Loop Transfer Recovery
LTR tunes an LQG design so the observer-based loop recovers the robustness margins of the ideal full-state feedback loop.
The LQG robustness gap
Full-state LQR feedback has excellent guaranteed margins: at least 60 degrees of phase margin and a gain reduction tolerance down to one half. When the state must be estimated by a Kalman filter, those guarantees can vanish, because the observer dynamics interact with the plant in ways the separation principle hides from the nominal performance calculation. Doyle's famous example showed an LQG loop with arbitrarily small margins.
Recovering the target loop
Loop transfer recovery closes the gap by deliberately detuning the estimator. As a scalar recovery parameter is increased, the process-noise intensity in the filter design is scaled up, pushing observer poles toward plant zeros and driving the loop transfer function at the plant input (or output) toward the target full-state loop. In the limit the recovered loop reproduces the robust LQR margins.
Trade-offs
Recovery is not free. Pushing the recovery parameter high amplifies measurement noise, because the filter is told to trust the model less and the sensor readings more, and it requires the plant to be minimum-phase for full recovery, since observer poles chase plant zeros. Non-minimum-phase zeros in the right half plane block complete recovery and cap the achievable robustness.
- Detune the Kalman filter with a recovery parameter
- Loop approaches the robust full-state feedback target
- Requires a minimum-phase plant for full recovery
- High recovery amplifies measurement noise
Recovery can be done at the input or the output; the choice depends on where robustness is most needed. Input recovery protects against actuator-side uncertainty, output recovery against sensor-side effects.
LTR is a structured way to trade estimator noise rejection for loop robustness, and it foreshadows the frequency-weighted trade-offs made explicit in H-infinity and loop-shaping design.