Global Exponential Observers for Two Classes of Nonlinear Systems

This paper develops sufficient conditions for the existence of global exponential observers for two classes of nonlinear systems: (i) the class of systems with a globally asymptotically stable compact set, and (ii) the class of systems that evolve on an open set. In the first class, the derived continuous-time observer also leads to the construction of a robust global sampled-data exponential observer, under additional conditions. Two illustrative examples of applications of the general results are presented, one is a system with monotone nonlinearities and the other is the chemostat system.

💡 Research Summary

The paper addresses the long‑standing problem of designing observers that guarantee global exponential convergence for nonlinear systems. It focuses on two important families of systems that are not covered by existing high‑gain, circle‑criterion, or transformation‑based observer techniques.

1. Systems with a globally asymptotically stable compact set.
The authors introduce three hypotheses. (H1) postulates the existence of a radially unbounded function V and a positive‑definite function W such that (\dot V\le -W) for all admissible inputs; this ensures that every trajectory enters a compact invariant set (S={x:V(x)\le b}) in finite time. (H2) provides a local exponential observer structure by requiring a positive‑definite matrix P, a decay rate μ, and a mapping (h_k) that links the measured output y with the observer output. Under these conditions the error dynamics satisfy (\dot ξ^{!T}Pξ\le -μ|ξ|^2) when the estimate lies inside S. (H3) introduces a correction term ϕ that becomes active when V exceeds a prescribed threshold b. The correction term is constructed from a smooth “switching” function p(V) that is zero for small V, linear for large V, and interpolates in between.

The main result (Theorem 2.2) shows that the observer
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