12.1: Introduction to Loopshaping

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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This section formalizes the notion of loopshaping for linear control system design. The loopshaping approach is inherently two-fold. First, we shape the open-loop transfer function (or matrix) \(P(s)C(s)\), to meet performance and robustness specifications. Once this is done, then the compensator must be computed, from from knowing the nominal product \(P(s)C(s)\), and the nominal plant \(P(s)\).

Most of the analysis here is given for single-input, single-output systems, but the link to multivariable control is not too difficult. In particular, absolute values of transfer functions are replaced with the maximum singular values of transfer matrices. Design based on singular values is the idea of \(L_2\)-control, or linear quadratic Gaussian (LQG) control and the loop transfer recovery (LTR).