  This paper derives for non-linear, time-varying and feedback linearizable
systems simple controller designs to achieve specified state-and timedependent
complex convergence rates. This approach can be regarded as a general
gain-scheduling technique with global exponential stability guarantee. Typical
applications include the transonic control of an aircraft with strongly Mach or
time-dependent eigenvalues or the state-dependent complex eigenvalue placement
of the inverted pendulum. As a generalization of the LTI Luenberger observer a
dual observer design technique is derived for a broad set of non-linear and
time-varying systems, where so far straightforward observer techniques were not
known. The resulting observer design is illustrated for non-linear chemical
plants, the Van-der-Pol oscillator, the discrete logarithmic map series
prediction and the lighthouse navigation problem. These results [23] allow one
to shape globally the state- and time-dependent convergence behaviour ideally
suited to the non-linear or time-varying system. The technique can also be used
to provide analytic robustness guarantees against modelling uncertainties. The
derivations are based on non-linear contraction theory [18], a comparatively
recent dynamic system analysis tool whose results will be reviewed and
extended.
