Automatic Loop-Shaping of H ∞ /μ Problem in QFT Using Interval Consistency Based Hybrid Optimization R.Jeyasenthil, P.S.V. Nataraj and Harsh Purohit 1 Introduction Quantitative feedback theory (QFT) [1] is a frequency-domain method of robust control. A key step in QFT is one of synthesizing the controller using loop-shaping method. The loop-shaping is a graphical method to design a controller. In this step, a controller is designed by adding the poles and/or zeros along with gain until the nominal loop transmission function satisfies the performance specification constraint at each frequency. Traditionally, the manual loop-shaping depends on the designer experience and skill, so automatic loop-shaping (ALS) is preferred. It offers the possibility of finding a controller faster and better than the manual one. Existing methods [2–4] attempt to solve this nonlinear and nonconvex problem using convex (or) linear programming techniques, which lead to conservative designs. An ALS based on reliable deterministic global optimization (namely, interval branch and bound) is proposed in [5]. To speed up this, a method based on hybrid optimization with geometric constraint propagation idea is presented in [6]. These methods, for the first time in the literature, find a global optimum for a particular cho- sen loop structure [7]. Recently, the QFT controller optimization problem has been formulated as an Interval Constraint Satisfaction Problem (ICSP) with performance specification inequality as constraints [8, 9]. This ICSP formulation uses interval consistency technique (Hull and Box Consistency) to remove the inconsistent values which are not part of the solution [10]. The ICSP formulation gives all the feasible controllers solution in which optimal one is picked up manually based on objec- tive function (e.g.minimum high frequency gain). The main drawback of this ICSP formulation is the computational demand of its search for all feasible solutions. R. Jeyasenthil (B ) · P.S.V. Nataraj · H. Purohit IDP in Systems and Control Engineering, Indian Institute of Technology (IIT), Bombay, India e-mail: jeya@sc.iitb.ac.in P.S.V. Nataraj e-mail: nataraj@sc.iitb.ac.in H. Purohit e-mail: harsh.purohit@sc.iitb.ac.in © Springer International Publishing AG 2018 M. Ceberio and V. Kreinovich (eds.), Constraint Programming and Decision Making: Theory and Applications, Studies in Systems, Decision and Control 100, DOI 10.1007/978-3-319-61753-4_12 89