2.9 Sensor Noise, Delay, and Sampling

Measurement quality is part of the architecture¶

A sampled measurement can be written

y_k=Cx(t_k-\tau_d)+v_k,

where $v_k$ is measurement noise and $\tau_d$ includes sensing, communication, filtering, computation, and actuation delay. Quantization, dropouts, bias, drift, asynchronous clocks, and calibration error may also matter.

Noise limits useful bandwidth¶

High-gain feedback can reject disturbances but also amplify measurement noise into actuator activity, structural vibration, heat, and wear. Filtering reduces noise but adds phase lag and delay. A physical design that requires aggressive feedback near the noise floor may be inferior to one with slightly better passive behavior.

Sensor precision therefore trades against actuator demand and plant design. A more accurate sensor may enable a lighter structure or smaller actuator. Conversely, improving physical damping may permit cheaper sensing.

Delay consumes phase margin¶

A pure delay contributes phase $-\omega\tau_d$ . As bandwidth increases, the same delay becomes more damaging. Delays can make an attractive centralized architecture unusable and favor local sensing and control, faster communication, lower bandwidth, or a different plant.

For a flexible structure, moving a mode to higher frequency may appear beneficial, but if the controller bandwidth follows while delay is fixed, phase margin can shrink. CCD must evaluate the complete sampled-data loop.

Aliasing and sampling¶

Sampling below twice a signal frequency causes aliasing, but merely satisfying the Nyquist inequality is not enough for control. Anti-alias filters, transient response, delay, and closed-loop bandwidth require faster sampling. Flexible modes above the intended bandwidth can still alias into measurements and destabilize feedback if filtering is inadequate.

Architecture consequences¶

Sensor choice affects more than $C$ :

location changes modal visibility;
technology changes bandwidth and noise;
filtering changes delay;
communication changes latency and dropout risk;
redundancy changes reliability, mass, and cost; and
sampling changes computation and energy.

Activity 2.9: delay budget¶

Activity 2.9: delay budget

A vision-based inverted-pendulum controller has bandwidth $\omega_b=12$ rad/s and a required delay-phase contribution no larger than $20^\circ$ . Its present end-to-end delay budget is

Stage	Delay
Camera exposure	6 ms
Image transfer	4 ms
Vision processing	12 ms
State estimation	3 ms
Control computation	1 ms
Communication	5 ms
Actuator response	8 ms

Use $\phi_d=\omega_b\tau_d(180/\pi)$ . The original camera has angular-noise standard deviation $0.40^\circ$ , and feedback gain $K=20$ N m/rad gives torque-noise estimate $\sigma_u=K\sigma_\theta$ .

The following upgrades are available:

Upgrade	New stage delay	Cost	Sensor noise
GPU vision processing	4 ms	USD 180	unchanged
Local communication	1 ms	USD 100	unchanged
Faster actuator	4 ms	USD 250	unchanged
Short-exposure camera	3 ms	USD 90	$0.80^\circ$
Global-shutter camera	3 ms	USD 220	$0.50^\circ$

The architecture must satisfy both $\phi_d\leq20^\circ$ and $\sigma_u\leq0.20$ N m.

Calculate the original total delay, delay phase, and remaining or exceeded phase budget.
Convert each camera-noise value to radians and calculate $\sigma_u$ . Identify camera options that violate the torque-noise constraint.
Evaluate all single upgrades and all two-upgrade combinations. Find the least-cost architecture satisfying both delay and noise constraints.
For the chosen architecture, report total delay, phase, torque noise, cost, and phase margin relative to the $20^\circ$ limit.
If the controller bandwidth increases to 18 rad/s, recompute the allowable total delay and determine whether the chosen architecture remains feasible.
Classify each delay term as sensing, computation, communication, estimation, control, or physical actuation. State which reductions require information-architecture changes and which require physical redesign.