This paper proposes a control strategy for complex and nonlinear systems, based on a parallel distributed compensation (PDC) controller. A solution is presented to solve a stability problem that arises when dealing with a Takagi-Sugeno discrete system with great numbers of rules. The PDC controller will use a classical controller like a PI, PID, or RST in each rule with a pole placement strategy to avoid causing instability. The fuzzy controller presented combines the multicontrol approach and the performance of the classical controllers to obtain a robust nonlinear control action that can also deal with time-variant systems. The presented method was applied to a small greenhouse to control its inside temperature by variation in ventilation rate inside the process. The results obtained will show the efficiency of the adopted method to control the nonlinear and complex systems.

In the last few decades the Takagi-Sugeno (TS) fuzzy systems have become an important means for both modeling and control, and their performance in these two domains is proved in the research especially in the application of complex and nonlinear systems. TS fuzzy systems use a multimodel approach by fitting enough local linear models each describing an operating functional zone of the process [

Such a tool can be used for control purposes; in fact when dealing with complex systems classical methods become inefficient and so emerges the need for other control techniques. The PDC controllers [

In the continuous domain, a solution was presented using a PD or a PID controller for each rule. The denominator of the closed loop system is treated as an uncertain polynomial with affine linear uncertainty structure where the stability can be verified using a frequency domain criterion [

There are different classes of fuzzy systems; the most often used are Mamdani fuzzy systems [

The fuzzy proposition “

Assume that all the subsystems considered are completely controllable and completely observable. Also, denote the following states and inputs variables:

The matrix (

The output level

Assuming that

On the other hand, the controller is a TS fuzzy system, having the same antecedent of the fuzzy model and differing from it in its consequent. A linear controller is developed for each rule, and the global control action is synthesized in the same way as the TS fuzzy model output [

The stability problem of (

The following theorem gives a sufficient stability condition for a system of the form of (

The closed-loop TS fuzzy system of the form of (

The problem arises when the TS fuzzy system has a great number of rules, for it is difficult to find matrix

The solution presented in this paper deals with a transfer function form of the TS fuzzy system. Combining (

Let

The transfer function of the closed-loop fuzzy system will be

From (

In fact, the cross-coupling and possible instability are created by the two multiplications

In order to fulfill this objective the chosen local controllers must have the same order as the local models. For instance, if the system is a first-order one, then the local controllers must be a PI, and if the system is a second-order one, then the correspondent local controllers should be a PID. For higher-order systems, one can choose an RST controller to each local model with an equivalent order having the polynomial

Also, assuming all stable zeros of the transfer function (

Here all the local controllers share the same polynomial

We have also

In the transfer function (

The presented method was applied to control the inside temperature of a small greenhouse (Figure

Greenhouse process.

The effect of the solar radiation is preponderant during daylight; in fact, the components of the greenhouse absorb the radiation energy and convert it to heat energy released in the air by heat transfer [

Difference between inside and outside temperatures during daylight.

Solar radiation intercepted during daylight.

All these factors result in a nonlinear evolution with time-varying parameters of the climate inside the greenhouse [_{2} concentration) [

But the complexity of the system makes it hard to achieve successful results, and so most of the farmers use an on-off control action between two boundary temperature values. But, this kind of control is unhealthy to the canopy. On the other hand, the time where the ventilation is effective is related to the weather. In fact, the possible constraints that oppose the use of this method all along the year are

the presence of solar radiation that will enhance the inside temperature and make it surpass the outside temperature,

the desired temperature of the climate inside the greenhouse that must be greater with several degrees than the outside temperature. The choice of the desired temperature depends on the requirement of the cultivation inside the greenhouse.

These conditions can be met in 3 to 4 months of the year. When the outside temperature is near the desired inside temperature, the farmers use the pad cooling system; it is a combination between fans and a wet pad that decreases the introduced air with several degrees [

In this paper, we focus on the evolution of the inside temperature by manipulating the ventilation rate to obtain a desired behavior of the process. As described in Section

The process is a greenhouse with the following geometrical characteristics: 1.5 m height, total width of 1 m, and total length of 1.5 m.

The inputs of the system are obtained from several sensors located inside and outside the greenhouse as follows.

The inside and outside temperatures were measured by two LM35 transistors; both have an AD 620 amplifier to amplify the delivered signal.

The inside humidity was measured using a humidity sensor (SY-HS-230BT).

The solar radiation was obtained by a pyranometer type LP-PYRA 03 module.

In order to control the ventilation rate, the greenhouse was equipped with two fans driven by two three-phase motors. These engines have a power supply delivered by a frequency converter (microdrive FC 51 Danfoss). The rotational speed of the fans is proportional to the frequency of the three-phase power supply and so the ventilation rate inside the greenhouse depends on the frequency of the power supply. Thus, the control input will be the frequency that will be computed by the fuzzy controller.

The communication between MATLAB, the sensors, and the frequency converter is carried by the data acquisition module (KUSB-3100).

The structure of the discrete TS fuzzy system that will be approximated will have a local model for every rule

The fuzzy identification is performed in two steps. The first one is the determination of the appropriate membership functions of each input to set the antecedent part of the TS fuzzy system, which is done using fuzzy clustering based on C-means algorithm. The membership functions were fixed and

Membership function of the inside temperature.

Membership function of the delivered frequency.

Membership function of the outside temperature.

Membership function of the inside humidity.

Membership function of the intercepted solar radiation.

The second step is the estimation of (

The estimated vector

Adaptive fuzzy control scheme.

The system is a first-order one so the local controller for each rule will be a PI controller. From the previous section, the transfer function of the PDC controller can be deduced as follows:

With this expression it is also possible to take into account the saturation of the control input since we deal with local classical controllers with recurrent form [

The desired output

Practical results: outside temperature (blue line) and inside temperature (black line).

Action control of the fuzzy controller.

Intercepted solar radiation.

Inside humidity.

Moreover, we can note from Figure

TS fuzzy systems can assimilate the evolution of complex and nonlinear process and act consequently to provide acceptable results in terms of modeling and control. However, a stability problem arises because of the great number of rules used to model such systems. This paper offers a solution using classical controllers like a PI, a PID, or an RST for each rule and a pole placement strategy. The fuzzy controller presented combines the multicontrol approach and the performance of the classical controllers. The result is a robust nonlinear control action that can also deal with time-variant systems.