Qualitative theory for dynamical systems with saturation nonlinearities

The authors consider general dynamical systems formed from linear dynamics, but with provisions for the control variables to be bounded and for the state variables and output variables to go into saturation. Such systems are, for example, common in digital filters and in hybrid digital feedback control strategies. Building upon results from the theory of neural networks, they recast they problem as a collection of linear, reduced order systems with linear constraints and make precise the meaning of solutions, including the various saturation modes. In addition, the existence and qualitative properties of equilibria in the various regions of the state space are discussed. When compared with classical analyses of these issues, the results are surprisingly sharp and simple. They appear to be an example of a situation in which an algebraic approach has pointed the way into a nonlinear problem area.