Output- and state-dependent Riccati equation: An output feedback controller design

Abstract

An output- and state-dependent Riccati equation (OSDRE) is investigated for a set of coupled k-th order ordinary differential equations (ODEs) in the framework of nonlinear optimal control, in terms of stability, optimality, and application. A state-feedback control is a dominant part of the SDRE in the literature in comparison with an output feedback design. The output feedback design removes outer layers of kinematics or coordinate transformations in some systems and directly controls the output dynamics. This enables efficient control over many systems in which there is an indirect relation between outputs and states such as robots, mobile manipulators, autonomous vehicles, unmanned aerial vehicles, etc. To present the OSDRE, the co-state vector of the optimal control must include both state and output vector; and then the optimality conditions result in a control law and a Riccati equation, which are derived based on the output vector. The application of the OSDRE is devoted to the control of free-floating space manipulators (FFSM). The end-effector position of the FFSM is a function of mobile base coordinates (position and orientation) that are not actuated due to the free-floating condition. Then the output feedback SDRE directly controls the end-effector and guides the mobile base to provide enough reaction force/moment to push the gripper towards the desired end-point. So, the workspace analysis is crucial since the motion of the end-effector depends on the distance and configuration between the initial and final conditions. Simulation results for a mathematical third-order coupled ODE and a three-link FFSM confirm success in the control of the output dynamics. It was shown that the output of the mathematical model successfully tracked the desired output regardless of the individual state regulation. The FFSM has also tracked the desired output position with an accuracy of 0.02% of the travel distance.