H-infinity Loop-Shaping


button2H-infinity Loop-Shaping

By: Song Yoong Siang (PhD Student)



                H-infinity loop-shaping is one of the most important example of a robust control technique. This method minimizes the sensitivity of a system over its frequency spectrum, and this guarantees that the system will not greatly deviate from expected trajectories when disturbances enter the system.

                H-infinity loop-shaping control incorporates loop shaping to obtain performance and robust stability trade off, and a particular H infinity optimization problem to guarantee closed-loop stability and a level of robust stability at all frequency.

                In loop shaping approach, designer selects a controller which achieves sufficiently high open-loop gain at low frequency to guarantee concerning closed loop performance can be made at these frequencies.

                In H infinity synthesis, designer specifies closed loop objectives in terms of requirements on the singular value of weighted closed loop transfer functions, and a stabilizing controller is obtained which satisfies these requirements.

                The H-infinity loop-shaping control has two main stages:

  1. Loop shaping is used to shape the nominal plant singular values to give desired open loop properties at frequencies of high and low loop gain.
  2. The normalized coprime factor H infinity problem is used to robustly stabilize the shaped plant.

                This control technique had the following benefits:

  1. Easy to apply: commercial software handles the hard math
  2. Easy to implement: standard transfer functions and state space methods can be used
  3. Plug and play: no need re-tuning on an installation by installation basis



  1. McFarlane, D. C., and Glover, K., 1992, A loop shaping design procedure using H synthesis. IEEE Transaction on Automatic Control, Vol-37, No. 6, 759-769.