control//controller design//LQR//Bryson's rule

Bryson's rule is a heuristic for choosing the first weights of an LQR by normalizing each state and each input by the largest value you are willing to tolerate, and it turns the abstract question *what are \(Q\) and \(R\)?* into one an engineer can answer from the requirements sheet. The weights are diagonal:

Bryson's rule is a heuristic for choosing the first weights of an LQR by normalizing each state and each input by the largest value you are willing to tolerate, and it turns the abstract question what are QQQ and RRR? into one an engineer can answer from the requirements sheet. The weights are diagonal:

Qii=1(max acceptable xi)2,Rjj=1(max acceptable uj)2.Q_{ii}=\frac{1}{(\text{max acceptable } x_i)^2},\qquad R_{jj}=\frac{1}{(\text{max acceptable } u_j)^2}.Qii​=(max acceptable xi​)21​,Rjj​=(max acceptable uj​)21​.

Each term of the cost then equals one when its variable reaches what you tolerate, so no state dominates the cost merely because it is measured in small units. A position in metres and an angle in radians become comparable: 10 cm of position error and 10° of tilt both count as one.

For the lateral axis of a small drone, accepting 0.1 m, 0.5 m/s, 10° and 90°/s, with 0.5 N·m of torque, the rule gives Q=diag⁡(100, 4, 32.8, 0.41)Q=\operatorname{diag}(100,,4,,32.8,,0.41)Q=diag(100,4,32.8,0.41) (the angles converted to radians) and R=4R=4R=4. The LQR built on those numbers settles a 1 m step in about a second; it also asks for 5 N·m in the first instant, which shows what the rule is: a starting point that makes the trade readable, never a promise that the limits will be respected.

It is a normalization, not a tuning. After the first design you scale individual weights while watching the simulated response: double a QiiQ_{ii}Qii​ if that state wanders too far, raise RRR if the motors work too hard. Since only the ratio between QQQ and RRR matters, a single overall factor (the aggressiveness slider of the planar drone) sweeps from lazy to nervous.

The maximum acceptable values come from the requirements, and writing them down forces the conversation that should happen anyway: how much drift is allowed during an inspection, how much torque the arms can take.

A hard limit is not a weight. Bryson's rule can make violations expensive, but the LQR's command still grows with the error; a limit that must hold needs MPC or a safety filter.

The same normalization appears on the estimation side, where RRR is the sensor's variance and its inverse is the weight a measurement deserves (measurement noise); there the numbers are measured instead of chosen (estimation-control duality).