On the nature and stability of differential-algebraic systems

Nonlinear dynamical systems described by a class of higher index differential-algebraic equations (DAE) are considered. A quantitative and qualitative analysis of their nature and of the stability properties of their solution is presented. Using tools from geometric control theory, higher index differential-algebraic systems are shown to be inherently unstable about their solution manifold. A qualitative geometric interpretation is given, and the consequences of this instability are discussed, in particular as they relate to the numerical solution of these systems. The paper concludes with ideas and directions for future research.

[1]  Ruey-Wen Liu,et al.  Existence of state equation representation of linear large-scale dynamical systems , 1973 .

[2]  D. Luenberger.  Dynamic equations in descriptor form , 1977 .

[3]  R. F. Sincovec,et al.  Solvability, controllability, and observability of continuous descriptor systems , 1981 .

[4]  T. Kailath,et al.  A generalized state-space for singular systems , 1981 .

[5]  L. Petzold.  Differential/Algebraic Equations are not ODE's , 1982 .

[6]  D. Cobb.  Controllability, observability, and duality in singular systems , 1984 .

[7]  W. Rheinboldt.  Differential-algebraic systems as differential equations on manifolds , 1984 .

[8]  A. Isidori.  Nonlinear Control Systems , 1985 .

[9]  M. Shayman,et al.  Singular systems: A new approach in the time domain , 1986, 1986 25th IEEE Conference on Decision and Control.

[10]  C. Pantelides.  The consistent intialization of differential-algebraic systems , 1988 .

[11]  Stephen L. Campbell,et al.  Descriptor systems in the 1990s , 1990, 29th IEEE Conference on Decision and Control.

[12]  W. J. Terrell,et al.  Duality, observability, and controllability for linear time-varying descriptor systems , 1991 .

[13]  P. Müller,et al.  On the observer design for descriptor systems , 1993, IEEE Trans. Autom. Control..

[14]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[15]  M. Boutayeb,et al.  Design of observers for descriptor systems , 1995, IEEE Trans. Autom. Control..

[16]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[17]  THE INTERACTION BETWEEN SIMULATION AND MODELING IN INFINITE DIMENSIONAL SYSTEMS , 1999 .

[18]  Brandon W. Gordon,et al.  State Space Modeling of Differential-Algebraic Systems Using Singularly Perturbed Sliding Manifolds , 1999, Dynamic Systems and Control.

[19]  Peter C. Müller,et al.  Observer design for descriptor systems , 1999, IEEE Trans. Autom. Control..

[20]  Rein Luus,et al.  Singular control problems , 2000, Iterative Dynamic Programming.

[21]  Guy Bessonnet,et al.  Optimal dynamics of constrained multibody systems. Application to bipedal walking synthesis , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).