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Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of noninteger order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.

This paper presents a new practical tuning method for fractional order proportional and integral (FO-PI) controller. The plant to be controlled is mainly first order plus delay time (FOPDT). The tuning is optimum in the sense that the load disturbance rejection is optimized yet with a constraint on the maximum or peak sensitivity. We generalized Ms constrained integral (MIGO) based controller tuning method to handle the FO-PI case, called F-MIGO, given the fractional order α. The F-MIGO method is then used to develop tuning rules for the FOPDT class of dynamic systems. The final developed tuning rules only apply the relative dead time τ of the FOPDT model to determine the best fractional order α and at the same time to determine the best FO-PI gains. Extensive simulation results are included to illustrate the simple yet practical nature of the developed new tuning rules. The tuning rule development procedure for FO-PI is not only valid for FOPDT but also applicable for other general class of plants.

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