On Geometric Properties of Triangularizations for Nonlinear Control Systems

We consider triangular decompositions for nonlinear control systems. For systems that are exactly linearizable by static feedback it is well known that a triangular structure exists in adapted coordinates using the Frobenius theorem to straighten out a nested sequence of involutive distributions. This triangular form is based on explicit ordinary differential equations from which it can be easily seen that exactly linearizable systems are also flat. We will analyze this triangularization also from a dual perspective using a Pfaffian system representation. This point of view allows the introduction of a triangular form corresponding to implicit ordinary differential equations. For systems that are flat but not exactly linearizable by static feedback, this modified triangular form turns out to be useful in setting up a constructive algorithm to compute so-called 1-flat outputs.