Nonlinear observer design for a class of nonlineardynamical systems using polynomial optimization

In this thesis, nonlinear observer design problem is examined for nonlinear dynamical systems having polynomial type terms using Sum Of Squares (SOS), Linear Matrix Inequalities (LMI) and Semidefinite programming (SDP) tools for state estimation problems. First, the stabiliy analysis problems are discussed in the context of optimization that is frequently in this study. A Lyapunov function is parameterized in a way that is suitable for formulating the problem of analysis as a problem of an optimization problem that contains LMI terms, affine equalities and can be solved by cone solvers such as MOSEK, PENLAB and SEDUMI. The available tools is utilized to structure the optimization problem into a form that can be solved by a cone solver. Since the problem in this form is a convex optimization problem due to the convexity of the inequalities and the affine equality constraints, this problem can be solved reliably. Using the tools that are used to construct the analysis problem as an optimization problem, the same techniques are applied to the problem of input-to-state stabiliy (ISS) analysis problem. Using the constraints that are imposed on the ISS-Lyapunov function, this analysis problem can also be formulated in a way that is similar to asymptotical stabiliy problems. ISS analysis problem is pertinent to this study in that there is a connection between observer synthesis problem and ISS analysis problem since the observer needs to satisfy constraints that are similar to the constraints imposed on an ISS-Lyapunov function. The observer synthesis problem can be directly formulated as a nonlinear control design problem with disturbance rejection requirements in that the disturbance in that new problem is state estimate terms. Using this, the problem can be expressed as an optimization problem having bilinear terms. Due to the structure present in this optimization problem, using differentiation through optimization problem algorithms, one can optimize the parameters of a nonlinear observer. This process is easy to express and straightforward to implement. The results of the study is given and stabiliy certificates are constructed using convex optimization algorithms.