Observer design for Lur'e-type systems via injection of a reconstructed nonlinear output
Observer design for Lur'e-type systems typically reduces to solving a linear matrix inequality (LMI). In certain cases, the observer gain may grow unbounded with the coupling strength. We propose reconstructing key nonlinear terms via a bank of second-order sliding-mode observers and injecting them into a Luenberger observer as additional measurements. This feeds a second correction channel that attenuates the nonlinear coupling in the error dynamics. The convergence of the combined observer is guaranteed under a new LMI which contains the classical one as a special case. We exhibit parameter regimes where classical designs require impractically large gains, while the proposed design maintains moderate gains. A stability analysis of the proposed observer characterizes the trade-off between the two designs and identifies the noise regimes in which the proposed design is preferable. We illustrate the approach on a Wilson-Cowan network showing the trade-off between the nonlinear coupling strength, observer gain, and noise.