Brief paperSampled-data adaptive observer for state-affine systems with uncertain output equation☆
Introduction
The problem of simultaneous state and parameter estimation, based on sampled measurements, is of great practical interest since most physical systems are continuous-time and subject to parameter uncertainty. It is also of great theoretical interest because the existing observer design and analysis methods are not applicable to specific nonlinear systems and their extension to wider classes constitutes new challenges. The first adaptive observers for nonlinear systems were not sampled-data, see e.g. Bastin and Gevers (1988), Besançon, De León-Morales, and Huerta-Guevara (2006), Marino and Tomei (1996) and Zhang (2002). Their strong nonlinearity makes their direct discretization a highly complex issue. In particular, there is no guarantee that the performances of the original continuous-time adaptive observers are preserved in their approximate discrete-time versions. The first sampled-data adaptive observers, for nonlinear systems, have been developed (Ahmed-Ali et al., 2009, Folin et al., 2016, Hann and Ahmed-Ali, 2012). In Ahmed-Ali et al. (2009), a class of state affine systems was considered where the unknown parameters come linearly in the state equation and the associated regressor is output-independent. Then, an adaptive observer has been developed using the so-called continuous-discrete design principle. Accordingly, online state estimation is performed using an (open-loop) estimator all the time, except for the sampling instants. At these instants, the state estimate trajectory is corrected using an observer (involving a feedback innovation term). The parameter estimates are only updated at the sampling instants (and kept constant on the rest of the time). It turns out that both the state and the parameter estimate trajectories are discontinuous. Nevertheless, the observer is exponentially convergent, under persistent excitation conditions, if the sampling interval is sufficiently small. A quite different adaptive observer has been proposed in Hann and Ahmed-Ali (2012) for an almost similar class of systems as in Ahmed-Ali et al. (2009). This adaptive observer involves an inter-sample output predictor that is reinitialized at each sampling instant using the output measurements. Its main feature is that the state (resp. the parameters) estimates are generated using all the time the same state estimator (resp. same parameter adaptive law). Therefore, the trajectories of both the state and the parameter estimates are continuous. Again, the observer exponential convergence is ensured under PE conditions. A common limitation of the (sampled-output) nonlinear adaptive observers proposed in Ahmed-Ali et al. (2009) and Hann and Ahmed-Ali (2012) is that they are not applicable to systems with output-injection, i.e. those where the regressor (entering the state equation) is output-dependent. This class of systems has been considered in Folin et al. (2016) where an adaptive observer, involving inter-sample output-estimator, has been proposed. Exponential convergence is established, under ad-hoc persistent excitation conditions, provided the sampling period is sufficiently small.
In this paper, the problem of adaptive observer design is considered for a different class of nonlinear systems. Specifically, the unknown parameters enter the output equation, while they entered the state equation in the previous works. Furthermore, the regressor (that is associated with the unknown parameter vector) is output-dependent. Consequently, the system affine nature with respect to the parameters is lost almost all the time, because of the output sampling. Another difficulty with this class of systems is that the output signal enters nonlinearly in the output equation making impossible the construction of dynamic inter-sample output-predictors like those in Folin et al. (2016) or Karafyllis and Kravaris (2009). Therefore, a quite different static (inter-sample output) predictor, reinitialized at sampling times, is designed in this paper. Furthermore, the proposed adaptive observer includes a state estimator and an adaptive parameter law featuring continuous state and parameter estimate trajectories. The observer exponential convergence is established, for small sampling intervals, under persistent excitation (PE) conditions guaranteeing system observability and identifiability. To the authors’ knowledge it is the first time that an exponentially convergent adaptive observer is developed for (nonlinear) systems with unknown parameters in the output equation.
The paper is organized as follows: the class of systems under study is described in Section 2 along with the observation objectives; the observer design and analysis are presented in Sections 3 Adaptive observer design, 4 Adaptive observer analysis, respectively; simulation results are provided in Section 5; technical proofs are appended.
Section snippets
Class of systems
The system under study is described by the following model: with, with arbitrary, where and denote the system input and output, respectively; is the state vector. All quantities in (2)–(4), including the integer and , are known except for the parameter vector . Furthermore, , , and are functions. The input signal is bounded and the mapping (defined
Adaptive observer design
To get online estimates and , of the state vector and the unknown parameter vector , we propose the adaptive observer of Table 1.
Clearly, the adaptive observer of Table 1 is composed of four main parts:
(i) the state observer (8)–(10) providing the state estimates ;
(ii) the parameter estimator (13)–(15) providing the parameter estimates (which is a least-squares with forgetting factor equal to 1);
(iii) the adaptive law (11) providing the observer matrix gain ;
(iv)
Adaptive observer analysis
Let us introduce the following estimation errors: One immediately gets from (9) and (2): Subtracting (1) from (8), one gets using (22):
Also, it readily follows from (13) and (22) that: Now, introduce the
Simulation
The proposed approach is illustrated on a Wiener system (see Remark 1). A noise-free case is considered first. Afterwards, a case with output measurement noise is considered.
The considered Wiener system is inspired on Example 1 in Wigren (1993), which considers a Wiener model that describes a valve for control of fluid flow. The linear dynamic part is a continuous-time equivalent (using zero-order-hold) of the discrete-time model presented in Wigren (1993). The continuous-time model has
Concluding remarks
The problem of sampled-data adaptive observer design has been addressed for the class of state- and parameter-affine systems described by (1)–(2). This class of systems is quite different from those in the existing works (on adaptive observers) because the unknown parameter vector comes presently into the output equation. Furthermore, the corresponding regressor depends on the output which is not accessible to measurements, except at the sampling times. Then, the affine
Acknowledgment
This work was done during a visit of K. Tiels and M. Schoukens to the Labaratoire d’Automatique de Caen, University of Caen Normandie, in 2016.
Tarek Ahmed-Ali was born in Algeria in 1972. In 1994 he received an electrical engineering degree from Ecole Nationale Polytechnique d’Alger. In 1998 he received his Ph.D. degree from the University of Orsay - Paris Sud within the L2S-CNRS. At 2009, he was appointed as full professor at Ecole Nationale Supérieure des Ingénieurs de Caen (ENSICAEN). His main recent research interests include observer design, performance and robustness issues in nonlinear, hybrid, networked and infinite
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Tarek Ahmed-Ali was born in Algeria in 1972. In 1994 he received an electrical engineering degree from Ecole Nationale Polytechnique d’Alger. In 1998 he received his Ph.D. degree from the University of Orsay - Paris Sud within the L2S-CNRS. At 2009, he was appointed as full professor at Ecole Nationale Supérieure des Ingénieurs de Caen (ENSICAEN). His main recent research interests include observer design, performance and robustness issues in nonlinear, hybrid, networked and infinite dimensional systems.
Koen Tiels received the degree of master in Electromechanical Engineering in July 2010 and the degree of Doctor in Engineering (Ph.D.) in March 2015, both from the Vrije Universiteit Brussel, Belgium. He was a post-doctoral researcher in the period 2015–2018 at the same university. In February 2018, he joined the Department of Information Technology at Uppsala University, Sweden, where he is currently a post-doctoral researcher. His research interests are in the field of nonlinear system identification.
Maarten Schoukens is an Assistant Professor in the Control Systems group of the Department of Electrical Engineering at the Eindhoven University of Technology. He received the master’s degree in electrical engineering and the Ph.D. degree in engineering from the Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 2010 and 2015 respectively. From 2015 to 2017, he has been a Post-Doctoral Researcher with the ELEC Department, VUB. In October 2017 he joined the Control Systems research group, TU/e, Eindhoven, The Netherlands as a Post-Doctoral Researcher, in 2018 he became an Assistant Professor in the same group. Maarten was awarded an FWO Ph.D. Fellowship in 2011, and a Marie Skłodowska-Curie Individual Fellowship in 2018. His main research interests include the measurement and data-driven modeling of linear parameter-varying and nonlinear dynamical systems using system identification and machine learning techniques.
Fouad Giri obtained the Ph.D. degree in automatic control and the Accreditation to Supervise Researches, both from Grenoble Institut National Polytechnique (INP), Grenoble, France. Since 1982, he has been Associate-Professor and Professor successively with the University of Rabat, Morocco and the Université de Caen Normandie, Caen, France. He is co-founding director of the Caen Control Laboratory (LAC). He has received the ‘Research and Doctoral Supervising’ award, granted by the French Ministry of High Education and Research, continuously from 1998 to now. He has been recipient of the 2018 French IFAC NMO Award. He has been serving as the Chair of the IFAC Technical Committee TC1.2 (Adaptive and Learning Systems) since 2014 and served as Vice-Chair of that TC during the triennial 2011–2014. He served as the General Chair of the 11th IFAC International Workshop on Adaptation and learning in Control and Signal processing (ALCOSP’2013) and the 5th IFAC International Workshop on Periodic Systems Control (PSYCO’2013) both held in Caen, France, in 2013. He has been Associate Editor for several journals and conferences including Control Engineering Practice (2008 to 2014), IEEE CSS Conference Editorial Board (CEB) (2010 to present), IEEE Transactions on Control System Technology (2012 to 2014), Automatica (2014 to present). He has also served as Technical Associate Editor of the IFAC World Congress 2014 and as Associate Editor for the IFAC World Congress 2017. He has published more than 100 scientific journal articles, over 200 conference papers, 15 book chapters. He has coauthored/coedited four books on Electric Motors Control (Wiley, 2013); Nonlinear System Identification (Springer, 2010; Feedback Systems Control (two books in French published by Eyrolles, 1993, 1994). He has supervised to completion more than 20 Ph.D. students.
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This work was supported in part by the ERC advanced grant SNLSID, under contract 320378. K. Tiels acknowledges the financial support by the Swedish Foundation for Strategic Research (SSF) via the project ASSEMBLE (contract number: RIT15-0012) and by the Swedish Research Council (VR) via the project NewLEADS—New Directions in Learning Dynamical Systems (contract number: 621-2016-06079). The material in this paper was partly presented at the IFAC Symposium on Systems Identification (SYSID), Stockholm, Sweden, 2018. This paper was recommended for publication in revised form by Associate Editor Brett Ninness under the direction of Editor Torsten Söderström.