Introduction
Food production in greenhouses with controlled environmental variables (temperature, humidity & CO2 content) is an alternative to achieve crops with high production rates, high quality, and low energy cost. In order to improve the efficiency of greenhouse crops, different strategies have been developed for temperature control since this variable strongly impacts the development of the plants 1. One of these strategies is based on the development of algorithms that allow mitigating the effects of dead time when it is dominant on the process dynamics 2, according to the Smith predictor structures type for predictive control 3, modified Smith Predictor 4 and multivariable controllers for greenhouses 5. With the use of these structures, the gain margin, the phase margin and the bandwidth restrictions imposed by dead time systems have been improved 6.
Thus, this development was oriented to the design of a robust controller based on the Quantitative Feedback Theory (QFT) and a structure as from a Smith predictor structure for dead-time system applied to temperature control of the greenhouse to scale with a heating system since this structure offers a high stability based on the gain margin, the phase margin, and the rejection of disturbances in the system output. Thereby, it was started from modeling of the temperature behavior inside a greenhouse was used to design a robust QFT controller (Section II), in which the system stability and controller´s behavior against external disturbances in contrast with a PID controller were validated. Likewise, it was inferred the proposed control strategy performance that showed the robust stability and rejection of disturbances with minimum effort of the control signal (Section III). Finally, conclusions were drawn regarding the study carried out (Section IV).
Methodology
Mathematical model identification
A mathematical model that related the temperature gradient of the greenhouse with the duty cycle applied to the AC-AC converter for a heating system was defined by (1). Besides, the parametric variation of the plant temperature and the system uncertainty space were quantified for the design of the QFT controller temperature.
Therefore, Fig.1 shows a random binary excitation signal (RBS), the greenhouse real system and the identified system response. Likewise, RBS signal related the input and output of the system and was configured with amplitude between [0.25 - 0.75] of the duty cycle of PWM signal, applied to the AC-AC ON-OFF converter 7, with a bandwidth BW= 0.00468 Hz, which was selected from the response of the temperature to a step input signal of 50% of the PWM duty cycle applied to the AC-AC converter. The sampling frequency was F s = 1 and the number of samples was 15000.
Considering RBS signal shown by Fig. 1 first-order transfer functions with dead time was identified, where dead time is L = 120.5 s, the system time constant is T = 213.9 s, and the static system gain is K = 75.4. This model is represented by (1), which related the temperature inside the greenhouse, with the duty cycle of the PWM signal applied to the AC-AC ON-OFF converter.
QFT controller design
The uncertainty space is one of the most relevant aspects and pillars for QFT controllers design 8. Hence, for the developed controller, an uncertainty interval was established for the static gain K, time constant T, and dead time L, listed in Table I, based on identification tests. Those tests are similar to Mathematical model identification made in before section, at different points of operation of the heating system at the greenhouse. Therefore, a family of plants was evaluated against a set of frequencies of interest between 0.0001 rad/s and 0.1 rad/s, taken into consideration the bandwidth of the system. Thus, a phase [°] - magnitude [dB] representation of the plants set on the Nichols chart was obtained for each frequency.
The performance of a controller with a conventional Smith predictor is affected by its sensitivity to the process parametric variation 9, and external disturbances 2. However, dead time compensation techniques based on a modified Smith predictor scheme has been used successfully in the tuning of PID controllers with two degrees of freedom 10, auto-tuning models of PID controllers 11 and in PID controllers with systems that present variable dead time 12.
Hence, robust design of a Smith predictor based on the consideration of the bandwidth and a quantitative approach of the compensator was proposed 13, taking on a structure grounded in the concept of the modified Smith predictor 14, for a system P r with dead time L. The structure uses an estimated plant without delay in an internal loop with an estimated pure delay, which allows mitigating the effects of dead time to facilitate the design of the controller G using a quantitative approach. In Fig. 2, a structure based on the Smith predictor concept for dead time is proposed.
Thus, transfer function H (s) is given by (2), the equivalent plant Peq (s) is given by (3) and the system input-output rate y (s) / r (s) is given according to (4).
Moreover, a QFT controller considering Smith predictor was designed for an uncertainty process. The choice of is a critical factor due to Q (s) degrades the system for each value that H (s) takes in the uncertainty space. So this, one first algorithm was proposed for a plant set selection such that | Q (s)| ≤ m d in the frequency range of interest of the controller 0 ≤ w ≤ ω BW , where m d is set to 3dB14, additionally from the second algorithm, a single plant of the set was selected that satisfied the first algorithm and allowed to minimize the cost function given by (5), where n ((equals the number of frequencies of interest. A∙(T eq (j ω)) represents the model template area and A∙(T (j ω)) represents nominal plant template P r .
Therefore, the transfer function given by (6), was calculated with the algorithms proposed 14 for the frequency range in the matter.
Since greenhouse is subject to external disturbances and presents variation in the parameters due to different environmental conditions, two performance specifications were defined based on the recommended minimum robust stability of 5dB for gain margin and 45° for phase margin given by (7) 15, and in the rejection of load disturbances in the temperature inside the greenhouse given by (8).
Hence, parameters δ u (ω) and δ S (ω) were quantified, either as constants or from transfer functions that represent the desired dynamics of the plant under closed loop 16, 17. The criterion used for robust stability was defined with δ u (ω) = 1.3 15. In this way, the rejection of disturbances of the greenhouse was defined from the parameter δ S (ω) given by (9) 18. Therefore, this was determined as a transfer function that represents the desired dynamics of the plant before a disturbance. Consequently, a settling time of 1500 s was chosen for the output before a step type disturbance, as a condition of the sensitivity function of the system. To define the transfer function δ S (ω), the pole assignment method was applied 16.
Firstly, an L (jω) value must be obtained which fits the inequalities established in the performance specifications, where L (jω) = G (j ω) ∙ P (jω), based on the controller performance specifications given by (7) and (8), in addditon, to the transfer functions that represent the parameters δ u (ω) and δ S (ω). Thus, the control problem focused on determining a unique G (jω) controller that meets all the performance specifications established from the plant with uncertainty P(jω) in the frequency range of interest (19).
In order to solve the control problem, a quadratic inequality was proposed for each performance specification 20, as shown by (10) and (11).
Loop-shaping technique introduces a G (s) controller that modifies the loop function L o until it complies with the constraints imposed by the contours of the performance specifications, this way the unique controller that complies is what manages to take the function of the loop L o on the contours of each specification 19. Fig. 3 shows the response in the frequency of interest. This was achieved by adding poles and zeros to the L o loop function until the desired response was reached 15. The transfer function of the QFT controller is given by (12).
The PID controller was designed from the transfer function P r (s) and performance affixed indices for the QFT controller design associated with its transient response. Since Control System Toolbox in Matlab, PID controller parameters were tuned, this is given by (13). An integrator, a complex zero at 0.00196 ± 0.00775j and a pole on P = -0.1 was added. Besides, the gain was set at K = 5.5×10-5. Proportional gain K p = 0.0028, integral gain K i = 5.184, derivative gain K d = 0.835, and derivative filter constant Nd = 0.183 21 was normalized on equation (14). This was based on parameters given by (13).
Fig. 4 shows the block diagram that allows implementing the QFT controller and PID controller. In this way, to implement the QFT controller, transfer function given by (12) was introduced on G (s) block, and to implement the PID controller, transfer function given by (13) was introduced on G (s) block.
Results and Discussion
To begin with, an experimental system for real-time data acquisition of the greenhouse was implemented, in which the control action was coded into a signal by pulse width modulation (PWM) to determine on and off times on the solid-state relay AC-AC converter. Likewise, in Matlab®, Simulink Desktop Real-Time, real-time control algorithm was implemented to interact physically with the process. Hence, Tests were carried out to validate the stability of the system and the performance of the controller against external disturbances in reference to the greenhouse temperature.
Taking a look at Fig. 5, the system response is displayed for 40°C, which it is observed that the system dynamic response presented an overshoot of less than 1%. Also, settling time was approximately 1000 s, and the control signal remained close to 15% of duty cycle.
Likewise, Fig. 6 represents a conventional PID and QFT controller with a modified Smith predictor response at 40°C. Therefore, it is observed that the QFT controller presented an overshoot of less than 2%, besides that, a lower effort in the control signal and a fast response was noticed in comparison with PID controller that presented an overshoot close to 3%, a greater effort in the control signal and a slower response. The settling time of the QFT controller was close to 1000 s in contrast to the PID controller that approached 1200 s. In addition, QFT controller presented high sensitivity to noise in the sensor, while the PID controller was more robust by the derivative filter. In the same way, both controllers showed an error in a steady state close to zero. Table II lists the performance indices for tests at 40°C and at 50°C.
QFT controller response to an external variation of the temperature inside the greenhouse was validated, which it was subjected to a disturbance at 4000 s. Temperature disturbance is based on a turbine activation that is connected to the greenhouse, which forced the external wind circulation, causing that the temperature inside the greenhouse sudden decrease. Fig. 7 shows the QFT and PID controller’s behavior.
Finally, it is appreciated that the QFT controller lasted 850 s to compensate for the disturbance, whereas the PID controller lasted 690 s. In addition, QFT controller presented an abrupt control action without straining the actuator, whereas the PID controller presented a smoother response. Both controllers showed an error in a steady state close to zero after compensating the disturbance. QFT controller control signal showed an increase of 10% to compensate for the temperature change, whereas the PID controller showed an increase of 13%. Table III shows indices performance for tests at 40°C of temperature.
Conclusions
The proposed controller applied to the range of uncertainty for the temperature system parameters quickly mitigated the effects of the dead time, which favored the system tuning and therefore its stability. Likewise, external disturbances effects and changes in the point of operation with minimum effort of the control signal were mitigated. It also kept within controller performance specifications such as settling time and the overshoot. Final results showed that the dynamic response of the QFT controller improved 12%, with a decrease of 1% in the overshoot and 3% in the effort of the control signal, compared to PID controller results. Lastly, implemented an experimental system for the acquisition of real-time data from the greenhouse allowed demonstrate high sensitivity to noise in QFT controller sensing, in contrast to the low sensitivity of the sensing in PID controller. This condition raised the need for a more exhaustive study to improve the sensitivity in QFT controllers.