Loop shaping with closed-loop magnitude contours on the Bode plot

A weakness of classical loop shaping (of open-loop frequency responses) is that the closed-loop frequency response is implicitly rather than explicitly shaped. This paper defines fsbode and ftbode contours which are added to open loop Bode plots. The contours are gain and phase valued functions of frequency, found using a frequency-dependent desired magnitude for closed loop sensitivity or complementary sensitivity functions. The closed loop magnitude attains this frequency-dependent desired value at frequencies where the open-loop response intersects the contour. The paper shows a design example and presents an extension of the concept to SISO components of more general (MIMO) feedback systems with H_∞ objectives.