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Volume : 44

Part : 2

 

 

Suppression of quasiperiodic thermoacoustic oscillations via genetic programming

 

Bo Yin1

The Hong Kong University of Science and Technology

Clear Water Bay, Hong Kong

 

Yu Guan2

The Hong Kong University of Science and Technology

Clear Water Bay, Hong Kong

 

Stephane Redonnet3

The Hong Kong University of Science and Technology

Clear Water Bay, Hong Kong

 

Vikrant Gupta4

Southern University of Science and Technology

Shenzhen, Guangdong Province, China

 

Larry K.B. Li5

The Hong Kong University of Science and Technology

Clear Water Bay, Hong Kong

 

ABSTRACT

 

We use genetic programming (GP) to discover data-driven control laws for the suppression of qua- siperiodic oscillations in a prototypical thermoacoustic system. We rank the control laws based on a predefined cost function that accounts for the pressure amplitude and the actuation effort. We then breed subsequent generations of control laws via a tournament process. We find that GP closed-loop control is more effective than GP open-loop control and conventional periodic forcing, producing a similarly high degree of amplitude suppression but with the lowest actuation effort. We also find that GP closed-loop control can identify unforeseen actuation mechanisms, providing new insight into the physical coupling between the heat release rate and pressure fields.

 

1. INTRODUCTION

 

Most control strategies used in thermoacoustics are designed around fixed-point or period-1 states [1, 2, 3]. However, aperiodic oscillations can arise as well [4-7]. In the context of controlling aperi- odic thermoacoustic oscillations, Kashinath et al . [8] have numerically examined the forced synchronization of a two dimensional slot-stabilized laminar premixed flame undergoing quasiperiodic and chaotic motion. They found a variety of complex nonlinear phenomena such as phase locking and asynchronous quenching. On the experimental front, Guan et al . [9] have examined the effect of periodic acoustic forcing on quasiperiodic thermoacoustic oscillations via open-loop mapping. They found that the thermoacoustic amplitude can be reduced significantly via asynchronous quenching.

 

In the present study, we use genetic programming (GP) [10] to weaken two frequency quasiperiodic thermoacoustic oscillations in a laminar premixed combustor. GP is a data-driven framework capable of discovering closed-loop control laws without access to a system model [11]. It has been successfully used to manipulate a mixing layer [12, 13, 14], to modify the recirculation zone behind a backward-facing step [15], to control flow separation in a turbulent boundary layer [16], to reduce drag on a car model and rotating cylinders [17, 18] and to optimize mixing in a turbulent jet [19, 20]. In our experiments, we apply GP in closed-loop form to weaken two-frequency quasiperiodic ther- moacoustic oscillations in a laminar premixed combustor.

 

2. GP CONTROL FRAMEWORK

 

The GP control framework consists of three main steps: (i) initializing a generation of control laws, called individuals, (ii) evaluating the generation, and (iii) breeding subsequent generations [11]. The control laws are represented as syntax trees, their input signal is provided by a probe microphone, and their output signal is used to actuate a loudspeaker. To evaluate the performance of each individ- ual and to benchmark it against other control strategies, we define a cost function 𝐽=(𝑆)𝑇𝑒 + 𝜉(|𝑏′|)𝑇𝑒 , where the operator (∙)𝑇𝑒 denotes time averaging over the evaluation time 𝑇𝑒 , 𝑆 ≡ |𝑝′|/|𝑝’∗| is the instantaneous amplitude of the pressure oscillations in the controlled system normalized by that in the uncontrolled system, and 𝑏 ′ is the voltage fluctuation into the loudspeaker. A penalization co-efficient 𝜉 determines the trade-off between amplitude reduction and actuation cost [11, 15]. The in- dividuals in the first generation are initialized randomly based on pre-defined functions and operators (e.g. + , − , × , /, sin, cos, exp and log). Filter functions are used to avoid control laws that are mathe- matically invalid or potentially damaging to the actuator. After an entire generation has been evalu- ated, all its individuals are ranked by their cost function values. The top-performing individuals, called elitism individuals, are then passed directly to the next generation. The remaining individuals in the next generation are bred via genetic operations such as replication, crossover and mutation (figure 1). Here replication involves direct forwarding of selected individuals, crossover involves the exchange of random branches of two individuals, and mutation involves the replacement of random branches of selected individuals. Crossover and mutation can facilitate exploitation and exploration of the search pool, improving genetic diversity and search efficiency. We repeat this breeding process until the average cost function value of the top performing individuals no longer varies significantly from one generation to the next. We then take the individual with the lowest cost function value from the elitism individuals as the optimal control law.

 

 

Figure 1: Genetic operations (replication, mutation, and crossover) of the GP algorithm. The individ- uals are represented as syntax trees. In the example shown, the mathematical expression of the indi- vidual undergoing replication is 𝑏= sin(𝑥)+ cos (exp (𝑥)× (1 + 𝑥) , where 𝑥 is the input to the control law. The branches selected for mutation and crossover are in red and green, respectively.

 

3. RESULTS AND DISCUSSION

 

3.1. Convergence of the GP algorithm

 

The GP convergence history is shown in figure 2. Figure 2(a) shows the sorted cost function values of the individuals. Only six out of twelve generations are shown here to highlight the key trends. As the number of generations increases, more and more individuals achieve the minimum cost function value. This indicates that the GP algorithm can discover an increasing number of elitism individuals, and that the ensemble performance of the individuals in subsequent generations is superior to that in previous generations. However, the performance of the elitism individuals of each generation remains relatively unchanged as more generations are bred, indicating rapid convergence starting from the first generation. In evolutionary algorithms, it is common practice to initialize the first generation with as many different individuals as possible to enhance diversity in the genetic pool. Thus, we use 100 individuals in the first generation and 60 individuals in all the subsequent generations. Figure 2(b) shows the average cost function of the top 15 individuals in each generation. The average cost function decreases monotonically until the fifth generation and then remains relatively constant (within ±3%) thereafter, confirming the quick convergence of the GP algorithm.

 

 

Figure 2: (a) Sorted cost function values of six selected generations, with the last generation shown in red, and (b) average cost function of the top 15 individuals in each generation.

 

3.2. Control performance

 

The performance of the optimal individual is shown in figure 3. On activation of control at t = 1 s, both the GP forcing signal and the acoustic pressure fluctuation signal ( 𝑝′ ) exhibit large-amplitude beating-like modulations (figure 3a). Following this transient stage, the pressure amplitude drops rapidly to a lower value (figure 3b: around 4 Pa), as does the forcing signal. On benchmarking GP closed-loop control against conventional open-loop control and GP open-loop control [9], we find that all three control strategies can achieve similar degrees of pressure amplitude reduction (around 87 to 93%), but GP closed-loop control requires the lowest actuation effort, yielding the lowest cost function value ( J = 0.17).

 

 

Figure 3: (a) Time traces of the acoustic pressure fluctuation 𝑝′ (in blue) and of the loudspeaker input voltage fluctuation 𝑏′ (in red). GP closed-loop control is applied at t = 1 s, as indicated by the vertical dash line. The time interval enclosed by the rectangle in panel (a) is magnified in panel (b).

 

4. CONCLUSIONS

 

In this experimental study, we have used genetic programming (GP) in closed-loop form to sup- press two-frequency quasiperiodic oscillations in a prototypical laminar thermoacoustic system. We find that the GP algorithm can discover more effective control laws than can conventional open-loop control as well as GP open-loop control. This study demonstrates the potential of using GP as an active-control strategy to weaken aperiodic oscillations in self-sustained thermoacoustic systems.

 

5. ACKNOWLEDGEMENTS

 

This work was supported by the Research Grants Council of Hong Kong (Grant Nos. 16210418, 16210419 and 16200220) and the Guangdong–Hong-Kong–Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications (Project No. 2020B1212030001).

 

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