A Frequency Domain Quantitative Technique for Robust Control System Design

Most control techniques require the use of a plant model during the design phase in order to tune the controller parameters. The mathematical models are an approximation of real systems and contain imperfections by several reasons: use of low-order descriptions, unmodelled dynamics, obtaining linear models for a specific operating point (working with poor performance outside of this working point), etc. Therefore, control techniques that work without taking into account these modelling errors, use a fixed-structure model and known parameters (nominal model ) supposing that the model exactly represents the real process, and the imperfections will be removed by means of feedback. However, there exist other control methods called robust control techniques which use these imperfections implicity during the design phase. In the robust control field such imperfections are called uncertainties, and instead of working only with one model (nominal model), a family of models is used forming the nominal model + uncertainties. The uncertainties can be classified in parametric or structured and non-parametric or non-structured. The first ones allow representing the uncertainties into the model coefficients (e.g. the value of a pole placed between maximum and minimum limits). The second ones represent uncertainties as unmodelled dynamics (e.g. differences in the orders of the model and the real system) (Morari and Zafiriou, 1989). The robust control technique which considers more exactly the uncertainties is the Quantitative Feedback Theory (QFT). It is a methodology to design robust controllers based on frequency domain, and was developed by Prof. Isaac Horowitz (Horowitz, 1982; Horowitz and Sidi, 1972; Horowitz, 1993). This technique allows designing robust controllers which fulfil some minimum quantitative specifications considering the presence of uncertainty in the plant model and the existence of perturbations. With this theory, Horowitz showed that the final aim of any control design must be to obtain an open-loop transfer function with the suitable bandwidth (cost of feedback) in order to sensitize the plant and reduce the perturbations. The Nichols plane is used to achieve a desired robust design over the specified region of plant uncertainty where the aim is to design a compensator C(s) and a prefilter F(s) (if it is necessary) (see Figure 1), so that performance and stability specifications are achieved for the family of plants. 17