Variations on a Flat Theme Jean L´ EVINE CAS, ´ Ecole des Mines de Paris Dedicated to Michel Fliess for his 60th birthday Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Some Historical Considerations Differential Flatness has been introduced by the gang of the four (Michel Fliess, Pierre Rouchon, Philippe Martin and myself) in 1991. About 400 citations of the 1995 Int. J. Control paper (according to Scholar Google). Major contributions to the development of this concept have also been made by Richard Murray, Jean-Baptiste Pomet, Joachim Rudolph, Hebertt Sira-Ramirez and many others. H appy birthday, Michel! Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion A tribute to the Gang of the 4 Celebration of Flatness 10th anniversary in Mexico, November 2003. Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Contents Some Historical Considerations 1 Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Contents Some Historical Considerations 1 On Flatness-based Design 2 Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Contents Some Historical Considerations 1 On Flatness-based Design 2 On Necessary and Sufficient Conditions 3 The Formalism of Manifolds of Jets of Infinite Order Polynomial Matrices Interpretation in terms of Exact Sequences of Modules Flatness Necessary and Sufficient Conditions Necessary and Sufficient Conditions and Generalized Moving Frames Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Contents Some Historical Considerations 1 On Flatness-based Design 2 On Necessary and Sufficient Conditions 3 The Formalism of Manifolds of Jets of Infinite Order Polynomial Matrices Interpretation in terms of Exact Sequences of Modules Flatness Necessary and Sufficient Conditions Necessary and Sufficient Conditions and Generalized Moving Frames Exogeneous Feedback Linearization 4 Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Contents Some Historical Considerations 1 On Flatness-based Design 2 On Necessary and Sufficient Conditions 3 The Formalism of Manifolds of Jets of Infinite Order Polynomial Matrices Interpretation in terms of Exact Sequences of Modules Flatness Necessary and Sufficient Conditions Necessary and Sufficient Conditions and Generalized Moving Frames Exogeneous Feedback Linearization 4 Conclusions and Perspectives 5 Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Contents Some Historical Considerations 1 On Flatness-based Design 2 On Necessary and Sufficient Conditions 3 The Formalism of Manifolds of Jets of Infinite Order Polynomial Matrices Interpretation in terms of Exact Sequences of Modules Flatness Necessary and Sufficient Conditions Necessary and Sufficient Conditions and Generalized Moving Frames Exogeneous Feedback Linearization 4 Conclusions and Perspectives 5 Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Recalls on Differentially Flat Systems Definition The nonlinear system ˙ x = f ( x , u ) , with x = ( x 1 , . . . , x n ) : state and u = ( u 1 , . . . , u m ) : control, m ≤ n . is (differentially) flat if and only if there exists y = ( y 1 , . . . , y m ) such that: y and its successive derivatives ˙ y , ¨ y , . . . , are independent, u , . . . , u ( r ) ) (generalized output), y = h ( x , u , ˙ Conversely, x and u can be expressed as: y , . . . , y ( α ) ) , y , . . . , y ( α + 1 ) ) x = ϕ ( y , ˙ u = ψ ( y , ˙ ϕ ≡ f ( ϕ, ψ ) . with ˙ The vector y is called a flat output . Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Main advantages of Flatness Direct open-loop trajectory computation, without integration nor 1 optimization. Local stabilization of any reference trajectory using the 2 equivalence between the system trajectories and those of y ( α + 1 ) = v . “Flatness-Based Control” = Trajectory Planning + Trajectory Tracking. Alternative approach to Predictive Control (see e.g. Fliess and Marquez 2001, Delaleau and Hagenmeyer 2006, Devos and L´ evine 2006). Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Consequence on motion planning • x = f (x , x ,u) To every curve t �→ y ( t ) enough differentiable, there corresponds a t → (x(t), u(t)) )) trajectory � x ( t ) � t �→ = u ( t ) (ϕ,ψ) (ϕ,ψ) Lie-B Lie-Bäcklund klund � � y ( t ) , . . . , y ( α ) ( t )) ϕ ( y ( t ) , ˙ y ( t ) , . . . , y ( α + 1 ) ( t )) ψ ( y ( t ) , ˙ that identically satisfies the system t → (y(t), . . . , y ( α +1) +1) (t)) )) +1) = v y ( α +1) equations. Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Example: Linear Motor with Auxiliary Masses (I) Single Mass Case Model: M ¨ x = F − k ( x − z ) − r (˙ x − ˙ z ) m ¨ z = k ( x − z ) + r (˙ x − ˙ z ) Aims: rail linear motor Rest-to-rest fast and high precision displacements. flexible beam mass Measurements: bumper x and ˙ x measured, In collaboration with Micro-Contrˆ ole. z not measured. Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Flat output: y = r 2 1 − r 2 � � z − r mkx + k ˙ z mk x = y + r y + m z = y + r k ˙ k ¨ y , k ˙ y � � y + r Mm ky ( 3 ) + ( M + m ) ky ( 4 ) F = ( M + m ) ¨ Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Videos rail linear motor flexible beam mass bumper Mass=disturbance Input filtering Flatness-based Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Example: Linear Motor with Auxiliary Masses (II) The Case of Two Masses Model: rail M ¨ x = F − k ( x − z ) − r (˙ x − ˙ z ) linear motor − k ′ ( x − z ′ ) − r ′ (˙ z ′ ) x − ˙ flexible beams m ¨ z = k ( x − z ) + r (˙ x − ˙ z ) z ′ = k ′ ( x − z ′ ) + r ′ (˙ masses m ′ ¨ z ′ ) x − ˙ bumpers In collaboration with Micro-Contrˆ ole. Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Flatness: k + r ′ k + m ′ k ′ + rr ′ � r � � m � x = y + y + ˙ ¨ y k ′ kk ′ � mr ′ + rm ′ y ( 3 ) + mm ′ � kk ′ y ( 4 ) + kk ′ k + r ′ � m ′ k ′ + rr ′ y + rm ′ � r � � kk ′ y ( 3 ) z = y + y + ˙ ¨ k ′ kk ′ � r k + r ′ � � m k + rr ′ � y + mr ′ z ′ = y + kk ′ y ( 3 ) y + ˙ ¨ k ′ kk ′ � r k + r ′ � � m Mm ′ M rr ′ � y ( 3 ) + M ′ + ¯ F = ˆ y + ˆ ¯ k ′ + ˆ y ( 4 ) M ¨ M k ′ kk ′ k � mr ′ Mrm ′ y ( 5 ) + Mmm ′ � M ′ + ¯ kk ′ ¯ y ( 6 ) + kk ′ kk ′ M ′ = M + m ′ . with ˆ M = ( M + m + m ′ ) , ¯ M = M + m and ¯ Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Videos rail linear motor flexible beams masses bumpers Masses=disturbance Input filtering Flatness-based Jean L´ EVINE Variations on a Flat Theme
Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Combined trajectory planning and output feedback The US Navy Crane Example (B. Kiss, J.L., P. M¨ ullhaupt 2000) Jean L´ EVINE Variations on a Flat Theme
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