Application of Pareto Approach to Improve the State Dependent Riccati Equation (SDRE) Controller Performance

The main objective of this work is to study the State Dependent Riccati Equation (SDRE) regulator; an adaptive Linear Quadratic Regulator (LQR) which allows to deal with the non linearities of the system to be controlled. In order to use this controller, a nonlinear mathematical model of a flexible rotatory beam is built through the Lagrangian formulation which can represent a rigid-flexible satellite. The flexible displacement is modelled using the assumed modes theory and a structural damping is added applying the Rayleigh technique. There are two main objectives related to control: the first one is to control the hub angular position and the second one is the need to minimize flexible displacements of the satellite panel. Doing computational simulations, it is possible to draw the performance map of the system which map all SDRE reachable performances. Then, a sorting algorithm enables to get the Paretos border which represents the set of optimal performances. On the other hand, analyzing the influence of the weight matrixes terms, it is shown that it is possible to get the Paretos border performances using only a few terms of the SDRE weight matrixes. On the basis of this analysis, a law enabling to get weight matrixes values in function of a required performance is developed. Last of all, state dependent weight matrixes are used to show that they can improve the system performance. Based on the results, it turned out that the SDREs performance is better than the LQRs one, not only because it can deal with non linearities, but also because its design is more flexible.