In the field of control engineering, observability of uncertain nonlinear systems is often neglected and not examined. This is due to the complex analytical calculations required for the verification. Therefore, the aim of this work is to provide an algorithm which numerically analyzes the observability of nonlinear systems described by finite-dimensional, continuous-time sets of ordinary differential equations. The algorithm is based on definitions for distinguishability and local observability using a rank check from which conditions are deduced. The only requirements are the uncertain model equations of the system. Further, the methodology verifies observability of nonlinear systems on a given state space. In case that the state space is not fully observable, the algorithm provides the observable set of states. In addition, the results obtained by the algorithm allows insight into why the remaining states cannot be distinguished.

This paper deals with the design of linear observer-based state feedback controllers with constant gains for a class of nonlinear discrete-time systems in the form of a quasi-linear representation in presence of stochastic noise. For taking into account nonlinearities in the design of linear observer-based state feedback controllers, a polytopic modeling approach is investigated. An optimization problem is formulated to reduce the sensitivity of the controlled system towards stochastic input, state, and output noise with a predefined covariance. Due to the nonlinearities, the separation principle does not hold, thus, the controller and the observer have to be designed simultaneously. For this purpose, a Lyapunov-based method is used, which provides, in addition to the controller and observer gains, a stability proof for the nonlinear closed loop in a predefined polytopic domain. In general, this leads to nonlinear matrix inequalities. To solve these nonlinear matrix inequalities efficiently, we propose an approach based on linear matrix inequalities (LMIs) with a superposed iteration rule. When using this iterative LMI approach, a minimization task can be solved additionally, which desensitizes the closed loop to stochastic noise. The proposed method additionally enables the consideration of different linear closed loop structures by a unified Lyapunov-based framework. The efficiency of the proposed approach is demonstrated and compared with a classical LQG approach for a nonlinear overhead traveling crane.

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