Matematika i Matematičeskoe Modelirovanie (Jan 2015)
An Orbital Feedback Linearization Approach to Solving Terminal Problems for Affine Systems with Vector Control
Abstract
State-feedback linearization is widely used to solve various problems of the control theory. An affine system is said to be state-feedback linearizable if there are a smooth change of variables in the space of states and an invertible change of controls, which transform the system to the system of a regular canonical form. However if a system is not state-feedback linearizable it yet can be orbitally feedback linearized, i.e. the system can be transformed to a regular canonical form after a change of the independent variable.The article solves the following terminal problem for multi-dimensional stationary affine systems: for given two states, find controls and a time T such that the corresponding trajectory of the system joins these states for the time T. We make an integrable change of the independent variable depending on controls. As a result, we obtain a non-stationary affine system, its dimension being one less than dimension of the original system. The new terminal problem with the restriction on controls is formulated for the transformed system. We prove the relation between solutions of the original terminal problem and solutions of the terminal problem for the transformed system. It is shown that to solve the original terminal problem it is sufficient to solve terminal problem for the transformed system. Then, we check whether the transformed system can be state-feedback linearized. For this purpose, we check the necessary and sufficient conditions of state-feedback linearization for non-stationary affine systems. If the conditions are met then we transform the system to a regular canonical form for which the concept of inverse dynamics problems can be used to solve terminal problems. However, due to the restriction on controls an additional check is necessary whether the found controls meet the restriction.An example of the terminal problem for the five-dimensional affine system with two controls is given. We prove that the system in question is not state-feedback linearizable on any open subset of the state space. However, the system can be transformed to a regular canonical form after the change of the independent variable depending on controls. The proposed method allows us to solve the terminal problem.
