Globally convergent observers are designed for a class of systems with monotonic nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a time-varying multivariable sector nonlinearity. Using LMI software, observer gain matrices are computed to satisfy the circle criterion and, hence, to drive the observer error to zero. In output-feedback design, the observer is combined with control laws that ensure input-to-state stability with respect to the observer error. Robustness to unmodeled dynamics is achieved with a small-gain assignment design, as illustrated on a jet engine compressor example.