Robust Coupling Linear State Observer- Controller Design for MIMO Systems with Mismatched and Unstructured Uncertainties

In paper [1] a new robust decoupling state observer including an extra non-linear term has been proposed by Gu and Poon. This extra term is used to overcome the difficulty due to the unstructured parameter perturbation. It was highlighted that the nonlinear observer scheme requires the solution to a quadratic matrix inequality which is not straightforward. This inequality was rewritten as a quadratic Riccati equation by introducing a parameter  which was successfully solved by proposed algorithms. However, the proposed observer possesses some shortages, for example, strictly speaking, the observer error dynamics is a function of error, plant state and extra term, then the origin e=0 is not equilibrium point because plant state and extra term can increase unboundedly. Therefore, selected Lyapunov function is not a good candidate and observer state error does not go to zero. In this paper, a simple linear Luenberger robust coupling state observer – controller design for the linear MIMO systems with mismatched and unstructured parameter perturbations is considered. The same observer problem mentioned above is solved only by using a simple linear Luenberger scheme without including any extra non-linear term. The simple linear Luenberger observer scheme is considered as a completely dual form to the linear multivariable controller and investigated together with controller. Then the design parameters of coupling closed-loop observer-controller system are selected such that the observer error dynamics and plant state equations are globally asymptotically stable. The stability conditions are formulated in terms of two quadratic Riccati equations and matrix inequality. Therefore, the coupling observer–controller laws can be constructed from the positive definite solutions of these algebraic Riccati equations and matrix inequality. Reduced design methodology of completely symmetric dual coupling closed loop observer-controller system is also presented. KeywordsState observer, coupling system, robustness, asymptotic stability, parameter perturbation.