Relative structure at infinity and nonlinear semi-implicit DAE'S

A notion of relative structure at infinity and the concept of relative output rank with respect to a subsystem are introduced. We introduce the problem of relative decoupling, showing that this problem is solvable if and only if the relative output rank ρ(z) coincides with dimz. The Relative Dynamic Extension Algorithm (RDEA) is presented and a geometric interpretation is also given, showing that it computes the relative structure at infinity. We develop necessary and sufficient conditions for testing if a given output z of system is relatively flat with respect to Y. As a byproduct, we obtain conditions for the decoupling problem and for flatness of a class of Differential Algebraic Equations (DAE's). It is shown the the RDEA may be used for constructing dynamic linearizing feedback laws and/or the solution of the dynamic input-output decoupling problem for implicit systems.

[1]  Joachim Rudolph,et al.  Well-formed dynamics under quasi-static state feedback , 1995 .

[2]  E. Delaleau,et al.  Algebraic necessary and sufficient conditions of input-output linearization , 2001 .

[3]  Philippe Martin.  Contribution a l'etude des systemes differentiellement plats , 1992 .

[4]  Peter J. Olver.  Geometrical Aspects of Partial Differential Equations (V. V. Zharinov) , 1993, SIAM Rev..

[5]  Relative structure at infinity and nonlinear semi-implicit DAE'S , 2003, ECC.

[6]  L. Xiaoping.  On Linearization of Nonlinear Singular Control Systems , 1993, 1993 American Control Conference.

[7]  N..  Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations , 2002 .

[8]  Jessy W. Grizzle,et al.  Rank invariants of nonlinear systems , 1989 .

[9]  S. Čelikovský,et al.  Feedback control of affine nonlinear singular control systems , 1997 .

[10]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[11]  Philippe Martin,et al.  A Lie-Backlund approach to equivalence and flatness of nonlinear systems , 1999, IEEE Trans. Autom. Control..

[12]  S. Campbell.  Singular Systems of Differential Equations , 1980 .

[13]  J. Rudolph,et al.  Control of flat systems by quasi-static feedback of generalized states , 1998 .

[14]  M. A. Akanbi,et al.  Numerical solution of initial value problems in differential - algebraic equations , 2005 .

[15]  H. Nijmeijer,et al.  Dynamic input-output decoupling of nonlinear control systems , 1988 .

[16]  S. Kawaji,et al.  Feedback linearization of a class of nonlinear descriptor systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[17]  J. Descusse,et al.  Dynamic decoupling for right-invertible nonlinear systems , 1987 .

[18]  M. Shayman,et al.  Dynamics and Control of Constrained Nonlinear Systems with Application to Robotics , 1992, 1992 American Control Conference.

[19]  Nadéia Aparecida Veloso Pazzoto,et al.  Output Rank and Flatness of Regular Implicit Systems , 2001 .

[20]  N. U. Ahmed,et al.  Approach to controllability problems for singular systems , 1991 .

[21]  M. Fliess,et al.  Linéarisation par bouclage dynamique et transformations de Lie-Bäcklund , 1993 .

[22]  M. Fliess.  Automatique et corps différentiels , 1989 .

[23]  P. S. P. Silva.  Flatness of nonlinear control systems and exterior differential systems , 2001 .

[24]  Paulo Sérgio Pereira da Silva,et al.  Relative Flatness and Flatness of Implicit Systems , 1998, SIAM J. Control. Optim..

[25]  M. Fliess,et al.  Index and Decomposition of Nonlinear Implicit Differential Equations , 1995 .

[26]  Emmanuel Delaleau,et al.  Filtrations in feedback synthesis: Part I – Systems and feedbacks , 1998 .

[27]  Emmanuel Delaleau,et al.  Filtrations in feedback synthesis: Part II – Input-output decoupling and disturbance decoupling , 1998 .

[28]  Werner C. Rheinboldt,et al.  On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations , 1991 .