Most approaches to feedback synthesis are numerical design techniques that produce their results in form of numerical values for the controller parameters. In contrast, analytical design techniques produce the controller parameters as explicit expressions that are a function of open-loop system parameters and the desired closed loop. Advantages include fast commissioning in terms of physical system parameters, gain scheduling without extra design effort and analytical sensitivity insights. So far, it has not been known how to derive analytical PID expressions for higher than second-order systems. The paper proposes a technique to address this problem, analyses its properties, demonstrates computational simplicity and shows that its performance is comparable to known numerical techniques.