Generalized optimal zero phase error tracking controller design

This paper presents a simple design method of discrete-time feedforward controllers that provide the overall transfer function with the following frequency characteristics, (i) The phase is equal to zero for all frequencies, (ii) The gain is equal to one at given frequencies. (iii) The error between the gain and unity for a given frequency range is minimized under given preview steps. The contributions of this paper are as follows. First, a new approach based on the spectral factorization is proposed and the class of all controllers satisfying the above conditions (i) and (ii) is parametrized using the solution of a Diophantine equation, i.e., the controller is obtained in an explicit form. With this explicit parametrization, the optimal feedforward controller is obtained by an ordinary least square method. The design method proposed in this paper is simple and straightforward, whereas the design method in previous result requires the solution of an optimization problem with troublesome inequality constraint and involves trial and error. Secondly, the frequencies at which the gain is made equal to unity can be chosen arbitrarily, while, in previous result, the frequency is restricted to zero. Finally, the effectiveness of the proposed controller is demonstrated by simulation.