DERIVATIVE-FREE KALMAN FILTER-BASED CONTROL OF NONLINEAR SYSTEMS WITH APPLICATION TO TRANSFEMORAL PROSTHESES

Derivative-free Kalman filtering (DKF) for estimation-based control of a special class of nonlinear systems is presented. The method includes a standard Kalman filter for the estimation of both states and unknown inputs, and a nonlinear system that is transformed to controllable canonical state space form through feedback linearization (FL). A direct current (DC) motor with an input torque that is a nonlinear function of the state is considered as a case study for a nonlinear single-input-single-output (SISO) system. A three degreeof-freedom (DOF) robot / prosthesis system, which includes a robot that emulates human hip and thigh motion and a powered (active) transfemoral prosthesis disturbed by ground reaction force (GRF), is considered as a case study for a nonlinear multi-input-multi-output (MIMO) system. A PD/PI control term is used to compensate for the unknown GRF. Simulation results show that FL can compensate for the system’s nonlinearities through a virtual control term, in contrast to Taylor series linearization, which is only a first-order linearization method. FL improves estimation performance relative to the extended Kalman filter, and in some cases improves the initial condition region of attraction as well. A stability analysis of the DKF-based control method, considering both estimation and unknown input compensation, is also presented. The error dynamics are studied in both frequency and time domains. The derivative of the unknown input plays a key role in the error dynamics and is the primary limiting factor of the closed-loop estimation-based control system stability. It is shown that in realistic systems the derivative of the unknown input is the primary determinant of the region of convergence. It is shown that the tracking error asymptotically converges to the derivative of the unknown input.

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