The three models for the greenhouse system include: cooling model, heating model and ventilating model. Ventilation is one of the most important components for a successful greenhouse. If there is no proper ventilation, greenhouses and their plants become prone to problems. The main purpose of ventilation is to regulate the temperature at the optimum level, to ensure movement of air and also to ensure the supply of fresh air for photosynthesis and plant respiration. Heating model is essential, when the inside temperature of the greenhouse is very low during winter climate and at night time. Whereas, cooling model is required when the outside temperature is very high which may affect the plants during the summer mode at day time. The functional block diagram of GHS is shown in Figure 1.

A simple greenhouse heating-cooling-ventilating model can be obtained (as given in Equation (1)) by considering the diff erential equations, which govern sensible and latent heat, as well as water balances on the interior volume [5] [6] :

where

T_in/T_out: Indoor/outdoor temperature (˚C)

H_in/H_out: Interior/exterior humidity (g[H₂O/kg[dry air])

Q_heater: Heat provide by the greenhouse heater (W)

Q_fog: Water capacity of fog system (g/s)

V_R: Ventilation rate (m³/s)

The greenhouse system model parameters are shown in the Table 1.

In general, the operating conditions of the ventilation/cooling process are rather dominated by solar radiation alone (i.e.), hence can be neglected. Suppose that, , and. The control variables can be normalized as follows:, and.

In summer mode operation, Q_heater is set to zero. The climate model provided have two variables to be regulated, namely the indoor air temperature (T_in) and the humidity ratio (H_in), through the process of ventilation V_R,%(t) and fogging Q_R,%(t). After normalizing the control variables, the differential equation for cooling model for greenhouse system is re-written as given in Equation (2)

In order to express the GHS in state-space form, the inside temperature and absolute humidity are defined as the dynamic state variables, x₁(t) and x₂(t), respectively; the ventilation rate and the water capacity of the fog system as the control (actuator) variables, u₁(t) and u₂(t), respectively; the intercepted solar radiant energy, the outside temperature, and the outside absolute humidity as the disturbances, v_i(t), i = 1, 2, 3. The Equation (2) can alternatively be written in the following state-space form:

From the state-space form the values of A, B and C is given by:

In this case relative gains are positive values. If 0 < λ₁₁ < 1, then an interactions exists. From the interaction analysis it is found that y₁ is paired withu₁ and y₂ is paired withu₂ to form two loops with minimum interaction. However, the disturbances at one loop will affect responses of other loop due to coupling. Hence, decoupling is necessary for GHS. The feedback-feed forward linearization technique is used for decoupling is discussed in the subsequent section.

One method for linearization and decoupling for systems with external disturbances is presented in [2] [3] which is based on the feedback-feed forward linearization technique proposed by Isidori [4] . In the particular case of GHS model expressed in Equations (3) and (4), the feedback-feed forward linearization and decoupling procedure described above is followed and compensator is designed. The compensator that delivers the process input control variables u(u₁, u₂), has the form given in Equations (5) and (6).

where Q(t) must be anon-zero quantity.

where,

temperature change;

humidity change.

The transfer functions derived from the linearized and decoupled GHS are as given in Equations (8) and (9)

In the Internal Model Control (IMC) formulation, the controller, q(s), is based directly on that part of the process transfer function which can be inverted. The IMC formulation generally results in only one tuning parameter, the closed loop time constant (λ, the IMC filter factor). The PI tuning parameters are then becomes a function of this closed-loop time constant. The selection of the closed-loop time constant is directly related to the robustness (sensitivity to model error) of the closed-loop system.

The feedback controller contains the internal model and internal model controller, q(s). Now, the IMC design procedure can be used to design a standard feedback controller. The standard feedback controller is a function of the internal model, and internal model controller, q(s) as given in Equation (10).

For a given first-order process in Equation (10), the values of the PI tuning parameters are obtained using Equations (11), (12).

The desired value for λ is defined as a trade-off between performance and robustness and its optimum value is obtained using genetic algorithm. The objective function considered here is Integral Square Error (ISE) and the decision variables obtained is λ, filter factor.

The controller parameter values for IMC-PI are calculated by using trial and error method and the parameter values are tabulated in Table 2.

Although PI parameters obtained using conventional method gives a good response, it is not satisfactory. In order to meet the desired specification PI controller parameters are tuned using optimization technique [7] [8] namely Genetic Algorithm (GA). The various steps involved in the GA based optimization of PI controller are listed below.

Step 1: Choose the string length, population size, probability of crossover, probability of mutation and number of generation.

Step 2: Initialize the population size K_c and K_I..

Step 3: Carry out selection operation. Check for acceptance criteria, if satisfied stop otherwise goto next step (Step 4).

Step 4: Perform crossover operation. Check for acceptance criteria, if satisfied stop otherwise goto next step (Step 5).

Step 5: Perform mutation operation. Check for acceptance criteria, if satisfied stop otherwise goto next step (Step 3).

The GA parameters used for tuning GA based IMC- PI scheme are given in Table 3.

The desired PI parameters, around an operating point, are generated in random as the initial population. Each of these PI settings is applied to the system and the fitness of each chromosome is calculated. From that, the chromosomes with the best fit value are taken to the next iteration whereas the chromosomes with the worst fit are discarded and are replaced with new chromosomes derived from the mutation and crossover of best parent chromosomes. The function of the crossover operator is to generate new or “child” chromosomes from two “parent” chromosomes by combining the information extracted from the parents. Mutation operates individually on each individual by probilistically perturbing each bit string. This process repeats until the desired performance index is met with.

The objective function considered here is Integral Square Error (ISE) and the decision variables obtained are PI controller parameters namely K_C and K_I. The controller parameter values, thus obtained, for IMC-PI using GA are tabulated in Table 4.

The IMC-PI parameters obtained using GA based optimization technique. This is reflected in the value of λ obtained using GA based optimization algorithm gives a good response but the reduction in error is minimum. In order to further minimize the error the PI parameter is tuned using an optimization technique namely Particle Swarm Optimization (PSO).

In the design of IMC-PI using PSO, the desired PI parameters, around an operating point, are generated through a bird flocking in two-dimension space. The position of each agent is represented by XY axis position and also the velocity is expressed by v_x (velocity of X axis) and v_y (velocity of Y axis). Each of these PI settings is applied to the system; the velocity and position are calculated. From that, the best values are taken and the agent position is realized by the position and velocity information. The objective function considered here is Integral Square Error (ISE). To implement the PSO algorithm the values to the variables of the PSO algorithm is given by sampling number N = 12, the local attractors and global attractors C₁ = 2, C₂ = 2, [9] . The steps involved in PSO based optimization are listed below.

Step 1: Initialize the population of particles with random positions and velocities.

Step 2: For each particle, evaluate the fitness.

Step 3: Compare the particles current fitness (X_i) with the particle best fitness (P_i). If current is the best then set P_i fitness equal to current values and set P_i location equal to current location otherwise go to step 4.

Step 4: Compare the fitness of the particle (P_i) with the population overall best P_g. If the current value of particle fitness is better than P_g, then P_g fitness equal to current value otherwise go to step 5.

Step 5: Change the velocity and position using the Equation (13).

V_i (new) = W_iV_i + C₁ rand (P_i − X_i) + C₂ rand (P_g − X_i) (13)

Step 6: Repeat step 2, 3 & 4 till we get the best fitness.

The controller parameter values for IMC-PI controller using PSO are tabulated in Table 5.

The GHS output responses namely the inside temperature and inside humidity obtained using three different control schemes for the linearized and decoupled of the greenhouse are shown in Figure 3(a) and Figure 3(b) respectively.

From Figure 3(a) and Figure 3(b), it is observed that the performance of the IMC- PI controller tuned using PSO is better. This is reflected in the value of λ that is computed using the PSO-based optimization algorithm.

The corresponding variation in control signals namely the ventilation rate and water capacity of the fog system, for the linearized and decoupled model of the GHS are shown in Figure 4(a) and Figure 4(b), respectively. It is observed that the variation in the manipulated variable is smooth.

It is observed that the variation in the manipulated variable is smooth.