Map Invariance and the State Reconstruction Problem for Nonlinear Discrete-time Systems

The role of map invariance is examined within the context of the dynamic state reconstruction problem for nonlinear discrete-time systems. In particular, the key notion of invariant manifold for maps in nonlinear discrete-time dynamics is shown to be conceptually insightful and technically quite effective to address important issues related to the deterministic observerbased nonlinear state estimation problem in the discrete-time domain. As a necessary first methodological step, the problem of quantitatively characterizing the asymptotic long-term behavior of nonlinear discrete-time systems with a skew-product structure using the notion of map invariance is revisited. The formulation of this problem can be naturally realized through a system of invariance functional equations (FEs), for which a set of existence and uniqueness conditions of a solution is provided. Under a certain set of conditions, it is shown that the invariant manifold computed attracts all system trajectories/orbits, and therefore, the asymptotic long-term dynamic behavior of the system is determined through the restriction of the discrete-time system dynamics on the invariant manifold. Within the above analytical framework, the nonlinear full-order observer design problem in the discrete-time domain is considered, appropriately formulated and an interpretation of previous work on the problem is attempted through the notion of invariant manifolds for maps. Furthermore, this framework allows the development of a new approach to the nonlinear reduced-order observer design problem for multiple-output systems in the discrete-time domain, which is also presented in the present work. Finally, the performance of the proposed nonlinear reduced-order discrete-time observer is assessed in an illustrative bioreactor example through simulations.