Actuator-Aware Observer Inputs for Cascade ADRC of Quadrotors Under Rotor Lag, Measurement Delay and Sensor Noise

Xuan-Quy Dao; Ngoc-Bich Le; Le-Minh Nguyen

cs.ROOct 29, 2026

Actuator-Aware Observer Inputs for Cascade ADRC of Quadrotors Under Rotor Lag, Measurement Delay and Sensor Noise

Authors: Xuan-Quy Dao, Ngoc-Bich Le, Le-Minh Nguyen

Organizations: Eastern International University, Binh Duong, Vietnam · Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Abstract

In sampled cascade linear active disturbance rejection control (LADRC) of quadrotors, the extended-state observer typically receives the instantaneous command, although the delayed measurement reflects an earlier, rotor-filtered acceleration. This paper proposes actuator-aware (AA) observer inputs: altitude and attitude observers use limited virtual inputs filtered by an assumed rotor model and aligned with the measurement delay, and horizontal observers use acceleration reconstructed from measured attitude. Fixed gains are selected on the discrete-time model under sensitivity and step-response requirements and stationary covariance ceilings on each channel's control variance and increment and on the noise-induced inner-loop increment of the full cascade. In paired set-point trials, PID-Budget, AA-Budget and AA-HalfGain reach mean RMSEs of 65.5, 19.2 and 16.9~~mm; under a stricter common cascade ceiling, the AA designs reach 29.9 and 26.4~~mm versus 72.3mm for a PID that uses roughly double their rotor-command increment. PX4 software-in-the-loop runs reproduce these results. Behind EKF2, a noise model fitted to the estimation errors predicts the roll/pitch and yaw increments within 5% on held-out and out-of-sample runs and the total rotor-command increment within 14%. Redesigned for this stack under the same sensitivity and response specification and estimator-aware ceilings, AA-Budget-E reduces the set-point RMSE from 59.1mm (PID-E) to 44.0~mm with a lower rotor-command increment in every paired set-point and lemniscate run (mean S6 rotor-command-increment reductions relative to PID-E: 45% before and 15% after output quantization); AA-HalfGain-E beats PID-E only on the lemniscate. The evidence is simulation-based, and the ceilings are nominal tuning constraints.

Figures & tables

Study	Method	Noise	Actuator dynamics	Delay	$\mathbf{\Delta u}$ limit	Cascade path	Tuning/design	Validation
[ 5 ]	LADRC/GPI cascade	–	–	–	–	–	observer poles	flight
[ 16 ]	ADRC	–	–	–	–	–	Kriging surrogate	HIL
[ 17 ]	LADRC/fuzzy	–	–	–	–	–	modified crayfish	simulation
[ 19 ]	TD3–ADRC	–	sim. a	–	– b	–	TD3	bench
[ 25 ]	model-based ESO	–	yes	–	–	–	actuator-model ESO	simulation
[ 26 ]	RADRC/RESO	–	–	yes e	–	–	delay synchronization	analysis

TABLE I: Selected Features of Related Methods

Fig. 1: Actuator-aware cascade LADRC (AA-Budget and AA-HalfGain). Online loop (dashed frame, every

T_{s}

): outer LADRC, tilt/thrust inversion with measured angles, inner LADRC, thrust/torque limits, rotor allocation with clipping, and the plant with rotor lag; blocks marked M read the delayed noisy measurement bus. Red blocks are the two observer-input paths: measured-attitude inputs for

x,y

and the delay-aligned actuator model driven by the pre-allocation normalized inputs for

z,\phi,\theta,\psi

. The offline design fixes the gains before the run and is not a feedback signal.

Parameter	Symbol	Value
Mass, arm length a	$m$ , $l$	0.468 kg, 0.225 m
Thrust coeff. a	$k$	$2.98{\times}10^{-6}$ N s 2
Drag-torque coeff. a	$b$	$1.14{\times}10^{-7}$ N m s 2
Inertia a	$I_{xx}{=}I_{yy}$ , $I_{zz}$	$(4.856,8.801){\times}10^{-3}$ kg m 2
Rotor inertia, linear drag a	$I_{M}$ , $A$	$3.357{\times}10^{-5}$ kg m 2 , 0.25 kg/s
Rotor time constant c	$\tau_{m}$	30 ms

TABLE II: Nominal Model and Simulation Parameters

Controller	Architecture	Common constraints	Increment ceiling	Tuning
PID-Budget	2-DOF PID	variance, $M_{s}$ , OS, $t_{s}$	none	multistart + joint cascade
AA-Budget	actuator-aware RO-LADRC	variance, $M_{s}$ , OS, $t_{s}$	PID local value	bandwidth/ratio selection
RO-HalfGain	commanded inner/altitude ESO	variance, $M_{s}$ , OS, $t_{s}$	PID local value	predictive half-gain + bandwidth
AA-HalfGain	actuator-aware half-gain ESO	variance, $M_{s}$ , OS, $t_{s}$	PID local value	predictive half-gain + bandwidth
A1-Commanded	commanded inner/altitude RO-ESO	same as AA-Budget	same	redesign
A2-Unaligned	lag-model RO-ESO, $\hat{n}_{d}=0$	same as AA-Budget	same	redesign

TABLE III: Controller Design Protocol. The full-cascade increment limit applies to the horizontal reselection of AA-Budget, AA-HalfGain and A1–A3 and to the strict and budget-curve searches; PID-Budget is retained and audited; RO-HalfGain and A1 are searched in all classes. AA internal model: 30-ms rotor lag, 5-ms alignment (A2: none). Limits:

M_{s}

target 1.98 (reporting 2), OS 10%, settling of Section IV-F .

Fig. 2: Controlled roll-channel sweep of the observer/controller ratio at fixed controller bandwidths

\omega_{c}=3

, 6 and 10 rad/s. Dashed: commanded input; solid: actuator-aware input. Peaks use 4000 frequencies plus local refinement; blank sensitivity segments correspond to non-Schur loops, whose loss of stability the spectral-radius panel shows.

Controller	$\sigma_{xy}$	$M_{s}$	RMSE	CI	Inc.	Fail
	[mm]		mean $\pm$ SD [mm]	[mm]	[rad/s]
PID-Budget	132.9	1.978	65.5 $\pm$ 7.0	2.62	9.91	0/30
AA-Budget	32.2	1.974	19.2 $\pm$ 1.5	0.58	7.46	0/30
RO-HalfGain	300.5	1.963	226.4 $\pm$ 13.4	5.00	0.80	0/30
AA-HalfGain	27.8	1.974	16.9 $\pm$ 1.3	0.50	7.58	0/30

TABLE V: Nominal Set-Point Performance (S6, 30 Paired Trials).

\sigma_{xy}

is the predicted horizontal tracking standard deviation of the linear model; RMSE is the measured 3-D nonlinear position error.

M_{s}

is the maximum over the four channel classes; RMSE is mean

\pm

SD and CI is the 95% half-width of the mean.

Controller	Pred. $\sigma_{\Delta u}$	Meas.	Ratio $u$	Ratio $\Delta u$
PID-Budget	1.380	1.382	0.998	1.003
AA-Budget	0.937	0.938	0.985	1.002
AA-HalfGain	1.023	1.024	0.982	1.001

TABLE VI: Cascade Noise Prediction and Hover Measurements. Ratios are measured/predicted variances of the inner virtual control

u

and its increment

\Delta u

; increment standard deviations are in normalized angular-acceleration units. Measurements: ten paired 60-s noise-only hover trials, window 20–60 s.

Design	Ceil.	Pred.	Meas.	S6 RMSE	Rotor inc.
	[rad/s 2 ]	$\mathbf{\sigma_{\Delta u}}$	$\mathbf{\sigma_{\Delta u}}$	[mm]	[rad/s]
PID-Budget	1.382	1.380	1.382	65.5 $\pm$ 7.0	9.91
AA-Budget	1.382	0.937	0.938	19.2 $\pm$ 1.5	7.46
AA-HalfGain	1.382	1.023	1.024	16.9 $\pm$ 1.3	7.58
PID-Budget-S	0.598	0.592	0.592	72.3 $\pm$ 7.5	9.23
AA-Budget-S	0.598	0.496	0.497	29.9 $\pm$ 2.6	4.67

TABLE VII: Cascade Activity Constraints and Performance. Predictions use the stationary linear covariance model; measured values are the larger pooled roll/pitch increment of ten paired 60-s noise-only trials (window 20–60 s); tracking metrics are from 30 paired S6 trials. The -S designs use the strict ceiling and also retune the inner-loop gains.

Design	RMSE [mm]	CI [mm]	Inc.	Paired change [mm]
A1-Commanded	176.6 $\pm$ 14.0	5.21	2.10	157.4 $\pm$ 4.9
A2-Unaligned	19.2 $\pm$ 1.5	0.58	7.70	0.0132 $\pm$ 0.0034
A3-NoIncrement	15.9 $\pm$ 1.3	0.47	8.25	-3.28 $\pm$ 0.11
A4, fixed gains a	129.1 $\pm$ 6.9	2.56	7.84	109.9 $\pm$ 2.7
A4, reselected	275.7 $\pm$ 42.2	15.75	7.16	256 $\pm$ 16

TABLE VIII: Observer and Budget Ablations (S6, 30 Paired Trials). Last column: paired RMSE change versus AA-Budget, mean

\pm

95% CI half-width, the CI rounded to two significant digits and the mean to the same decimal place. A4: AA-Budget with commanded-tilt horizontal observer inputs. a Sampled horizontal loop not Schur stable; the nonlinear runs stay bounded.

Fig. 3: Local budget–performance trade-off. Channel-local increment ceilings are multiplied by

\gamma

, while the variance, sensitivity, overshoot and main cascade ceilings stay fixed; thirty pairs per candidate, error bars are 95% CIs of mean RMSE. Left: imposed local multiplier; right: measured total rotor-command increment, which includes disturbance-rejection effort. The PID points at

\gamma=0.5

and 0.75 are redesigned under the response requirement (Appendix B ); at

\gamma=0.75

the best PID candidate found keeps the attitude, horizontal and yaw gains of

\gamma=0.5

Condition	PID-Budget	AA-Budget	RO-HalfGain	AA-HalfGain
	RMSE [mm] [rotor-command increment, rad/s]
Reference	68.3 $\pm$ 7.6 [9.91]	19.6 $\pm$ 1.3 [7.45]	227.4 $\pm$ 19.4 [0.79]	17.2 $\pm$ 1.1 [7.57]
Noise $0.5\times$	68.4 $\pm$ 7.6 [4.96]	19.6 $\pm$ 1.3 [3.73]	227.5 $\pm$ 19.4 [0.40]	17.2 $\pm$ 1.1 [3.79]
Noise $2\times$	68.3 $\pm$ 7.6 [19.84]	19.7 $\pm$ 1.3 [14.92]	227.3 $\pm$ 19.3 [1.59]	17.3 $\pm$ 1.1 [15.15]
Noise $4\times$	68.3 $\pm$ 7.7 [39.76]	20.0 $\pm$ 1.2 [29.91]	227.2 $\pm$ 19.1 [3.18]	17.7 $\pm$ 1.0 [30.39]
Delay 0–10 ms	68.3 $\pm$ 7.6 [9.86]	19.6 $\pm$ 1.3 [7.42]	227.8 $\pm$ 19.4 [0.79]	17.2 $\pm$ 1.1 [7.53]

TABLE IX: Noise and Latency Stress Tests (S6, Ten Paired Trials per Condition). RMSE entries are mean

\pm

SD; brackets contain rotor-command increments. RO-HalfGain: design with

\omega_{o}/\omega_{c}\geq 1

Fig. 4: Noise-mismatch effect in S6. Tracking and measured rotor-command increments are shown separately for PID-Budget and the two actuator-aware variants.

Fig. 5: Fixed nominal gains under true lag and measurement-delay mismatch; internal model stays fixed. Left: true rotor time constant varied at the nominal 5-ms delay; right: true delay varied at the nominal 30-ms time constant. Missing sensitivity values are non-Schur sampled loops. The horizontal line is the reporting limit 2.

Controller	Lag [ms]	Runs	Fail	Threshold
PID-Budget	15–60	120	0	0
PID-Budget	75–120	120	30	0
AA-Budget	15–60	120	0	0
AA-Budget	75–120	120	7	93
AA-HalfGain	15–60	120	0	0
AA-HalfGain	75–120	120	31	79

TABLE X: Fixed-Gain Mismatch Outcomes (Twelve Plants per Lag Range, Ten Paired S6 Trials per Plant). Threshold counts bounded runs with rotor fluctuation above 20 rad/s; failures are counted separately.

Fig. 6: Per-plant counts out of ten paired S6 trials: failure plus bounded rotor-fluctuation threshold exceedance; each nonzero cell is annotated F/E with the failures F and the bounded exceedances E, which are disjoint events. This diagnostic does not label every threshold exceedance a self-excited oscillation.

Controller	RMSE	95% CI	Inc.	Max. error	Fail
	[mm]	[mm]	[rad/s]	[mm]
PID-Budget	276.5 $\pm$ 7.7	2.88	9.92	464.0 $\pm$ 32.0	0/30
AA-Budget	22.7 $\pm$ 1.7	0.64	7.46	94.9 $\pm$ 14.5	0/30
RO-HalfGain	217.3 $\pm$ 16.2	6.04	0.79	641.2 $\pm$ 86.6	0/30
AA-HalfGain	20.4 $\pm$ 1.5	0.55	7.58	91.6 $\pm$ 13.2	0/30

TABLE XI: Lemniscate Tracking Performance (S5, 30 Paired Trials, Window 5–45 s). Entries are mean

\pm

SD; maximum error is the per-trial maximum 3-D position-error norm.

Case	Design	MATLAB	PX4 SITL
S6	PID-Budget	65.5 $\pm$ 7.0	65.0 $\pm$ 6.7
	AA-Budget	19.2 $\pm$ 1.5	20.5 $\pm$ 1.3
	RO-HalfGain	226.4 $\pm$ 13.4	227.1 $\pm$ 12.9
	AA-HalfGain	16.9 $\pm$ 1.3	18.3 $\pm$ 1.3
S6 strict	PID-Budget-S	72.3 $\pm$ 7.5	72.2 $\pm$ 7.2
	AA-Budget-S	29.9 $\pm$ 2.6	30.8 $\pm$ 2.6

TABLE XII: MATLAB Simulation and PX4 SITL (S6, Window 10–40 s; S5, 5–45 s). Position RMSE in mm, mean

\pm

SD over the same paired seeds (S6: 91001–91030; S5 and ablations: 91001–91010). RO-HalfGain and A1-Commanded: designs with

\omega_{o}/\omega_{c}\geq 1

; PID-Budget-S: redesign under the response requirement; PID-Budget: set-point weights of Section IV-F , runs of the redesigned-controller campaign. No failures in either simulation.

Group	Runs	Skip	Steps (max)	Par.
Hover 60 s, -E designs (refit)	30	7	9 (3)	24
S6, -E designs	90	10	11 (2)	0
S5, -E designs	30	2	2 (1)	0
Hover 300 s, -E designs	30	11	16 (3)	0
Hover 300 s, primary and strict	60	35	68 (9)	3
S6, PID-Budget-S	30	9	22 (6)	9

TABLE XIII: Skipped Lockstep Steps in the SITL Runs of the Redesigned Controllers. All runs are reported; runs: total runs of the group, skip: runs with at least one skipped step, steps: total skipped steps (maximum per run), par.: runs overlapping a parallel computation on the host.

	(a) per design		(b) pooled		(c) design stage
Quantity	fit	h.o.	fit	h.o.	h.o.
$\sigma_{\Delta u}$ , $\phi\theta$	0.98–1.00	0.98–1.00	0.96–1.01	0.95–1.01	0.97–1.01
$\sigma_{\Delta u}$ , $\psi$	1.00	1.00–1.01	0.99–1.00	1.00–1.01	1.01–1.02
$\sigma_{\Delta u}$ , $z$	0.98–1.06	0.96–1.06	0.83–1.07	0.81–1.07	0.83–1.07
$J$	0.98–1.03	0.97–1.03	0.87–1.04	0.86–1.04	0.93–1.04
$\sigma_{u}$ , $\phi\theta$	0.81–0.97	0.81–0.97	0.81–0.97	0.81–0.97	0.87–0.93

TABLE XIV: Noise-Model Validation in 300-s Noise-Only Hover Behind EKF2 (Window 20–300 s, Nine Designs). Ratio of predicted to measured standard deviation, minimum–maximum over the designs; fit seeds 97001–97005, held-out (h.o.) seeds 97006–97010. (a) Per-design model; (b) model pooled over the nine designs; (c) design-stage model of the three -E designs, held-out seeds (out of sample).

\sigma_{\Delta u}

: inner increment (roll/pitch: cascade output);

\sigma_{u}

: control;

\sigma_{e}

: tracking error;

J

: rotor-command increment.

Design	$M_{s}$	$\sigma_{\Delta u}$ cas.	$\sigma_{\Delta u_{\psi}}$	$\sigma_{\Delta u_{z}}$	$J$
	$xy/z/\phi\theta/\psi$	pred./meas.	pred./meas.	meas.	pred./meas.
PID-Budget	1.94/1.81/2.04/1.99	0.0065/0.0066	4.75/4.74	5.0	–/0.0259
PID-E	1.98/1.81/1.98/1.98	0.0064/0.0067	4.22/4.21	4.8	0.0236/0.0254
AA-Budget-E	1.98/1.46/1.96/1.97	0.0059/0.0059	4.31/4.24	7.0	0.0302/0.0290
AA-HalfGain-E	1.98/1.58/1.81/1.97	0.0064/0.0064	3.99/3.93	7.4	0.0317/0.0305

TABLE XV: Estimator-Aware Designs: Sensitivity and Noise-Only Hover (EKF2 Input, Ten 60-s Runs, Seeds 96001–96010).

M_{s}

per class with the 6-ms rate lag (design target 1.98); ceilings

B^{E}_{\mathrm{cas}}=0.0065

B^{E}_{\psi}=4.75\times 10^{-4}

; predictions with the design-stage pooled model (out of sample);

\sigma_{\Delta u_{\psi}}

and

\sigma_{\Delta u_{z}}

10^{-4}

J

in rad/s. PID-Budget is the nominal design, not redesigned.

	S6					S5
Design	RMSE	vs PID-B.	vs PID-E	$J$	$J_{q}$	RMSE	$J$	$J_{q}$
	[mm]	[mm]	[mm]	[rad/s]	[rad/s]	[mm]	[rad/s]	[rad/s]
PID-Budget	65.0 $\pm$ 6.7	–	–	0.084	0.224	278.4 $\pm$ 8.3	0.222	0.273
PID-E	59.1 $\pm$ 6.3	$-5.9\pm 1.0$ (30)	–	0.089	0.231	335.8 $\pm$ 6.6	0.228	0.294
AA-Budget-E	44.0 $\pm$ 4.0	$-21.0\pm 1.3$ (30)	$-15.0\pm 1.1$ (30)	0.049	0.196	49.5 $\pm$ 5.6	0.111	0.205
AA-HalfGain-E	84.7 $\pm$ 8.2	$+19.7\pm 1.0$ (0)	$+25.6\pm 1.1$ (0)	0.047	0.192	90.6 $\pm$ 11.9	0.097	0.194

TABLE XVI: Estimator-Aware Designs in PX4 SITL: Tracking and Rotor-Command Increment. S6: 30 paired trials; S5: ten. RMSE mean

\pm

SD; paired change with 95% CI half-width and number of pairs in which the design is better;

J

: rotor-command increment logged before the 1/1000 output quantizer;

J_{q}

: computed from the quantized actuator outputs returned by PX4. No failures. PID-Budget: set-point weights of Section IV-F , runs made with the same build and activity logging as the -E runs on the same seeds (also used in Table XII ).

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