Fractional Order Control System And Controllers Design

Fractional order control is established on the idea of fractional order operators and theory of fractional order differential equations and its most significant and recent practical applications in automatic control has been extended robust control, iterative learning and adaptive control, path planning and motion control, systems identification, etc. In this paper, the fundamentals and applications of fractional order control were introduced.Fractional order Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. The very basic concepts regarding fractional order calculus were addressed and a numerical method for solution of the fractional order differential equations were determined.This analysis, as usual for the systems of integer order, will take into account the models or representations of these systems in different domains (time, Laplace and Z) to study their performance, discussing the conditions of stability, controllability and observability criteria and determining the error static coefficients. BODE's ideal transfer function and its model was introduced.Starting from the formulation of some possible models of fractional order systems, several approximations are discussed. For continuous models, some methods for obtaining an approximated rational function using the continued fractions expansion technique and curve fitting techniques are studied. For discrete models, approximations using the continued power series expansion and the continued fractions expansion technique in the Z domain are studied.A more general structure for the classic PID controller is proposed by using fractional integral and differential operators. A synthesis of fractional order PI~λD~μ controllers, analysis of their behavior and simulation methods in their closed loop behavior were pointed out.