Lay summary

The aim of this project is to investigate two different yet complementary iterative methods of decomposing the original nonlinear system into smaller simpler parts from which a control design is performed. This control design proceeds then from this decomposition either directly, or by proceeding by another iterative process starting from the transformed system back to the original system. The project splits into two subparts.In Subpart A), iterative methods are used to determine successive approximations of the original system and to a suitable set of Lyapunov functions. Power series generated from the data of the polynomial system are resorted to, together with iterative algorithms for determining their coefficients. When truncated, a power series becomes a polynomial and the classical tools of computer algebra can be used. As for the Lyapunov construction, both sum-of-squares decomposition-like and Groebner bases based techniques are investigated. A somewhat more specific class of systems, namely differentially flat system and approximately differentially flat systems are investigated using this polynomial paradigm. If a system is flat, it can be put in correspondence with a linear system. From this linear system, one can either directly determine a stabilizing control law, or a Control Lyapunov Function (CLF) for the original system. From this CLF other control laws can be found with sometimes better qualities compared with the direct approach.Subpart B) A nonlinear control design scheme using quotient-manifold methods is investigated. The design scheme keeps as much knowledge as necessary on both the initial manifold and the vectorfields and does not require them to be polynomial. Taking a quotient along an input vector field gives the directions that are not directly affected by the inputs. This allows reducing the dynamical system step by step through the generalization of the classical orthogonal projection used in linear algebra. The challenge lies in the manner in which both the reduction should be undertaken and the partial quotients should be used so as to design a stabilizing feedback.