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, etc.) and also by many other practical constraints (actuators, moistening cycle, etc.). In recent years, various advanced control techniques and related strategies, such as predictive control [1–3], adaptive control [4], nonlinear feedback control [5], fuzzy control [6–8], robust control [9], optimal control [10,11] and compatible control [12] are widely proposed for different types of greenhouse environment control. These studies are important to real-world engineering application in greenhouse production. However, most of these approaches are either theoretically complex or difficult to implement in the actual greenhouse production, and the controller designs in the greenhouse engineering application mostly adopt the conventional proportional, integral, and derivative (PID) controllers owing to the simple architecture, easy implementation and excellent performance. Even so, they do not generally give an overall consideration of various properties, such as strong interactions among variables, nonlinearities, multiple constrains and conflicting objectives, that may exist in greenhouse climate control systems. Naturally, it is difficult for such controllers to achieve a satisfactory control effect, and the tuning of such controllers is still a challenge to process engineers and operators in the greenhouse production.

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 [13–16], which include designs based on guess-and-check (such as trial and error tuning method), linear control theory (such as Ziegler-Nichols (Z-N) and Cohen-Coon methods (C-C)), etc. Nevertheless, it is hard for these conventional tuning methods to achieve the desired performance of the controlled greenhouse because there is nearly no effective analytical way of finding the optimal set of gain parameters, and they are mostly based on linear models, which are usually adjusted around operating points. Empirical methods such as Z-N, which can be used for the tuning a simple problem, nearly fail to deal with the complex systems like greenhouse environment, owing to the lack of empirical data for a wide range of problems. Hence, new designs for tuning the PID parameters have to be explored to regulate the greenhouse environment.

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 [13] proposed a modified crossover formula in genetic algorithms (GAs) to determine PID controller gains for multivariable processes. Arruda et al. [14] proposed an automatic tool to tune PID controllers in a MIMO process based on a Rank Niching Genetic Algorithm. The proposed scheme can be applied into a coupled MIMO process with several control loops, which are tuned in a unified way as a full MIMO controller. Herrero et al. [15] proposed a tuning scheme for robust PID controllers by adopting a specific multi-objective evolutionary algorithm. The scheme considers parametric uncertainty and can deal with model uncertainty by using a robust identification method. Ayala et al. [16] presented the design and the tuning of two PID controllers through Non-dominated Sorting Genetic Algorithm-II (NSGA-II) [17]. It is simple but effective to implement the robust solutions providing a good reference tracking performance in closed loop. These works mentioned above have obtained a certain effect by taking different approaches to different characteristics of their respective problems.

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.

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) (5) u ( t ) = K p ( e ( t ) + 1 T i ∫ 0 t e ( τ ) d τ + T d d e ( t ) d t ) (6) u ( t ) = K p e ( t ) + K i ∫ 0 t e ( τ ) d τ + K d d e ( t ) d twhere K_p is the proportional gain, T_i is the integral time constant, T_d is the derivative time constant, K_i = K_p/T_i is the integral gain and K_d = K_pT_d is the derivative gain. And e(t) is the current error signal, which is defined as (7) e ( t ) = r ( t ) − y ( t )where r(t) and y(t) are the reference signal and process output, respectively.

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 J₁ and J₂. That is, J₁ and J₂ 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 [17], Strength Pareto Evolutionary Approach 2 (SPEA2) [26] and the Pareto Archived Evolution Strategy (PAES) [27], appear to be very promising ways to approximate Pareto fronts. Considering newly developed and versatile multi-objective evolutionary algorithms, we adopt NSGA-II to optimize the performance criteria in this work, because it is a computationally efficient algorithm implementing the idea of a selection method based on classes of dominance of all the solutions. Each individual ind in the evolutionary algorithm represents the gain parameters of PID controllers. Therefore, the problem can be formulated as follows:

Find ind = [K_p1, K_i1, K_d1, K_p2, K_i2, K_d2]

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 (i.e., J₁ and J₂) 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 (t_r) and settling time (t_s), that is to say, the system is unstable, we formulate the penalty for the objective functions J₁ and J₂ as follows: J 1 = { ∞ , if   t s = = ∞ | | t r = = ∞ , J 1 , otherwise , J 2 = { ∞ , if   t s = = ∞ | | t r = = ∞ , J 2 , otherwise , where the performance measures t_r and t_s are given with infinity, respectively, when they are not found. Therefore, the infeasible solutions are immediately discarded upon creation, and not considered further during evolution, owing to the infinity of objective functions. Conversely, there exist no influence on the normal situation.

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 m² and a height of 4 m. The greenhouse has a shading screen that reduces the incident solar radiation energy by 60%. The maximum water capacity of the fogging system is 26 g[H2O]min⁻¹m⁻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 V_TH = 0.65 V. The greenhouse model parameters (shown in Table 1) are presented by [5] through identifying method, which are expressed per square meter (m²) 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 Figure 3. We consider the real-coded NSGA-II for the optimization. The corresponding operators and parameters are shown in Table 2. Simulations are performed under the environment of Matlab R2010a (version 7.10.0.499) and Microsoft Windows XP Professional SP2 OS. Computer used for these simulations is Intel(R) Core(TM)2 Duo CPU, 1.83 GHz, 2 GB of RAM.

Step signals are given for the inputs of controllers, and it takes about 11 minutes for each simulation. Figure 4 shows the Pareto front of simulation results using NSGA-II, and the results indicate that the performance objectives are conflicting decision criteria with each other. As can be seen from the Figure, the value of the index J₂ drops sharply as that of the index J₁ increases on the left of point A, whereas the index J₂ remains almost unchanged with the increase of the index J₁ on the right of point B. It suggests the Pareto Front AB is the optimal desired region for most situations. Therefore, a satisfactory tradeoff may be found from the region, and Decision Maker (DM) can select the best final solution according to his or her preference.

Figure 5 shows the effectiveness of step responses with PID control at each individual. The top plot shows the response from 32 °C to 25 °C for the inside air temperature, and the curve distribution is centralized and it reveal similar responses for each individual of population. The bottom plot shows the humidity change from 12 k/kg to 21 k/kg, and the rising time and setting time of some responses are longer than others and it shows the different response characters. Besides, both figures show the steady-state error is very small and reveal that they have good steady state performance. Moreover, the tuning scheme can achieve very good control performance as a whole. The corresponding control signals are illustrated in Figure 6. There are hardly oscillating process and the control signals are smooth. To achieve the desired climatic condition, the fogging systems should operate at high power, while low power for ventilation systems in the initial stage of control process. Then, the fogging systems maintain low power to operate, while high power for ventilation systems.

Figures 7 and 8 show the distributions of PID gain parameters at each individual. From the results, the controller I has small derivative gain, or even it may be implemented by adopting PI loop under the most conditions, whereas the proportional gains are large for two control loops. Moreover, the gain parameters tend to the same value with the evolution process, which reveals evolution completely during optimization processes and can find the optimal gain parameters.

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. Table 3 shows the maximum, minimum, mean and standard deviation of these specifications in the time domain. As can be seen, the maximum values of overshoot, rise time, settling time and steady state error are 3.6475%, 11.0482 minutes, 15.4252 minutes and 0.0285, respectively. Compared with other systems, the rise time and settling time are large, but the values are small enough for greenhouse production. It is worth notice that the values of minimum, mean and standard deviation are 0.0618%, 0.998% and 1.0429% in the column of overshoot, respectively. Nevertheless, there exist many curves without overshoot in the two responses of Figure 5, which implies that at least one of the two loops has overshoot. On the whole, these specifications shown in Table 3 reveal the system has good static-dynamic performance, and are completely acceptable for greenhouse climate control system.

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.