A sequential LMI approach to design a BMI-based multi-objective nonlinear observer

An observer for a nonlinear system may be required to satisfy multiple performance criteria such as minimum convergence rate and disturbance rejection, in addition to asymptotic stability. In such cases, the observer can no longer be designed using a linear matrix inequality. A bilinear matrix inequality (BMI) is needed instead and involves a non-separable product of the observer gain matrices and the Lyapunov positive definite matrix. This paper develops a technique to solve a BMI for such a multi-objective observer design problem. The BMI design condition is transformed into an eigenvalue problem and a convex-concave based sequential linear matrix inequality (LMI) optimization method is used to find a feasible solution to the BMI. The developed observer design method is applied to a robust automotive slip angle estimation problem, where the L₂ gain from the disturbance to the observer error is restricted be lower than 0.2 and the estimated states converges to the neighborhood of the real states within 0.3 s in the presence of uncertain vehicle dynamics.