Robust H∞ PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis

PID controller has been extensively used in industries since 1940s and still the most often implemented controller today. The PID controller can be found in many application areas: petroleum processing, steam generation, polymer processing, chemical industries, robotics, unmanned aerial vehicles (UAVs) and many more. The algorithm of PID controller is a simple, single equation relating proportional, integral and derivative parameters. Nonetheless, these provide good control performance for many different processes. This flexibility is achieved through three adjustable parameters of which values can be selected to modify the behaviour of the closed loop system. A convenient feature of the PID controller is its compatibility with enhancement that provides higher capabilities with the same basic algorithm. Therefore the performance of a basic PID controller can be improved through judicious selection of these three values. Many tuning methods are available in the literature, among with the most popular method the Ziegler-Nichols (Z-N) method proposed in 1942 (Ziegler & Nichols, 1942). A drawback of many of those tuning rules is that such rules do not consider load disturbance, model uncertainty, measurement noise, and set-point response simultaneously. In general, a tuning for high performance control is always accompanied by low robustness (Shinskey, 1996). Difficulties arise when the plant dynamics are complex and poorly modeled or, specifications are particularly stringent. Even if a solution is eventually found, the process is likely to be expensive in terms of design time. Varieties of new methods have been proposed to improve the PID controller design, such as analytical tuning (Boyd & Barrat, 1991; Hwang & Chang, 1987), optimization based (Wong & Seborg, 1988; Boyd & Barrat, 1991; Astrom & Hagglund, 1995), gain and phase margin (Astrom & Hagglund, 1995; Fung et al., 1998). Further improvement of the PID controller is sought by applying advanced control designs (Ge et al., 2002; Hara et al., 2006; Wang et al., 2007; Goncalves et al., 2008). In order to design with robust control theory, the PID controller is expressed as a state feedback control law problem that can then be solved by using any full state feedback robust control synthesis, such as Guaranteed Cost Design using Quadratic Bound (Petersen et al., 2000), H∞ synthesis (Green & Limebeer, 1995; Zhou & Doyle, 1998), Quadratic Dissipative Linear Systems (Yuliar et al., 1997) and so forth. The PID parameters selection by