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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper investigates the issue of tuning the Proportional Integral and Derivative (PID) controller parameters for a greenhouse climate control system using an Evolutionary Algorithm (EA) based on multiple performance measures such as good static-dynamic performance specifications and the smooth process of control. A model of nonlinear thermodynamic laws between numerous system variables affecting the greenhouse climate is formulated. The proposed tuning scheme is tested for greenhouse climate control by minimizing the integrated time square error (ITSE) and the control increment or rate in a simulation experiment. The results show that by tuning the gain parameters the controllers can achieve good control performance through step responses such as small overshoot, fast settling time, and less rise time and steady state error. Besides, it can be applied to tuning the system with different properties, such as strong interactions among variables, nonlinearities and conflicting performance criteria. The results implicate that it is a quite effective and promising tuning method using multi-objective optimization algorithms in the complex greenhouse production.

The greenhouse environment control problem is to create a favorable environment for the crop in order to reach predetermined results for high yield, high quality and low costs. It is a very difficult control problem to implement in practice due to the complexity of the greenhouse environments. For example, they are highly nonlinear, strong coupled and Multi-Input Multi-Output (MIMO) systems, they present dynamic behaviors and they are largely perturbed by the outside weather (wind velocity, outside temperature and humidity,

Furthermore, about 95% of the regulatory controllers of the process control, motor drives, automotive, fight control and instrumentation industries have PID structures. In spite of this widespread usage, the effectiveness is often limited owing to poor tuning, and tuning PID controllers efficiently is up to this time an interesting research. A lot of tuning methods have been presented in the extant literatures [

Recently, an optimal tuning method of PID controller by employing Evolutionary Algorithms (EAs) has been proposed and successfully used in a wide range of plants. For example, Chang [

The main objective of this work is to develop an intelligent tuning method for greenhouse climate control with two PID loops of a MIMO process, which is characterized by strong interactions among process variables, nonlinearities and conflicting performance criteria. The problem is stated as a multi-objective optimization problem where the two PID controllers are simultaneously tuned based on different and possibly conflicting specifications, such as good static-dynamic performance, as smooth control signals as possible. The populations are encoded the gain parameters and the corresponding objective (cost) functions or fitness functions are formulated based on the specifications of disturbances rejections, static-dynamic performance and smooth control.

The rest of the paper is organized as follows. Section 2 describes the considered greenhouse climate model and the corresponding nonlinear differential equations. In Section 3, the PID controller structure is proposed based on performance criteria. Section 4 describes the multi-objective evolutionary algorithm used in this work. The simulations and results are presented in Section 5. Finally, a conclusion and prospects are given in Section 6.

The greenhouse environment is a complex dynamical system. Over the past decades, people have gained a considerable understanding of greenhouse climate dynamics, and many methods describing the dynamic process of greenhouse climate have been proposed. Traditionally, there are two different approaches to describe it; one is based on energy and mass flows equations describing the process [_{2} supply flux,

In order to effectively validate the performance of the proposed algorithm below, the considered greenhouse analytic expression is based on the heating/cooling/ventilating model in this work, which can be obtained from many extant literatures [_{2} supply systems do not have an extensive use, therefore the related variables are not taken into account in this work. To simplify the model, we consider only some primary disturbance variables, such as solar radiation, outside temperature and humidity. After normalizing the control variables, we consider the greenhouse dynamic model presented in [

_{in}/T_{out}

_{in}/ H_{out}_{2}^{−}^{1}[dry air]),

^{−}^{1}),

_{0} = _{p}V_{TH}_{p}^{−}^{3}) and the specific heat of air (1,006 ^{−}^{1}^{−}^{1}), respectively. _{TH}_{TH}^{3}) of the greenhouse.

_{i}

^{−}^{1}) and
^{−}^{1}),

_{H}^{−}^{1}, here

_{v}

The ventilation rate _{R}_{R,}_{%}, we define _{%}_{,fog}

The climate model provided above can be used in summer operation, and two variables have to be regulated, namely the indoor air temperature (_{in}_{in}_{R,}_{%}(_{%}_{,fog}

In order to effectively express the state-space form, we define the inside temperature and absolute humidity as the dynamic state variables, _{1}(_{2}(_{1}(_{2}(_{i}

Owing to the complexity appearing as the cross-product terms between control and disturbance variables,

A typical structure of a PID controller involves three separate elements: the proportional, integral and derivative values. The proportional value determines the reaction to the current error, the integral value determines the reaction based on the sum of recent errors, and the derivative value determines the reaction based on the rate at which the error has been changing. The mathematical description of its control law is generally written in the ideal form in (5) or in the parallel form in (6)
_{p}_{i}_{d}_{i}_{p}/T_{i}_{d}_{p}T_{d}

Note that the greenhouse dynamic system mentioned above is a two-input and two-output continuous time nonlinear system. We consider a multi-variable PID control structure as shown in

Integral error is the most commonly used as a good measure for system performance. All kinds of such performance criteria, such as integrated absolute error (IAE), integrated square error (ISE), integrated time square error (ITSE) and integrated time absolute error (ITAE), are often employed in control system design. Killingsworth

In addition, the performance criterion for one loop of MIMO process may be different from the other and some outputs may have the highest priority among the others. In this work, we only consider the same priority for the same class of performance criteria. Consequently, to be convenient for a digital simulation of the system, the performance index in (9) for the MIMO system in the greenhouse is rewritten as follows:
^{th}^{th}

Another aspect, the seriously oscillatory of control signals can do great damage to the actuators, and it is not acceptable in the design of controllers. Therefore, to avoid such case and perform a smoothing operation, we consider another performance index defined as

It is worthy to notice that minimizing the first performance index _{1} will provide good static-dynamic performance and better disturbance rejection, while minimizing the second _{2} will perform a smoothing operation and avoid the serious oscillating of the actuators.

Evolutionary algorithms simulate the survival of the fittest in biological evolution by means of algorithms, and they are becoming increasingly valuable in solving real-world engineering problems. Compared with single-objective evolutionary algorithms, multi-objective techniques have many advantages, especially for the problem with multiple conflicting objectives. For example, they can search for a set of solutions with different trade-offs from a family of equivalent solutions, which are superior to other solutions and are considered equal from the perspective of simultaneous optimization of multiple competing objective functions. Such solutions are generally called non-inferior, non-dominated or Pareto optimal solutions.

The design of the controller usually need to satisfy multiple performance requirements such as good static-dynamic performance and smoothing control operation. However, it is almost impossible to attend the above requirements simultaneously. For instance, in many cases, seeking for some dynamic specifications usually causes the large variance of control law and the serious oscillating of the actuators. Consequently, a satisfactory tradeoff must be found and a set of optimal solutions must be provided by minimizing the performance index _{1} and _{2}. That is, _{1} and _{2} are taken as two objective (cost) functions or fitness functions for the optimization process.

Multi-objective optimization methods are used to obtain Pareto solutions for multiple conflicting objectives. Many algorithms, such as NSGA-II [

Find _{p}_{1}, _{i}_{1}, _{d}_{1}, _{p}_{2}, _{i}_{2}, _{d}_{2}]

to optimize _{1}(_{2}(

subject to _{i}_{i}_{j}_{j}_{j}

for

_{i}

_{i}

_{j}

_{j}

The specifications in the time domain, such as overshoot, rise time, settling time and steady-state error, are incorporated into NSGA-II and are calculated at each iteration, just the same as the objective functions (_{1} and _{2}) of the optimization process. When the obtained performance measures are infeasible during the process of step response, for example, there are not enough data to find rise time (_{r}_{s}_{1} and _{2} as follows:
_{r}_{s}

In the present section, a simulation experiment is presented to demonstrate the validity of the proposed method. For this example, we consider a greenhouse of surface area 1,000 ^{2} and a height of 4 ^{−}^{1}^{−}^{3}. The maximum ventilation rate corresponds to 20 air changes per hour.

The active mixing air volume of the temperature and humidity is given as _{TH}^{2}) of greenhouse area. Moreover, the initial values of indoor air temperature and humidity ratio are 32 °C and 12 g[H2O]/kg[air], respectively. The set points of indoor air temperature and humidity ratio are 25 °C and 21 g[H2O]/kg[air](the corresponding relative humidity about 70%), respectively. The external disturbances are shown in

Step signals are given for the inputs of controllers, and it takes about 11 minutes for each simulation. _{2} drops sharply as that of the index _{1} increases on the left of point _{2} remains almost unchanged with the increase of the index _{1} on the right of point

In order to demonstrate the effectiveness of the proposed tuning method, we consider other performance criteria such as overshoot, rise time, settling time and steady-state error. For simplification of analysis, a simple mean for the respective type performance criteria at each individual is implemented.

PID controllers have been extensively used in the greenhouse production process owing to their simple architecture, easy implementation and excellent performance. However, the tuning of several controllers in the complex greenhouse environment is a challenge to process engineers and operators. Many controllers are poorly tuned in practice due to the complexity of the controlled greenhouse such as the dynamical behavior of greenhouse climate and control requirements, which present strong interactions among variables, non-linearities, multiple constrains and conflicting objectives.

This paper presented the tuning of two PID controllers through NSGA-II based on multiple performance measures such as good static-dynamic performance and smooth control signals. The proposed tuning scheme has been tested for greenhouse climate control by minimizing ITSE and control increment or rate in a simulation experiment. Results show the effectiveness and usability of the proposed method for step responses. The obtained gains are applied in PID controllers and can achieve good control performance such as small overshoot, fast settling time, and less rise time and steady state error.

The results suggest that the proposed tuning scheme using multi-objective optimization algorithms is a quite promising method and it presents the following features: (i) it can be applied in the cases that the empirical methods fail to be used; (ii) it can effectively solve the strong interactions among process variables; (iii) it can be applied into certain strong nonlinear control system including nonconvex problems due to adopting global optimization algorithms.

It should be noted that this study has only examined the analytic greenhouse model. We have to point out that it is not suitable using such an approach if an analytic model is not provided, because there is nearly no effective way to formulate the objective (cost) functions or fitness functions of evolutionary algorithms required. Besides, this approach is time-consuming and heavily dependent on the computation time. It is not suitable for an online real time control requirement. Not withstanding its limitation, this study does suggest that the online optimal operation for greenhouse production process will be further studies.

Certainly, the method is not limited to greenhouse applications, but could easily be extended to other applications, and we expect that it will become more widely used in the future for other types of systems and controllers.

The authors would like to express their appreciation to the referees for their helpful comments and suggestions, and we are very grateful to Kalyanmoy Deb for providing the source code for the NSGA-II on the

Greenhouse climate dynamic model.

The diagram of a greenhouse climate control system.

Changes of outdoor air temperature, humidity ratio and solar radiation.

Pareto Front of performance index _{1} and _{2}.

Step responses with PID control at each individual.

The corresponding control signals.

PID gain parameters of 1^{th}

PID gain parameters of 2^{th}

Identified greenhouse model parameters.

Parameters name | unit expression | values |
---|---|---|

_{0} |
^{−1} |
−324.67 |

^{−1} |
29.81 | |

_{v} |
3.41 | |

465 | ||

^{−3}min^{−1}W^{−1} |
0.0033 | |

1/ |
^{−3}min^{−1} |
13.3 |

Operators and parameters of real-coded NSGA-II.

Description | values |
---|---|

Population size | 80 |

Number of generations | 50 |

Probability of crossover of real variable | 0.9 |

Probability of mutation of real variable | 0.5 |

Distribution index for crossover | 10 |

Distribution index for mutation | 20 |

Lower limits of the gain parameters(_{i} |
[0, 0, 0, 0, 0, 0] |

Upper limits of the gain parameters(_{i} |
[0.5,0.1,0.1,0.2,0.1,0.1] |

Lower limits of the control inputs(_{j} |
[0, 0] |

Upper limits of the control inputs(_{j} |
[1, 1] |

Sampling time (min) | 0.2 |

The corresponding performance criteria (mean of two loops).

Performance criteria | Overshoot (%) | Rise time (min) | Settling time (min) | Steady-state error |
---|---|---|---|---|

Description | ||||

Maximum | 3.6475 | 11.0482 | 15.4252 | 0.0285 |

Minimum | 0.0618 | 3.2781 | 4.6522 | 0.0018 |

Mean | 0.9980 | 5.2858 | 7.6943 | 0.0110 |

Standard deviation | 1.0429 | 2.1129 | 2.9504 | 0.0054 |