Brief paperUnknown input observer for LPV systems☆
Introduction
Since in practical situations, all process inputs are rarely known, unknown input observers (UIO) are essential in process control and diagnosis and have received a great attention. An UIO provides the system state estimate, even in the presence of unknown inputs (UI) encompassing faults, perturbations, modeling uncertainties, etc. The several proposed approaches can be split into two classes. In the first one, the state estimation is decoupled from the UI thanks to structural conditions (Darouach, Zasadzinski, & Xu, 1994) or by some state transformations Aldeen and Sharma, 2008, Hou and Muller, 1994, Tan et al., 2008. In the second one, the system state is usually augmented with the UI and, under some assumptions on the time variations of the UI, an augmented observer may provide simultaneous state and UI estimation. The so-called PI observer, proposed by Wojciechowski (1978) for linear systems affected by constant UI, has been extended to descriptor and nonlinear systems affected by time polynomial UI Gao and Ho, 2004, Ichalal et al., 2009, Koenig, 2006. The original linear UIO proposed by Darouach et al. (1994) has been extended to nonlinear polytopic systems by duplicating the polytopic structure of the system in the nonlinear UIO. Therefore, the structure of the UIO is fixed a priori and asymptotic convergence of the state estimation error towards zero is ensured by solving Lyapunov based LMI conditions Chadli and Karimi, 2013, Marx et al., 2007. Despite the appealing simplicity of this approach, the duplication of the polytopic structure of the system in the UIO drastically reduces the system class for which such an observer can be designed Ichalal and Mammar, 2015, Ichalal et al., 2015. For example, the UIO design may fail even if the system is strongly algebraically observable or at least strongly detectable.
In this paper a new LPV UIO is presented. Its polytopic structure does not necessarily mimic the system structure. It allows to design UIO for a larger class of systems by avoiding restrictive decoupling conditions (as those of Chadli and Karimi (2013), Hassanabadi, Shafiee, and Puig (2016) and Marx et al. (2007)). This result is obtained by postponing the use of the polytopic transformation of a general LPV form into a polytopic one. It results in more degrees of freedom in the UIO design by not searching for a common solution to several equality constraints imposed by the decoupling between state the estimation and the UI (Chadli & Karimi, 2013). Moreover, all the published works on UIO for polytopic or LPV systems consider linear time invariant output equations and cannot be applied to systems with LPV outputs Chadli and Karimi, 2013, Hassanabadi et al., 2016, Heemels et al., 2010, Marx et al., 2007. The proposed UIO structure allows to overcome this limitation and provides a solution for systems with LPV output equations. In the presence of disturbances, the UIO is designed to ensure perfect fault decoupling and asymptotic convergence in an origin centered ball which radius is minimized. If no disturbances affect the system, then the state estimation error asymptotically converges to zero and the fault estimation is also provided.
The paper is organized as follows. The problem statement and some useful materials are provided in Section 2. The UIO design for LPV systems affected by fault UI and disturbance UI is detailed in Section 3. Before concluding, some illustrative examples are provided in Section 4.
Section snippets
Notations
In the remainder of the paper, the following notations are used. For any square matrix, (resp. ) means that is positive (resp. negative) definite and stands for . The minimal and maximal eigenvalues of are respectively denoted and . For any matrix (not necessarily square), and respectively denote the transpose and the left pseudo inverse of . is the identity matrix. The Euclidean norm of a vector is denoted , and denotes the
Unknown input observer design
In this section, the design of the UIO (5) for (4) is detailed. The first objective of the UIO design is to perfectly decouple the state estimation from the faults . Secondly, the influence of the disturbance is minimized by ensuring the state estimation error to asymptotically converge in an origin-centered ball, using the ISS. The ball radius is minimized when computing the observer gains by LMI optimization. Thirdly, the proposed UIO design brings some relaxation thanks to the use of a
Illustrative examples
Three examples are provided to illustrate the proposed UIO contributions. In Example 1 it is shown that even with LTI output equation our UIO design outperforms the ones of Chadli and Karimi (2013) and of Marx et al. (2007) that cannot be applied. Example 2 illustrates the fact that, contrarily to the existing literature, an UIO can be designed for systems with LPV output. Example 3 presents a concrete application to the Lorenz chaotic circuit estimation for secure communications.
Example 1 Let us
Conclusion
This paper investigated the problem of unknown input observer design for LPV systems and brought two main contributions. The first one is to relax the known rank conditions needed to UIO design for LPV systems to decouple the estimation and the UI. The second one is that, contrarily to the existing literature, the output equation of the LPV system is not restricted to be linear time invariant. The proposed UIO designs rely on LMI optimization in order to compute the observer gains. In the
Benoît Marx obtained the PhD and HDR (habilitation à diriger des recherches) degrees in automatic control respectively, in 2003 from the Grenoble Institute of Technology (France) and in 2016 from the Université de Lorraine (France). Since 2004, he has been an associate professor at the Université de Lorraine and a member of the Research Centre for Automatic Control of Nancy. His research interests include state estimation, fault diagnosis and fault tolerant control of singular and/or nonlinear
References (28)
- et al.
Observer design for Takagi-Sugeno descriptor models: An LMI approach
Automatica
(2015) On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems
Annual Reviews in Control
(2009)- et al.
Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem
Theme: Modelling and Control
Fuzzy Sets and Systems
(2007) - et al.
On characterizations of the input-to-state stability property
Systems & Control Letters
(1995) - et al.
Extended results on robust state estimation and fault detection
Automatica
(2008) - et al.
Estimation of states, faults and unknown disturbances in non-linear systems
International Journal of Control
(2008) - et al.
Robust Observer Design for Unknown Inputs Takagi-Sugeno Models
IEEE Transactions on Fuzzy Systems
(2013) - et al.
Synchronization of Lorenz-based chaotic circuits with applications to communications
IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing
(1993) - et al.
Full-order observers for linear systems with unknown inputs
IEEE Transactions on Automatic Control
(1994) A survey on analysis and design of model-based fuzzy control systems
IEEE Transactions on Fuzzy Systems
(2006)
Non-linear estimation is easy
International Journal of Modelling, Identification and Control
Proportional multiple-integral observer design for descriptor systems with measurement output disturbances
IEE Proceeding Control Theory and Application
UIO design for singular delayed LPV systems with application to actuator fault detection and isolation
International Journal of Systems Science
Observer-based control of discrete-time LPV systems with uncertain parameters
IEEE Transactions on Automatic Control
Cited by (0)
Benoît Marx obtained the PhD and HDR (habilitation à diriger des recherches) degrees in automatic control respectively, in 2003 from the Grenoble Institute of Technology (France) and in 2016 from the Université de Lorraine (France). Since 2004, he has been an associate professor at the Université de Lorraine and a member of the Research Centre for Automatic Control of Nancy. His research interests include state estimation, fault diagnosis and fault tolerant control of singular and/or nonlinear systems, with particular attention paid to nonlinear systems represented by polytopic models.
Dalil Ichalal received his Master degree from Paul Cezanne Aix Marseille 3 University (France) in 2006, and a Ph.D. degree from the National Polytechnic Institute of Lorraine (France) in 2009. In 2010, he joined the Department of Electrical Engineering of the University of Evry Val d’Essonne (France) and the IBISC (Informatique, Biologie Integrative et systémes complexes) Laboratory. His research interests include state estimation and observer design, as well as fault diagnosis and fault tolerant control of nonlinear systems with application to vehicles and motorcycle.
José Ragot, obtained an engineering degree in automatic control, in 1969 from Ecole Centrale de Nantes (France). Since 1985, he has been a Professor in Automatic Control at the University of Lorraine (France) and is a member of the Research Center for Automatic Control of Nancy. His major research fields include process diagnosis, fault detection and isolation, data validation and reconciliation, fault-tolerant control using either model-based and data-driven approaches. He has successfully advised 70 Ph.D. students and his research results have been published in over 150 papers in leading journals and 370 communications in international conferences, 4 books and 14 chapters in collective volumes. His research is strengthened by industrial links in various fields such as mineral and metallurgical processing, chemical engineering, water treatment, aerospace, environmental processes.
Didier Maquin received Ph.D. degree in electrical engineering from university of Nancy in 1987 and a Habilitation from the National Polytechnic Institute of Nancy in 1997. Since 2003, he is a full Professor of Automatic Control at the University of Lorraine, France. He is a member of the Research Center for Automatic Control of Nancy (CRAN) which is funded by the National Center for Scientific Research (CNRS) and the University of Lorraine since 1984. His research fields include process fault detection and isolation and fault-tolerant control using either model-based (with a particular focus on Takagi–Sugeno models) and data-driven approaches (Kernel PCA or related approaches). Didier Maquin co-supervized 30 Ph.D. students and co-authored 60 journal articles and more than 150 international conference communications.
Saïd Mammar (M’97,SM’11) received the Dipl. Ing. degree from the École Supérieure d’Électricité, Gif-sur-Yvette, France, in 1989, the Ph.D. degree in automatic control from the Université Paris XI (Supelec), Orsay, France, in 1992, and the H.D.R. degree from the University of Evry, Evry, France, in 2001. From 1992 to 1994, he held a research position with the French National Institute on Transportation Research and Safety, Versailles, France, where he was involved with research on traffic network control. From 1994 to 2002, he was an Assistant Professor with the University of Evry, where he has been a Professor since 2002. From 2006 to September 2009, he was a Scientific and University Attaché with the French Embassy, The Hague, The Netherlands. From 2010 to 2014, he has been the Head of the laboratory Informatique, Biologie Intégrative et Systèmes Complexes, University of Evry, where he has also been the Vice President of International Affairs from 2012 to 2015. He is the Dean of the Faculty of Science and Technology and the Vice-President in charge of Finance. His research interests include observer design, robust control, vehicle longitudinal and lateral control for driving assistance, and intelligent transportation systems.
- ☆
The material in this paper was partially presented at the 9th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes, SAFEPROCESS, September 2–4, 2015, Paris, France. This paper was recommended for publication in revised form by Associate Editor Mario Sznaier under the direction of Editor Richard Middleton.