|
We consider an nth-order SISO continuous nonlinear system of the form x equals f(x) + g(x).u with output y equals h(x) where x and u represent the states and the input, the objective is to transform the system in the state-apace using nonlinear feedback and a diffeomorphism (nonlinear transformation) to a linear system from input to output. This is the so-called Input- Output Feedback Linearization (IOFL) problem. In general, we assume the equations describing the system are not in normal (canonical) form. Assuming that the states x are measurable and the relative degree of the system is equal to the system order (n), a diffeomorphism z equals (phi) (x) can be found such that the system can be linearized in new coordinates z using only a nonlinear state feedback. Based on the traditional approaches in feedback linearization theory, the computation of the IOFL control law requires the exact knowledge of f, g, and h. If these are not known or partially known, an adaptive network can be used to learn such a control law. We employ high-order three layer networks to learn the IOFL control law based on a new method inspired by direct adaptive control architecture, details of which have been published elsewhere. The weights of the network (the controller) are adjusted using generalized delta rule algorithm. A linear reference model of order n is used as a teacher such that the closed-loop response of the resulting system follows the linear system response when both are excited with the same input. The diffeomorphism is implemented using a state variable filter (Paynter filter). Here we intend to report the experimental results obtained for a single link manipulator with flexible joint that is a feedback linearizable system. The transformation of nonlinear systems to linear ones is of interest since linear systems have been widely studied and there is a rich theoretical background as well as sophisticated design tools for controller synthesis. This new approach enables the discovery of IOFL control law without the exact knowledge of the system equations.
|
