# Performance Analysis of Data-Driven and Model-Based Control Strategies Applied to a Thermal Unit Model

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

^{®}and Simulink

^{®}environments and exploiting their standard toolboxes or free software tools. Note that some control strategies proposed in this work were already successfully applied to nonlinear models of energy conversion systems as shown, e.g., in [26,27,28].

## 2. Thermal Unit Mathematical Description

^{2}) is the cross-sectional area of the duct, ${\rho}_{a}$ (kg/m

^{3}) denotes the air density, $q\left(t\right)$ represents the supplied constant heat gain, whilst U (J/m

^{2}·K) indicates the heat transfer coefficient. Note that in Equations (1)–(3), the term ${\left(UA\right)}_{h}$ (J/K) indicates the product of the heat transfer coefficient U (J/m

^{2}·K) with the efficient surface area, A (m

^{2}), through which the heat is transmitted and regarding the TU module. On the other hand, ${\left(UA\right)}_{\mathrm{int}}$ (J/K) indicates the coefficient with reference to the inner duct wall, whilst ${\left(UA\right)}_{\mathrm{ext}}$ (J/K) denotes the same term regarding the outer duct wall.

^{∘}C for the whole measurement range. The airflow has been measured using a Hot Wire Thermo-Anemometer with a declared accuracy of $5\%$. For more details regarding the TU module, which is beyond the scope of this paper, the interested reader is referred to [9].

## 3. Control Designs for the TU Module

#### 3.1. Standard PID Controller Design

^{®}environment.

#### 3.2. Fuzzy Controller Design

^{®}toolbox [38].

- A TS prototype structure with order n, the membership functions ${\mu}_{{A}_{i}}(.)$ and an appropriate number of rules K are assumed;
- The input and output data sampled from the process under control are exploited by the ANFIS tool for providing the TS model parameters ${a}_{i}$ and ${b}_{i}$ according to a selected error criterion;
- By varying the design parameters n and K with a trial and error procedure, the optimal values of the parameters ${a}_{i}$ and ${b}_{i}$ are obtained in order to achieve the minimisation of the selected error criterion.

^{®}environment [43]. This tool allows one to obtain in an easy and straightforward way the parameters of the TS fuzzy structure of Equation (11). Moreover, the FMID strategy provides the controller model simply using a data-driven approach scheme addressed in [43]. This approach exploits again the estimation of the rule-based fuzzy model parameters and requires only the input-output data sampled from the controlled process. In particular, the FMID scheme uses the Gustafson–Kessel clustering methodology to partition the input-output data into suitable regions, denoted again as ${R}_{i}$, the so-called clusters [43]. For each i-th cluster, the parameters ${a}_{i}$ and ${b}_{i}$ of the affine models of Equation (9) with their membership function ${\mu}_{{A}_{i}}(.)$ are derived. The estimation of the TS fuzzy model in the form of Equation (11) is based on the choice of a suitable model structure n and a number of rules K (usually equal to the number of clusters). The selection of these parameters is performed in order to minimise a prescribed cost function usually related with the closed-loop system performance [43].

#### 3.3. Adaptive Controller Design

^{®}environment [45].

^{®}and Simulink

^{®}environments [45].

#### 3.4. Model Predictive Controller Designs

^{®}environment, which can require the knowledge of a state-space LTI model of the controlled process of Equation (4). A continuous-time LTI description of this dynamic process can be obtained by means of the linearisation of Equation (4), which leads to the state-space model in the form of Equation (18):

**A**,

**B**and

**C**are defined by the linearisation at the operating point corresponding to the equilibrium state ${x}_{e}={\left(\right)}^{1.0093}T$ and inputs ${u}_{e}={\left(\right)}^{19.2351}T$:

^{®}environment, which uses a state-space LTI description of the controlled process of Equation (4). A continuous-time model of this plant can be obtained by means of an identification procedure, for example based on the System Identification Toolbox in the MATLAB

^{®}environment. In this way, the subspace identification (N4SID) procedure has led to the state-space matrices in Equation (20) [41]:

^{®}and Simulink

^{®}environments. Therefore, this simulation code is able to implement the moving horizon estimation, dynamic optimisation and simulation, thus solving the nonlinear MPC problems [29]. The nonlinear input-output dynamic model used in simulation has been obtained again by exploiting the System Identification Toolbox in the MATLAB

^{®}environment. In particular, this estimation procedure performed via a prediction error method (PEM) has provided a nonlinear regression model with two inputs and one output, with standard regressors corresponding to the orders ${n}_{a}={n}_{b}=2$ for both the inputs and the output, without dead-times (${n}_{k}=1$) [41]. Moreover, the nonlinearity has been modelled via a sigmoidal network with 10 neurons. Therefore, this nonlinear regression model is able to fit the identification data with an accuracy higher than 90% [41]. Section 4 will show and compare the results achieved with the different models implementing both the linear and nonlinear MPC strategies.

## 4. Simulation Results

^{®}and Simulink

^{®}environments using the most appropriate development tools. These control solutions will be compared in terms of a performance index represented by the mean sum of squared error ($MSSE\%$) computed via Equation (21):

^{®}block. The proportional, integral and derivative gains have been determined as ${K}_{p}=1.4465$, ${K}_{i}=0.0339$ and ${K}_{d}=0.4228$, respectively. The derivative filter time–constant has been estimated as ${T}_{f}=4.4034$.

^{®}environment are summarised in Figure 9. In this situation, the set-point $r\left(t\right)$ (blue continuous line) is tracked with an $MSSE\%=1.14\%$. On the other hand, the step transient response presents a settling time ${T}_{s}=3.98$ s and a maximum overshoot $S\%=41.65\%$. Finally, it is worth nothing that the high overshoot at the beginning of the simulation in Figure 9 is due to the initial conditions of the delay blocks of the fuzzy controller represented in Figure 9 that are zero.

^{®}PID block in order to optimise its parameters. This feature can represent an important aspect when a good trade-off between control performance and implementation simplicity is required. In general, the standard PID control law is usually based on the process controlled variable and not on the knowledge of the underlying process behaviour. However, on the one hand, an automatic tuning of its parameters allows the PID controller to manage general control requirements, also in terms of step response rise time, closed-loop bandwidth, maximum overshoot and system oscillation amplitude. On the other hand, PID controllers cannot guarantee any control optimality and, in some situations, the overall system stability, but provides a viable and easy-to-use tool for providing a simple control law with acceptable performance.

#### 4.1. Control Solution Sensitivity Evaluation

^{®}and Simulink

^{®}software tools, in order to automate the overall design and simulation phases. These feasibility and reliability studies are of paramount importance for real application of control strategies once implemented for future air conditioning system installations.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Sample Availability:**The software simulation codes for the proposed control strategies and the simulated thermal unit model are available from the authors in the MATLAB and Simulink environments.

**Figure 4.**The inflow air temperature ${T}_{i}\left(t\right)$ considered as a disturbance $d\left(t\right)$ acting on the controlled system.

**Figure 5.**Controlled outlet temperature ${T}_{o}$ with the PID regulator obtained via the auto-tuning procedure.

**Figure 6.**Diagram of the TU simulated model with the Adaptive Neuro-Fuzzy Inference System (ANFIS) fuzzy regulator.

**Figure 9.**TU outflow air temperature with the Takagi–Sugeno (TS) fuzzy regulator derived via the FMID toolbox.

**Figure 13.**TU module outlet air temperature compensated by the linear MPC with and without disturbance $d\left(t\right)$ compensation.

**Figure 14.**TU module outlet air temperature compensated by the identified linear and the nonlinear MPC solutions.

**Table 1.**Estimated model parameters with their accuracy [9].

Parameter | ${\widehat{\mathit{\alpha}}}_{1}$ | ${\widehat{\mathit{\alpha}}}_{2}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ |

Value | 55.026 ± 2.011 | −92.835 ± 3.398 | 9.9837 ± 0.2512 | 7.7661 ± 0.2835 |

Parameter | ${\widehat{\mathit{\eta}}}_{\mathbf{1}}$ | ${\widehat{\mathit{\eta}}}_{\mathbf{2}}$ | $\widehat{\mathit{\epsilon}}$ | |

Value | −8.5689 ± 0.2070 | −8.3067 ± 0.2984 | −122.95 ± 4.426 |

Controller Type | Settling Time ${\mathit{T}}_{\mathit{s}}$ | Overshoot $\mathit{S}\%$ | Tracking Error $\mathit{MSSE}\%$ |
---|---|---|---|

Auto-tuning PID | $2.17$ s | $36.14\%$ | $1.65\%$ |

ANFIS Fuzzy | $2.21$ s | $38.22\%$ | $1.07\%$ |

FMID Fuzzy | $3.98$ s | $41.65\%$ | $1.14\%$ |

Adaptive | $3.65$ s | $40.18\%$ | $1.18\%$ |

Linear MPC with disturbance compensation | $1.85$ s | $35.51\%$ | $0.41\%$ |

Linear MPC w/o disturbance compensation | $1.97$ s | $37.64\%$ | $1.27\%$ |

Linear MPC with identified model | $1.82$ s | $35.02\%$ | $0.33\%$ |

Nonlinear MPC with identified neural model | $1.79$ s | $29.73\%$ | $0.14\%$ |

Control Scheme | $\mathit{MSSE}\%$ Best Case | $\mathit{MSSE}\%$ Worst Case | $\mathit{MSSE}\%$ Average Value |
---|---|---|---|

Auto-tuning PID | $1.44\%$ | $3.14\%$ | $1.65\%$ |

ANFIS Fuzzy | $1.03\%$ | $2.33\%$ | $1.07\%$ |

FMID Fuzzy | $1.10\%$ | $2.47\%$ | $1.14\%$ |

Adaptive | $1.06\%$ | $1.78\%$ | $1.18\%$ |

Linear MPC with disturbance compensation | $0.38\%$ | $0.81\%$ | $0.41\%$ |

Linear MPC w/o disturbance compensation | $0.97\%$ | $1.98\%$ | $1.27\%$ |

Linear MPC with identified model | $0.28\%$ | $0.67\%$ | $0.33\%$ |

Nonlinear MPC with identified neural model | $0.09\%$ | $0.31\%$ | $0.14\%$ |

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**MDPI and ACS Style**

Turhan, C.; Simani, S.; Zajic, I.; Gokcen Akkurt, G.
Performance Analysis of Data-Driven and Model-Based Control Strategies Applied to a Thermal Unit Model. *Energies* **2017**, *10*, 67.
https://doi.org/10.3390/en10010067

**AMA Style**

Turhan C, Simani S, Zajic I, Gokcen Akkurt G.
Performance Analysis of Data-Driven and Model-Based Control Strategies Applied to a Thermal Unit Model. *Energies*. 2017; 10(1):67.
https://doi.org/10.3390/en10010067

**Chicago/Turabian Style**

Turhan, Cihan, Silvio Simani, Ivan Zajic, and Gulden Gokcen Akkurt.
2017. "Performance Analysis of Data-Driven and Model-Based Control Strategies Applied to a Thermal Unit Model" *Energies* 10, no. 1: 67.
https://doi.org/10.3390/en10010067