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## Flatness and defect of nonlinear systems: Introductory theory and examples (1995)

Venue: | INTERNATIONAL JOURNAL OF CONTROL |

Citations: | 345 - 23 self |

### Citations

2865 |
Algebraic geometry
- Hartshorne
- 1977
(Show Context)
Citation Context ...ial k-isomorphism between them (a differential k-isomorphism commutes with d/dt and preserves every element of k). 4 It is a standard fact in classic commutative algebra and algebraic geometry (c.f. (=-=Hartshorne 1977-=-)) that one needs prime ideals for interpreting “concrete” equations in the language of field theory. In our differential setting, we of course need differential prime ideals (see (Kolchin 1973) and a... |

2493 |
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, volume 42 of Applied mathematical sciences
- Guckenheimer, Holmes
- 1983
(Show Context)
Citation Context ...23). Since the averaged system admits a hyperbolic asymptotically stable equilibrium, the perturbed system admits an hyperbolic asymptotically stable limit cycle around (α, p, z) = (α sp , 0, z sp ) (=-=Guckenheimer and Holmes 1983-=-, theorem 4.1.1, page 168): such control maintains (z,α) near (z sp ,α sp ). Moreover this control method is robust in the following sense: the existence and the stability of the limit cycle is not de... |

387 |
[Linear Systems ],
- Kailath
- 1980
(Show Context)
Citation Context ...nstant linear controllable system ˙ξ = Fξ + Gv. Up to a static state feedback and a linear invertible change of coordinates, this linear system may be written in Brunovsky canonical form (see, e.g., (=-=Kailath 1980-=-)), ⎧ ⎪⎨ ⎪⎩ y (ν1) 1 y (νm) m = v1 . = vm where ν1, ..., νm are the controllability indices and (y1,...,y (ν1−1) 1 sis of the vector space spanned by the components ofξ. Set Y = (y1,...,y (ν1−1) 1 ,..... |

379 |
Differential Algebra and Algebraic Groups
- Kolchin
- 1973
(Show Context)
Citation Context ...91), where trajectories play a key role. We start with a brief review of differential fields (see also (Fliess 1990a, Fliess and Glad 1993)) and we refer to the books of Ritt (Ritt 1950) and Kolchin (=-=Kolchin 1973-=-) and Seidenberg’s paper (Seidenberg 1952) for details. Basics on the customary (non-differential) field theory may be found in (Fliess 1990a, Fliess and Glad 1993) as well as in the textbook by Jacob... |

363 | Nonholonomic motion planning: steering using sinusoids,’’
- Murray, Sastry
- 1993
(Show Context)
Citation Context ...990)). A considerable amount of realistic models are indeed flat. We treat here two case-studies, namely the crane (D’Andréa-Novel and Lévine 1990, Marttinen et al. 1990) and the car with n trailers (=-=Murray and Sastry 1993-=-, Rouchon et al. 1993a). Notice that the use of a linearizing output was already known in the context of static state feedback (see (Claude 1986) and (Isidori 1989, page 156)). One major property of d... |

273 |
Paradigms and puzzles in the theory of dynamical systems,
- Willems
- 1991
(Show Context)
Citation Context ... flatness is best defined by not distinguishing between input, state, output and other variables. The equations moreover might be implicit. This standpoint, which matches well with Willems’ approach (=-=Willems 1991-=-), is here taken into account by utilizing differential 3salgebra which has already helped clarifying several questions in control theory (see, e.g., (Diop 1991, Diop 1992, Fliess 1989, Fliess 1990a, ... |

162 |
A motion planner for nonholonomic mobile robots,”
- Laumond, Jacobs, et al.
- 1994
(Show Context)
Citation Context ...on dPn ) via κn and its first n-derivatives with respect to dsn sn. Such considerations can be of some help in finding a curve avoiding collisions. More details on obstacle avoidance can be found in (=-=Laumond et al. 1993-=-) where a car without trailer is considered. The multi-steering trailer systems considered in (Bushnell et al. 1993), (Tilbury et al. 1993), (Tilbury and Chelouah 1993) are also flat: the flat output ... |

110 |
Sussmann Controllability of Nonlinear Systems
- Jurdjevic, J
- 1972
(Show Context)
Citation Context ... single-input dynamics which is linearizable by a dynamic feedback is also linearizable by a static one. This implies the existence of non-flat systems which verify the strong accessibility property (=-=Sussmann and Jurdjevic 1972-=-). We introduce a nonnegative integer, the defect, which measures the distance from flatness. These new concepts and mathematical tools are providing the common formalism and the underlying structure ... |

85 | Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories.
- Sussmann, Liu
- 1991
(Show Context)
Citation Context ...on point oscillates rapidly. Notice that some related calculations may be found in (Baillieul 1993). For the use of high-frequency control in different contexts see also (Bentsman 1987, Meerkov 1980, =-=Sussmann and Liu 1991-=-). (Acheson 1993, Stephenson 1908) 5.1 The Kapitsa pendulum z α Figure 7: The Kapitsa pendulum: the suspension point oscillates rapidly on a vertical axis. l 24 m gsThe notation are summarized on figu... |

65 | der Schaft - Nijmeijer, van - 1982 |

50 |
An algebraic approach to linear and nonlinear control
- Fliess, Glad
- 1993
(Show Context)
Citation Context ...), is here taken into account by utilizing differential 3salgebra which has already helped clarifying several questions in control theory (see, e.g., (Diop 1991, Diop 1992, Fliess 1989, Fliess 1990a, =-=Fliess and Glad 1993-=-)). Flatness might be seen as another nonlinear extension of Kalman’s controllability. Such an assertion is surprising when having in mind the vast literature on this subject (see (Isidori 1989, Nijme... |

41 |
On dynamic feedback linearization
- Charlet, Lévine, et al.
- 1989
(Show Context)
Citation Context ...e motion planning of the car with n-trailers. They are all treated within the framework of dynamic feedback linearization which, contrary to the static one, has only been investigated by few authors (=-=Charlet et al. 1989-=-, Charlet et al. 1991, Shadwick 1990). Our point of view will be probably best explained by the following calculations where all vector fields and functions are real-analytic. Consider ˙x = f (x, u) (... |

40 |
Stable Average Motions of Mechanical Systems Subject to Periodic Forcing,” Dynamics and Control of Mechanical Systems: The Falling Cat and
- Baillieul
- 1993
(Show Context)
Citation Context ...rd personal microcomputer under MATLAB. The control of the three non-flat systems is based on high frequency control and approximations by averaged and flat systems (for other approaches, see, e.g., (=-=Baillieul 1993-=-, Bentsman 1987, Meerkov 1980)). We exploit here an idea due to the Russian physicist Kapitsa (Bogaevski and Povzner 1991, Landau and Lifshitz 1982) for stabilizing these three systems in the neighbor... |

40 |
Controllability of a multibody mobile robot
- Laumond
- 1993
(Show Context)
Citation Context ...ngines are powerful with respect to the masses of the trolley and the load. The motion planning of the car with n-trailer is perhaps the most popular example of path planning of nonholonomic systems (=-=Laumond 1991-=-, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, Tilbury et al. 1993, Martin and Rouchon 1993, Rouchon et al. 1993b). It is a flat system where the linearizing output is ... |

39 |
An introduction to motion planning under multirate digital control:’
- Monaco, Normand-Cyrot
- 1992
(Show Context)
Citation Context ...e masses of the trolley and the load. The motion planning of the car with n-trailer is perhaps the most popular example of path planning of nonholonomic systems (Laumond 1991, Murray and Sastry 1993, =-=Monaco and Normand-Cyrot 1992-=-, Rouchon et al. 1993a, Tilbury et al. 1993, Martin and Rouchon 1993, Rouchon et al. 1993b). It is a flat system where the linearizing output is the middle of the axle of the last trailer. Once the li... |

38 |
Generalized controller canonical forms for linear and nonlinear dynamics.
- Fliess
- 1990
(Show Context)
Citation Context ... (Willems 1991), is here taken into account by utilizing differential 3salgebra which has already helped clarifying several questions in control theory (see, e.g., (Diop 1991, Diop 1992, Fliess 1989, =-=Fliess 1990-=-a, Fliess and Glad 1993)). Flatness might be seen as another nonlinear extension of Kalman’s controllability. Such an assertion is surprising when having in mind the vast literature on this subject (s... |

34 |
Elimination in control theory
- Diop
- 1991
(Show Context)
Citation Context ...matches well with Willems’ approach (Willems 1991), is here taken into account by utilizing differential 3salgebra which has already helped clarifying several questions in control theory (see, e.g., (=-=Diop 1991-=-, Diop 1992, Fliess 1989, Fliess 1990a, Fliess and Glad 1993)). Flatness might be seen as another nonlinear extension of Kalman’s controllability. Such an assertion is surprising when having in mind t... |

34 |
Principle of vibrational control: Theory and applications,”
- Meerkov
- 1980
(Show Context)
Citation Context ... MATLAB. The control of the three non-flat systems is based on high frequency control and approximations by averaged and flat systems (for other approaches, see, e.g., (Baillieul 1993, Bentsman 1987, =-=Meerkov 1980-=-)). We exploit here an idea due to the Russian physicist Kapitsa (Bogaevski and Povzner 1991, Landau and Lifshitz 1982) for stabilizing these three systems in the neighborhood of quite arbitrary posit... |

34 | Necessary condition and genericity of dynamic feedback linearization,”
- Rouchon
- 1994
(Show Context)
Citation Context ...geometric properties. This explains why flat systems are so often encountered in spite of the non-genericity of dynamic feedback linearizability in some customary mathematical topologies (Tchoń 1994, =-=Rouchon 1994-=-). We hope to have convinced the reader that flatness and defect bring a new theoretical and practical insight in control. We briefly list some important open problems: • Ritt’s work (Ritt 1950) shows... |

30 | Some basic structural properties of generalized linear systems - Fliess - 1990 |

28 | A differential geometric setting for dynamic equivalence and dynamic linearization - Pomet - 1995 |

27 |
Sufficient conditions for dynamic state feedback linearization
- Charlet, Lévine, et al.
- 1991
(Show Context)
Citation Context ...the car with n-trailers. They are all treated within the framework of dynamic feedback linearization which, contrary to the static one, has only been investigated by few authors (Charlet et al. 1989, =-=Charlet et al. 1991-=-, Shadwick 1990). Our point of view will be probably best explained by the following calculations where all vector fields and functions are real-analytic. Consider ˙x = f (x, u) (x∈R n , u ∈ R m ), (1... |

27 |
Automatique et corps différentiels
- Fliess
- 1989
(Show Context)
Citation Context ...ems’ approach (Willems 1991), is here taken into account by utilizing differential 3salgebra which has already helped clarifying several questions in control theory (see, e.g., (Diop 1991, Diop 1992, =-=Fliess 1989-=-, Fliess 1990a, Fliess and Glad 1993)). Flatness might be seen as another nonlinear extension of Kalman’s controllability. Such an assertion is surprising when having in mind the vast literature on th... |

26 | Diflerential Equations. - Tikhonov, Vasil’eva, et al. - 1985 |

24 | Rank invariants of nonlinear systems, - Benedetto, Moog - 1989 |

23 | motion planning and trailer systems - Flatness - 1993 |

22 |
A pendulum theorem,
- Acheson
- 1993
(Show Context)
Citation Context ...idea is closely related to a curiosity of classical mechanics that a double inverted pendulum (Stephenson 1908), and even the N linked pendulums which are inverted and balanced on top of one another (=-=Acheson 1993-=-), 4scan be stabilized in the same way. Closed-loop stabilization around reference averaged trajectories becomes straightforward by utilizing the endogenous feedback equivalence to linear controllable... |

22 | Dynamic Feedback Linearization of Nonholonomic wheeled mobile robot”, Robotics and Automation - d’Andrea-Novel, Bastin, et al. - 1992 |

22 |
Absolute equivalence and dynamic feedback linearization,
- Shadwick
- 1990
(Show Context)
Citation Context ...rs. They are all treated within the framework of dynamic feedback linearization which, contrary to the static one, has only been investigated by few authors (Charlet et al. 1989, Charlet et al. 1991, =-=Shadwick 1990-=-). Our point of view will be probably best explained by the following calculations where all vector fields and functions are real-analytic. Consider ˙x = f (x, u) (x∈R n , u ∈ R m ), (1) ∂ f where f (... |

18 | A multi-steering trailer system: conversion into chained form using dynamic feedback”,
- Tilbury, Sordalen, et al.
- 1993
(Show Context)
Citation Context ...ning of the car with n-trailer is perhaps the most popular example of path planning of nonholonomic systems (Laumond 1991, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, =-=Tilbury et al. 1993-=-, Martin and Rouchon 1993, Rouchon et al. 1993b). It is a flat system where the linearizing output is the middle of the axle of the last trailer. Once the linearizing output is determined, the path pl... |

17 | Linearized control systems and application to smooth stabilization,”
- Coron
- 1994
(Show Context)
Citation Context ...ontag 1988) showed that strong accessibility implies the existence of controls such that the linearized system around a trajectory passing through a point a of the state-space is controllable. Coron (=-=Coron 1994-=-) and Sontag (Sontag 1992) demonstrated that, for any a, those controls are generic. The above considerations with those of section 2.3 and 2.4 lead in our context to the following definition of contr... |

15 |
Some basic theorems in differential algebra (characteristic p arbitrary
- Seidenberg
- 1952
(Show Context)
Citation Context ...e. We start with a brief review of differential fields (see also (Fliess 1990a, Fliess and Glad 1993)) and we refer to the books of Ritt (Ritt 1950) and Kolchin (Kolchin 1973) and Seidenberg’s paper (=-=Seidenberg 1952-=-) for details. Basics on the customary (non-differential) field theory may be found in (Fliess 1990a, Fliess and Glad 1993) as well as in the textbook by Jacobson (Jacobson 1985) and Winter (Winter 19... |

13 |
al.,”A Simplified Approach of Crane Control via a Generalized State-Space Model”,
- Fliess, et
- 1991
(Show Context)
Citation Context ...tion of flatness, which is of geometric nature. The last two sections are devoted respectively to the flat and non-flat examples. First drafts of various parts of this article have been presented in (=-=Fliess et al. 1991-=-, Fliess et al. 1992b, Fliess et al. 1992a, Fliess et al. 1993b, Fliess et al. 1993c). 2 The algebraic framework We consider variables related by algebraic differential equations. This viewpoint, whic... |

13 | Remarks on equivalence and linearization of nonlinear systems, - Jakubczyk - 1992 |

12 |
On differential systems with quadratic impulses and their applications to Lagrangian mechanics
- Bressan, Rampazzo
- 1993
(Show Context)
Citation Context ...s test of the high-frequency control for the inverted pendulum. 27s5.2 The variable-length pendulum gravity O Figure 9: pendulum with variable-length. Let us consider the variable-length pendulum of (=-=Bressan and Rampazzo 1993-=-). The notations are summarized on figure 9. We assume as in (Bressan and Rampazzo 1993) that the velocity ˙u = v is the control. The equations of motion are: ⎧ ⎨ ⎩ u q ˙q = p ˙p = −cos u + qv 2 ˙u = ... |

11 |
l’intégration de certains systèmes indéterminés d’équations différentielles
- Cartan, Sur
- 1915
(Show Context)
Citation Context ...ut integrating any differential equation, in terms of the flat output and a finite number of its derivatives. This general idea can be traced back to works by D. Hilbert (Hilbert 1912) and E. Cartan (=-=Cartan 1915-=-) on under-determined systems of differential equations, where the number of equations is strictly less than the number of unknowns. Let us emphasize on the fact that this property may be extremely us... |

11 |
The structure of fields
- Winter
- 1974
(Show Context)
Citation Context ...ems. 2.4 Differentials and tangent linear systems Differential calculus, which plays such a role in analysis and in differential geometry, admits a nice analogue in commutative algebra (Kolchin 1973, =-=Winter 1974-=-), which has been extended to differential algebra by Johnson (Johnson 1969). To a finitely generated differential extension L/K , associate a mapping dL/K : L → �L/K , called (Kähler) differential 7 ... |

10 |
Algebraic methods in nonlinear perturbation theory
- Bogaevski, Povzner
- 1987
(Show Context)
Citation Context ...control and approximations by averaged and flat systems (for other approaches, see, e.g., (Baillieul 1993, Bentsman 1987, Meerkov 1980)). We exploit here an idea due to the Russian physicist Kapitsa (=-=Bogaevski and Povzner 1991-=-, Landau and Lifshitz 1982) for stabilizing these three systems in the neighborhood of quite arbitrary positions and trajectories, and in particular positions which are not equilibrium points. This id... |

10 | On differentially flat nonlinear systems - Martin, Rouchon - 1992 |

10 | Flatness and motion planning: the car with n-trailers. - Martin - 1993 |

9 |
A remark on Willems’ trajectory characterization of linear controllability
- Fliess
- 1992
(Show Context)
Citation Context ... ,...,ym,...,y (νm−1) m ,...,ym,...,y (νm−1) m (2) ) is another ba); 1 See (Li and Feng 1987) for a definition of this concept via the structure algorithm. See (Di Benedetto et al. 1989, Delaleau and =-=Fliess 1992-=-) for a connection with the differential algebraic approach. 2sthus Y = Tξ where T is an invertible (n + q) × (n + q) matrix. Otherwise stated, Y = T�(x, z). The invertibility of � yields � � x = � z ... |

9 | Generalized State Variable Representation for a Simplified Crane Description,” - Fliess, Levine, et al. - 1993 |

9 |
Finite dimensional open-loop control generators for nonlinear systems
- Sontag
- 1988
(Show Context)
Citation Context ...lability Sussmann and Jurdjevic (Sussmann and Jurdjevic 1972) have introduced in the differential geometric setting the concept of strong accessibility for dynamics of the form ˙x = f (x, u). Sontag (=-=Sontag 1988-=-) showed that strong accessibility implies the existence of controls such that the linearized system around a trajectory passing through a point a of the state-space is controllable. Coron (Coron 1994... |

8 |
Differential-algebraic decision methods and some applications to system theory
- Diop
- 1992
(Show Context)
Citation Context ...l with Willems’ approach (Willems 1991), is here taken into account by utilizing differential 3salgebra which has already helped clarifying several questions in control theory (see, e.g., (Diop 1991, =-=Diop 1992-=-, Fliess 1989, Fliess 1990a, Fliess and Glad 1993)). Flatness might be seen as another nonlinear extension of Kalman’s controllability. Such an assertion is surprising when having in mind the vast lit... |

8 |
On an induced stability,”
- Stephenson
- 1908
(Show Context)
Citation Context ...arbitrary positions and trajectories, and in particular positions which are not equilibrium points. This idea is closely related to a curiosity of classical mechanics that a double inverted pendulum (=-=Stephenson 1908-=-), and even the N linked pendulums which are inverted and balanced on top of one another (Acheson 1993), 4scan be stabilized in the same way. Closed-loop stabilization around reference averaged trajec... |

7 |
Nonlinear control systems, 2nd edn.,
- Isidori
- 1989
(Show Context)
Citation Context ...zing or flat output. The terminology flat is due to the fact that y plays a somehow analogous role to the flat coordinates in the differential geometric approach to the Frobenius theorem (see, e.g., (=-=Isidori 1989-=-, Nijmeijer and van der Schaft 1990)). A considerable amount of realistic models are indeed flat. We treat here two case-studies, namely the crane (D’Andréa-Novel and Lévine 1990, Marttinen et al. 199... |

7 |
Contribution àl’étude des systèmes diffèrentiellement plats
- Martin
- 1992
(Show Context)
Citation Context ...e polynomials over k. The two dynamic feedbacks corresponding to (13) are called endogenous as they do not necessitate the introduction of any variable that is transcendental over D and ˜D (see also (=-=Martin 1992-=-)). If we know ˜x (resp. x), we can calculate u (resp. ũ) from ũ (resp. u) without integrating any differential equation. The relationship with general dynamic feedbacks is given in appendix B. Remark... |

7 |
Control Study with a Pilot Crane
- Marttinen, Virkkunen, et al.
- 1990
(Show Context)
Citation Context ... e.g., (Isidori 1989, Nijmeijer and van der Schaft 1990)). A considerable amount of realistic models are indeed flat. We treat here two case-studies, namely the crane (D’Andréa-Novel and Lévine 1990, =-=Marttinen et al. 1990-=-) and the car with n trailers (Murray and Sastry 1993, Rouchon et al. 1993a). Notice that the use of a linearizing output was already known in the context of static state feedback (see (Claude 1986) a... |

6 |
Global stabilization for controllable systems without drift
- Coron
- 1992
(Show Context)
Citation Context ...nd connections with control). Can flatness and defect be determined by this kind of procedures ? • great progress have recently been made in nonlinear time-varying feedback stabilization (see, e.g., (=-=Coron 1992-=-, Coron 1994)). Most of the examples which were considered happen to be flat (see, e.g., (Coron and D’Andréa-Novel 1992)). The utilization of this property is related to the understanding of the notio... |

6 |
Algorithme de structure, filtrations et découplage
- Delaleau, Fliess
- 1992
(Show Context)
Citation Context .....,y (ν1−1) 1 ,...,ym,...,y (νm−1) m ,...,ym,...,y (νm−1) m (2) ) is another ba); 1 See (Li and Feng 1987) for a definition of this concept via the structure algorithm. See (Di Benedetto et al. 1989, =-=Delaleau and Fliess 1992-=-) for a connection with the differential algebraic approach. 2sthus Y = Tξ where T is an invertible (n + q) × (n + q) matrix. Otherwise stated, Y = T�(x, z). The invertibility of � yields � � x = � z ... |

6 | Universal nonsingular controls.
- Sontag
- 1992
(Show Context)
Citation Context ...strong accessibility implies the existence of controls such that the linearized system around a trajectory passing through a point a of the state-space is controllable. Coron (Coron 1994) and Sontag (=-=Sontag 1992-=-) demonstrated that, for any a, those controls are generic. The above considerations with those of section 2.3 and 2.4 lead in our context to the following definition of controllability, which is inde... |

5 |
Systems without drift and flatness
- Martin, Rouchon
- 1993
(Show Context)
Citation Context ...n-trailer is perhaps the most popular example of path planning of nonholonomic systems (Laumond 1991, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, Tilbury et al. 1993, =-=Martin and Rouchon 1993-=-, Rouchon et al. 1993b). It is a flat system where the linearizing output is the middle of the axle of the last trailer. Once the linearizing output is determined, the path planning problem becomes pa... |

4 |
Steering chained form nonholonomic systems using sinusoids: the firetruck example
- Bushnell, Tilbury, et al.
- 1993
(Show Context)
Citation Context ...ding a curve avoiding collisions. More details on obstacle avoidance can be found in (Laumond et al. 1993) where a car without trailer is considered. The multi-steering trailer systems considered in (=-=Bushnell et al. 1993-=-), (Tilbury et al. 1993), (Tilbury and Chelouah 1993) are also flat: the flat output is then obtained by adding to the Cartesian coordinates of the last trailer, the angles of the trailers that are di... |

4 | Modelling and nonlinear control of an overhead crane - d'Andrea-Novel, Levine - 1990 |

4 |
Basic algebra II, 2nd edn
- Jacobson
- 1989
(Show Context)
Citation Context ... Seidenberg’s paper (Seidenberg 1952) for details. Basics on the customary (non-differential) field theory may be found in (Fliess 1990a, Fliess and Glad 1993) as well as in the textbook by Jacobson (=-=Jacobson 1985-=-) and Winter (Winter 1974) (see also (Fliess 1990a, Fliess and Glad 1993)); they will not be repeated here. 2.1 Basics on differential fields An (ordinary) differential ring R is a commutative ring eq... |

4 |
A Treatise on the Analytical Dynamics of Particules and Rigid Bodies (4th edition
- Whittaker
- 1937
(Show Context)
Citation Context ...y of mass m around a rotation axle � of constant direction is investigated. This system is flat as a consequence of the following considerations. According to an old result due to Huygens (see, e.g. (=-=Whittaker 1937-=-, p. 131–132)), the equations describing the motion are equivalent to those of a pendulum of the same mass m and of length l = J where d �= 0 is the vertical distance between the mass center G and the... |

3 |
Everything you always wanted to know about linearization but were afraid to ask
- Claude
- 1986
(Show Context)
Citation Context ...en et al. 1990) and the car with n trailers (Murray and Sastry 1993, Rouchon et al. 1993a). Notice that the use of a linearizing output was already known in the context of static state feedback (see (=-=Claude 1986-=-) and (Isidori 1989, page 156)). One major property of differential flatness is that, due to formulas (5) and (6), the state and input variables can be directly expressed, without integrating any diff... |

3 | Smooth stabilizing time-varying control laws for a class of nonlinear systems. Application to mobile robots - Coron, d’Andrea-Novel - 1993 |

3 |
Questioning the classic state space description via circuit examples
- Fliess, Hasler
- 1990
(Show Context)
Citation Context ...,u (αn) ) = 0 B1(y1, x, u, ˙u,...,u (β1) ) = 0 . Bp(yp, x, u, ˙u,...,u (βp) ) = 0 where the Ai’s and Bj’s are polynomial over k. The integer n is the dimension of the dynamics D/k < u >. We refer to (=-=Fliess and Hasler 1990-=-, Fliess et al. 1993a) for a discussion of such generalized state-variable representations (8) and their relevance to practice. Example (continued) Set u1 = x3 and u2 =˙x4. The extension D/R < u > is ... |

3 |
Theory, 4th edn
- Landau, Lifschiz, et al.
- 1975
(Show Context)
Citation Context ...y averaged and flat systems (for other approaches, see, e.g., (Baillieul 1993, Bentsman 1987, Meerkov 1980)). We exploit here an idea due to the Russian physicist Kapitsa (Bogaevski and Povzner 1991, =-=Landau and Lifshitz 1982-=-) for stabilizing these three systems in the neighborhood of quite arbitrary positions and trajectories, and in particular positions which are not equilibrium points. This idea is closely related to a... |

3 |
An intrinsic condition for regular decoupling
- Martin
- 1993
(Show Context)
Citation Context ...amples which were considered happen to be flat (see, e.g., (Coron and D’Andréa-Novel 1992)). The utilization of this property is related to the understanding of the notion of singularity (see, e.g., (=-=Martin 1993-=-) for a first step in this direction and the references therein). • the two averaged systems associated to high-frequency control are flat. Can this result be generalized to a large class of devices ?... |

3 |
Steering a three-input nonholonomic using multirate controls
- Tilbury, Chelouah
- 1993
(Show Context)
Citation Context ...on obstacle avoidance can be found in (Laumond et al. 1993) where a car without trailer is considered. The multi-steering trailer systems considered in (Bushnell et al. 1993), (Tilbury et al. 1993), (=-=Tilbury and Chelouah 1993-=-) are also flat: the flat output is then obtained by adding to the Cartesian coordinates of the last trailer, the angles of the trailers that are directly steered. This generalization is quite natural... |

2 |
Kahler differentials and differential
- Johnson
- 1969
(Show Context)
Citation Context ...hich plays such a role in analysis and in differential geometry, admits a nice analogue in commutative algebra (Kolchin 1973, Winter 1974), which has been extended to differential algebra by Johnson (=-=Johnson 1969-=-). To a finitely generated differential extension L/K , associate a mapping dL/K : L → �L/K , called (Kähler) differential 7 and where �L/K is a finitely generated left L � � d -module, such that dt �... |

2 |
Towards robustness and genericity of dynamic feedback linearization
- Tcho'n
- 1994
(Show Context)
Citation Context ...ical and/or geometric properties. This explains why flat systems are so often encountered in spite of the non-genericity of dynamic feedback linearizability in some customary mathematical topologies (=-=Tchoń 1994-=-, Rouchon 1994). We hope to have convinced the reader that flatness and defect bring a new theoretical and practical insight in control. We briefly list some important open problems: • Ritt’s work (Ri... |

1 |
Vibrational control of a class of nonlinear multiplicative vibrations
- Bentsman
- 1987
(Show Context)
Citation Context ...ocomputer under MATLAB. The control of the three non-flat systems is based on high frequency control and approximations by averaged and flat systems (for other approaches, see, e.g., (Baillieul 1993, =-=Bentsman 1987-=-, Meerkov 1980)). We exploit here an idea due to the Russian physicist Kapitsa (Bogaevski and Povzner 1991, Landau and Lifshitz 1982) for stabilizing these three systems in the neighborhood of quite a... |

1 | Modeling and feedback control of wheeled mobile robots - Campion, D’Andréa-Novel, et al. - 1992 |

1 |
Free Rings and their Relations. second edn
- Cohn
- 1985
(Show Context)
Citation Context ...general non-commutative case, k � � d still is a principal ideal ring and the most important properties of left dt k � � d -modules mimic those of modules over commutative principal ideal rings (see (=-=Cohn 1985-=-)). dt Let M be a left k � � d - module. An element m ∈ M is said to be torsion if, and only if, there exists dt π ∈ k � � d , π �= 0, such that π · m = 0. The set of all torsion elements of M is a su... |

1 | 1992b. Full dynamic feedback linearization of a class of mechanical systems - Martin, Sépulchre |

1 | Rouchon 1992b. Sur les systèmes non linéaires différentiellement plats - Martin, P |

1 | Rouchon 1993b. Défaut d’un système non linéaire et commande haute fréquence - Martin, P |

1 | Rouchon 1993c. Differential flatness and defect: an overview - Martin, P |

1 | Rouchon 1993d. Linéarisation par bouclage dynamique et transformations de Lie-Bäcklund - Martin, P |

1 | Rouchon 1993e. Towards a new differential geometric setting in nonlinear control - Martin, P |

1 |
Über den Begriff der Klasse von
- Hilbert
- 1912
(Show Context)
Citation Context ... be directly expressed, without integrating any differential equation, in terms of the flat output and a finite number of its derivatives. This general idea can be traced back to works by D. Hilbert (=-=Hilbert 1912-=-) and E. Cartan (Cartan 1915) on under-determined systems of differential equations, where the number of equations is strictly less than the number of unknowns. Let us emphasize on the fact that this ... |

1 |
Functional reproducibility of general multivariable analytic nonlinear systems
- Li, Feng
- 1987
(Show Context)
Citation Context ...rollability indices and (y1,...,y (ν1−1) 1 sis of the vector space spanned by the components ofξ. Set Y = (y1,...,y (ν1−1) 1 ,...,ym,...,y (νm−1) m ,...,ym,...,y (νm−1) m (2) ) is another ba); 1 See (=-=Li and Feng 1987-=-) for a definition of this concept via the structure algorithm. See (Di Benedetto et al. 1989, Delaleau and Fliess 1992) for a connection with the differential algebraic approach. 2sthus Y = Tξ where ... |

1 | Prime differential ideals in nonlinear rational control systems - Moog, Perraud, et al. - 1989 |

1 | 17) Analogous calculations on the generalized state variable equation R ¨θ =−2 ˙R ˙θ − ¨D cos θ − g sin θ given in (Fliess et al. 1991, Fliess et al. 1993a) lead to a prime differential ideal. Another more direct way for obtaining the differential field c - Crane |