On state observers for nonlinear systems

For linear systems, it is known that there exists a state observer if and only if the system is detectable, or in other words, asymptotically stabilizable by output injection. In this paper, it is shown that asymptotic stabilizability by output injections is neither necessary nor sufficient for the construction of a nonlinear observer which may not have an exponential error decay rate. The concept of uniform observers is introduced and necessary conditions and sufficient conditions are given for the construction of uniform observers.