Optimal FOPID dataset is generated by optimizing the FOPID controller design for a range of FOPDT systems FOPDT₁, FOPDT₂, …FOPDT_n. The CMA-ES algorithm is used for optimization with seven different maximum sensitivity values, M_s = [1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0] under two design objectives, SPT and LDR. The FOPDT system is in the form of G_p as given in Eq. (1) with process gain ‘K’, time constant ‘T’, dead time ‘L’ and also, normalized dead time, τ = L/ (L + T). The τ is associated with the dynamic behavior of a FOPDT system.

(1) Gp(s)=KTs+1e−Ls

Hence, the ‘n’ number of different FOPDT systems (FOPDT₁, FOPDT₂, … FOPDT_n) are produced by varying τ as τ₁,τ₂,…τ_n between the range [0.3, 0.8]. The structure of FOPID controller used is given in Eq. (2) and Oustaloup approximation in Eq. (3)

(2) Gc(s)=Kp(Tisλ+1Tisλ)(Tdsμ+1TdNs+1),0≤λ,μ≤2

(3) sv≅k∏n=1N1+sωz,n1+sωp,n,ν>0

The FOPID controller parameters are obtained for ‘n’ different FOPDT systems by minimizing the Integral Absolute Error (IAE). CMA-ES [25,26] optimization algorithm is used to minimize IAE in Eq. (4) for the two performance objectives namely, SPT to obtain minimum IAE_sp and LDR to obtain minimum IAE_ld as in Eq. (5).

(4) min⏟θIAEsp/ld=∫0∞|esp/ld(t)|dt

subjectto,Ms==M

where,M=[1.4,1.5,1.6,1.7,1.8,1.9,2.0]

(5) Ms=min⏟ω|11+L(jω)|,

The dataset generated from the optimization block will be of the form as given in Tab. 1. Both SPT and LDR objectives are optimized for 160 numbers of different FOPDT systems and hence 160 numbers of SPT and LDR data points are obtained respectively and this is provided in the last column of Tab. 1. The variables [τ, M_s] are chosen as input parameters while [K_p, T_i, T_d, λ, μ] are chosen as the output parameters.

To design MLOT, an MLA that can handle many inputs with many output parameters is required. Before using such MLA, the inter-variable correlation between the output parameters must be determined to obtain an efficient machine learning model. Also, the dataset generated from the optimization block may contain few data points that may differ from other observations termed outliers. Outliers may be due to measurement error which must be discarded to avoid misleading of modeling. Data analysis using R studio^® is carried out in this work to check and discard any outliers, and also to analyze the relationship between the output parameters of the generated dataset.

Data analysis results confirm a high cross-correlation among output parameters. Two MLAs, FFBPNN and MLSSVR have been identified to support the highly correlated dataset. FFBPNN and MLSSVR are employed to design the proposed MLOT using the three datasets obtained from the data analysis block.

In FFBPNN, the Levenberg-Marquardt function with Mean Square Error (MSE) minimization is considered. In MLSSVR multi-tasking is achieved by using weight vector, w_i = w₀ + v_i, where, w_i ∈ R. w₀ is the regular weight vector that determines the output while vi is dedicated for correlation in output variable. Radial basis function is used as the kernel function and grid search technique in conjunction with Leave one out cross-validation is used to tune the parameters in MLSSVR.Additionally, a Multi-Variate Regression (MVR) model which is an extension of the linear regression model is developed to compare the performances of the developed MLOTs.

The three MLAs are compared based on their statistical parameters such as Correlation Coefficient (CC), Mean Square Error (MSE), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE). The results of MLOTs are also compared with the existing FOPID tuning rule. The MATLAB^® program to evaluate FOPID controller parameters using the proposed MLOT is available with the authors. This MATLAB^® program can be used to determine FOPID controller parameters for any FOPID and higher-order systems.

The FOPDT system parameters K and T are set to 1. Different FOPDT systems are considered for dataset generation with different τ ϵ [0.3, 0.8].

The optimum values of IAE_sp obtained from CMA-ES for various FOPDT sample systems are compared with the results obtained from ZN rules and TR given by Padula and Visioliin Tab. 3. The comparison is given for three example systems. It is observed from Tab. 3, IAE_sp is minimum for the CMA-ES method, in all three FOPDT sample systems.

The graphs in Fig. 2 confirm that the optimal controller parameters obtained using the CMA-ES algorithm are found to vary smoothly for τ in both the design objectives.

The input and output variables of the SPT and LDR dataset are given in Table I for developing the MLAs. Based on Cook’s distance, outliers present in the Kp parameter of the SPT dataset are identified and denoted by the red colour ‘+’ (plus) symbol as shown in Fig. 3a data points after removing outliers are given in Fig. 3b. Similarly, outliers are removed in LDR dataset also.

The cross-correlation matrix of the four output variables for the SPT dataset after removing outliers is given in Tab. 4. The diagonal plots in Fig. 4 represent the histogram of the data distribution for each output variable. The lower diagonal plots represent the variation between the output variables. The upper diagonal represents their cross-correlation values similar to that tabulated in Tab. 4. The font size is larger for large cross-correlation values while font size is smaller for lower values of cross-correlation in Fig. 4. The red-colored ‘***’ represents very large cross-correlation values among all.

The three MLAs, MLOT-FFBPNN, MLOT-MLSSVR, MLOT-MVR algorithms for SPT, LDR, and CMB datasets are used to develop totally nine MLOT models. The MLOT models are tested using randomly generated data points named testing data points that are not involved in the training and validation phase. Figs. 5a and 5b shows the variation of T_d^ld, μ^ld with respect to τ^ld for MLOT-LDR.

Also, it can be observed that the results of testing data points in Fig. 6 follow the graphs of parameter variations similar to the validation data points in Fig. 5. Also, the results from TR show that controller parameters from FFBPNN and MLSSVR are closer to the optimum controller parameters than the results from TR.

In graph μ^ldVsτ^ld of Fig. 6b for M_s = 2.0, the point A indicates the value of μ^ld from MLSSVR (blue color) and point B from FFBPNN (green color) for the same value of τ^ld = 0.15. But, point A and point B clearly indicate that FFBPNN (green color) does not follow the pattern as given in Fig. 5b when compared to MLSSVR (blue color). This indicates that the MLSSVR algorithm produces better results than FFBPNN.

The statistical performances of proposed MLOTs are better than TR and MVR as given in Tab. 5 for K_p and Tab. 6 for T_d.

The statistical results of controller parameters are given as spider graphs in Figs. 7 and 8. For a better machine learning model, the CC values must be very large and the MSE, RMSE, MAE values must be the least. The CC values for all the output parameters in MLSSVR are larger compared to all other methods. This is shown by the outermost web denoted by red colour in Fig. 7a. Also, the MSE values of MLSSVR results are the least and it is given by the innermost web given in Fig. 7b denoted in red colour.

The CC values for the CMB dataset are compared among MLOT-MLSSVR, MLOT-FFBPNN, MLOT-MVR models in Fig. 8a. The MLOT-MLSSVR takes the outermost spider web (red colour) denoting larger correlations among the generated CMA-ES controller parameters and the controller parameters from validation data. Similarly, the results from MLOT-MLSSVR have the least MSE while using the CMB dataset. This is denoted by the innermost spider web given in red colour as shown in Fig. 8b.