Fuzzy fractional-order PID control for two-wheeled self-balancing robots on inclined road surface

Figures & data

Figure 1. Model of the TWSBR.

Table 1. Robot parameters.

Parameter	Definition	Numerical value
$m_w$	Mass of wheels	1.551 kg
$m_p$	Mass of the IB	1.6 kg
$I_w$	Moment of inertia of wheel	0.005 kg.m${}^2$
$I_p$	Moment of inertia of IB	0.0027 kg.m${}^2$
l	Distance between centre of IB and chassis	0.13 m
r	The radius of the road	0.08 m
g	Acceleration of gravity	9.81 m/s${}^2$

Figure 2. Diagram of balance state.

Figure 3. Schematic diagram of FFOPID controller.

Figure 4. Membership function – error signal, fractional order of error.

Table 2. Rule based for $K_P$/ $K_I$/ $K_D$.

	$K_P$/ $K_I$/ $K_D$
	e
$\frac{{\mathrm{d}}^{\mu }}{\mathrm{d}{t}^{\mu }}e$	NS1	ZO1	PS1
NS2	M1/B2/M3	S1/S2/B3	M1/B2/M3
ZO2	B1/M2/S3	M1/M2/B3	B1/B2/S3
PS2	M1/M2/M3	S1/S2/B3	M1/M2/M3

Table 3. Optimal parameters of controller.

	Parameters
Controller Type	$G_E$/$K_P$	$G_{EC}$/$K_I$	$G_{PD}$/$K_D$	$G_{PI}$	λ	µ
FFOPID	1.9718	0.5059	0.7009	0.5949	0.9983	0.8021
FPID	1.4431	1.9718	0.5057	0.4190	1	1
FOPID	0.4829	0.4011	2.5947	2.6722	0.9843	0.3001
PID	2.2566	1.6125	0.2663	–	–	–

Figure 5. When $\phi =5^{\circ }$, the robot's position and balance performance on the inclined plane. (a) Position control of the robot. (b) Balance control of the robot and (c) The output of the FFOPID controller.

Table 4. The performance index result under the condition of $\phi =5^{\circ }$.

	x			θ
	IAE	ISE	ITAE	IAE	ISE	ITAE
PID	0.8001	0.2698	1.3073	0.4658	0.0497	1.1616
FPID	0.6788	0.2385	0.9788	0.4507	0.0603	0.9183
FOPID	0.8047	0.2745	1.0615	0.3713	0.0669	0.5498
FFOPID	0.6666	0.3589	0.7166	0.2147	0.0112	0.5303

Figure 6. When $\phi ={10}^{\circ }$, the robot's position and balance performance on the inclined plane. (a) Position control of the robot. (b) Balance control of the robot and (c) The output of the FFOPID controller.

Table 5. The performance index result under the condition of $\phi ={10}^{\circ }$.

	x			θ
	IAE	ISE	ITAE	IAE	ISE	ITAE
PID	1.4647	0.5687	2.9777	0.5553	0.0535	2.0755
FPID	1.3569	0.5459	2.3036	0.4513	0.0490	1.2590
FOPID	1.2563	0.4734	2.1013	0.3027	0.0226	0.8019
FFOPID	0.9665	0.3589	1.2566	0.2147	0.0112	0.5303

Figure 7. After adding $0.05\ast randn$ disturbance, the robot's position and balance performance on the inclined plane. (a) $\phi =5^{\circ }$ and (b) $\phi ={10}^{\circ }$.

Table 6. When $\phi =5^{\circ }$, performance index under the condition of $0.05\ast randn$ disturbance.

	x			θ
	IAE	ISE	ITAE	IAE	ISE	ITAE
PID	1.0929	0.2822	5.5358	0.6265	0.0523	3.7875
FPID	0.8351	0.2473	3.1351	0.5170	0.0575	2.1137
FOPID	1.0628	0.2844	4.2739	0.5128	0.0681	2.2224
FFOPID	0.8073	0.2522	3.0071	0.4665	0.0725	1.7539

Table 7. When $\phi ={10}^{\circ }$, performance index under the condition of $0.05\ast randn$ disturbance.

	x			θ
	IAE	ISE	ITAE	IAE	ISE	ITAE
PID	1.8066	0.5382	7.8453	0.8121	0.0629	5.4014
FPID	1.5587	0.5344	5.4277	0.5664	0.0545	2.8683
FOPID	1.4492	0.4983	4.5960	0.4137	0.0276	2.0727
FFOPID	1.2576	0.3850	4.3972	0.2875	0.0112	1.6375

Figure 8. After adding 0.5 rad interference, the robot's position and balance performance on the inclined plane. (a) $\phi =5^{\circ }$ and (b) $\phi ={10}^{\circ }$.

Table 8. When $\phi =5^{\circ }$, performance index under the condition of 0.5 rad disturbance.

	x			θ
	IAE	ISE	ITAE	IAE	ISE	ITAE
PID	1.2806	0.3201	6.9639	0.9494	0.1116	6.8602
FPID	1.0893	0.2736	5.8265	0.9681	0.1377	6.7171
FOPID	1.5340	0.5055	5.0775	0.7816	0.1783	4.6738
FFOPID	0.8612	0.2649	2.7833	0.7715	0.1699	4.5413

Table 9. When $\phi ={10}^{\circ }$, performance index under the condition of 0.5 rad disturbance.

	x			θ
	IAE	ISE	ITAE	IAE	ISE	ITAE
PID	1.9452	0.6204	8.7103	1.0598	0.1203	8.0035
FPID	1.6506	0.5711	5.5636	0.9582	0.1390	6.7372
FOPID	1.5340	0.5055	5.0393	0.7160	0.1336	4.9672
FFOPID	1.1202	0.3731	2.8652	0.4385	0.0662	2.7836

Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.