Finite Dimensional Methods for Differential Flatness

In Control System Theory, the study of continuous-time, finite dimensional, underdetermined systems of ordinary differential equations is an important topic. Classification of systems in different categories is a natural initial step to the analysis of a given control problem. Systems of equations can often be “transformed” into other “equivalent” ones. Then, a control system is associated to a set of equivalent equations. In this setting, a property of a control system can be defined as a property that has to be satisfied by some arbitrary system of equations in the set representing the control system. Assessing such a property can be a difficult task. In this thesis, we review and study a number of ways to determine whether a multi-input nonlinear system is flat, i.e. whether it is equivalent to a linear system after some dynamical extension and change of coordinates. This is a difficult as well as a well studied problem. Therefore, coming up with some altogether new approach or solution is to a certain extent illusory. A substantial part of the text is devoted to describing existing approaches and sometimes to propose either an original alternative or an original point of view. Another part of the thesis is dedicated to the study of a drastically reduced version of the problem, where more can be said in an algorithmic way. Nonlinear control systems are first modeled as the embedding of some fibered bundle to the first jet of the time-and-state-variables manifold. The exterior system, or Pfaffian system, corresponding to the ODE then arises naturally. Input prolongations are then introduced as lifts of the previously mentioned embedding. Various filtration techniques and their applications to static feedback linearization are discussed next. A full chapter is devoted to the infinite dimensional approach involving matrix differential operators. A now classical theorem, linking integrability of the basis of a differential module after application of one such an operator, and the flatness property is discussed. The relations obtained can be decomposed in an equivalent set of differentially closed equations. We state a version of the resulting theorem where the “curvature equations” are trivial. A subproblem that has attracted the attention of researchers is the question whether a given system — subject to some state constraints — is flat. In this setting, useful concepts are that of a covering of a system by another one and the accompanying result stating that a flat system can cover only a flat system. Hence, if a “large” linear system is given together with a set of nonlinear constraints, flatness of the constrained system is assessed if the unconstrained system can be shown to cover the constrained one. Starting with the classical notion of controlled invariance and a generalized notion coined dynamic controlled invariance, sufficient conditions are discussed which also involve the notion of right invertibility and the dynamic extension algorithm. Modeling of mechanical systems by free moving point masses subject to some control forces and quadratic constraints is often effective. The resulting unconstrained equations are linear and bilinear in the state and control/Lagrangian multiplier variables. We propose a relative derived flag that leads to a filtration with guaranteed integrability at each stage. This leads to a very effective sufficient condition for the flatness of the unconstrained model. The algorithm, together with the test described in the previous section, is used to show flatness of some generalized pendulum-like equations. They are also shown to specialize to the non-holonomic car equations and to the VTOL/pendulum equations when some specific parameters are suitably chosen.