PIλDμPIλDμ 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.