Shadowed Type2 Fuzzy Sets in Dynamic Parameter Adaption in Cuckoo Search and Flower Pollination Algorithms for Optimal Design of Fuzzy FaultTolerant Controllers
Abstract
:1. Introduction
2. General Type2 Fuzzy Sets and Shadowed Sets
General Type2 Fuzzy Sets with Shadowed Sets
3. Background of Optimization Algorithm and Mathematical Formulation
3.1. Cuckoo Search (CS) Algorithm
 An egg, kept in a nest, represents a solution. An artificial cuckoo can lay only one egg at a time [70].
 The cuckoo bird searches for the most suitable nest to lay its eggs in to maximize the survival rate of its eggs (solution). Because of an exclusivist selection strategy, only highquality eggs (best solutions around the optimal value) that are more similar to the host bird’s eggs have a chance to develop (next generation) and become mature cuckoos [70].
 The population (number of host nests) remains constant. The host bird can find the alien egg with a probability of ${p}_{a}\in [0,1]$ (worse solutions away from the optimal value), and these eggs are thrown away or the nest is neglected and a new nest is established in a different location. Otherwise, the egg matures and lives to the next generation. Lèvy flights around the best current solutions aid in the selection of new eggs (solutions) laid by a cuckoo [70,72].
Algorithm 1: Pseudo code for CS Algorithm [70,72] 

3.2. Flower Pollination
Algorithm 2: Pseudo code for FP Algorithm [73,74] 

3.3. Mathematical Modeling of FPA
4. TwoTank Conical Frustum NonInteracting Level System with Mathematical Model
Modeling of Coupled Frustum Tank Level Control Process
5. Proposal and Methods
 If iteration is low, then Lèvy flight (P) is low;
 If iteration is moderate, then Lèvy flight (P) is moderate;
 If iteration is high, then Lèvy flight (P) is high.
 If iteration is low, then switching Probability (P’) is low;
 If iteration is moderate, then switching Probability (P’) is moderate;
 If iteration is high, then switching Probability (P’) is high.
6. Simulation Results
Comparative Analysis and Result Discussion
Simulation Results with Noise and Intermittent Fault
 (1)
 Interval type2 fuzzy controller with random noise with magnitude $M=0.5$ (Gaussian random number);
 (2)
 Interval type2 fuzzy controller with random noise and TTCFNLC system with intermittent actuator fault in main actuator $CV$ with magnitude of ${M}_{1}=50\%$ and ${M}_{2}=60\%$ at time $t=4$ s and $t=6$ s respectively;
 (3)
 Interval type2 fuzzy controller with random noise and TTCFNLC system with intermittent system component (leak) fault in bottom of the frustum tank 2 (additional flow rate ${f}_{o2}$) with magnitude of ${M}_{1}=50\%$ and ${M}_{2}=60\%$ at time $t=4$ s and $t=6$ s, respectively.
7. Statistical Results and Analysis
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
ANNs  Artificial Neural Networks 
ACO  Ant Colony Optimisation 
CS  Cuckoo Search 
CV  Control Valve 
CPS  CyberPhysical Systems 
DE  Differential Evolution 
DMs  Decision Makers 
DOF  Degrees of Freedom 
DTs  Decision Threes 
EA  Elevation Area 
FF  Feedforward 
FP  Flower Pollination 
FSs  Fuzzy Sets 
FLC  Fuzzy Logic Controller 
FISs  Fuzzy Inference Systems 
FTS  Frustum Tank System 
GA  Genetic Algorithm 
GT2FS  General Type2 Fuzzy Sets 
GT2FIS  General Type2 Fuzzy Inference System 
GOA  Grasshopper Optimization Algorithm 
GSO  Galactic Swarm Optimization 
LFM  Lower Membership Function 
LPFNs  Linguistic Pythagorean fuzzy numbers 
MFs  Membership Functions 
MSE  Mean Square Error 
MPPT  Maximum Power Point Tracking 
NN  Neural Network 
PID  Proportional Integral Derivative 
PSO  Particle swarm optimisation 
PV  Photo Voltaic 
RTAC  Rotational/Translational ProofMass Actuator 
RA  Reduction Area 
RL  Reinforcement Learning 
RMSE  Root Mean Square Error 
SA  Shadowing Area 
SISO  Single Input Single Output 
ST2FIS  Shadowed Type2 Fuzzy Inference System 
SFC  Stochastic Fractal Search 
SVMs  Support Vector Machines 
TTCFNLCS  TwoTank Conical Frustum NonInteracting Level System 
T1FLCs  Type1 Fuzzy Logic Controllers 
${t}_{fr}$  Fault recovery time 
HS  Harmony Search 
IP  Inverted Pendulum 
IAE  Integral Absolute Error 
ICS  Improved Cuckoo Search 
ISE  Integral Square Error 
ITAE  Integral Time Absolute Error 
ITSE  Integral Time Absolute Error 
IT2FLCs  Interval Type2 Fuzzy Logic Controllers 
IT2FIS  Interval Type2 Fuzzy Inference System 
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Control Problem  Input  Output  Total Size of Vector  

Total  Type of MFs  Total  Type of MFs  
TTCFNLC Process  3  2—IT2 Trapezoidal and 1—IT2 Triangular in each Input  3  3—IT2 Triangular in each output  80 
Algorithm  Performance Index  Simulation Scenarios  

Without Fault  With Actuator Fault 1  
CS  ITAE  $1.481\times {10}^{2}$  $2.612\times {10}^{2}$ 
ITSE  $3.418\times {10}^{2}$  $3.933\times {10}^{2}$  
IAE  $0.891\times {10}^{1}$  $0.989\times {10}^{1}$  
ISE  $1.793\times {10}^{2}$  $3.156\times {10}^{2}$  
FP  ITAE  $3.213\times {10}^{3}$  $4.773\times {10}^{3}$ 
ITSE  $7.422\times {10}^{3}$  $8.1014\times {10}^{3}$  
IAE  $1.971\times {10}^{2}$  $2.262\times {10}^{2}$  
ISE  $3.9135\times {10}^{3}$  $4.828\times {10}^{4}$ 
Algorithm  Performance Index  Simulation Scenarios  

Without Fault  With Actuator Fault 1  
CS  BEST  $1.1725\times {10}^{4}$  $1.2027\times {10}^{2}$ 
WORST  0.1671  0.2315  
AVERAGE  $2.958\times {10}^{2}$  $9.0912\times {10}^{2}$  
STANDARD DEVIATION  $4.4234\times {10}^{2}$  $4.74513\times {10}^{2}$  
FP  BEST  $2.3412\times {10}^{3}$  0.2923 
WORST  0.2161  0.3019  
AVERAGE  $7.3812\times {10}^{2}$  $1.4091\times {10}^{1}$  
STANDARD DEVIATION  $5.6124\times {10}^{2}$  $5.2341\times {10}^{2}$ 
Algorithm  Performance Index  Simulation Scenarios  

Without Fault  With Actuator Fault  
CS  BEST  $1.0913\times {10}^{2}$  $1.4230\times {10}^{1}$ 
WORST  $3.8561\times {10}^{1}$  $4.4671\times {10}^{1}$  
AVERAGE  $1.3123\times {10}^{1}$  $2.9125\times {10}^{1}$  
STANDARD DEVIATION  $1.1901\times {10}^{1}$  $8.5209\times {10}^{2}$  
FP  BEST  $4.7816\times {10}^{2}$  $2.6091\times {10}^{1}$ 
WORST  $4.2891\times {10}^{1}$  $4.9891\times {10}^{1}$  
AVERAGE  $2.6543\times {10}^{1}$  $3.8342\times {10}^{1}$  
STANDARD DEVIATION  $1.2671\times {10}^{1}$  $10.02813\times {10}^{2}$ 
Type of Uncertainties  Fault Magnitude  Nature of Uncertainties  Time of Occurrence in Second  Metaheuristic Algorithms  

Fuzzy CS  Fuzzy FP  
$\left({\mathit{t}}_{\mathbf{fr}}\right)$ in Second  
Actuator Fault 1  50%  Abrupt  4  0.47  0.54 
Actuator Fault 2  60%  6  0.52  0.57  
Leak Fault 1  50%  Abrupt  4  0.43  0.48 
Leak Fault 2  60%  6  0.49  0.56 
Algorithm  Performance Index  Simulation Scenarios  

With Noise  With Noise and Actuator Fault 1 and 2  With Noise and Leak Fault 1 and 2  
CS  BEST  $7.16\times {10}^{2}$  $9.78\times {10}^{2}$  $8.37\times {10}^{2}$ 
WORST  0.2667  0.2767  0.2709  
AVERAGE  0.1415  0.1607  0.1447  
SD  $5.24\times {10}^{2}$  $5.35\times {10}^{2}$  $5.52\times {10}^{2}$  
FP  BEST  $9.73\times {10}^{2}$  $1.034\times {10}^{1}$  $9.87\times {10}^{2}$ 
WORST  0.3093  0.3477  0.3118  
AVERAGE  0.2154  0.2390  0.2179  
SD  $6.67\times {10}^{2}$  $8.36\times {10}^{2}$  $6.89\times {10}^{2}$ 
Algorithm  Performance Index  Simulation Scenarios  

With Noise  With Noise and Actuator Fault 1 and 2  With Noise and Leak Fault 1 and 2  
CS  BEST  0.2675  0.3127  0.2893 
WORST  0.4934  0.5260  0.5204  
AVERAGE  0.37  0.3957  0.3737  
SD  $6.17\times {10}^{2}$  $6.51\times {10}^{2}$  $6.23\times {10}^{2}$  
FP  BEST  0.3142  0.3215  0.3138 
WORST  0.5561  0.5896  0.5574  
AVERAGE  0.4583  0.4812  0.4610  
SD  $7.39\times {10}^{2}$  $8.82\times {10}^{2}$  $7.44\times {10}^{2}$ 
Type of Uncertainties  Fault Magnitude  Nature of Uncertainties  Time of Occurrence in Second  Metaheuristic Algorithms  

Fuzzy CS  Fuzzy FP  
$\left({\mathit{t}}_{\mathbf{fr}}\right)$ in Second  
Actuator Fault 1  50 %  Abrupt  4  0.49  0.64 
Actuator Fault 2  60 %  6  0.53  0.61  
Leak Fault 1  50 %  Abrupt  4  0.44  0.53 
Leak Fault 2  60 %  6  0.51  0.59 
Parameter  Value 

${H}_{0}$  ${\mu}_{1}\ge {\mu}_{2}$ 
${H}_{a}$  ${\mu}_{1}<{\mu}_{2}$ (Claim) 
Level of significance  95 % 
A  0.05 
Critical value  −1.645 
CS  FP  zValue  

Average  Std  Average  Std  
FCS ST2FIS  FPA ST2FIS  
$3.7981\times {10}^{1}$  $3.3761\times {10}^{1}$  $7.9023\times {10}^{1}$  $4.5021\times {10}^{1}$  $5.081\times {10}^{+01}$ 
FCS Using IT2FIS vs. FPA Using IT2FIS  

Test Statistic  pValue 
16.13333  0.000061 
$Q=\frac{12n}{k(K+1)}\left[{\sum}_{j}^{}{R}_{j}^{2}\frac{k{(K+1)}^{2}}{3}\right]$  
Q = $0.066666\times 4292270$  
Q = 16.13333 
FCS Using T1FIS vs. FPA Using T1FIS  

Test Statistic  pValue 
16.13333  0.0000589 
$Q=\frac{12n}{k(K+1)}\left[{\sum}_{j}^{}{R}_{j}^{2}\frac{k{(K+1)}^{2}}{3}\right]$  
Q = $0.066666\times 4292270$  
Q = 16.13333 
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Patel, H.R.; Shah, V.A. Shadowed Type2 Fuzzy Sets in Dynamic Parameter Adaption in Cuckoo Search and Flower Pollination Algorithms for Optimal Design of Fuzzy FaultTolerant Controllers. Math. Comput. Appl. 2022, 27, 89. https://doi.org/10.3390/mca27060089
Patel HR, Shah VA. Shadowed Type2 Fuzzy Sets in Dynamic Parameter Adaption in Cuckoo Search and Flower Pollination Algorithms for Optimal Design of Fuzzy FaultTolerant Controllers. Mathematical and Computational Applications. 2022; 27(6):89. https://doi.org/10.3390/mca27060089
Chicago/Turabian StylePatel, Himanshukumar R., and Vipul A. Shah. 2022. "Shadowed Type2 Fuzzy Sets in Dynamic Parameter Adaption in Cuckoo Search and Flower Pollination Algorithms for Optimal Design of Fuzzy FaultTolerant Controllers" Mathematical and Computational Applications 27, no. 6: 89. https://doi.org/10.3390/mca27060089