Simulation of Active Control Using Fuzzy Logic Applied to a Pulse Combustor

Simulation of Active Control Using Fuzzy Logic Applied to a Pulse Combustor

C. A. Botura

G. Botura Jr.

J. A. Carvalho Jr.

L. Mesquita

Universidade Estadual Paulista

Campus de Guaratinguetá

Av. Ariberto Pereira da Cunha, 333

12516-410 Guaratinguetá, SP. Brazil

M. A. Ferreira

Instituto Nacional de Pesquisas Espaciais

Rod. Presidente Dutra, km 40

12630-000 Cachoeira Paulista, SP. Brazil

This work analyzes an active fuzzy logic control system in a Rijke type pulse combustor. During the system development, a study of the existing types of control for pulse combustion was carried out and a simulation model was implemented to be used with the package Matlab and Simulink. Blocks which were not available in the simulator library were developed. A fuzzy controller was developed and its membership functions and inference rules were established. The obtained simulation showed that fuzzy logic is viable in the control of combustion instabilities. The obtained results indicated that the control system responded to pulses in an efficient and desirable way. It was verified that the system needed approximately 0.2 s to increase the tube internal pressure from 30 to 90 mbar, with an assumed total delay of 2 ms. The effects of delay variation were studied. Convergence was always obtained and general performance was not affected by the delay. The controller sends a pressure signal in phase with the Rijke tube internal pressure signal, through the speakers, when an increase the oscillations pressure amplitude is desired. On the other hand, when a decrease of the tube internal pressure amplitude is desired, the controller sends a signal 180º out of phase.

Keywords: Fuzzy logic, pulse combustion, rijke tubes

Introduction

The occurrence of combustion generated acoustic oscillations is very common in rocket chambers and in industrial furnaces, and such oscillations can lead to disastrous consequences. This type of process is known as pulsating combustion. However, in several cases, there are benefits associated with pulsating combustion, as improvement of fuel and oxidizer mixing, larger heat transfer rates, higher reaction temperatures, and more efficient combustion.

The Rijke tube consists of a vertical tube with a heat source placed in its lower half (Lord Rayleigh, 1945). The device, with an appropriate instrumentation, allows the study and control of the oscillations. A Rijke type combustor is obtained if the heat source is a combustion process. Figure 1 shows a scheme of such combustor for liquified petroleum gas (Momma, 1993).

An important scope of research in combustion is centered in combustion instabilities, its influences and mechanisms of attenuation. To attenuate those oscillations, several control schemes were developed. The most practical and successful were those based in a closed-loop control system to produce the control signal. Control is accomplished by generating a pressure wave exactly 180 degrees out of phase with the pressure wave in the tube. To implement this type of control it is necessary to know the existing pressure wave and use the pressure signal as the input to realize the control. Heckl (1988) developed an algorithm based on a feedback network to control and eliminate oscillation instabilities inside a Rijke tube combustor. The test results showed an attenuation of more than 40 dB in the oscillation level. Poinsot et al. (1989) also used the destructive interference and feedback concept to suppress instabilities. The theory was tested in a 250 kW combustor, achieving a sound pressure attenuation of more than 20 dB. McManus et al. (1993) conducted several studies using open and closed-loop control systems and adaptive control mechanisms for oscillation attenuation.

Oscillation stabilization was the objective in the three studies above. However, literature neglects the use of active control in practical systems, which are influenced by a pressure field with a determined frequency. Heckl (1988), Poinsot et al. (1989), and McManus et al. (1993) showed the viability of active controller implementation in a pulsating device through a relatively simple instrumentation. The speed of the computer, an important parameter for real time system control, allowed fast data acquisition and processing of the collected data and the execution of a large number of interruption instructions. This together with the digital controller accuracy, and sophisticated sensors and actuators allowed the application of active control in combustion instabilities.

Without control, a combustor may not oscillate at an expected pressure level. Ferreira (1997) could not attain repeatability in his values of acoustic pressure amplitude in a gaseous fueled Rijke combustor.

Direct control is one of the methods to change the pressure in a Rijke tube, related to modulation of the acoustic pressure field with speakers. Another method uses fuel modulation to change the acoustic field indirectly through an oscillatory heat liberation process (Dubey, 1998).

Figure 2 shows a block diagram of the dynamic controller and the speaker position in the Rijke tube for modulation of the acoustic pressure field. The position of the speakers has vital importance because the pressure modes cannot be controlled if they are positioned in pressure knots.

A method to conduct direct control of the pressure wave is described in this work. The method uses fuzzy logic, which was developed by Zadeh (1965), and further utilized in control of several mechanical and thermal systems (Umbers and King, 1980; Miyamoto et al., 1998; Bortolet et al., 1999; Caputo and Pelagagge, 2000; Li and Chang, 2000).

Fuzzy Logic Considerations

Fuzzy logic sets the control system operational laws in linguistic terms, instead of using mathematical equations, as it is done in the classical methodology. Fuzzy logic is applied in systems with high degree of complexity, in which modeling using mathematical equations leads to imprecise control schemes. The use of fuzzy logic in complicated systems takes the advantages of the linguistic terms to represent them correctly. These terms are frequently expressed in the form of logical implications, such as rules "IF-THEN." A fuzzy controller is in general composed of three blocks, each one corresponding to the following phases: 1. fuzzification, 2. inference, 3. defuzzification.

A fuzzy controller operates repeating a cycle of the three above phases. First all the important variables that represent the process to be controlled are measured. The variables used in the present work are: 1. input variables: error and error variation, 2. output variable: amplitude. After this, the measured signals are transformed in an appropriate fuzzy set by membership functions. This step is called fuzzification. The membership functions used in the control that represents a fuzzy subset have been labeled: NB: Negative Big, N: Negative, NS: Negative Small, Z: Zero, PS: Positive Small, P: Positive, PB: Positive Big.

The fuzzy set is then used by the inference engine, which is based in the control rules, to define the overall fuzzy output for defuzzification. Figure 3 shows the block diagram of the fuzzy controller. Table 1 shows the 49 inference rules utilized in the developed system.

Graphs of the membership functions for the input variables "error" and "error variation" are presented in Figure 4. These functions are identical for both variables. Graphs of the membership functions for the output variable "amplitude" are presented in Figure 5. The formats utilized for such membership functions are triangular and trapezoidal. This is explained for the case in which the variable membership degrees can be obtained, stored and efficiently handled in terms of the real time needed by the inference phase (Driankov et al, 1996).

The combination of individual subsets obtained from the inference engine is then transformed, in the defuzzification phase, in a crisp value. This value represents actions to be taken by the fuzzy controller in cycles of individual control. The action developed in this control system is the generation in the tube of a sinusoidal pressure wave with a determined RMS amplitude value. In the work of this paper, the centroid method was used to perform defuzzification.

Rijke Tube Modeling

The Rijke tube is ruled by Rayleigh criterium (Lord Rayleigh, 1945), which states that when heat is transferred to a flowing gas in phase with a pressure disturbance, the pressure amplitude will increase. Pressure amplitude is limited by energy balance between the energy absorbed by the acoustic wave and that dissipated by the tube.

Developed Model

The Rijke tube is represented by the block diagram of Figure 6. The developed modes uses the characteristics established by the Rayleigh criterion. The tube acoustic waves are essentially sinusoidal, oscillating in one of the tube’s fundamental frequencies.

The block "Internal pressure generator" provides a sinusoidal wave from input amplitude, frequency and phase values. In the case of a Rijke tube excited with speakers, the amplitude will be maintained at a pre-set value, either amplifying or attenuating the tube natural acoustic wave. The program utilized by the simulator was developed from this block diagram.

Control System Model

The control output supplied is a voltage signal that will be amplified and supplied to the speaker. Figure 7 presents the control system block diagram.

The Rijke tube model, presented and discussed in the item "Developed model", is now represented by the "Rijke tube" block. This block receives as input the sinusoidal pressure signal from the "Speaker" block. In the real Rijke tube system this corresponds to the signal provided by the speaker. The values established by the system operator are represented by the "Set-point" block. The "Comparator" block executes the comparison between the RMS pressure amplitude value, received from Rijke tube block, and the signal in RMS value received from the "Set-point" block. Its output is the difference between these values. The difference corresponds to the error. In the real system, the tube pressure amplitude is obtained from pressure transducers coupled to charge amplifiers. The block "Memory" memorizes the error of the output from block "Comparator". The "Subtractor" block executes the difference between the error value, obtained in the output from the block "Comparator" and the output from the block "Memory". With this difference the variation of the error between two consecutive instants of control is obtained. The obtained value is the control variable error variation ("errorvar").

The control is performed by the "Fuzzy Logic Controller", whose internal constitution is presented in Figure 3. It receives the variables "error", from the "Comparator" block, and "errorvar", from "Subtractor" block, as inputs. The output of the "Fuzzy Logic Controller" block furnishes an RMS value for the sinusoidal pressure signal, generated in the "Sine wave generator" block, which is sent to the "Speaker" block. This pressure wave is responsible for the alteration of the tube internal pressure.

Results and Discussions

To perform simulation and obtain the results for analysis, the tube was taken as oscillating with an internal pressure wave whose RMS amplitude was 30 mbar. The required variation of pressure with time was considered to be that shown in Figure 8, which was put into the "Set-point" block. The total delay time is 2 ms.

In this example case, control starts at 0.1 s and the tube remains under the action of the previously set variables until the start of control. In the instant 0.1 s, the required RMS pressure amplitude is 90 mbar, remaining in this value until 0.5 s, when a 30 mbar RMS pressure amplitude value is selected in the "Set-point" block.

Figure 9 presents the sinusoidal wave resulting from the pressure wave inside the tube and the action of the controller and Figure 10 presents the RMS value of the wave.

It can be noted that the natural sinusoidal wave inside the tube remains with constant amplitude until 0.1 s. In this instant, when the system is turned on, the reference value for the pressure is 90 mbar. Starting from this instant, there is amplification in the pressure wave amplitude as time goes on, due to the action of the pressure signal introduced by the speaker. Figure 11 presents the error signal in the controller input. In approximately 0.4 s, the tube internal wave reaches the previously defined value in the "Set-point", with an error of 0.03 mbar. From the instant 0.5 s, the value in the "Set-point" is set in 30 mbar. It can be verified that the pressure starts decreasing, reaching the new required value in approximately 0.8 s, with an error of 0.01 mbar. Figure 12 presents the pressure signal supplied by the speaker. The signal from the "Sine wave generator" block decreases as the error converges to zero. This occurs in both cases: from 30 to 90 mbar and from 90 to 30 mbar.

With Figure 13 it is possible to evaluate the phase between the generated signal and the tube internal pressure signal, through the error of "Comparator" block. If the error is positive, an increase of the tube internal pressure is required. With the detail on the left side of the figure, it is seen that if the signal in the "Sine wave generator" block output is in phase with the tube pressure sinusoidal signal, the latter is amplified until reaching the set point value. On the other hand, if the error is negative, it is necessary to decrease the tube pressure amplitude. By generating, in the "Sine wave generator" block output, a sinusoidal signal 180º out of phase, a decrease of the pressure amplitude is obtained. This is observed in the detail on the right of Figure 13. The figure shows that the controller generates signals in phases of 0º and 180º in relation to the tube pressure and these signals are capable of executing the intended function. The amplitude variation of the generated signal with the decrease of the error value can also be observed. This amplitude variation decreases as the pressure approaches the value defined as set point.

Finally, Figure 14 shows the response of the tube pressure wave for different computation processing and power amplifier total time delays. In this case, the pressure transducer and charge amplifier delays were kept as 1 ms.

Conclusion

This work discussed a fuzzy logic controller system for a Rijke tube type combustor. First, it was necessary to develop a model to represent the tube operation to be controlled. The necessary control variables were obtained, as well as the membership functions and necessary rules for the fuzzy controller.

The simulation showed that the use of fuzzy logic control is viable. The developed control system presented excellent results with an example pulse inserted in the "Set-point" block. It was verified that the control needed approximately 0.2 second to increase the tube internal pressure amplitude from 30 to 90 mbar, considering a total delay of 2 ms.

The delay variation effects were investigated. It was verified that, even requiring different stabilisation times for different delays, convergence was always obtained. The general control system actuation was not affected by the delay.

It was verified that the controller sends a pressure signal in phase with the internal tube pressure signal, through the speaker, when an increase the oscillation pressure amplitude is required. The phase is 180º when a decrease of the internal pressure amplitude is required.

The use of fuzzy logic in the control of a Rijke tube and other acoustic devices does not require a well established mathematical model, which is necessary and, often very difficult to obtain, when a conventional control system is developed.

Acknowledgment

The authors are grateful to FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) for the support of this work through process number 1997/7308-6.

Presented at COBEM 99 - 15th Brazilian Congress of Mechanical Engineering, 22-26 November 1999, São Paulo. SP. Brazil. Technical Editor: José Roberto F. Arruda

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Publication Dates

• Publication in this collection
  21 Aug 2002
• Date of issue
  May 2002