Decentralized stable and robust fault-tolerant PI plus fuzzy control of MIMO systems: a quadruple tank case study

Sr. no.	Description	Value
1	Area of the tanks A₁, A₃, A₂, and A₄	32 cm²
2	Area of outlet pipes ^a₁ and a₃	0.071 cm²
3	Area of outlet pipes ^a₂ and ^a₄	0.057 cm²
4	Constant k	0.50 V/cm
5	Gravitational constant g	981 cm/s²

Uncertainty domain specified by working points.

Table 2.

Operating parameters of minimum phase and non-minimum phase system.

Parameters	Operating point minimum phase	Operating point non-minimum phase
$h_{1}^{0}$ , $h_{2}^{0}$	12.76, 13.1	12.3, 12.7
$h_{3}^{0}$ , $h_{4}^{0}$	2.1, 1.8	5.1, 5.7
$υ_{1}^{0}$ , $υ_{2}^{0}$	3.33, 3.36	3.14, 3.31
^k₁^{, k}₂	3.33, 3.38	3.14, 3.33
^λ₁^{, λ}2	0.7, 0.6	0.43, 0.34

Open-loop response of quadruple system with non-minimum phase configuration.

Open-loop response of quadruple system with minimum phase configuration.

Schema of simplest decentralized control of TITO system (Schmidt, 2002).

Decentralized control structure for minimum phase system with two fuzzy and two PI controllers. f_sys, f_a, and f_s denotes system component (leak), actuator, and sensor faults, respectively.

Table 3.

Rule base for type-1 FLC 1 loop 1.

^f₁, e₁ and ė₁ ${\dot{e}}_{1}$	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PS	PM	PB
PM	NS	ZR	PS	PS	PM	PB	PB
PB	ZR	PS	PS	PM	PB	PB	PB

Table 4.

Rule base for type-1 FLC 2 loop 2.

^f₂, e₂ and ė₂	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PM	PB	PB
PM	NS	ZR	PS	PM	PM	PB	PB
PB	ZR	PS	PM	PB	PB	PB	PB

Table 5.

Parameters for FLC.

Parameter	Parameter value
No. of input variables	2
No. of output variables	1
No. of linguistic variables per MF	7
No. of rules	49
Membership function (MF)	Triangular
Defuzzification methods	Center of gravity method

Stability analysis for minimum phase configuration.

Step responses: minimum phase stable system subject to process disturbances.

Step responses: minimum phase stable system subject to system component (leak) fault.

Step responses: minimum phase stable system subject to actuator fault.

Step responses: minimum phase stable system subject to sensor fault.

Error comparison for minimum phase configuration.

Step responses: non-minimum phase, unstable system.

Stability analysis for non-minimum phase configuration.

Step responses: non-minimum phase stable system subject to process disturbances.

Step responses: non-minimum phase stable system subject to system component fault.

Step responses: non-minimum phase stable system subject to actuator fault.

Step responses: non-minimum phase stable system subject to sensor fault.

Error comparison for non-minimum phase configuration.

References

Authors

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Articles in this issue

DOI: https://doi.org/10.21307/ijssis-2019-004 | Journal eISSN: 1178-5608

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Language: English

Page range: 1 - 20

Submitted on: Dec 19, 2018

Published on: Sep 5, 2019

Published by: International Journal on Smart Sensing and Intelligent Systems

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

System component fault,

Decentralized control,

Robust stability,

Robust performance

Related subjects:

Engineering,

Introductions and overviews,

Engineering, other

© 2019 Himanshukumar R. Patel, Vipul A. Shah, published by International Journal on Smart Sensing and Intelligent Systems
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 12 (2019): Issue 1 (January 2019)

Decentralized stable and robust fault-tolerant PI plus fuzzy control of MIMO systems: a quadruple tank case study

Figures & Tables

Figure 1:

Table 1.

Figure 2:

Table 2.

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Figure 4:

Figure 5:

Figure 6:

Table 3.

Table 4.

Table 5.

Figure 7:

Figure 8:

Figure 9:

Figure 10:

Figure 11:

Figure 12:

Figure 13:

Figure 14:

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Figure 16:

Figure 17:

Figure 18:

Figure 19:

Paradigm

My account

^f₁, e₁ and ė₁ ${\dot{e}}_{1}$	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PS	PM	PB
PM	NS	ZR	PS	PS	PM	PB	PB
PB	ZR	PS	PS	PM	PB	PB	PB

^f₂, e₂ and ė₂	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PM	PB	PB
PM	NS	ZR	PS	PM	PM	PB	PB
PB	ZR	PS	PM	PB	PB	PB	PB

^f₁, e₁ and ė₁ ${\dot{e}}_{1}$	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PS	PM	PB
PM	NS	ZR	PS	PS	PM	PB	PB
PB	ZR	PS	PS	PM	PB	PB	PB

^f₂, e₂ and ė₂	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PM	PB	PB
PM	NS	ZR	PS	PM	PM	PB	PB
PB	ZR	PS	PM	PB	PB	PB	PB

^f₁, e₁ and ė₁ ${\dot{e}}_{1}$	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PS	PM	PB
PM	NS	ZR	PS	PS	PM	PB	PB
PB	ZR	PS	PS	PM	PB	PB	PB

^f₂, e₂ and ė₂	NB	NM	NS	ZR	PS	PM	PB
NB	NB	NB	NB	NM	NS	NS	ZR
NM	NB	NB	NM	NS	NS	ZR	PM
NS	NB	NM	NS	ZR	PS	PM	PB
ZR	NM	NM	NS	ZR	PS	PM	PB
PS	NM	NS	ZR	PS	PM	PB	PB
PM	NS	ZR	PS	PM	PM	PB	PB
PB	ZR	PS	PM	PB	PB	PB	PB