Flexibility and robust performance have made the FOPID (Fractional Order PID) controllers a better choice than PID (Proportional, Integral, Derivative) controllers. But the number of tuning parameters decreases the usage of FOPID controllers. Using synthetic data in available FOPID tuners leads to abnormal controller performances limiting their applicability. Hence, a new tuning methodology involving real-time data and overcomes the drawbacks of mathematical modeling is the need of the hour. This paper proposes a novel FOPID controller tuning methodology using machine learning algorithms. Feed Forward Back Propagation Neural Network (FFBPNN), Multi Least Squares Support Vector Regression (MLSSVR) chosen to design Machine Learning based Optimal Tuner (MLOT) can handle the interdependency between the controller parameters and multiple outputs for multiple inputs.The proposed tuner finds application in the control of power and energy systems. It can accomplish tracking, disturbance-rejection, and robustness controller performances, thus making FOPID controller design easier and accurate. Comparisons with existing FOPID tuning rules show better controller performances and easy tuning. Thus, this paper addresses a unique, real-time, model-free, easily tunable FOPID tuning methodology satisfying plant requirements.

In most of the process control industries, control loops are of Proportional Integral Derivative (PID) type [

Due to design flexibility, the FOPID controllers find applications especially in systems having nonlinear dynamics [

Even though the FOPID controller is found to outperform the conventional PID controllers, design of FOPID controller can be more difficult as it involves five tuning parameters namely, proportional gain constant ‘

Determination of FOPID controller parameters is achieved using many optimization algorithms [

In last two decades, tuning rules such as Ziegler-Nichols (ZN), Cohen-Coon and Kappa–Tau are the classical empirical tuning rules for FOPID control parameters. FOPID tuning rules based on optimal load disturbance rejection [

In addition to the above said drawbacks, these FOPID tuning rules are devised using very little information on system dynamics, also require prior assumptions, approximations and fail to include robustness.Hence, a more accurate, flexible, easily accessible tuning rule, without the need of any mathematical formulations or initial estimation of parameter set is need of the hour.

Hence to overcome the above-said drawbacks, Machine Learning based Optimal Tuner (MLOT) is proposed in this paper. Machine Learning Algorithm (MLA) is advantageous since, it is the strongest predictive modeling for linear as well as nonlinear patterns, with lesser assumptions supplying a single predictive model [

In this proposed work, the optimal FOPID controller dataset is generated to achieve two-different performance specifications, Set-Point Tracking (SPT) and Load Disturbance Rejection (LDR) for various First Order Plus Dead Time (FOPDT) systems using Covariance Matrix Adaptive Evolutionary Strategy (CMA-ES) [

Data analytics is performed on SPT and LDR datasets to remove outliers if any and to identify the most suitable MLA. Thus, identified MLA accomplishes the task of MLOT. Two MLAs, Feed Forward Back Propagation Neural Network (FFBPNN) [

The proposed MLOT is justified in terms of performance specifications, statistical variations in controller parameters obtained from MLAs by comparing with the Tuning Rules (TR) given by Padula and Visioli.

The authors claim, the following points as the novelty of this proposed work.

A Universal tuner for FOPID controllers using MLA is proposed.

Data analysis with R studio^{®} is used to identify appropriate MLAs.

Among the two MLAs, MLSSVR is chosen and verified using statistical analysis.

Better tracking, disturbance rejection, and robustness performances were achieved.

Proposed MLOT is applicable to the FOPDT system and also to any higher-order systems.

The organization of the paper is as follows. Section 2 describes the proposed methodology. Results and discussions are placed in Section 3 and conclusions are given in Section 4.

The flow diagram of the proposed methodology is given in _{s}). Optimal SPT dataset and LDR dataset are obtained as K_{p}^{sp}, T_{i}^{sp}, T_{d}^{sp}, λ^{sp}, μ^{sp} and K_{p}^{ld}, T_{i}^{ld}, T_{d}^{ld}, λ^{ld}, μ^{ld} respectively.

The generated dataset is analyzed using R studio for removing outliers and identification of suitable MLAs. MLOT-FFBPNN and MLOT-MLSSVR are formulated with the dataset from the data analysis block which is applicable to any process control system.

Optimal FOPID dataset is generated by optimizing the FOPID controller design for a range of FOPDT systems FOPDT_{1}, FOPDT_{2}, …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 _{p} as given in

Hence, the ‘n’ number of different FOPDT systems (FOPDT_{1}, FOPDT_{2}, … FOPDT_{n}) are produced by varying _{1}_{2},…_{n} between the range [0.3, 0.8]. The structure of FOPID controller used is given in

The FOPID controller parameters are obtained for ‘n’ different FOPDT systems by minimizing the Integral Absolute Error (_{sp} and LDR to obtain minimum _{ld} as in

The dataset generated from the optimization block will be of the form as given in _{s}] are chosen as input parameters while [K_{p}, T_{i}, T_{d}, λ, μ] are chosen as the output parameters.

Dataset | Input parameters | Output parameters | Number of data points |
---|---|---|---|

SPT | [K_{p}^{sp}, T_{i}^{sp}, T_{d}^{sp}, λ^{sp}, μ^{sp}] |
160 | |

LDR | [K_{p}^{ld}, T_{i}^{ld}, T_{d}^{ld}, λ^{ld}, μ^{ld}] |
160 |

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.

For a multiple input multiple output model, some data may be suitable in a single dimension but they may become an outlier in a multi-dimension. The Cook’s distance identifies the presence of outliers. These outliers are then removed from the dataset by identifying the data points having standard deviations greater than the mean value.

The SPT and LDR dataset contains four output variables. The cross-correlation matrix of these four output variables for the SPT dataset after removing outliers is obtained. Positive/Negative large cross-correlation values denote high linear interrelationship among the variables. A Smaller cross-correlation value does not mean that the corresponding parameter is an independent one. For such cases, the cross-correlation plots have to be examined for non-linear relations. Hence, in this work, both the cross-correlation matrix values and the cross-correlation plots have been analyzed for determining the interrelationship among the output parameters before modeling with MLAs.

The SPT and LDR dataset are combined together to obtain the third dataset, the CMB dataset from the generated data points as given in ^{c} = [τ^{sp}, τ^{ld}] while M_{s}^{c} = [M_{s}^{sp}, M_{s}^{ld}]. Also, K_{p}^{c} = [K_{p}^{sp}, K_{p}^{ld}], T_{i}^{c} = [T_{i}^{sp}, T_{i}^{ld}], T_{d}^{c} = [T_{d}^{sp}, T_{d}^{ld}], λ^{c} = [λ^{sp}, λ^{ld}], μ^{c} = [μ^{sp}, μ^{ld}].

Dataset | Input parameters | Output parameters | Number of data points |
---|---|---|---|

SPT | [τ^{sp}, M_{s}^{sp}] |
[K_{p}^{sp}, T_{i}^{sp}, T_{d}^{sp}, λ^{sp}, μ^{sp}] |
150 |

LDR | [τ^{ld}, M_{s}^{ld}] |
[K_{p}^{ld}, T_{i}^{ld}, T_{d}^{ld}, λ^{ld}, μ^{ld}] |
144 |

CMB | [τ^{c}, M_{s}^{c}] |
[K_{p}^{c}, T_{i}^{c}, T_{d}^{c}, λ^{c}, μ^{c}] |
294 |

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, _{i} = _{0} _{i}, _{i} ∈ _{0} is the regular weight vector that determines the output while

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.

An Intel^{®} Core™ i7-3632 QM CPU with 2.2 GHz speed and 8 GB RAM computer with 8 logical processors is used to develop the proposed MLOT in this paper. The optimization is carried out with the help of the MATLAB^{®} toolbox while the data analysis is performed using R studio^{®}.

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 _{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 _{sp} is minimum for the CMA-ES method, in all three FOPDT sample systems.

τ = 0.35 | ||||||
---|---|---|---|---|---|---|

Sample system 1 | ||||||

ZN | 0.5528 | 0.9536 | 0.3797 | 1.3607 | 1.0828 | 1.8327 |

TR | 1.0281 | 0.9829 | 0.1698 | 1 | 1.2000 | 1.4736 |

CMA-ES | 1.3571 | 1.0544 | 0.1701 | 1 | 1.1006 | 0.7868 |

Sample system 2 | τ = 0.4212 | |||||

ZN | 0.6762 | 1.1426 | 0.3619 | 1.3198 | 1.0841 | 2.7109 |

TR | 0.5534 | 0.9746 | 0.3566 | 1 | 1.2000 | 2.6390 |

CMA-ES | 0.7867 | 1.1045 | 0.3150 | 1 | 1.1106 | 1.4562 |

Sample system 3 | τ = 0.6710 | |||||

ZN | 0.9315 | 2.0084 | 0.4172 | 1.3244 | 0.9105 | 4.7749 |

TR | 0.2911 | 0.9676 | 0.7492 | 1 | 1.2000 | 4.8330 |

CMA-ES | 0.5107 | 1.2859 | 0.5364 | 1 | 1.1160 | 2.6707 |

The graphs in

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

The cross-correlation matrix of the four output variables for the SPT dataset after removing outliers is given in

Output variables | ||||
---|---|---|---|---|

1.0000 | –0.6503 | –0.7461 | 0.0666 | |

–0.6503 | 1.0000 | 0.8020 | 0.0609 | |

–0.7461 | 0.8020 | 1.0000 | –0.0047 | |

0.0666 | 0.0609 | –0.0047 | 1.0000 |

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. _{d}^{ld}, ^{ld} with respect to ^{ld} for MLOT-LDR.

Also, it can be observed that the results of testing data points in

In graph ^{ld}^{ld }of _{s} = 2.0, the point A indicates the value of ^{ld} from MLSSVR (blue color) and point B from FFBPNN (green color) for the same value of ^{ld} = 0.15. But, point A and point B clearly indicate that FFBPNN (green color) does not follow the pattern as given in

The statistical performances of proposed MLOTs are better than TR and MVR as given in _{p} and _{d.}

Variables | _{p} |
|||
---|---|---|---|---|

Method | TR | MVR | FFBPNN | MLSSVR |

MLOT-SPT | ||||

CC | 0.6307 | 0.6992 | 0.9999 | |

MSE | 1.6063 | 6.2 × 10^{−6} |
0.0003 | 0.0004 |

RMSE | 1.2674 | 0.0056 | 0.0172 | 0.3522 |

MAE | 0.8444 | 0.0033 | 0.0106 | 0.2801 |

MLOT-LDR | ||||

CC | 0.6433 | 0.5974 | 0.9883 | |

MSE | 0.6540 | 0.0339 | 0.0009 | 0.0003 |

RMSE | 0.2584 | 0.1842 | 0.0292 | 0.0162 |

MAE | 0.8125 | 0.1671 | 0.0168 | 0.0113 |

MLOT-CMB | ||||

CC | – | 0.6892 | 0.9914 | |

MSE | – | 0.0579 | 0.0015 | 0.0010 |

RMSE | – | 0.2406 | 0.0386 | 0.0311 |

MAE | – | 0.2049 | 0.0263 | 0.0236 |

Variables | _{d} |
|||
---|---|---|---|---|

Method | TR | MVR | FFBPNN | MLSSVR |

MLOT-SPT | ||||

CC | 0.692 | 0.6956 | 0.9977 | 0.9981 |

MSE | 0.093 | 0.0005 | 0.0009 | 0.0004 |

RMSE | 0.305 | 0.0218 | 0.0302 | 0.2268 |

MAE | 0.2061 | 0.009 | 0.0161 | 0.1818 |

MLOT-LDR | ||||

CC | 0.7849 | 0.7501 | 0.9951 | 0.9969 |

MSE | 0.0993 | 0.0171 | 0.0004 | 0.0002 |

RMSE | 0.2650 | 0.1309 | 0.0202 | 0.0158 |

MAE | 0.3561 | 0.1042 | 0.0128 | 0.0096 |

MLOT-CMB | ||||

CC | – | 0.7073 | 0.9888 | 0.9905 |

MSE | – | 0.0416 | 0.0016 | 0.0013 |

RMSE | – | 0.2039 | 0.0404 | 0.0365 |

MAE | – | 0.1676 | 0.0277 | 0.0227 |

The statistical results of controller parameters are given as spider graphs in

The CC values for the CMB dataset are compared among MLOT-MLSSVR, MLOT-FFBPNN, MLOT-MVR models in

This paper proposes a novel FOPID controller tuning methodology using machine learning algorithms. Feed Forward Back Propagation Neural Network (FFBPNN), Multi Least Squares Support Vector Regression (MLSSVR) chosen to design Machine Learning based Optimal Tuner (MLOT) can handle the interdependency between the controller parameters and multiple outputs for multiple inputs. The statistical analysis values reveal that, the CC values of MLSSVR is 0.999, 0.9966, 0.9944 for MLOT-SPT, MLOT-LDR, MLOT-CMB for K_{p}. The MLOT-MLSSVR using the CMB dataset perfectly captures variations among controller parameters. The graphical analysis of parameter variations and statistical analysis confirms the better results for the proposed MLOT over the FOPID Tuning Rule (TR) given by Padula and Visioli. The proposed MLOT is also applicable for dead time-dominated systems and lag-dominated systems. Pre-processing with large amount of dataset consumes computational time which can be overcome using cloud computing and big data analysis in future.

The author with a deep sense of gratitude would thank the supervisor for his guidance and constant support rendered during this research.