Robust PID control using generalized KYP synthesis: direct open-loop shaping in multiple frequency ranges

In this paper, a method for designing PID controllers to meet multiple frequency-domain inequalities (FDI) specifications on the open-loop transfer function in finite and semi-infinite frequency ranges is presented. The method, which is based on the generalized Kalman-Yakubovich-Popov (GKYP) lemma, enables direct loop shaping through LMI optimization without frequency gridding or weights. The resulting synthesis condition is nonconservative; its infeasibility implies that no PID controller exists to meet the specifications. The design method extends to systems with parametric uncertainties using Lyapunov function, which depends on the parameter in a linear fractional manner. The robust PID control design is also reduced to an LMI optimization problem, albeit with potential conservatism. The effectiveness of the GKYP synthesis has been demonstrated through several numerical examples. In addition to PID control design, robust GKYP synthesis applies to open-loop shaping and filtering problems, where the poles of the controller or filter are fixed.