The present chapter1 is devoted to the flatness-based control design for one of the major control applications of the 20th century, namely ight control.
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Keywords
- Model Predictive Control
- Reference Trajectory
- Nonlinear Control System
- Control Letter
- Nonlinear Observer
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References
R. Abraham and J. Marsden. Foundations of Mechanics. Benjamin/Cummings Pub. Co., Reading Mass., 2nd edition, 1978.
R. L. Anderson and N. H. Ibragimov. Lie-Bäcklund Transformations in Applications. SIAM, Philadelphia, 1979.
B. D’ Andréa-Novel and J. Lévine. Modelling and nonlinear control of an overhead crane. In M.A. Kashoek, J.H. van Schuppen, and A.C.M. Rand, editors, Robust Control of Linear and Nonlinear Systems, MTNS’89, volume II, pages 523–529. Birkhäuser, Boston, 1990.
D.V. Anosov and V.I. Arnold. Dynamical Systems, I., volume 1 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, New York, 1980.
A.C. Antoulas. On canonical forms for linear constant systems. Int. J. Control, 33(1):95–122, 1981.
E. Aranda-Bricaire, C.H. Moog, and J.-B. Pomet. A linear algebraic framework for dynamic feedback linearization. IEEE Trans. Automat. Contr., 40(1):127–132, 1995.
V.I. Arnold. Équations Différentielles Ordinaires. MIR, Moscou, 1974.
V.I. Arnold. Chapitres Supplémentaires de la Théorie des Équations Différentielles Ordinaires. MIR, Moscou, 1980.
D. Avanessoff and J.-B. Pomet. Flatness and Monge parameterization of two-input systems, control-affne with 4 states or general with 3 states. ESAIM. Contrôle, Optimisation et Calcul des Variations, 13(2):237–264, 2007.
L. Bers. On Hilbert’s 22nd problem. In Browder F., editor, Mathematical Developments Arising From Hilbert Problems, Proceedings of Symposia in Pure Mathematics, pages 559–609. American Mathematical Society, Providence, Rhode Island, 1976.
D. Bestle and M. Zeitz. Canonical form design for nonlinear observers with linearizable error dynamics. Int. J. of Control, 23:419–431, 1981.
L. Bitauld, M. Fliess, and J. Lévine. A flatness based control synthesis of linear systems and application to windshield wipers. In Proc. ECC’97, Brussels, July 1997.
W. Boothby. An introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, New York, 1975.
P. Brunovský. A classication of linear controllable systems. Kybernetica, 6:176–178, 1970.
J. Carr. Application of Center Manifold Theory. Springer, 1981.
É. Cartan. Sur l’équivalence absolue de certains systèmes d’équations différentielles et sur certaines familles de courbes. Bull. Soc. Math. France, 42:12–48, 1914. reédité in Oeuvres Complètes, part II, vol 2, pp. 1133–1168, CNRS, Paris, 1984.
H. Cartan. Calcul Différentiel. Hermann, Paris, 1967.
B. Charlet, J. Lévine, and R. Marino. On dynamic feedback linearization. Systems & Control Letters, 13:143–151, 1989.
B. Charlet, J. Lévine, and R. Marino. Sufficient conditions for dynamic state feedback linearization. SIAM J. Control Optimization, 29(1):38–57, 1991.
S.S. Chern, W.H. Chen, and K.S. Lam. Lectures on Differential Geometry, volume 1 of Series on University Mathematics. World Scientic, 2000.
V.N. Chetverikov. New flatness conditions for control systems. In Proceedings of NOLCOS’01, St. Petersburg, pages 168–173, 2001.
C. Chevalley. Theory of Lie Groups. Princeton University Press, 1946.
Y.K. Chin, A. Kade, J. Kowalik, and D. Graham. Electronic windshield wiper system i: modelling and validation, ii: control and sensitivity study. Int. J. of Vehicle design, 12(2):175–196, 1991.
Y. Choquet-Bruhat. Géométrie différentielle et systèmes extérieurs. Dunod, Paris, 1968.
P.M. Cohn. Free Rings and Their Relations. Academic Press, London, 1985.
O. Dahl. Path-constrained robot control with limited torques–experimental evaluation. IEEE Trans. on Robotics and Automation, 10(5):658–669, 1994.
O. Dahl and L. Nielsen. Torque-limited path following by on-line trajectory time scaling. IEEE Trans. on Robotics and Automation, 6(5):554–561, 1990.
J. De Dona, F. Suryawan, M. Seron, and J. Lévine. A flatness-based iterative method for reference trajectory generation in constrained NMPC. In D. Raimondo L. Magni and F. Allgöwer, editors, Proc. of the International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control, NMPC’08, Pavia, Italy, 2008.
E. Delaleau and V. Hagenmeyer. Commande prédictive non linéaire fondée sur la platitude différentielle. In D. Dumur, editor, La commande prédictive : Avancées et perspectives. Hermès, 2006.
E. Delaleau and J. Rudolph. Control of flat systems by quasi-static feedback of generalized states. Int. J. Control, 71(5):745–765, 1998.
M. Demazure. Bifurcations and Catastrophes, Geometry of Solutions to Nonlinear Problems. Springer, Universitext, 2000.
Th. Devos. Étude et comparaison de plusieurs lois de commande de grues. PhD thesis, Université Paris-Sud, 2009.
Th. Devos and J. Lévine. A flatness-based nonlinear predictive approach for crane control. In M. Alamir and F. Allgöwer, editors, Proc. of the IFAC Conf. on Nonlinear Model Predictive Control for Fast Systems, NMPC’06, Grenoble, France, 2006.
J. Dieudonné. Fondements de l’Analyse Moderne. Gauthier-Villars, Paris, 1960.
B. Etkin. Dynamics of flight-Stability and control. John Wiley, 2nd edition, 1982.
N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equations, 31:53–98, 1979.
A.F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht, Boston, London, 1988.
R. Findeisen and F. Allgöwer. An introduction to nonlinear model predictive control. In Proc. 21st Benelux Meeting on Systems and Control, Veldhoven, The Netherlands, 2002.
M. Fliess. Some basic structural properties of generalized linear systems. Systems & Control Letters, 15:391–396, 1990.
M. Fliess. A remark on Willems’ trajectory characterization of linear controllability. Systems & Control Letters, 19:43–45, 1992.
M. Fliess and R. Marquez. Continuous-time linear predictive control and flatness : a module-theoretic setting with examples. International Journal of Control, 73:606–623, 2000.
M. Fliess, J. Lévine, and P. Rouchon. A simplied approach of crane control via a generalized state-space model. In Proc. 30th IEEE Control Decision Conf., Brighton, pages 736–741, 1991.
M. Fliess, J. Lévine, Ph. Martin, and P. Rouchon. Sur les systèmes non linéaires différentiellement plats. C.R. Acad. Sci. Paris, I–315:619–624, 1992a.
M. Fliess, J. Lévine, Ph. Martin, and P. Rouchon. On differentially flat nonlinear systems. In Proc. IFAC-Symposium NOLCOS’92, Bordeaux, pages 408–412, 1992b.
M. Fliess, J. Lévine, and P. Rouchon. A generalized state variable representation for a simplied crane description. Int. J. Control, 58:277–283, 1993.
M. Fliess, J. Lévine, Ph. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control, 61 (6):1327–1361, 1995.
M. Fliess, J. Lévine, Ph. Martin, and P. Rouchon. A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44(5):922–937, 1999.
E. Fossas and J. Franch. Linearization by prolongations: a new bound for three input systems. In Proceedings of the 14th IFAC World Congress, Beijing, 1999.
J. Franch. Flatness, Tangent Systems and Flat Outputs. PhD thesis, Universitat Politècnica de Catalunya Jordi Girona, 1999.
F.R. Gantmacher. Théorie des Matrices, t. I, II. Dunod, Paris, 1966.
R.B. Gardner and W.F. Shadwick. The GS algorithm for exact linearization to Brunovsky normal form. IEEE. Trans. Automat. Contr., 37:224–230, 1992.
C. Godbillon. Géométrie Différentielle et Mécanique Analytique. Hermann, 1969.
J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.
V. Hagenmeyer and E. Delaleau. Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example. International Journal of Control, 81(10):1645–1663, 2008.
D. Hilbert. Über den Begriff der Klasse von Differentialgleichungen. Math. Ann., 73:95–108, 1912. Also in Gesammelte Abhandlungen, Vol. 3, pages 81–93, Chelsea, New York, 1965.
D. Hilbert. Mathematische probleme. Archiv für Mathematik und Physik, 1: 44–63 and 213–237, 1901. Also in Gesammelte Abhandlungen, Vol.3, pages 290–329, Chelsea, New York, 1965.
M.W. Hirsch and S. Smale. Differential Equations, Dynamical Systems and Linear Algebra. Acamedic Press, New-York, 1974.
M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds. Lecture Notes in Mathematics. Springer, 1977.
L.R. Hunt, R. Su, and G. Meyer. Design for multi-input nonlinear systems. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory, pages 268–298. Birkhäuser, Boston, 1983a.
L.R. Hunt, R. Su, and G. Meyer. Global transformations of nonlinear systems. IEEE Trans. Automat. Control, 28:24–31, 1983b.
N. H. Ibragimov. Transformation Groups Applied to Mathematical Physics. Reidel, Boston, 1985.
A. Isidori. Nonlinear Control Systems. Springer, New York, 3rd edition, 1995.
B. Jakubczyk. Invariants of dynamic feedback and free systems. In Proc. ECC’93, Groningen, pages 1510–1513, 1993.
B. Jakubczyk and W. Respondek. On linearization of control systems. Bull. Acad. Pol. Sci. Ser. Sci. Math., 28(9–10):517–522, 1980.
T. Kailath. Linear Systems. Prentice Hall, Englewood Cliffs, NJ, 1980.
H. K. Khalil. Nonlinear Systems. Prentice Hall, Englewood Cliffs, NJ, 1996.
B. Kiss. Planication de trajectoires et commande d’une classe de systèmes mécaniques plats et liouvilliens. PhD thesis, École des Mines de Paris, 2001.
B. Kiss, J. Lévine, and Ph. Müllhaupt. Modelling, flatness and simulation of a class of cranes. Periodica Polytechnica, 43(3):215–225, 1999.
B. Kiss, J. Lévine, and Ph. Müllhaupt. A simple output feedback PD controller for nonlinear cranes. In 39th IEEE Conference on Decision and Control, Sydney, Australia, 2000a.
B. Kiss, J. Lévine, and Ph. Müllhaupt. Modelling and motion planning for a class of weight handling equipment. J. Systems Science, 26(4):79–92, 2000b.
B. Kiss, J. Lévine, and Ph. Müllhaupt. Control of a reduced size model of US Navy crane using only motor position sensors. In A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, editors, Nonlinear Control in the Year 2000, volume 2, pages 1–12. Springer, 2000c.
S. Kobayashi and K. Nomizu. Foundations of Differential Geometry, volume I. John Wiley & Sons, 1996.
P.V. Kokotović, H.K. Khalil, and J. O’Reilly. Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London, 1986.
M.V. Kondratieva, A.V. Mikhalev, and E.V. Pankratiev. On Jacobi’s bound for systems of differential polynomials. Algebra, Moscow Univ. Press, Moscow, pages 79–85, 1982. in russian.
A.I. Kostrikin and I.R. Shafarevich. Algebra, I., volume 11 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, New York, 1980.
I. S. Krasil’shchik, V. V. Lychagin, and A. M. Vinogradov. Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach, New York, 1986.
A.J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Lett., 3:47–52, 1983.
A.J. Krener and W. Respondek. Nonlinear observers with linear error dynamics. SIAM J. Control Optimiz., 23:197–216, 1985.
M. Krstić, I. Kanellakopoulos, and P. Kokotović. Nonlinear and Adaptive Control Design. Wiley, New York, 1995.
J.P. LaSalle and S. Lefschetz. Stability by Lyapunov’s Direct Method With Application. Academic Press, New York, 1961.
J. Lévine. Are there new industrial perspectives in the control of mechanical systems? In P.M. Frank, editor, Advances in Control, Highlights of ECC’99, pages 197–226. Springer, London, 1999.
J. Lévine. Static and dynamic state feedback linearization. In A. Fossard and D. Normand-Cyrot, editors, Nonlinear Systems, volume 3, pages 93–126. Chapman & Hall, 1997.
J. Lévine. On necessary and suffcient conditions for differential flatness.http://www.arxiv.org, arXiv:math.OC/0605405, 2006.
J. Lévine. On the synchronization of a pair of independent windshield wipers. IEEE Trans. Control Systems Technology, 12(5):787–795, 2004.
J. Lévine. On necessary and suffcient conditions for differential flatness. In Proc. of IFAC NOLCOS 2004 Conference, Stuttgart, 2004.
J. Lévine and D.V. Nguyen. Flat output characterization for linear systems using polynomial matrices. Systems & Control Letters, 48:69–75, 2003.
J. Lévine and P. Rouchon. Quality control of binary distillation columns based on nonlinear aggregated models. Automatica, 27(3):463–480, 1991.
J. Lévine and P. Rouchon. An invariant manifold approach for robust control design and applications. In U. Helmke, R. Mennicken, and J. Saurer, editors, Systems and Networks: Mathematical Theory and Applications, Proc. MTNS-93, Regensburg, Germany, volume 2, pages 309–312. Akademie Verlag, Berlin, 1994.
J. Lévine, J. Lottin, and J. C. Ponsart. A nonlinear approach to the control of magnetic bearings. IEEE Trans. Control Systems Technology, 4(5):524–544, 1996.
J. Lévine, P. Rouchon, G. Yuan, C. Grebogi, B.R. Hunt, E. Kostelich, E. Ott, and J. Yorke. On the control of US navy cranes. In Proc. ECC’97, Brussels, paper N. 717, 1997.
A. Liapounoff. Problème Général de la Stabilité du Mouvement. Annales de la Faculté des Sciences de Toulouse, deuxième série. Edouard Privat, 1907. reprint J. Gabay, Paris, 1988.
R. Marino. On the largest feedback linearizable subsystem. Systems & Control Letters, 6:345–351, 1986.
R. Marino and P. Tomei. Nonlinear Control Design. Prentice Hall, Englewood Cliffs, NJ, 1995.
Ph. Martin. Aircraft control using flatness. In Proc. CESA Multiconference, Lille, pages 194–199, 1996.
Ph. Martin. Endogenous feedbacks and equivalence. In U. Helmke, R. Mennicken, and J. Saurer, editors, Systems and Networks: Mathematical Theory and Applications, Proc. MTNS-93, Regensburg, Germany, volume 2, pages 343–346. Akademie Verlag, Berlin, 1994.
Ph. Martin. Contribution à l’étude des systèmes diffèrentiellement plats. PhD thesis, école des Mines de Paris, 1992.
Ph. Martin and P. Rouchon. Systems without drift and flatness. In Proc. MTNS 93, Regensburg, Germany, August 1993.
Ph. Martin, R.M. Murray, and P. Rouchon. Flat systems. In G. Bastin and M. Gevers, editors, Plenary Lectures and Minicourses, Proc. ECC 97, Brussels, pages 211–264, 1997.
D. McLean. Automatic Flight Control Systems. Prentice Hall, 1990.
D.T. McRuer, I.L. Ashkenas, and D.C. Graham. Aicraft Dynamics and Automatic Control. Princeton University Press, 1973.
J. Milnor. Topology from the Differential Viewpoint. University Press of Virginia, Charlottesville, 1978.
M. Morari and J.H. Lee. Model predictive control: past, present and future. Computers & Chemical Engineering, 23:667–682, 1999.
P. MÜllhaupt. Introduction à l’Analyse et à la Commande des Systèmes Non Linéaires. Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, 2009.
P. MÜllhaupt, B. Srinivasan, J. Lévine, and D. Bonvin. A toy more difficult to control than the real thing. In Proc. ECC’97, Brussels, July 1997.
P. MÜllhaupt, B. Srinivasan, J. Lévine, and D. Bonvin. Cascade control of the toycopter. In Proc. ECC’99, Karslruhe, 1999.
P. MÜllhaupt, B. Srinivasan, J. Lévine, and D. Bonvin. Control of the toycopter using a flat approximation. IEEE Trans. Control Systems Technology, 16(5):882–896, 2008.
Ju.I. Neĭmark and N.A. Fufaev. Dynamics of Nonholonomic Systems. American Mathematical Society, Providence, Rhode Island, 1972.
H. Nijmeijer and A.J. van der Schaft. Nonlinear Dynamical Control Systems. Springer, New York, 1990.
F. Ollivier. Standard bases of differential ideals. In Proceedings of AAECC8, volume 508 of Lecture Notes In Computer Science, pages 304–321. Springer, 1990.
F. Ollivier and S. Brahim. La borne de Jacobi pour une diffété dénie par un système quasi régulier (Jacobi’s bound for a diffiety dened by a quasiregular system). Comptes rendus Mathematique, 345(3):139–144, 2007.
P.J. Olver. Equivalence, Invariants, and Symmetry. Cambridge University Press, 1995.
P. S. Pereira da Silva. Flatness of nonlinear control systems : a Cartan-Kähler approach. In Proc. Mathematical Theory of Networks and Systems (MTNS’2000), Perpignan, pages 1–10, 2000.
P.S. Pereira da Silva and C. Corrêa Filho. Relative flatness and flatness of implicit systems. SIAM J. Control and Optimization, 39(6):1929–1951, 2001.
N. Petit, Y. Creff, L. Lemaire, and P. Rouchon. Minimum time constrained control of acid strength on a sulfuric acid alkylation unit. Chemical Engineering Science, 56:2767–2774, 2001.
F. Pham. Géométrie et Calcul Différentiel sur les Variétés. InterEditions, Paris, 1992.
H. Poincaré. Sur l’uniformisation des fonctions analytiques. Acta Mathematica, 31:1–63, 1907. Also in Œuvres de Henri Poincaré, t. 4, pages 70–139, Gauthier-Villars, Paris, 1950.
J.W. Polderman and J.C. Willems. Introduction to Mathematical System Theory: a Behavioral Approach. Springer-Verlag, Berlin, 1997.
J.-B. Pomet. A differential geometric setting for dynamic equivalence and dynamic linearization. In B. Jakubczyk, W. Respondek, and T. Rzežzuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, pages 319–339. Banach Center Publications, Warsaw, 1993.
J.-B. Pomet. On dynamic feedback linearization of four-dimensional affine control systems with two inputs. ESAIM-COCV, 1997. http://www.emath.fr/Maths/Cocv/Articles/articleEng.html.
J.B. Pomet, C. Moog, and E. Aranda. A non-exact Brunovsky form and dynamic feedback linearization. In Proc. 31st. IEEE Conf. Decision Cont., pages 2012–2017, 1992.
J.F. Pommaret. Partial Differential Control Theory. Kluwer academic publishers, 2001.
J.F. Pommaret and A. Quadrat. Localization and parametrization of linear multidimensional control systems. Systems & Control Letters, 37:247–269, 1999.
L. Pontriaguine. Équations différentielles Ordinaires. MIR, Moscou, 1975.
M. Rathinam and R.M. Murray. Conguration flatness of Lagrangian systems underactuated by one control. SIAM J. Control and Optimization, 36(1): 164–179, 1998.
J. F. Ritt. Jacobi’s problem on the order of a system of differential equations. Annals of Mathematics, 36:303–312, 1935.
H.H. Rosenbrock. Multivariable and State-Space Theory. Wiley, New York, 1970.
P. Rouchon. Necessary condition and genericity of dynamic feedback linearization. J. Math. Systems Estim. Control, 4(2):257–260, 1994.
J. Rudolph. Flatness based control of distributed parameter systems. Shaker Verlag, Aachen, 2003.
J. Rudolph, J. Winkler, and F. Woittenek. Flatness based control of distributed parameter systems: examples and computer exercises from various technological domains. Shaker Verlag, Aachen, 2003.
D. Ruelle. Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, 1989.
S. Sastry. Nonlinear Systems, Analysis, Stability and Control. Springer, New York, 1999.
J.A. Saunders and F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems. Springer, 1987.
K. Schlacher and M. Schöoberl. Construction of at outputs by reduction and elimination. In Proc. 7th IFAC Symposium on Nonlinear Control Systems, Pretoria, South Africa, pages 666–671, August 2007.
W.F. Shadwick. Absolute equivalence and dynamic feedback linearization. Systems Control Letters, 15:35–39, 1990.
H. Sira-Ramirez and S.K. Agrawal. Differentially at systems. Marcel Dekker, New York, 2004.
J.J.E. Slotine and W. Li. Applied Nonlinear Control. Prentice Hall, Englewood Cliffs, NJ, 1991.
W.M. Sluis. A necessary condition for dynamic feedback linearization. Systems Control Letters, 21:277–283, 1993.
W.M. Sluis and D.M. Tilbury. A bound on the number of integrators needed to linearize a control system. Systems & Control Letters, 29(1):43–50, 1996.
R. Sommer. Control design for multivariable non-linear time-varying systems. Int. J. Control, 31:883–891, 1980.
E.D. Sontag. Mathematical Control Theory, Deterministic Finite Dimensional Systems. Springer, New York, 2nd edition, 1998.
H.J. Sussmann. A general theorem on local controllability. SIAM J. Control and Optimiz., 25:158–194, 1987.
A. Tannenbaum. Invariance and System Theory: Algebraic and Geometric Aspects. Springer, New York, 1980.
R Thom. Stabilité Structurelle et Morphogénèse. InterEdition, Paris, 2nd edition, 1977.
A. Tikhonov, A. Vasil’eva, and A. Sveshnikov. Differential Equations. Springer, New York, 1980.
H.L. Trentelman. On at systems, behaviors and observable image representations. Systems & Control Letters, 21:51–55, 2004.
M. van Nieuwstadt, M. Rathinam, and R.M. Murray. Differential flatness and absolute equivalence. In Proc. 33th IEEE Conf. Decision Control, Lake Buena Vista, Fl., pages 326–332, 1994.
M. van Nieuwstadt, M. Rathinam, and R.M. Murray. Differential flatness and absolute equivalence of nonlinear control systems. SIAM J. Control Optim., 36(4):1225–1239, 1998.
M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, Englewood Cliffs, NJ, 2nd edition, 1993.
J. von Löowis, J. Rudolph, J. Thiele, and F. Urban. Flatness-based trajectory tracking control of a rotating shaft. In 7th International Symposium on Magnetic Bearings, Züurich, pages 299–304, 2000.
M. Vukobratović and R. Stojić. Modern Aircraft Flight Control. Springer-Verlag, 1988.
J.C. Wanner. Dynamique du vol et pilotage des avions. Technical report, école Nationale Supérieure de l’Aéronautique et de l’Espace, Toulouse, France, 1984.
E.T. Whittaker. A Treatise on the Analytical Dynamics of Particules and Rigid Bodies. Cambridge University Press, Cambridge, 4th edition, 1937.
W.A.Wolovich. Linear Multivariable Systems, volume 11 of Series in Applied Mathematical Sciences. Springer, New York, 1974.
W.M. Wonham. Linear Multivariable Control: a Geometric Approach. Springer, 1974.
X.H. Xia and W.B. Gao. Nonlinear observer design by observer error linearization. SIAM J. Control Optimiz., 27:199–216, 1989.
V.V. Zharinov. Geometrical Aspect of Partial Differential Equations. World Scientic, Singapore, 1992.
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Lévine, J. (2009). Automatic Flight Control Systems. In: Analysis and Control of Nonlinear Systems. Mathematical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00839-9_14
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