sufficient controllability condition for multidimensional affine systems

The controllability problem is one of the most important problems of the control theory. This problem is well studied for linear stationary systems. A linear stationary system is known to be controllable if and only if the system is feedback linearizable. For affine systems, relations between controllability and feedback linearizability are more difficult: if a system is feedback linearizable, then it is controllable; the inverse statement is wrong.

In the present paper, multidimensional affine systems which cannot be feedback linearized are considered. In order to prove controllability of such system we use a special approach based on checking an existence of a terminal problem solution. If a terminal problem for the system in question has a solution for all boundary conditions and for arbitrary finite time of control, then the system is controllable. In the present paper it is supposed that the system in question by a smooth nonsingular change of variables can be transformed into a special canonical form | a regular quasicanonical form defined in the whole state space. It is also supposed that a zero dynamics subsystem of the system of a quasicanonical form is one-dimensional. As a result, the terminal problem for the original system is transformed into the equivalent terminal problem for the system of a regular quasicanonical form. The necessary and sufficient condition of an existence of a terminal problem solution is well-known for such systems. In the present paper, this condition is used to prove an existence of a terminal problem solution for the system of a regular quasicanonical form with one-dimensional zero dynamics subsystem for all boundary conditions and for arbitrary finite time of control. Thereby, the sufficient condition of controllability is proven for such systems. An example of the five-dimensional affine system with two-dimensional control is given to illustrate the proposed condition.

Obtained results may be used to solve control problems for various technical systems.