On the relation between analysis and synthesis conditions for discrete-time systems with saturation nonlinearities

This paper considers linear time-invariant (LTI) discrete-time systems with saturation and/or dead-zone nonlinearities, and proposes analysis and synthesis methods of a regional l
2
performance and/or a pole placement based on a quadratic Lyapunov function via a generalized sector and a polytopic approach. In particular, a new domain of l
2
performance is defined by a region of initial states of the system considering the l
2
performance and/or the pole placement. For analysis, the problems based on the two approaches can be recast as linear matrix inequality (LMI) optimization ones respectively, and in the special case of single saturation or single dead-zone nonlinearity, it is proved that the generalized sector approach is exactly the same as the polytopic approach. Similarly, for synthesis, the problem based on the generalized sector approach can be recast as an LMI optimization problem where the outputs of the nonlinearities are assumed to be available for control. Next, the relation is clarified that the analysis and synthesis conditions can be reduced to the corresponding conditions for the continuous-time systems as the sampling period goes to zero. Finally, it is pointed out that our LMI-based approach is helpful through a numerical example designing anti-windup control systems.