Contents Differentially Flat Nonlinear Control Systems: Overview of the Theory and Applications, and Differential Algebraic Aspects. Jean L´ EVINE CAS, Unit´ e Math´ ematiques et Syst` emes, MINES-ParisTech, France. DART-IV, Beijing, October 27–30, 2010 Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Contents Contents Introduction: Basic Notions of System Theory 1 Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Contents Contents Introduction: Basic Notions of System Theory 1 Recalls on Differential Flatness 2 Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Contents Contents Introduction: Basic Notions of System Theory 1 Recalls on Differential Flatness 2 Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation Flatness Necessary and Sufficient Conditions 3 Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Contents Introduction: Basic Notions of System Theory 1 Recalls on Differential Flatness 2 Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation Flatness Necessary and Sufficient Conditions 3 Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Introduction: Basic Notions of System Theory Consider a smooth n -dimensional manifold X and a family of vector fields x �→ f ( x , u ) ∈ T x X for all x ∈ X , indexed by u ∈ R m , control � � ∂ f input , with m ≤ n , and rank = m in a suitable open dense set, ∂ u and the (ordinary) differential equation in X x = f ( x , u ) ˙ (1) with initial state x 0 ∈ X at time t = 0. System (explicit representation) The system associated to (1) is the pair ( X , f ) . Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions After elimination of u (implicit function theorem), we get the equivalent set of n − m implicit equations F ( x , ˙ x ) = 0 (2) � ∂ F � where F : TX → R n − m , satisfies rank = n − m in a suitable ∂ ˙ x open dense set, and with initial state x 0 ∈ X at time t = 0. System (implicit representation) The system associated to (2) is the pair ( X , F ) . The two notions, explicit and implicit, are indeed equivalent! System (Differential algebraic definition for F polynomial) Let K = R be the ground field and D / K the differential field generated by the variables x 1 , . . . , x n and the relation (2), assumed polynomial. The system associated to (2) is the differential field D / K . Its differential transcendence degree is diff tr d ◦ D / K = n − m . Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Reachability (see e.g. Sussmann and Jurdjevic, J. Diff. Eq. 1972, Sussmann, SIAM J. Control, 1987) The reachable set at time T > 0, noted R T ( x 0 ) , is the set of points x T ∈ X such that there exists u piecewise continuous on [0, T] and an integral curve t �→ X t ( x 0 ; u ) of the system, satisfying X T ( x 0 ; u ) = x T . We say that the system is locally reachable if for every neighborhood V of x T in X , R T ( x 0 ) ∩ V has non empty interior. For linear systems , reachability is equivalent to controllability . Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Motion Planning Problem: Given x 0 ∈ X : initial state, x T ∈ X : final state, and T > 0: prescribed duration, find reference trajectories t �→ x ∗ ( t ) and t �→ u ∗ ( t ) satisfying x ∗ ( t ) = f ( x ∗ ( t ) , u ∗ ( t )) such that x ∗ ( 0 ) = x 0 , x ∗ ( T ) = x T . ˙ Reference Trajectory Tracking: Given a family of perturbations f p ∈ TX of f and a reference trajectory t �→ x ∗ ( t ) of ( X , f ) , find a feedback law x �→ u ( x ) such that, noting e = x − x ∗ , the error equation e ( t ) = f p ( e ( t ) + x ∗ ( t ) , u ( e ( t ) + x ∗ ( t ))) − f ( x ∗ ( t ) , u ∗ ( t )) ˙ is asymptotically stable for all perturbations. Remark: These problems haven’t yet received a general answer. Our aim is to show that, for differentially flat systems , a simple solution may be obtained. Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions A simple example: Fast rest-to-rest displacements of a pendulum without measuring the pendulum position. rot a t i o n axis pe n dulu m v e rt i c a l b u m pe r p o si t i o n in d i c a tor motor PID control on motor position input filtering flatness-based Experiment realized thanks to Micro-Controle/Newport (France). Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Other simple examples: Fast rest-to-rest displacements of a linear motor with perturbating oscillating masses rail rail linear motor linear motor flexible beams flexible beam masses mass bumpers bumper Mass=perturbation input filtering Masses=perturbation input filtering flatness-based flatness-based Experiments realized thanks to Micro-Controle/Newport (France). Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions More difficult: Fast rest-to-rest displacements of a US Navy crane (reduced scale model – Kiss, L´ evine and M¨ ullhaupt, 2000, Devos and L´ evine, 2006) Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions More and more difficult: 2kPi: Juggling robot (Lenoir, Martin and Rouchon, CDC 98) Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions An infinite dimensional example: Well-head / riser underwater connexion (reduced scale model – CAS / French Institute of Petroleum (IFP)) actuators motors simulating the wave excitation flexible water tank riser synchronized digital well-head cameras Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of Contents Introduction: Basic Notions of System Theory 1 Recalls on Differential Flatness 2 Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation Flatness Necessary and Sufficient Conditions 3 Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of Recalls on Differential Flatness (Fliess, L´ evine, Martin and Rouchon, 1991) Definition (informal) The nonlinear system x = f ( x , u ) ˙ is said (differentially) flat if and ony if there exists y = ( y 1 , . . . , y m ) such that: y and successive derivatives ˙ y , ¨ y , . . . , are independent, u , . . . , u ( r ) ) (generalized output), y = h ( x , u , ˙ conversely, x and u are given by: y , . . . , y ( s ) ) , y , . . . , y ( s + 1 ) ) x = ϕ ( y , ˙ u = ψ ( y , ˙ with ˙ ϕ ≡ f ( ϕ, ψ ) . The vector y is called a flat output . Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of In the mathematical literature, this concept may be traced back to two different trends: solution of underdetermined differential equations (Monge, Goursat, Zervos, etc.) and in particular Hilbert (“umkerhrbar, integrallos transformationen”, 1912) and Cartan (absolute equivalence, 1914. See also Shadwick, 1990); parameterization and uniformization in the sense of Hilbert’s 22nd Problem (Poincar´ e 1907). Jean L´ EVINE Flat Systems, Differential Algebraic Aspects
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