Loop Shaping
Loop shaping designs a controller by directly molding the open-loop frequency response to meet performance and robustness specifications.
Shaping the loop gain
Loop shaping designs the compensator so that the open-loop transfer function L equal to the plant times the controller has a desired magnitude and phase across frequency. The specifications translate cleanly: high loop gain at low frequency gives good tracking and disturbance rejection, low loop gain at high frequency gives noise rejection and robustness, and a well-behaved crossover in between sets bandwidth and stability margins. The designer sculpts L to hit these targets.
The crossover region
The gain-crossover frequency, where the loop magnitude passes through one, sets the closed-loop bandwidth and hence speed. Near crossover the slope of the magnitude and the phase are constrained by Bode's gain-phase relationship: a magnitude falling at 20 decibels per decade corresponds to about minus 90 degrees of phase, leaving healthy phase margin, while a 40-decibel-per-decade slope crowds the phase toward instability. Good loop shape rolls off gently through crossover, steeply elsewhere.
Classical and modern
Classically, lead compensators add phase near crossover to boost margin, lag compensators raise low-frequency gain for accuracy, and lead-lag combines both. The modern H-infinity loop-shaping procedure of McFarlane and Glover formalizes this: pre- and post-compensators shape the plant's singular values to a desired loop, then a robust stabilizing controller is synthesized for the shaped plant, combining classical intuition with a guaranteed robustness margin.
- High loop gain low, low loop gain high, clean crossover
- Crossover frequency sets closed-loop bandwidth
- Bode gain-phase relation constrains crossover slope
- H-infinity loop shaping adds a guaranteed robustness margin
Loop shaping is intuitive because each specification maps to a region of the frequency axis, and it scales from single loops to multivariable design through the singular values of the loop matrix.
For a design-stage plasma-shaping loop, loop shaping would set low-frequency accuracy and high-frequency robustness with an explicit crossover, then be robustified via the H-infinity procedure, all in simulation. It is the bridge from classical to modern frequency-domain design.