Observer Error Reduction via Direct State Reconstruction

State observers inherently suffer from error peaking in the face of unknown initial states and un-modeled disturbances. This issue can make their state estimates useless during transient periods and whenever disturbances follow in quick succession. In this article, we propose an approach to fighting observer error peaking at its root. By directly reconstructing the system state from the output and its derivatives, we obtain an approximation of the true state that is used to drastically reduce the transient error, even if the output measurement is noisy. We provide proofs of stability of the resulting nonlinear modification to the standard linear observer structure and illustrate its effectiveness by means of simulation. We prove that the principle of separation holds for this observer structure and discuss a rigorous design procedure which guarantees stability. In particular, we show that the matrix for feedback of the direct state reconstruction can be chosen in a convex cone in the space of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}^{n\times n}$</tex-math></inline-formula> matrices. This cone can easily be calculated explicitly.