PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions

This paper deals with the design of fractional PID controller for integer and fractional plants. A new analytic method is proposed, the developments are based on the expansion of the control loop signals as well as a chosen reference model input and output over a piecewise orthogonal functions, namely, Block pulse, Walsh and Haar wavelets. The generalized operational matrices of differentiation related to these bases which are fitting the Riemann-Liouville definition accurately are used to replace the fractional differential calculus by an algebraic one easier to solve. Thereafter, the controller tuning is elaborated simply with a matrix manipulation manner. At first, a least square is drawn to find only the controller gains, then a nonlinear function defined as a matrix norm is minimized to optimize the whole parameters. A variety of examples covering both integer and fractional systems and reference models are presented to show the validity of the technique. (C) 2009 Elsevier B.V. All rights reserved.