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

Venue: | International Journal of Control |

Citations: | 206 - 17 self |

### BibTeX

@ARTICLE{Fliess95flatnessand,

author = {Michel Fliess and Jean Lévine and Pierre Rouchon},

title = {Flatness and defect of nonlinear systems: Introductory theory and examples},

journal = {International Journal of Control},

year = {1995},

volume = {61},

pages = {1327--1361}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman’s controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems ’ standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of planar curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from nonlinear physics. A high frequency control strategy is proposed such that the averaged systems become flat. ∗This work was partially supported by the G.R. “Automatique ” of the CNRS and by the D.R.E.D. of the “Ministère de l’Éducation Nationale”. 1 1

### Citations

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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... |

1534 |
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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... |

280 | Nonholonomic motion planning: steering using sinusoids
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(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... |

254 |
Differential Algebra and Algebraic Groups
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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... |

250 |
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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 ,..... |

204 |
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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, ... |

146 |
A Motion Planner for Nonholonomic Mobile Robots
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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 ... |

76 |
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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 ... |

69 | Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories
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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... |

46 | der Schaft - Nijmeijer, van - 1990 |

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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... |

36 |
An introduction to motion planning under multirate digital control
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(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... |

35 |
Stable average motions of mechanical systems subject to periodic forcing. In: Dynamics and Control of Mechanical Systems: The Falling Cat and Related Problems
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(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... |

33 |
On dynamic feedback linearization
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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) (... |

32 |
Controllability of a multibody mobile robot
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(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 ... |

28 |
Elimination in Control Theory
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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... |

26 | Necessary condition and genericity of dynamic feedback linearization
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- 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... |

25 |
Generalized controller canonical form for linear and nonlinear dynamics
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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... |

24 | Some Basic Structural Properties of Generalized Linear Systems - Fliess - 1990 |

22 |
Automatique et Corps Differentiels
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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... |

20 |
Sufficient conditions for dynamic state feedback linearization
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(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... |

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

18 |
Absolute equivalence and dynamic feedback linearization
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(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 (... |

17 |
Principle of vibrational control: theory and applications
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(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... |

16 |
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... |

16 | Differential equations - Tikhonov, Vasil’eva, et al. - 1980 |

15 | Dynamic feedback linearization of nonholonomic wheeled mobile robots - &Andrea-Novel, Bastin, et al. - 1992 |

14 | Rank invariants of nonlinear systems - DiBenedetto, Grizzle, et al. - 1989 |

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

12 | Linearized control systems and application to smooth stabilization
- Coron
- 1994
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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... |

12 | Remarks on equivalence and linearization of nonlinear systems - Jakubczyk |

12 | A multisteering trailer system: Conversion into chained form using dynamic feedback
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(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... |

10 |
A simplified approach of crane control via a generalized state-space model
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(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... |

9 |
Algebraic Methods in Nonlinear Perturbation THeory (Springer-Verlag
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- 1991
(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... |

9 |
Some basic theorems in differential algebra (characteristic p arbitrary
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(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... |

9 |
Finite dimensional open-loop control generators for nonlinear systems
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- 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 |
On differential systems with quadratic impulses and their applications to Lagrangian mechanics
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(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 = ... |

8 |
Sur l'int'egration de certains syst`emes ind'etermin'es d"equations diff'erentielles
- Cartan
- 1953
(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... |

8 | On differentially flat nonlinear system - Martin, Rouchon - 1992 |

8 | Flatness and motion planning: The car with n-trailers - Martin - 1993 |

8 |
The structure of fields
- Winter
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(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 ... |

7 |
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 ... |

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... |

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 |
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... |

6 |
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... |

5 |
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 ... |

5 |
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... |

5 | A generalised state variable representation for a simplified crane description - Fliess, Lévine, et al. - 1993 |

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... |