Non-linear estimation is easy

Non-linear state estimation and some related topics like parametric estimation, fault diagnosis and perturbation attenuation are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with online estimations, illustrate our viewpoint.

[1]  M. Fliess,et al.  A revised look at numerical differentiation with an application to nonlinear feedback control , 2007, 2007 Mediterranean Conference on Control & Automation.

[2]  Yacine Chitour,et al.  Time-varying high-gain observers for numerical differentiation , 2002, IEEE Trans. Autom. Control..

[3]  J. Rudolph,et al.  Control of flat systems by quasi-static feedback of generalized states , 1998 .

[4]  M. Fliess,et al.  Compression différentielle de transitoires bruités , 2004 .

[5]  Salim Ibrir,et al.  Online exact differentiation and notion of asymptotic algebraic observers , 2003, IEEE Trans. Autom. Control..

[6]  Luc Jaulin,et al.  Applied Interval Analysis , 2001, Springer London.

[7]  J. Gauthier,et al.  Deterministic Observation Theory and Applications , 2001 .

[8]  Mamadou Mboup,et al.  Analyse et représentation de signaux transitoires : application à la compression, au débruitage et à la détection de ruptures , 2005 .

[9]  Emmanuel Delaleau,et al.  Filtrations in feedback synthesis: Part I – Systems and feedbacks , 1998 .

[10]  Pierre Rouchon,et al.  Generalized state variable representation for a simplified crane description , 1993 .

[11]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[12]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[13]  Leon M. Tolbert,et al.  A differential-algebraic approach to speed estimation in an induction motor , 2006, IEEE Transactions on Automatic Control.

[14]  Jean-Baptiste Pomet.  On dynamic feedback linearization of four-dimensional affine control systems with two inputs , 1997, ESAIM: Control, Optimisation and Calculus of Variations.

[15]  R. Martinez-Guerra,et al.  Diagnosis for a class of non-differentially flat and Liouvillian systems , 2007, IMA J. Math. Control. Inf..

[16]  Arie Levant,et al.  Higher-order sliding modes, differentiation and output-feedback control , 2003 .

[17]  Torkel Glad,et al.  Using Differential Algebra to Determine the Structure of Control Systems , 2000 .

[18]  M. Fliess,et al.  Une interprétation algébrique de la transformation de laplace et des matrices de transfert , 1994 .

[19]  Jessy W. Grizzle,et al.  Interpolation and numerical differentiation for observer design , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[20]  Michel Fliess,et al.  Analyse non standard du bruit , 2006, ArXiv.

[21]  H. T. H. PIAGGIO,et al.  The Operational Calculus , 1943, Nature.

[22]  C. Moog,et al.  Nonlinear Control Systems: An Algebraic Setting , 1999 .

[23]  Salim Ibrir,et al.  Linear time-derivative trackers , 2004, Autom..

[24]  S. Diop,et al.  A numerical procedure for filtering and efficient high-order signal differentiation , 2004 .

[25]  吉田 耕作.  Operational calculus: A theory of hyperfunctions , 1984 .

[26]  Emmanuel Delaleau,et al.  Filtrations in feedback synthesis: Part II – Input-output decoupling and disturbance decoupling , 1998 .

[27]  Maria Pia Saccomani,et al.  Parameter identifiability of nonlinear systems: the role of initial conditions , 2003, Autom..

[28]  M. Fliess,et al.  Correcteurs proportionnels-intégraux généralisés , 2002 .

[29]  R. Murray,et al.  Differential Flatness and Absolute Equivalence of Nonlinear Control Systems , 1998 .

[30]  M. Fliess,et al.  Corps de Hardy et observateurs asymptotiques locaux pour systèmes différentiellement plats , 1997 .

[31]  Romeo Ortega,et al.  Global regulation of flexible joint robots using approximate differentiation , 1994 .

[32]  M. Fliess,et al.  Variations sur la notion de contrôlabilité , 2000 .

[33]  M. Mariton,et al.  Control of complex systems , 1991 .

[34]  H. Khalil,et al.  Discrete-time implementation of high-gain observers for numerical differentiation , 1999 .

[35]  Philippe Martin,et al.  Feedback linearization and driftless systems , 1994, Math. Control. Signals Syst..

[36]  Cédric Join,et al.  Flatness-based fault tolerant control of a nonlinear MIMO system using algebraic derivative estimation , 2007 .

[37]  Hebertt Sira-Ramírez,et al.  Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques , 2007 .

[38]  Eric Busvelle,et al.  On determining unknown functions in differential systems, with an application to biological reactors. , 2003 .

[39]  Frank L. Lewis,et al.  Intelligent Fault Diagnosis and Prognosis for Engineering Systems , 2006 .

[40]  S. Diop,et al.  FROM THE GEOMETRY TO THE ALGEBRA OF NONLINEAR OBSERVABILITY , 2002 .

[41]  Martin Hasler,et al.  Questioning the Classic State-Space Description Via Circuit Examples , 1990 .

[42]  M. Fliess,et al.  Reconstructeurs d'état , 2004 .

[43]  Fliess Michel,et al.  Control via state estimations of some nonlinear systems , 2004 .

[44]  Avrie Levent,et al.  Robust exact differentiation via sliding mode technique , 1998, Autom..

[45]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[46]  J. McConnell,et al.  Noncommutative Noetherian Rings , 2001 .

[47]  V. Chetverikov,et al.  A Nonlinear Spencer Complex for the Group of Invertible Differential Operators and Its Applications , 2004 .

[48]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[49]  Murat Arcak,et al.  Observer design for systems with multivariable monotone nonlinearities , 2003, Syst. Control. Lett..

[50]  Sunil K. Agrawal,et al.  Differentially Flat Systems , 2004 .

[51]  Joseph Johnson,et al.  Kahler Differentials and Differential Algebra , 1969 .

[52]  Luca Zaccarian,et al.  Current Trends in Nonlinear Systems and Control , 2009 .

[53]  M. Fliess,et al.  An algebraic framework for linear identification , 2003 .

[54]  J. Rudolph,et al.  Some Examples and Remarks on Quasi-Static Feedback of Generalized States , 1998, Autom..

[55]  J. Grizzle,et al.  On numerical differentiation algorithms for nonlinear estimation , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[56]  Sette Diop,et al.  Elimination in control theory , 1991, Math. Control. Signals Syst..

[57]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[58]  Peter C. Müller,et al.  A simple improved velocity estimation for low-speed regions based on position measurements only , 2006, IEEE Transactions on Control Systems Technology.

[59]  Nahum Shimkin,et al.  Nonlinear Control Systems , 2008 .

[60]  Philippe Martin,et al.  Any (controllable) driftless system with 3 inputs and 5 states is flat , 1995 .

[61]  M. Fliess,et al.  Nonlinear observability, identifiability, and persistent trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[62]  Witold Respondek,et al.  Lowering the Orders of Derivatives of Controls in Generalized State Space Systems , 1995 .

[63]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[64]  J. Harnad.  The trouble with Physics: The rise of string theory, the fall of a Science, and what comes next , 2008 .

[65]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

[66]  François Ollivier,et al.  Algorithmes efficaces pour tester l'identifiabilité locale , 2002 .

[67]  Jie Chen,et al.  Robust Model-Based Fault Diagnosis for Dynamic Systems , 1998, The International Series on Asian Studies in Computer and Information Science.

[68]  Lennart Ljung,et al.  On global identifiability for arbitrary model parametrizations , 1994, Autom..

[69]  Philippe Martin,et al.  A Lie-Backlund approach to equivalence and flatness of nonlinear systems , 1999, IEEE Trans. Autom. Control..

[70]  Cédric Join,et al.  Robust residual generation for linear fault diagnosis: an algebraic setting with examples , 2004 .

[71]  Jean-Pierre Barbot,et al.  An algebraic framework for the design of nonlinear observers with unknown inputs , 2007, 2007 46th IEEE Conference on Decision and Control.

[72]  Michèle Basseville,et al.  Fault Detection and Isolation in Nonlinear Dynamic Systems: A Combined Input-Output and Local Approach , 1998, Autom..

[73]  H. Nijmeijer,et al.  New directions in nonlinear observer design , 1999 .

[74]  Petr Mandl,et al.  Numerical differentiation and parameter estimation in higher-order linear stochastic systems , 1996, IEEE Trans. Autom. Control..

[75]  Vincent Fromion,et al.  A global exponential observer based on numerical differentiation , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[76]  Sette Diop,et al.  Differential-Algebraic Decision Methods and some Applications to System Theory , 1992, Theor. Comput. Sci..

[77]  M. Fliess,et al.  Deux applications de la géométrie locale des diffiétés , 1997 .

[78]  Janos Gertler,et al.  Fault detection and diagnosis in engineering systems , 1998 .

[79]  M. Fliess.  Automatique et corps différentiels , 1989 .

[80]  Michel Fliess,et al.  An Algebraic State Estimation Approach for the Recovery of chaotically Encrypted Messages , 2006, Int. J. Bifurc. Chaos.

[81]  H. Sira-Ramírez,et al.  Robust algebraic state estimation of chaotic systems , 2006, 2006 IEEE International Conference on Control Applications.

[82]  Alexandre Sedoglavic.  A probabilistic algorithm to test local algebraic observability in polynomial time , 2001, ISSAC '01.

[83]  Alan S. I. Zinober,et al.  Nonlinear and Adaptive Control , 2003 .

[84]  M. V. Iordache,et al.  Diagnosis and Fault-Tolerant Control , 2007, IEEE Transactions on Automatic Control.

[85]  S. Diop,et al.  Diagnosis of nonlinear systems using an unknown-input observer: an algebraic and differential approach , 2004 .

[86]  Warren Bower.  New directions , 1937 .

[87]  Cédric Join,et al.  Estimation des dérivées d'un signal multidimensionnel avec applications aux images et aux vidéos , 2005 .

[88]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .