  In this paper, we propose a new approach to design globally convergent
reduced-order observers for nonlinear control systems via contraction analysis
and convex optimization. Despite the fact that contraction is a concept
naturally suitable for state estimation, the existing solutions are either
local or relatively conservative when applying to physical systems. To address
this, we show that this problem can be translated into an off-line search for a
coordinate transformation after which the dynamics is (transversely)
contracting. The obtained sufficient condition consists of some easily
verifiable differential inequalities, which, on one hand, identify a very
general class of "detectable" nonlinear systems, and on the other hand, can be
expressed as computationally efficient convex optimization, making the design
procedure more systematic. Connections with some well-established approaches
and concepts are also clarified in the paper. Finally, we illustrate the
proposed method with several numerical and physical examples, including
polynomial, mechanical, electromechanical and biochemical systems.
