Stability of the feedback linearization of bilinear plants with bounded inputs

Many nonlinear plants can be better approximated and controlled using bilinear models instead of linear ones. For many bilinear models, a feedback linearization can be achieved, forcing the plant to behave like a linear system, with all known advantages. In practical applications, however, this means that the dynamic range of the applied control input is enlarged, increasing the danger of running into bounds, due, for example, to power saturation. Running into bounds can cause problems even for linear systems (e.g. instability) but in general, if the open-loop system is stable, not too conservative conditions can be given to insure stability [1]. In bilinear systems the issue is more complex, as the control input and the state are connected multiplicatively. This makes the methods used in the linear case very conservative. However, they suggest a very simple analytical approach, as the saturated bilinear system actually becomes a linear system with different dynamics. These dynamics can be influenced by the choice of the saturation levels. Starting from this key idea, the paper shows that the stability problem is actually a multi-model linear problem, and gives sufficient conditions for the global stability of the closed-loop system. An algorithmic approach is given based on the work presented in [2], and an alternative testing procedure is derived from the approach of [3]. The results are applied to a hydraulic system to show the practical aspects. A discussion of the implications of the results on the control system design closes the paper.