Observer Design for Bilinear Systems Using Semidefinite Programming

This paper proposes a dynamic bilinear observer design methodology for a bilinear system. The main idea is to replace the constant observer gain with a dynamic observer system. As a result, it gives an extra degree of freedom in designing bilinear observers. The dynamic observer includes a state transition matrix and constant gain matrices, which pave the way for more control over the observer's transient properties. The proposed design method uses the Lyapunov stability approach and casts the design problem into a semidefinite programming problem. The semidefinite programming problem is solved using linear matrix inequality (LMI) tools. Semidefinite programming is used not to optimize an objective function but instead to find a feasible solution. The novelty of the proposed design method is that despite being a nonlinear observer, its stability does not depend on the input. In contrast, a nonlinear observer's stability generally depends on the input signal. A numerical example is given to illustrate the proposed design method.