Implementation of FOPID controller with modified harmony search optimization for precise modelling and auto-tuning of nonlinear systems

Figures & data

Table 1. System physical specifications.

Equipment	Details
Conical tank	Built of Stainless casing with 70 cm height, Top to the bottom radius of 17.5 and 1.25 cm respectively.
Differential Pressure Transmitter (DPT)	The level is measured using DPT with one end open to the air.
Centrifugal Pump	Capacity of 800LPH
Final Control Actuator/ Element (FCA)	Air to release, input 3-15 PSI, pneumatically actuated
Flow meter	Rota meter capacity of 0-600 LPH

Figure 1. Reaction of the open-loop plant/system.

Table 2. The transfer function of the system based on the height of the tank.

Model No.	Range in cm	Height of the tank	The transfer function of the process
1	0–5	5	Drain
2	6–25	20	${\displaystyle \frac{8.414}{0.06608\text{ s}+1}}{}^{\ast }\mathrm{e}\left(-1.05\text{s}\right)$
3	26–45	40	${\displaystyle \frac{14.57}{1.03\text{ s}+1}}{}^{\ast }\mathrm{e}\left(-1.05\text{s}\right)$
4	46–65	60	${\displaystyle \frac{18.81}{3.684\text{ s}+1}}{}^{\ast }\mathrm{e}\left(-1.05{}^{\ast }\mathrm{s}\right)$
5	66–70	5	Vent

Figure 2. Settings of the system.

Figure 3. Response of the open loop with input.

Figure 4. Step input to system.

Figure 5. Response of the PID controller.

Figure 6. Fitness function of HSA with iteration.

Figure 7. (a) Optimal response of HSA (b) Response of HSA with various set points (c) Error performance of the system.

Figure 8. Output response of FOPID with system.

Figure 9. HSA convergence curve.

Table 3. Optimized gain for FOPID.

Sl. No.	Constraint	Value
1	KP	24.904799659226462
2	KI	0.047159861740317
3	KD	6.531214849178121
4	Μ	0.054520288198813
5	Λ	0.059107294824940

Table 4. Evaluation of the envisioned HAS-based system's dynamic responsiveness.

Time domain Parameters/ Optimization method	HAS- FOPID	ABC (Artificial Bee Colony)- FOPID	BBO (Biogeography-based optimization)- FOPID	BOA (butterfly optimization algorithm)- FOPID	FF (Fire Fly)- FOPID	PSO (Particle swarm Optimization)- FOPID
Rise Time (tr)	6.2416e^−04	4.0025	2.2301e^−04	6.3660e^−04	8.2126e^−04	4.2640e^−04
Settling Time (ts)	1.2001	1.3061	1.4062	1.3071	1.4901	1.3001
Overshoot (%Mp)	5.4627e^−05	2.2377e^−06	3.2091e^−05	8.0278e^−05	3.2088e^−07	1.5994e^−04
Peak Value	9.9981	12.9943	10.9969	14.9961	11.9973	14.9955
Peak Time (tp)	1.0002	1.9236	1.7132	1.4235	1.6002	1.1403

Figure 10. Bode diagram of conical tank between phase, magnitude, and frequency.

Table 5. Tank Transfer function.

Model No.	The transfer function of the Tank	PM	GM	ωpc	ωgc
1	${\displaystyle \frac{8.414}{0.06608\text{ s}+1}}{}^{\ast }\mathrm{e}(-1.05{}^{\ast }\mathrm{s})$	109.8544	Inf	NaN (Not a Number)	41.9092
2	${\displaystyle \frac{8.414}{0.06608\text{ s}+1}}{}^{\ast }\mathrm{e}(-1.05{}^{\ast }\mathrm{s})$	Inf	101.3109	NaN	4.8539
3	${\displaystyle \frac{8.414}{0.06608\text{ s}+1}}{}^{\ast }\mathrm{e}(-1.05{}^{\ast }\mathrm{s})$	Inf	98.7383	NaN	1.7660

Figure 11. (a,b) Effect of supply disturbance in conical tank system.

Figure 12. (a) Bode plot of model no.1 transfer function conical tank system (b) Bode plot of model no.2 transfer function conical tank system (c) Bode plot of model no.3 transfer function conical tank system (d) Pole-Zero Map of the conical tank design.

Figure 13. (a) Robust analysis plot (b) Robust analysis plot disturbance point.