Robust H_infinity Filter Design for Lipschitz Nonlinear Systems via Multiobjective Optimization

In this paper, a new method of H_infinity observer design for Lipschitz nonlinear systems is proposed in the form of an LMI optimization problem. The proposed observer has guaranteed decay rate (exponential convergence) and is robust against unknown exogenous disturbance. In addition, thanks to the linearity of the proposed LMIs in the admissible Lipschitz constant, it can be maximized via LMI optimization. This adds an extra important feature to the observer, robustness against nonlinear uncertainty. Explicit bound on the tolerable nonlinear uncertainty is derived. The new LMI formulation also allows optimizations over the disturbance attenuation level (H_infinity cost). Then, the admissible Lipschitz constant and the disturbance attenuation level of the H_infinity filter are simultaneously optimized through LMI multiobjective optimization.

💡 Research Summary

The paper addresses the design of an H∞ observer (filter) for nonlinear systems that satisfy a Lipschitz condition, formulating the problem entirely in terms of linear matrix inequalities (LMIs). The authors start by modeling the plant as (\dot x = A x + B u + \Phi(x) + D w), (y = C x), where (\Phi(\cdot)) is a nonlinear mapping with Lipschitz constant (\gamma) and (w) denotes an unknown exogenous disturbance. The observer is chosen in the classic Luenberger form (\dot{\hat x}=A\hat x + B u + L(y-C\hat x)). Defining the estimation error (e = x-\hat x) and a quadratic Lyapunov function (V = e^{!T} P e) ((P>0)), the authors derive a differential inequality that simultaneously enforces (i) an exponential decay rate (\alpha) (i.e., (\dot V + 2\alpha V) term), (ii) an H∞ performance bound (\mu) on the transfer from disturbance (w) to the error output, and (iii) the Lipschitz bound on the nonlinear term. By applying the S‑procedure and the Lipschitz inequality (|\Phi(x_1)-\Phi(x_2)| \le \gamma|x_1-x_2|), the inequality is recast as an LMI that is linear in the decision variables (P), (L), the scalar (\gamma), and the attenuation level (\mu).

A crucial observation is that (\gamma) appears linearly in the LMI, which allows it to be treated as an optimization variable rather than a fixed design parameter. Consequently, the maximum admissible Lipschitz constant (\gamma_{\max}) – a measure of robustness against nonlinear uncertainty – can be directly maximized within the same convex framework that yields the observer gain and the Lyapunov matrix.

The paper then introduces a multi‑objective optimization (MOO) layer. Two competing objectives are identified: (1) minimizing the H∞ attenuation level (\mu) to improve disturbance rejection, and (2) maximizing (\gamma) to enlarge the set of nonlinearities that the filter can tolerate. The authors propose a weighted‑sum scalarization (\min ; w_1 \mu + w_2 / \gamma) subject to the LMIs, where the weights (w_1, w_2) allow the designer to trade off between disturbance attenuation and nonlinear‑uncertainty robustness. Alternatively, a Pareto frontier can be generated by sweeping the weights, giving a clear picture of the achievable trade‑offs. The decay rate (\alpha) is also incorporated as a design constraint, guaranteeing exponential convergence of the estimation error.

Numerical examples illustrate the methodology. In a two‑dimensional system with (\Phi(x) =