Normal Forms and Bifurcations of Control Systems

Research supported in part by AFOSR-49620-95-1-0409 and by NSF 9970998. To be presented at the IEEE CDC 2000, Sydney.We present the quadratic and cubic normal forms of a nonlinear control system around an equilibrium point. These are the normal forms under change of state coordinates and invertible state feedback. The system need not be linearly controllable. A control bifurcation of a nonlinear system occurs when its linear approximation loses stabilizability. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations

Controllability on infinite-dimensional manifolds

Following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable.Comment: 19 pages, 1 figur

Geometric Approach to Pontryagin's Maximum Principle

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page

Second-order necessary conditions in optimal control: Accessory-problem results without normality conditions

An optimal control problem, which includes restrictions on the controls and equality/inequality constraints on the terminal states, is formulated. Second-order necessary conditions of the accessory-problem type are obtained in the absence of normality conditions. It is shown that the necessary conditions generalize and simplify prior results due to Hestenes (Ref. 5) and Warga (Refs. 6 and 7).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45219/1/10957_2004_Article_BF00934437.pd

A Pseudospectral Observer for Nonlinear Systems

The article of record as published may be located at http://dx.doi.org/10.2514/6.2005-5845Proceedings of AIAA Guidance, Navigation, and Control Conference ; Paper no. AIAA-2005-5845, San Francisco, California, Aug. 15-18, 2005We present a method for designing an observer for nonlinear systems based on Pseudospectral discretization and a moving horizon strategy. The observer has a low computational burden, fast convergence rate and an ability to handle measurement noise. Our observer can also be applied to nonlinear systems governed by deferential-algebraic equations (DAE) which is very did_cult to deal with by other designs like the unscented Kalman filter. The performance of the proposed observer is demonstrated by numerical experiments on a time-varying chaotic nonlinear system with unknown parameters and also a nonlinear circuit with singularity-induced bifurcation.NAApproved for public release; distribution is unlimited