This paper decomposes non-linear 2nd-order dynamics in two decoupled lst-order complex component dynamics. The stability of the 2nd-order dynamics can then be assesed based on the two decomposed non-linear 1st-order complex dynamics using the tools of contraction theory. Both diagonal and Jordan decompositions are performed. This result allows to place exact state-and time-dependent complex exponential convergence or contraction rates in a controller or observer design, ideally suited to the non-linear or time-varying continuous system. It hence extends the well established LTI eigenvalue placement of a Luenberger observer or Ackermann controller~(10,14) to the time-varying or non-linear case. Typical applications include the transonic control of an aircraft with strongly Mach or angle-of-attack dependent eigenvalues or the state-dependent eigenvalue placement of the inverted pendulum. The corresponding observer design is illustrated for a Mach dependent vertical channel dynamics of a navigation system, non-linear chemical plants or the Van-der-Pol oscillator. Simple extensions to higher-order systems composed of cascades of 2nd or lst-order dynamics are also discussed. The derivations are based on non-linear contraction theory, a comparatively recent dynamic system analysis tool whose results will be reviewed.
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