by weighting functions. considerable attention and have been the subject of intensive research research during during the the last last decades decades (Li (Li et et al., al., 2018). 2018). represent time varying models interpolated by weighting tensive complex nonlinear systems by functions a set functions. of linearThese convex models consider scheduling based time varying models interpolated by weighting functions. Model-based fault diagnosis (FD) techniques have been considerable attention and have been the subject of intensive research during the last decades (Li et al., 2018). time varying models interpolated by weighting functions. convex models consider scheduling functions based tensive research during the last decades (Li et have al., 2018). Model-based fault diagnosis (FD) techniques have been These These convex models consider scheduling functions based Model-based fault (FD) techniques been time varying models interpolated by (Hamdi weighting functions. on exogenous measurable parameters et al., 2019). These convex models consider scheduling functions based widely recognized recognized asdiagnosis powerful approaches that have2018). been tensive research during the last decades (Li et have al., Model-based fault diagnosis (FD) techniques These convex models consider scheduling functions based on exogenous measurable parameters (Hamdi et al., 2019). Model-based fault diagnosis (FD) techniques have widely as powerful approaches that have been on exogenous measurable parameters (Hamdi et al., 2019). widely recognized as powerful approaches that have been These convex models consider scheduling functions based However, in general, these functions include the input, Model-based fault diagnosis (FD) techniques have been on exogenous measurable parameters (Hamdi et al., 2019). successfully applied in many practical systems, such as widely recognized as powerful approaches that have on exogenous measurable parameters (Hamdi et al., 2019). However, in general, these functions include the input, widely recognized as in powerful approaches that have been successfully applied in many practical practical systems, such as However, in general, these functions include the input, successfully applied many systems, such as on exogenous measurable parameters (Hamdi et al., 2019). output, and states of the system (Casavola and Gagliardi, However, in general, these functions include the input, unmanned aerial vehicles (L´ o pez-Estrada et al., 2016), widely recognized as powerful approaches that have been successfully applied in practical systems, as However, in states general, these functions include the input, and of the system (Casavola and Gagliardi, successfully aerial applied in many many practical systems, such as output, unmanned aerial vehicles (L´ pez-Estrada et al., al.,such 2016), output, and of the system (Casavola and Gagliardi, unmanned vehicles (L´ oo pez-Estrada et 2016), in states general, these functions include the input, 2015). Note that when the nonlinear sector approach output, and states of the system (Casavola and Gagliardi, electric vehicles vehicles (Djeziri et al., 2013), DC motors motors (Casavola successfully applied in et many practical systems, such as However, unmanned aerial vehicles (L´ o pez-Estrada et al., 2016), output, and states of the system (Casavola and Gagliardi, 2015). Note that when the nonlinear sector approach unmanned aerial vehicles (L´ o pez-Estrada et al., 2016), electric (Djeziri al., 2013), DC (Casavola 2015). Note that when the nonlinear sector approach electric vehicles (Djeziri et al., 2013), DC motors (Casavola output, and states of the system (Casavola and Gagliardi, (Ohtake et al., 2003) is applied to a nonlinear model, 2015). Note that2003) whenis the the nonlinear sector approach approach and Gagliardi, Gagliardi, 2015), among others. unmanned aerial vehicles (L´ o2013), pez-Estrada et al., 2016), 2015). electric vehicles (Djeziri et DC (Casavola that when nonlinear sector (OhtakeNote et al., al., applied to aa nonlinear nonlinear model, electric vehicles (Djeziri et al., al., 2013), DC motors motors (Casavola and 2015), among others. (Ohtake et is applied to model, and Gagliardi, 2015), among others. 2015). Note that2003) when the nonlinear sector approach both qLPV systems and Takagi-Sugeno Systems are (Ohtake et al., 2003) is applied to aa (TS) nonlinear model, electric vehicles (Djeziri et al., 2013), DC motors (Casavola and Gagliardi, 2015), among others. (Ohtake et al., 2003) is applied to nonlinear model, both qLPV systems and Takagi-Sugeno (TS) Systems are and Gagliardi, 2015), among others. both qLPV systems and Takagi-Sugeno (TS) Systems are  (Ohtake et al., 2003) is applied to a nonlinear model, equivalent (Rotondo et al., 2016). both qLPV systems and Takagi-Sugeno (TS) Systems are This work is supported by Tecnol´ o gico Nacional de M´ e xico and and Gagliardi, 2015), among others.  both qLPV systems and Takagi-Sugeno (TS) Systems are equivalent (Rotondo et al., 2016). work is supported by Tecnol´ o gico Nacional de M´ e xico and  This equivalent (Rotondo et al., 2016). This work is supported by Tecnol´ o gico Nacional de M´ e xico and both qLPV systems and Takagi-Sugeno (TS) Systems are  theThis National Council of Science and oTechnology (CONACyT). The equivalent (Rotondo et al., 2016). work is supported by Tecnol´ gico Nacional de M´ e xico and  equivalent (Rotondo et al., 2016). the National Council of Science and Technology (CONACyT). The Most of the proposed approaches for FD based on convex This work is supported by Tecnol´ o gico Nacional de M´ e xico and theThis National Council of Science andThis Technology (CONACyT). The  support is gratefully acknowledged. work has been also partially equivalent (Rotondo et al., 2016). Most of the proposed approaches for FD based on convex work is supported by Tecnol´ o gico Nacional de M´ e xico and the National Council of Technology (CONACyT). The Most of the proposed approaches for FDvariables based onare convex support is gratefully work has been also the National Councilacknowledged. of Science Science and andThis Technology The systems consider that the scheduling persupport is acknowledged. This work has (CONACyT). been also partially partially Most of proposed approaches for based convex funded bygratefully the Spanish State and Research Agency (AEI) and The the Most of the the proposed approaches for FD FDvariables based on onare convex the National Council of Science Technology (CONACyT). systems consider that the scheduling persupport is gratefully acknowledged. This work has been also partially systems consider that the scheduling variables are perfunded by the Spanish State Research Agency (AEI) and the support is gratefully acknowledged. This work has been also partially Most of the proposed approaches for FD based on convex funded is byRegional the Spanish State Research Agency (AEI) and the fectly measurable. In practical applications, the scheduling systems consider that the scheduling variables are European Development Fund (ERFD) through the projects support gratefully acknowledged. This work has been also partially funded by the Spanish State Research Agency (AEI) and the systems consider that the scheduling variables are perperfectly measurable. In practical applications, the scheduling European Regional Development Fund (ERFD) through the projects funded by the Spanish State Research Agency (AEI) and the fectly measurable. In practical applications, the scheduling European Regional Development Fund (ERFD) through the projects systems consider that the scheduling variables are pervariables are not measurable or are measured with uncerSCAV (ref. MINECO DPI2017-88403-R) and DEOCS (ref. MINECO fectly measurable. In applications, the scheduling funded byRegional the Spanish State Research Agency (AEI) and the European Development Fund through the projects fectly measurable. In practical practicalor applications, thewith scheduling variables are not measurable are measured uncerSCAV (ref. MINECO DPI2017-88403-R) and DEOCS (ref. European Regional Development Fund (ERFD) (ERFD) through theMINECO projects variables are not measurable or are measured with uncerSCAV (ref. MINECO DPI2017-88403-R) and DEOCS (ref. MINECO fectly measurable. In practical applications, the scheduling DPI2016-76493), and by the AEI through the Maria de Maeztu Seal tainties due to measurement noise, offset, low-resolution variables are not measurable or are measured with uncerEuropean Regional Development Fund (ERFD) through the projects SCAV (ref. and DEOCS (ref. MINECO variables are to notmeasurement measurable or are measured with uncerDPI2016-76493), and DPI2017-88403-R) by the the Maria de Maeztu Seal tainties due noise, offset, low-resolution SCAV (ref. MINECO MINECO DPI2017-88403-R) (ref. MINECO DPI2016-76493), by the AEI AEI through throughand theDEOCS Maria de Maeztu Seal tainties due to noise, offset, low-resolution of Excellence to and IRI DPI2017-88403-R) (MDM-2016-0656) and the grant Juan de la variables arecalibration, notmeasurement measurable or are measured with uncersensors, bad indirect measurements, and other SCAV (ref. MINECO and DEOCS (ref. MINECO DPI2016-76493), and by the the de Seal tainties due to measurement noise, offset, low-resolution of Excellence to IRI (MDM-2016-0656) the grant Juan de la DPI2016-76493), and by the AEI AEI through throughand the Maria Maria de Maeztu Maeztu Seal tainties due to measurement noise, offset, low-resolution sensors, bad calibration, indirect measurements, and other of Excellence to IRI (MDM-2016-0656) and the grant Juan de la sensors, bad indirect measurements, and other Cierva-Formacion (FJCI-2016-29019). DPI2016-76493), and by the AEI throughand the Maria de Maeztu Seal tainties due calibration, to measurement noise, offset, low-resolution of Excellence to IRI (MDM-2016-0656) the grant Juan de la sensors, bad calibration, indirect measurements, and Cierva-Formacion (FJCI-2016-29019). of Excellence to IRI (MDM-2016-0656) and the grant Juan de la sensors, bad calibration, indirect measurements, and other other Cierva-Formacion (FJCI-2016-29019). Cierva-Formacion (FJCI-2016-29019). of Excellence to IRI (MDM-2016-0656) and the grant Juan de la sensors, bad calibration, indirect measurements, and other Cierva-Formacion (FJCI-2016-29019). Cierva-Formacion 2405-8963 © 2019, (FJCI-2016-29019). IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.355



S. Gómez-Peñate et al. / IFAC PapersOnLine 52-28 (2019) 100–105

factors (Zhang et al., 2016). In this case, it is necessary to consider the inexact scheduling variables in order to design a reliable and effective FD system. For instance, in Chandra et al. (2017), a sliding mode observer scheme to reconstruct actuator and sensor when the scheduling parameters are imperfectly known was presented. In Hassanabadi et al. (2017), a polytopic proportional-integral (PI) unknown input observer to address the problem of actuator fault estimation for singular delayed LPV systems was proposed. Zhu and Zhao (2017) proposed a methodology for simultaneous fault detection and control of switched LPV systems with inexact scheduling parameters. However, despite the few works reported in the literature, problem remains open and is of both practical and theoretical importance. In this work, the design of a PI observer for qLPV systems is proposed in order to estimate system states, actuator and sensor faults. The proposed approach considers inexact scheduling variables, the faults are considered as time-variants and the performance criterion is chosen to describe robustness to sensor noise and measurement uncertainty on the scheduling variables by solving a set of linear matrix inequalities (LMIs), which are obtained through a Lyapunov formulation. The rest of the paper is organized as follows: the problem formulation and preliminaries are given in Section 2. In Section 3, the design of a proportional-integral qLPV observer for the fault estimation in actuators and sensors considering inexact scheduling functions is presented. Simulation results are given in Section 4. Finally, the paper finishes with the conclusions in Section 5. 2. PROBLEM FORMULATION AND PRELIMINARIES Consider a qLPV system subject to sensor noise, actuator and sensor faults described by the equations: x(t) ˙ =A(α)x(t) + B(α)u(t) + Fa (α)fa (t), y(t) =Cx(t) + Fs fs (t) + Dw(t), (1) where x(t) ∈ Rn , u(t) ∈ Rnu , y(t) ∈ Rny , fa (t) ∈ Rnfa , and fs (t) ∈ Rnfs are the state, input output, actuator faults, and sensor faults vectors, respectively. w(t) ∈ Rnw is the sensor noise. A(α) ∈ Rn×n denotes the state matrix, B(α) ∈ Rn×nu is the input matrix, C ∈ Rny ×n is the output matrix, Fa (α) ∈ Rn×nfa and Fs ∈ Rny ×nfs are the actuator and sensor fault distribution matrices, respectively, D ∈ Rny ×nw is the disturbance matrix, and αi is weighting functions. Assume that the matrices Fa (α) and Fs are full rank. Suppose that the numbers of scheduling variables is q and the scheduling variables are independent with each other. Then, if the bounds of scheduling variables are known and measurable, the system can be described by a polytopic qLPV system with 2q vertices, such that the system matrix set S = (A(α), B(α), F (α)) can be expressed as:   2q 2q   S = Ω|Ω = αi Ωi ; 0 ≤ αi ≤ 1; αi = 1 (2) i=1

i=1

where Ωi = (Ai , Bi , Fa,i ) and the value of matrix set for each vertex is known. Since the scheduling variables are measurable online, the value of weighting functions αi for each vertex can be determined online.

101

Note that, in the case of perfectly measured scheduling variables, the weighting factor can be used directly in the design of components of a control system, such as observers or controllers. However, in the case of inexact scheduling variables there exist mismatches between the real and the measured weighting factors that can deteriorate or destabilize the observer or controller. In this case, it is necessary to use a robust approach that considers these mismatches in order to guarantee stability and good performance. In this work, the weighting factors are: αi = λi (t)α ˆi, (3) where α ˆ i are the uncertain weighting factors due to an inaccurate measurement of the scheduling variables; λi (t) is the uncertain factor, whose minimum and maximum values are given by λi and λi , respectively, such that: 2q 2q   A(α) = λi (t)α ˆ i Ai (4) α i Ai = i=1

i=1 q

2 

α ˆ i (Ai i=1 2q ˆi = i=1 α

= with yields:

q

+ (λi (t) − 1)Ai ) =

2 

α ˆ i (Ai + ∆Ai (t)) ;

i=1

1, and following the above procedure

B(α) = Fa (α) =

q

2  i=1 2q 

α ˆ i (Bi + ∆Bi (t)) ; α ˆ i (Fa,i + ∆Fa,i (t)) ;

and

(5) (6)

i=1

with: ∆Ai (t) = (λi (t) − 1)Ai , ∆Bi (t) = (λi (t) − 1)Bi , (7) (8) ∆Fa,i (t) = (λi (t) − 1)Fa,i . Then, the system (1) can be rewritten as an uncertain system as: x(t) ˙ = (A(α ˆ ) + ∆A(α)) ˆ x(t) + (B(α ˆ ) + ∆B(α ˆ )) u(t) + (Fa (α ˆ ) + ∆Fa (α ˆ )) fa (t), (9) y(t) =Cx(t) + F fs (t) + Dw(t). In order to estimate simultaneously the actuator and sensor faults, the qLPV system (9) is transformed by using a new state z(t) ∈ Rny that is a filtered version of y(t) (Youssef et al., 2017), defined by z(t) ˙ = −E (z(t) − y(t)), where E is a stable matrix. The augmented system can be represented as follow: ˙ ¯α ¯α ¯ α ¯ α X(t) =(A( ˆ ) + ∆A( ˆ ))X(t) + (B( ˆ ) + ∆B( ˆ ))u(t) ¯ ¯ ¯ + (F (α ˆ ) + ∆F (α ˆ ))f (t) + Hw(t), ¯ Y (t) =CX(t), (10) T

where X(t) = [ x(t) z(t) ] ∈ Rn¯ and f (t) = [ fa (t) fs (t) ] ∈ Rnf with n ¯ = n + ny , nf = nfa + nfs and     A(α ˆ) 0 ∆A(α ˆ) 0 ¯α ¯α A( ˆ) = , ∆A( ˆ) = , 0 EC −E   0 B(α ˆ) ∆B(α ˆ) ¯ α ¯ α B( ˆ) = , ∆B( ˆ) = , 0 0     ∆Fa (α ˆ) 0 ˆ) 0 F a (α , ∆F¯ (α ˆ) = , F¯ (α ˆ) = 0 EF 0 0 s     0 ¯ = H , C¯ = 0 Iny . ED

T

In the augmented system (10) the sensors faults appear as actuator faults and the matrix F¯ (α ˆ ) is full column rank.

102

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The faults f (t) are assumed to be time-varying signals whose kth time derivatives are bounded by f0 . The following notation is used: f˙(t) =f1 (t), f˙1 (t) =f2 (t), .. . f˙k−1 (t) =fk (t), fk (t) ≤f0 .

(11)

3. DESCRIPTION OF THE PROPORTIONAL-INTEGRAL OBSERVER Under the assumption that the uncertain system (10) is locally observable, the proposed PI observer estimates simultaneously the state and the faults in spite of inexact scheduling variables, and is given by the following: ˆ˙ ¯ α) ˆ + B(ˆ ¯ α)u(t) + F¯ (ˆ X(t) =A( ˆ X(t) α)fˆ(t)   ˆ ) Y (t) − Yˆ (t) + ϕx (ˆ α), + KP (α ˆ Yˆ (t) =C¯ X(t),   ˙ ˆ ) Y (t) − Yˆ (t) + fˆ1 (t) + ϕf (α ˆ ), fˆ(t) =KI (α   ˙ ˆ ) Y (t) − Yˆ (t) + fˆj+1 (t) + ϕfj (ˆ α), fˆj (t) =KIj (α

(12)

ˆ ), KI (ˆ α) and KIj (α) ˆ for j = 1, . . . , k − 1, where KP (α represent the proportional and integral gains, respectively, which are to be designed. The signals ϕx (α ˆ ), ϕf (α ˆ ) and ϕfj (ˆ α) are introduced in order to compensate the effect due to inexact scheduling variables, as shown later in Theorem 1. Based on (11), the augmented form of the LPV model (10) and the PI observer (12) are given, respectively, by: ¯˙ ¯ + (B(ˆ X(t) =(A(α ˆ ) + ∆A(ˆ α))X(t) α) + ∆B(ˆ α))u(t) + Gw(t) + Rfk (t), ¯ ¯ Y (t) =C X(t), (13) and ˆ ˆ ¯˙ ¯ + B(ˆ ¯ (t)) + ϕ(α X(t) =A(α ˆ )X(t) α)u(t) + K(α ˆ )(Y¯ (t) − Yˆ ˆ ), ˆ ¯ (t) =C X(t), ¯ Yˆ where T ¯ X(t) = [ X(t) f (t) f1 (t) . . . fk−1 (t) ] ,  T ˆ ¯ ˆ X(t) = X(t) fˆ(t) fˆ1 (t) . . . fˆk−1 (t) ,

(14)

T

ˆ ) ϕf ( α ˆ ) ϕf1 (α ˆ ) . . . ϕfk−1 (ˆ α) ] , ϕ(α ˆ ) = [ ϕx ( α n n ¯ α ) ∈ R k , nk = n ¯ + k × nf and with X(t) ∈ R k , ϕ(ˆ ˆ ˆ ¯ ¯ ¯ ¯ e¯(t) = X(t) − X(t), e¯y (t) = Y (t) − Y (t),  ¯   ¯  A(α) ˆ F¯ (α ˆ) 0 0 . . . 0 B(α ˆ) 0 I nk 0 . . . 0   0  0   0   0  0 0 I . . . 0    n k A(ˆ α) =  , B(α ˆ) =    . , .. .. .. .. ..   ...  ..  . . . . . 0 0 0 0 ... 0 0 T  K(ˆ α ) = K P (α ˆ ) KI ( α ˆ ) KI1 (α ˆ ) . . . KIk−1 (ˆ α) ,     ¯ 0 0 . . . 0 T , R = 0 0 0 . . . I nf T , G= H



¯α ∆A( ˆ ) ∆F¯ (α ˆ) 0 0 . . . 0 0 0 ...  0  0 0 0 0 ...  ∆A(α ˆ) =  .. .. .. .. ..  . . . . . 0 0 0 0 ...  T ¯ α ˆ) 0 0 . . . 0 , ∆B(α ˆ ) = ∆B(   C = C¯ 0 0 0 . . . 0 .

 0 0 0 , ..  .

0

In order to facilitate the observer design, using (7) and (8) the uncertainties are bounded as follows: (15) ∆Ai  ≤ ζ1,i , ∆Bi  ≤ ζ2,i , (16) with positives scalars ζ1,i and ζ2,i . therefore, the following is valid: 2q 2q 2q    α ˆ i ||∆Ai || ≤ α ˆ i ζ1,i , α ˆ i ∆Ai || = ||∆A(α ˆ )|| =|| i=1

i=1

i=1

q

||∆B(α ˆ )|| ≤

2 

α ˆ i ζ2,i ,

(17)

i=1

The dynamics of the augmented state estimation error, denoted as e¯˙ (t), is represented in the following form:   ¯ + W v(t) e¯˙ (t) = A(α ˆ ) − K(α ˆ )C e¯(t) + ∆A(α) ˆ X(t) + ∆B(α ˆ )u(t) − ϕ(α ˆ ),

(18)

T

where v(t) = [ fk (t) w(t) ] and W = [ R G ] . The error dynamics (18) of the augmented system is ¯ associated with the state vector X(t), the input u(t), the noise w(t), and the function ϕ(t). Since the main criterion for selecting the gain K(α ˆ ) is to make the estimation error system stable such that the estimation error would converge to zero in absence of uncertainties, we define a new variable as: ξ(t) = L¯ e(t), (19) with a constant matrix L. Then, the challenges and objectives are to tune the observer gain K(α ˆ ) and discontinue function ϕ(t), such that the dynamical estimation error in (18) is asymptotically stable in absence of uncertainty and noise, and the effect from the external input v(t) to the signal ξ(t) is constrained as: (20) ξ(t)2 < γv(t)2 , where  · 2 denotes the 2-norm of a L2 -bounded signal and γ is the H∞ performance index.

The following theorem provides the conditions for the asymptotic stability and the H∞ performance of the estimation error in (18). Theorem 1. Given the LPV system (13) and the state observer (14), the estimation error (18) is asymptotically stable with H∞ performance and attenuation level γ > 0, if there exist a matrix P > 0, matrices Mi and positive scalars ψ6 and ψ2 , such that following LMI constraints are feasible:   Λi P W P ζ1,i LT  ∗ −γ 2 I 0 0 0    (21)  ∗ ∗ −ψ6 I 0 0  < 0,  ∗ ∗ ∗ −ψ2 I 0  ∗ ∗ ∗ ∗ −I



S. Gómez-Peñate et al. / IFAC PapersOnLine 52-28 (2019) 100–105

with: Λi = ATi P +P Ai −C T MiT −Mi C, e¯y (t) = Y¯ (t)−Yˆ¯ (t), if |¯ ey (t)| ≥ , then q

ϕ(α ˆ) =

2 

2 α ˆ i ψ3 ζ1,i

i=1

+

ˆ ˆ ¯ T X(t) ¯ X(t) P −1 C T e¯y (t) 2¯ ey (t)T e¯y (t)

q

2 

2 α ˆ i ψ5 ζ2,i

i=1

if |¯ ey (t)| < , then

u(t)T u(t) P −1 C T e¯y (t), 2¯ ey (t)T e¯y (t)

ϕ(α ˆ ) = 0,

where ψ2−1 = ψ1 (1 + ψ4−1 ) and ψ3 = ψ1 (1 + ψ4 ). Using the previous procedure one gets: ˆ )T P e¯(t) + e¯(t)T P ∆B(α ˆ )u(t) u(t)T ∆B(α ≤ ψ5−1 (P e¯(t))T P e¯(t) + ψ5 u(t)T ∆B(α ˆ )T ∆B(α ˆ )u(t)

(22)

(23)

where: ψ2 ψ 6 ψ4 , ψ3 = , ψ4 − ψ2 ψ6 (ψ4 + 1) ψ2 ψ4 and  are positive scalars arbitrarily fixed; and if the PI observer parameters are computed as: (24) Ki = P −1 Mi . ψ5 =

Proof The performance criterion (19) is equivalent to: (Zhang et al., 2016) J (t) := V˙ (t) + ξ(t)T ξ(t) − γ 2 v(t)T v(t) < 0, (25)

where V (t) is the Lyapunov function which is selected as V = e¯(t)T P e¯(t), with P > 0, such that: J (t) :=e¯˙ (t)T P e¯(t) + e¯(t)T P e¯˙ (t) + ξ(t)T ξ(t) − γ 2 v(t)T v(t).

Then, by considering (18), the following inequality is obtained:   α)−K(ˆ α)C) e¯(t) ˆ )−K(α ˆ )C)T P +P (A(ˆ J (t) := e¯(t)T (A(α + e¯(t)T P W v(t) + v(t)T W T P e¯(t)

¯ T ∆A(α ¯ + X(t) ˆ )T P e¯(t) + e¯(t)T P ∆A(ˆ α)X(t) + u(t)T ∆B(α) ˆ T P e¯(t) + e¯(t)T P ∆B(ˆ α)u(t)

− 2¯ e(t)T P ϕ(α ˆ ) + e¯(t)T LT L¯ e(t) − γ 2 v(t)T v(t) < 0. (26) Lemma 1. (Ichalal et al., 2010) For matrices X and Y with appropriate dimensions, the following properly holds for any positive scalar ψ: X T Y + Y T X ≤ ψX T X + ψ −1 Y T Y. Hence, by applying Lemma 1: ¯ ¯ T ∆A(α ˆ )T P e¯(t) + e¯(t)T P ∆A(ˆ α)X(t) X(t) ¯ ¯ T ∆A(α ≤ ψ1−1 (P e¯(t))T P e¯(t) + ψ1 X(t) ˆ )T ∆A(α ˆ )X(t) considering (15), the following relationship is established: ¯ ¯ T ∆A(α ˆ )T ∆A(α ˆ )X(t) ψ1−1 (P e¯(t))T P e¯(t) + ψ1 X(t) ¯ T X(t), ¯ ≤ ψ1−1 e¯(t)T P 2 e¯(t) + ψ1 ζ1 (ˆ α)2 X(t) (27) ˆ ¯ ¯ where X(t) = e¯(t) + X(t), then, the expression (27) becomes: ¯ T X(t) ¯ ˆ 2 X(t) ψ1−1 e¯(t)T P 2 e¯(t)+ψ1 ζ1 (α) = ψ1−1 e¯(t)T P 2 e¯(t)   ˆ ˆ ˆ ˆ¯ ¯ T e¯(t)+¯ ¯ ¯ T X(t) ˆ )2 e¯(t)T e¯(t)+X(t) e(t)T X(t)+ X(t) . +ψ1 ζ1 (α Using again the Lemma 1, the last expression can be rewritten as follows:  ˆ ¯ T e¯(t) ψ1−1 e¯(t)T P 2 e¯(t) + ψ1 ζ1 (ˆ α)2 e¯(t)T e¯(t) + X(t)  ˆ ˆ ˆ ¯ + X(t) ¯ T X(t) ¯ ≤ ψ1−1 e¯(t)T P 2 e¯(t) + e¯(t)T X(t) ˆ ˆ ¯ T X(t), ¯ ˆ )2 e¯(t)T e¯(t) + ψ3 ζ1 (ˆ α)2 X(t) + ψ2−1 ζ1 (α

103

(28)

≤ ψ5−1 e¯(t)T P 2 e¯(t) + ψ5 ζ2 (α ˆ )2 u(t)T u(t). (29) If ey (t) is zero, since each subsystem is observable, the estimation error is zero. If ey (t) is non-zero, in order to cancel the effect of the uncertainties on the dynamics of the output system, ϕ(t) is selected as in the equation (22) and by substituting the expression (22) in (26): ˆ¯ T X(t) ˆ¯ X(t) P −1 C T e¯y (t) ˆ ) =2¯ e(t)T P ψ3 ζ1 (α ˆ )2 2¯ e(t)T P ϕ(α 2¯ ey (t)T e¯y (t) u(t)T u(t) P −1 C T e¯y (t ˆ )2 +2¯ e(t)T P ψ5 ζ2 (α 2¯ ey (t)T e¯y (t) ˆ¯ T X(t) ˆ¯ 2¯ e(t)T P ϕ(α ˆ ) = ψ 3 ζ1 ( α ˆ )2 X(t) + ψ5 ζ2 ( α ˆ )2 u(t)T u(t), T

T

(30)

T

with e¯y (t) = C¯ e(t) and e¯y (t) = e¯(t) C .

Such that the performance criteria J (t) is: q

J (t) ≤

q

2  i=1

α ˆi

2  i=1



where



 α ˆ i e¯(t)T Γi e¯(t) + e¯(t)T P W v(t)

 + v(t)T W T P e¯(t) − γ 2 v(t)T v(t) ≤ 0

e(t)T v(t)T

    Γi P W e(t) ≤ 0, v(t) ∗ −γ 2 I

Γi = (Ai − Ki C)T P + P (Ai − Ki C)

2 + ψ6−1 P 2 + ψ2−1 ζ1,i + LT L.

with

ψ6−1

=

ψ1−1

+

(31)

(32)

ψ5−1 .

The analysis prove that (31) holds if:   Γi P W < 0. (33) ∗ −γ 2 I Given that (33) is nonlinear, a change of variable Mi = P Ki is performed in order to obtain a LMI representation. Finally, the Schur complement is considered to obtain the LMI given in Theorem 1, which can be easily solved with specialized software. This completes the proof.  In the practical implementation the magnitude of ϕ(α ˆ) increases without limit due to the fact that the estimation error ey (t) tends to zero. This problem can be overcome considering that ey (t) not converge asymptotically to zero, but to keep it in a small neighborhood of zero depending on the magnitude of ,  is a small positive scalar, as is considered in Theorem 1. It necessary to mention that the H∞ performance index γ indicates the effect of the disturbance v(t) to the signal ξ(t). It is desired that the performance index γ is as small as possible. 4. SIMULATION EXAMPLE In this section, a Van de Vusse reactor is used to illustrate the method proposed for fault estimation. A Van de Vusse

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104

reactor is known to be a highly nonlinear process (Rabaoui et al., 2017). In this reactor, a product A is converted into the desired product B in an isothermal continuous stirred tank reactor (CSTR) which has exothermic reaction instability with a prolonged cooling jacked temperature above 305K and the product B is also converted to a product C: k1 k2 B −→ C. In addition to this consecutive reaction, A −→ a high order parallel reaction occurs and A is converted k3 D. Where the concentration of into product D, 2A −→ product A and B is denoted by CA and CB respectively, the following mass balance equation is obtained: Fu dCA = −k1 CA − k3 CA 2 + (CAf − CA ) , dt V Fu dCB = k1 CA − k2 CB + (−CB ) , (34) dt V where CAf the concentration of the reactant A in the inlet flow, fu is the inlet flow rate and V is the constant volume of the CTRS. Considering x1 = CA and x2 = CB as state variables and u = fu /V as the input, the state-space model is given by: x˙ 1 (t) = − k1 x1 (t) − k3 x1 2 (t) + (CAf − x1 (t))[u(t) + fa (t)]; x˙ 2 (t) =k1 x1 (t) − k2 x2 (t) + (−x2 (t))[u(t) + fa (t)]; y1 (t) =x1 (t) + fs (t) + w(t); y2 (t) =x2 (t) + w(t); (35) where CAf the concentration of the reactant A in the inlet flow, fu is the inlet flow rate and V is the constant volume of the CTRS. The kinetic parameters are chosen to be k1 = 100/h, k2 = 50/h, k3 = 10/h, CAf = 10mol/h and V = 1L. fa is the actuator fault, fs is the sensor fault and w1 (t), w2 (t) are the measurement noises. The nonlinear model (35) can be converted into a qLPV representation by embedding the nonlinearities within the varying parameters such as: x(t) ˙ =A(α(ρ(t)))x(t) + B(α(ρ(t)))u(t) + Fa (α(ρ(t)))fa (t) y(t) =Cx(t) + Fs fs (t) + Dw(t) (36) where the matrices are defined as:   −k1 − k3 ρ1 (t) 0 ; A(α(ρ(t))) = k1 −k2   CAf − ρ1 (t) B(α(ρ(t))) = ; Fa (α(ρ(t))) = B(α(ρ(t))); ρ (t)    2   1 10 1 Fs = ;C= and D = , with the scheduling 0 01 1 parameters ρ1 (t) = x1 (t) and ρ2 (t) = x2 (t) vary in a hyper-cube such that: 0.1 ≤ ρ1 (t) ≤ 1 and 0.2 ≤ ρ2 (t) ≤ 1.2. Finally, we can rewrite the proposed form (36) in the following polytopic form: ˙ x(t) =

4 

αi [Ai x(t) + Bi u(t) + Fa,i fa (t)] ;

i=1

y(t) =Cx(t) + Fs fs + Dw(t);

(37)

where convex weighing functions are defined as: ρ1 (t) − ρ1 ρ2 (t) − ρ2 , α2 (ρ(t)) = 1 − α1 (ρ(t)), α1 (ρ(t)) = ρ1 − ρ 1 ρ2 − ρ 2 ρ − ρ1 (t) ρ2 (t) − ρ2 , α4 (ρ(t)) = 1 − α3 (ρ(t)), α3 (ρ(t)) = 1 ρ1 − ρ 1 ρ2 − ρ 2 and the vertex matrices are described as follows:



 −k1 − k3 ρ1 0 ; k1 −k2   −k1 − k3 ρ1 0 ; A3 = A4 = k1 −k2     CAf − ρ1 CAf − ρ1 B1 = ; B2 = ; −ρ2 −ρ2     CAf − ρ1 CAf − ρ1 B3 = ; ; B4 = −ρ2 −ρ2

A1 = A2 =

Fa,1 = B1 ; Fa,2 = B2 ; Fa,3 = B3 ; Fa,4 = B4 . 4.1 Actuator and sensor fault estimation Solving the LMI (21) in Theorem 1, the unknown gains of the PIO are obtained. The constants are selected as E = 80,  = 10−5 , ψ4 = 10 and since the main objective is to estimate the state variables and the faults, the value of matrix L is selected  as  1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 L= . 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 In this case, the uncertain factors are delimited as follows: λi ∈ [0.9 1.1], for i = 1, . . . , 4. (38) In consequence, it is possible to compute ζ1,i and ζ2,i from (15) and (16). The simulation results are carried out with level attenuation γ = 7.64, the obtained constants ψ6 = 4.82 × 104 , ψ2 = 1.38 × 105 and matrices Ki (i = 1, . . . , 4), are used to construct the PIO (14) which is implemented in simulation. The initial conditions are: ¯ 0 = [ 0.5 0.4 0.51 0.41 0 0 0 0 ]T and X ˆ ¯ 0 = [ 0.1 0.2 0.1 0.2 0 0 0 0 ]T . X The input flow rate Fu is considered variable such that the input signal u(t) will expressed by:  5 + 2 sin(0.07t) f or t ≥ 0 u(t) = , (39) 0 otherwise and the actuator and sensor fault is assumed as a variant time signal and its second-derivative norm-bounded such that:  0.2 sin(2t − 2) f or 1 ≤ t ≤ 4.142 fa (t) = (40) 0 otherwise  0.05 sin(3t − 18) f or 6 ≤ t ≤ 9.157 fs (t) = (41) 0 otherwise

The measurement noise w(t) in the output, is a centered Gaussian noise with variance 0.0001. The scheduling variables are also affected by the measurement noise, in order to consider the uncertainty in the weighting functions, these weighting functions with uncertainty are shown in Fig. 1. In this case, the measurements of the scheduling variables are considered free of faults. Fig. 2 shows states estimation and Fig. 3 shows timevarying sinusoidal actuator and sensor faults and their estimates. These simulation results demonstrate the applicability of the method for estimating actuator and sensor faults for qLPV systems with weighting functions with uncertainty.



S. Gómez-Peñate et al. / IFAC PapersOnLine 52-28 (2019) 100–105

0.6

α ˆ 1 (t) 0.5

α ˆ 2 (t) α ˆ 3 (t)

Magnitude

0.4

α ˆ 4 (t)

0.3 0.2 0.1 0.0 0

2

4

6

8

10

Time (s)

Fig. 1. Inexact weighting factors

Magnitude (mol/l)

1.1 1.0

x1 (t)

0.9

xˆ1 (t)

0.8

x2 (t)

0.7

xˆ2 (t)

0.6 0.5 0.4 0.3 0

2

4

6

8

10

Time (s)

Fig. 2. State variables and their estimates fa (t) fˆa (t)

Magnitude

0.4 0.2 0.0 −0.2

Magnitude

0.10

0

2

4

6

8

10

6

8

10

(s) fTime s (t) fˆs (t)

0.05 0.00 −0.05

0

2

4

Time (s)

Fig. 3. Faults and their estimates 5. CONCLUSION In this work, a polytopic PIO for state, actuator and sensor faults estimation was proposed. It was considered that the qLPV system was affected by noise measurement in the scheduling variables and output of the system. The used strategy was based on the H∞ performance criteria to be robust against sensor noise and uncertainty induced by inexact scheduling variables. Furthermore, it was demonstrated that the proposed approach is suitable to estimate system states and actuator and sensors faults by a qLPV Proportional-Integral observer. Finally, a numerical example of the Van de Vusse reactor model was presented to show the effectiveness and applicability of the proposed approach. Future work will be done to extend the method to fault tolerant control. Note that a reconfigurable controller can be designed in order to maintain stability, acceptable dynamic performance and steady state of the system, in the event of a fault, based on the the proposed fault estimation method. REFERENCES Casavola, A. and Gagliardi, G. (2015). Fault detection and isolation of electrical induction motors via LPV fault observers: A case study. In International Journal of Robust and Nonlinear Control.

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