Fig 1.
Overall representation of Cyber-Physical System (CPS).
Table 1.
The table of nomenclature.
Fig 2.
The control framework of CPS with real-time setpoints and noise in model’s feedback.
The environmental factors would be predicted and applied as a real-time setpoint and anomaly in sensor is estimated in feedback loop. Gain parameters and order parameters in FOPID controller are tuned to be robust against source of variability.
Fig 3.
The block diagram of robust FOPID control in CPS framework with real-time setpoints and noise in model’s feedback.
Real-time setpoint is estimated by approximation function of environmental factors (). Anomaly in sensor’s feedback is function of uncertain variable
. FOPID gain parameters and order parameters are tuned robustly in such a way to make CPS insensitive against sources of variability in system.
Fig 4.
Crossing two sets of DOE dealing with uncertainty in a model, one DOE (l samples) over decision variables of the model and second DOE (m samples) over uncertain variables in the model.
Fig 5.
Algorithmic representation of proposed approach for hybrid GP-PSO based robust simulation-optimization under uncertainty.
Fig 6.
Five bar linkage robot manipulator.
Table 2.
Numeric values of the parameters of the five-bar manipulator dynamics.
Table 3.
Robust FOPID optimal results using proposed algorithm for 10 repetitions for θ = 0.25 (the results obtained over 9 different uncertainty scenarios).
Table 4.
Robust FOPID optimal results using proposed algorithm for 10 repetitions for θ = 0.5 (the results obtained over 9 different uncertainty scenarios).
Table 5.
Robust FOPID optimal results using proposed algorithm for 10 repetitions for θ = 0.75 (the results obtained over 9 different uncertainty scenarios).
Fig 7.
EI criterion magnitudes and best SNR obtained by sequential expected improvement over 10 different repetition of proposed algorithm for θ = 0.25, θ = 0.5 and θ = 0.75.
Two stopping rules are adjusted, EI value becomes smaller than 0.01 or reach 15 sequential iterations.
Fig 8.
Mean and Std of overall function (OF) related to best point so far (smaller SNR) obtained by sequential expected improvement over 10 different repetition of proposed algorithm for θ = 0.25, θ = 0.5 and θ = 0.75.
Two stopping rules are adjusted, EI value becomes smaller than 0.01 or reach 15 sequential iterations.
Fig 9.
The step responses of the robot manipulator with 9 different uncertainty scenarios ( and
) for θ = 0.25, θ = 0.5 and θ = 0.75.
Fig 10.
Sensitivity analysis via 50 bootstrapped GP surrogate and 95% Confidence Intervals (CIs) over robust optimal point obtained by original GP surrogate for θ = 0.25.
Augmented parametric bootstrapping is performed using on hand set of input/output data provided among original optimization program.
Fig 11.
Sensitivity analysis via 50 bootstrapped GP surrogate and 95% Confidence Intervals (CIs) over robust optimal point obtained by original GP surrogate for θ = 0.5.
Augmented parametric bootstrapping is performed using on hand set of input/output data provided among original optimization program.
Fig 12.
Sensitivity analysis via 50 bootstrapped GP surrogate and 95% Confidence Intervals (CIs) over robust optimal point obtained by original GP surrogate for θ = 0.75.
Augmented parametric bootstrapping is performed using on hand set of input/output data provided among original optimization program.
Table 6.
Comparison results for FOPID tuning using different methods over 100 different uncertainty scenarios for θ = 0.25.
Table 7.
Comparison results for FOPID tuning using different methods over 100 different uncertainty scenarios for θ = 0.5.
Table 8.
Comparison results for FOPID tuning using different methods over 100 different uncertainty scenarios for θ = 0.75.
Table 9.
p-values of the t-test and the Wilcoxon signed rank test for pairwise comparison of proposed algorithm with three common stochastic optimizers over 10 repetitions.
Fig 13.
The performance comparison of proposed algorithm with other three solvers in the literature for tuning of stochastic FOPID controller.
The performance criterion Rp,s measured based on two terms, accuracy of solution (lower objective function) and number of function evaluations (computational cost), see Eq (19).