A time lag between a change of an input and the corresponding change of the output that real dynamical systems often show has a whole range of causes. For needs of mathematical modelling, it is aggregated into a total phenomenon called time delay or dead time . In process control, one often encounters systems that can be described by transfer functions with time delays . If a dynamical system with time delay is modelled as a time invariant linear system, its transfer function (rational function) becomes due to time delay a transcendental function . For design and analysis purposes, these delays are usually approximated by rational transfer functions . This is usually carried out using delay approximation methods Viz. Taylor series expansion, Padé.
PI and PID controllers have been at the heart of control engineering practice for seven decades . The use of the PI or PID controller is ubiquitous in industry. It has been stated, for example, that in process control applications, and more than 95% of the controllers are of PI or PID type [1, 4, 5]. PID controllers can assure satisfactory performances with a simple algorithm for a wide range of processes [6, 7].
The Internal Model Control (IMC) provides a progressive, effective, natural, generic, unique, powerful, and simple framework for analysis and synthesis of control system performance [8-11]. The easiness and enhanced performance of the IMC based tuning rule, and the analytically derived IMC-PID tuning techniques have appealed the attention of the industrial users, in the past decade [10, 11]. The well-known IMC-PID tuning rule provides a clear compromise in the midst of closed loop performance and robustness to model uncertainties, and is achieved by only one user-defined tuning parameter, which is directly related to the closed-loop time constant[3, 8, 10 - 13].
IMC – PID Design:
The widely used approximate or predictive models of the process, more specifically chemical processes are the FOPTD, Equation (1). Garcia and Morari [12, 14] introduced internal model control; IMC it is characterized as a controller where the process model is explicitly an integral part of the controller. The Design process of IMC involves factorizing the predictive plant model as invertible and non-invertible parts depicted in Equation (2) by simple factorization or all pass factorization [8, 12, 13, 15, 16]. The Internal model controller in Equation (3) is the inverse of the invertible portion of the plant model , IMC filter , Equation (4) is used to realize the controller.
The design of the IMC controller is