Observability in systems with extreme multistability

In control engineering and dynamical systems, obtaining accurate measurements of a system’s state vector can be challenging due to the inaccessibility of certain internal states or the prohibitive cost of measurement. To address these issues, observers are employed to reconstruct the system’s state. While linear observers have proven effective for linear systems, applying them to nonlinear systems presents additional challenges and opportunities. In this paper, we explored the design and implementation of a linear observer applied on a 6-variable dynamical system, namely extreme multistable Rössler system (EMRS). Our proposed methodology leverages observability indices and graphical representations to enhance the linearization process at points of maximum observability. This approach allows for generating the gain set for a Luenberger observer, thereby improving its performance in estimating the states of a complex nonlinear system such as the EMRS. Our numerical and experimental results demonstrate the effectiveness of linearization using observability coefficients for the design of a linear observer. This observer can efficiently track and reconstruct the dynamics of a multi-stable nonlinear system, enabling the use of a linear observer on a nonlinear system.