Automation and Robots for Disaster Response

Surrogate Optimal Fractional Control for Constrained Operational Service of UAV Systems

By inergency On Apr 5, 2024

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Author Contributions

Conceptualization, A.M.M. and M.B.A.; methodology, A.M.M., M.B.A. and K.M.M.; software, A.M.M., M.B.A. and K.M.M.; validation, A.M.M. and M.B.A.; investigation, M.M. (Mohammed Moness) and M.M. (Moataz Mohamed); resources, M.M. (Mohammed Moness) and M.M. (Moataz Mohamed); writing—original draft preparation, K.M.M.; writing—review and editing, A.M.M., M.B.A. and M.M. (Moataz Mohamed); supervision, M.M. (Mohammed Moness) and M.M. (Moataz Mohamed). All authors have read and agreed to the published version of the manuscript.

Figure 1.
Challenge complexity of intersecting decision levels and constraints of UAV operation.

Figure 2.
Design optimization criterion with its reflection in a set of operational requirements.

Figure 3.
Frequency responses for a sample of FOPID with unity gains.

Figure 4.
The quadrotor model frames.

Figure 5.
The quadrotor waypoint navigation FOPID control structure.

Figure 6.
The twin-rotor copter physical setup.

Figure 7.
The twin-rotor copter cross-coupled FOPID control.

Figure 8.
Stability boundaries in the $K_{I}$ – $K_{D}$ plane for arbitrarily selected values of $K_{P}, λ, μ$ .

Figure 9.
Flowchart of the surrogate optimization algorithm.

Figure 10.
Conceptual plot of the surrogate optimization algorithm.

Figure 11.
Position step responses for the quadrotor.

Figure 12.
Simulation of the mission scenario for the quadrotor.

Figure 13.
Zoomed 2D planar view for the quadrotor mission trajectory at transitional regions (A, B, and C).

Figure 14.
Zoomed 2D planar view for the quadrotor mission trajectory at transitional regions (D, E, and F).

Figure 15.
Histogram of control actions for the quadrotor.

Figure 16.
Acceleration power spectral density for the quadrotor mission.

Figure 17.
Visualization of the optimal FOPID controllers for the twin-rotor copter.

Figure 18.
Convergence of surrogate optimization for the FOPID controllers of the twin-rotor copter.

Figure 19.
Real-time orientation step responses for the twin-rotor copter.

Figure 20.
Real-timecontrol actions of the step commands for the twin-rotor copter.

Figure 21.
Real-time orientation sinusoidal responses for the twin-rotor copter.

Figure 22.
Real-time control actions of the sinusoidal commands for the twin-rotor copter.

Figure 23.
Real-time orientation sawtooth responses for the twin-rotor copter.

Figure 24.
Real-time control actions of the sawtooth commands for the twin-rotor copter.

Table 1.
The quadrotor model parameters [67].

Parameter	Value
$K_{P, θ}$ (s⁻²)	3402.97
$K_{D, θ}$ (s⁻¹)	116.67
$K_{P, ϕ}$ (s⁻²)	3402.97
$K_{D, ϕ}$ (s⁻¹)	116.67
$K_{P, ψ}$ (s⁻¹)	1950
$K_{t h}$ (s⁻¹)	3900
m (kg)	0.1

Table 2.
Statistical measures of the quadrotor controller optimization.

Axis	Controller	Objective Function					Computation Time (min)
Axis	Controller	Min. Value	Mean	Median	Standard Deviation	Average Simulations Count	Mean	Median	Standard Deviation
x	PID-GA	664.2483	665.0124	664.3411	1.1576	2450	14.0791	14.0254	1.4368
	FOPID-GA	672.5661	679.6325	678.4195	6.8963	2517	26.3810	26.3464	3.1829
	FOPID-Surrogate	672.0003	683.4908	680.4865	8.3657	1000	11.1183	10.9377	0.5672
y	PID-GA	664.2483	665.0124	664.3411	1.1576	2450	7.0880	7.0214	0.9581
	FOPID-GA	673.3710	680.4129	676.7823	8.0841	2497	21.9964	20.4116	4.2519
	FOPID-Surrogate	677.0695	682.9058	681.5786	5.5742	1000	10.4262	10.0240	0.7907
z	PID-GA	274.7040	274.7089	274.7041	0.0125	2450	9.5922	9.3179	1.4732
	FOPID-GA	286.9981	289.5838	288.8558	3.1095	2450	25.4097	23.9121	5.1696
	FOPID-Surrogate	287.9273	289.4379	289.1571	1.2448	1000	19.3086	19.3221	1.0178

Table 3.
PID and FOPID gains for the quadrotor controllers.

Axis	Controller	$K_{P}$	$K_{I}$	$λ$	$K_{D}$	$μ$
x	PID-GA	0.4598	0	1	0.3061	1
	FOPID-GA	0.0696	0.0089	0.9053	0.2196	0.7202
	FOPID-Surrogate	0	0.0205	0.6440	0.2480	0.6292
y	PID-GA	0.4598	0	1	0.3061	1
	FOPID-GA	0.0574	0.0244	0.5658	0.2389	0.7197
	FOPID-Surrogate	0.0376	0.0544	0.4275	0.2375	0.7272
z	PID-GA	8.6983	5.8476	1	1.3159	1
	FOPID-GA	0.7844	7.6204	0.8063	2.1821	0.6411
	FOPID-Surrogate	0.6607	7.9986	0.7507	2.2529	0.6387

Table 4.
Time-domain characteristics for quadrotor step responses.

Axis	Controller	Rise Time (s)	Overshoot	Settling Time (s)	Settling Error
x	PID-GA	0.9164	4.5587	2.6726	1.5332 $\times 10^{- 7}$
	FOPID-GA	0.7591	13.7547	2.9700	0.0177
	FOPID-Surrogate	0.6958	17.7479	3.0561	0.0161
y	PID-GA	0.9164	4.5587	2.6726	1.5332 $\times 10^{- 7}$
	FOPID-GA	0.7444	13.0912	2.9666	0.0131
	FOPID-Surrogate	0.7329	16.4277	3.4230	0.0101
z	PID-GA	0.2402	4.3196	0.6700	3.1186 $\times 10^{- 8}$
	FOPID-GA	0.2283	4.3987	0.6157	0.0042
	FOPID-Surrogate	0.2163	6.4502	0.6352	0.0059

Table 5.
Statistical measures of the twin-rotor copter controller optimization.

Controller	Objective Function			Computation Time (min)
Controller	Minimum Value	Mean	Standard Deviation	Mean	Standard Deviation
$F P I D_{m} θ$	184.8821	207.2058	26.6892	9.8651	0.2540
$F P I D_{m} ψ$	4.9927	5.4286	0.3315	14.2360	0.6657
$F P I D_{t} θ$	0.6314	0.7114	0.0508	15.1317	0.3795
$F P I D_{t} ψ$	21.2900	23.3846	1.5300	10.2655	1.1018

Table 6.
FOPID gains for the twin-rotor copter controller.

Controller	$K_{P}$	$K_{I}$	$λ$	$K_{D}$	$μ$
$F P I D_{m θ}$	$10.6718$	$2.8072$	$0.7189$	$- 11.1744$	$3.5408 \times 10^{- 4}$
$F P I D_{m ψ}$	$0.0433$	$- 0.8524$	$0.5716$	$- 0.2683$	$0.4986$
$F P I D_{t θ}$	$- 0.4207$	$0.4692$	$0.5032$	$0.4454$	$0.5111$
$F P I D_{t ψ}$	$0.8572$	$0.4179$	$0.6026$	$1.9416$	$0.9928$

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