1. Introduction

Rijke-type instability is a widely investigated example of a thermoacoustic phenomenon, which describes the generation of potentially unstable pressure oscillations that results from the dynamic coupling between unsteady heat transfer and acoustics [1, 2]. The resulting oscillations in Rijke-type systems can degrade performance and even cause structural damage in combustion systems. Based on this fact, thermoacoustic instability is a primary challenge that must be addressed in the design and manufacture of land-based gas turbines and aircraft engines [3–11]. Other applications for which thermoacoustic oscillations are a concern include boilers, furnaces, ramjet engines, and rocket motors. The myriad practical engineering applications impacted by Rijke-type instability necessitate the design of reliable control systems to regulate the potentially catastrophic effects of thermoacoustic oscillations.

Control design methods for thermoacoustic oscillation suppression systems can be separated into two main categories: passive control and active control approaches. Passive control methods [12–18] can employ acoustic dampers, such as Helmholtz resonators [13] or acoustic liners [12], or they can be achieved by physically redesigning the system by changing the location of the heat source, for example. Passive approaches offer the virtues of simplicity and inexpensive maintenance; however, the performance of passive control methods can only be ensured over a relatively narrow range of operating conditions [4]. To expand the usable range of operating conditions, active control methods offer the capability to automatically adjust the level of control actuation in response to sensor stimuli.

Active control methods are usually implemented in closed-loop configurations, where sensor measurements are utilized in a feedback loop to automatically drive the input signal to the actuators. Figure 1 provides an example of functional schematic of a closed-loop thermoacoustic oscillation suppression system. The two primary strategies for achieving closed-loop active control of thermoacoustic oscillations include (1) using a monopole-like acoustic source such as a loudspeaker to control the acoustic field [19] or (2) using a secondary fuel injector to control the unsteady heat release rate [20, 21]. Several active control approaches to suppress thermoacoustic oscillations have been presented in recent research literature.

Physically speaking, thermoacoustic oscillation suppression can be achieved by disrupting the inherent dynamic coupling between the unsteady heat release and the acoustic waves. By designing active control systems to alter the interaction between the acoustic waves and the unsteady heat release, the amplitude of the thermoacoustic oscillations can be forced to decrease, instead of increase. Additional challenges in designing control systems for thermoacoustic oscillations can be incurred as a result of parametric model uncertainty and unmodeled operating conditions. The recent result in [26] presents a nonlinear active control method, which is proven to asymptotically regulate thermoacoustic oscillations in Rijke-type systems that do not include parametric model uncertainty and unmodeled nonlinearities. The design of active closed-loop control systems for thermoacoustic oscillation suppression that achieve reliable performance over a wide range of operating conditions and model uncertainty remains very much an open problem.

Figure 1. A functional implementation diagram of a thermoacoustic oscillation control system, including a microphone

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems

http://dx.doi.org/10.5772/intechopen.70683

163

for sensing and a loudspeaker for actuation.

In this chapter, an observer-based nonlinear active closed-loop control method is presented, which achieves asymptotic suppression of self-excited thermoacoustic oscillations in a Rijke-type

Standard linear control systems for thermoacoustic oscillation suppression are based on stabilizing the closed-loop system through causing the dominant eigenmodes to exponentially decay. However, for realistic thermoacoustic systems where the eigenmodes are nonorthogonal, controlling only the dominant eigenmode can result in the excitation of other modes as a result of the coupling between the acoustic modes. To address this challenge, a transient growth controller is presented in [22–24], which achieves strict dissipativity. Experimental or numerical empirical methods for thermoacoustic oscillation control have been widely considered, but more systematic approaches such as robust and adaptive control have gained popularity in more recent research. Active linear control methods have been widely investigated for applications considering simplified thermoacoustic dynamic models [1, 25]. However, by leveraging the tools of nonlinear control, effective suppression control of thermoacoustic oscillations can be achieved over a wide range of operating conditions and dynamic model uncertainty.

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems http://dx.doi.org/10.5772/intechopen.70683 163

1. Introduction

162 Adaptive Robust Control Systems

Rijke-type instability is a widely investigated example of a thermoacoustic phenomenon, which describes the generation of potentially unstable pressure oscillations that results from the dynamic coupling between unsteady heat transfer and acoustics [1, 2]. The resulting oscillations in Rijke-type systems can degrade performance and even cause structural damage in combustion systems. Based on this fact, thermoacoustic instability is a primary challenge that must be addressed in the design and manufacture of land-based gas turbines and aircraft engines [3–11]. Other applications for which thermoacoustic oscillations are a concern include boilers, furnaces, ramjet engines, and rocket motors. The myriad practical engineering applications impacted by Rijke-type instability necessitate the design of reliable control systems to

Control design methods for thermoacoustic oscillation suppression systems can be separated into two main categories: passive control and active control approaches. Passive control methods [12–18] can employ acoustic dampers, such as Helmholtz resonators [13] or acoustic liners [12], or they can be achieved by physically redesigning the system by changing the location of the heat source, for example. Passive approaches offer the virtues of simplicity and inexpensive maintenance; however, the performance of passive control methods can only be ensured over a relatively narrow range of operating conditions [4]. To expand the usable range of operating conditions, active control methods offer the capability to automatically adjust the

Active control methods are usually implemented in closed-loop configurations, where sensor measurements are utilized in a feedback loop to automatically drive the input signal to the actuators. Figure 1 provides an example of functional schematic of a closed-loop thermoacoustic oscillation suppression system. The two primary strategies for achieving closed-loop active control of thermoacoustic oscillations include (1) using a monopole-like acoustic source such as a loudspeaker to control the acoustic field [19] or (2) using a secondary fuel injector to control the unsteady heat release rate [20, 21]. Several active control approaches to suppress thermoacoustic

Standard linear control systems for thermoacoustic oscillation suppression are based on stabilizing the closed-loop system through causing the dominant eigenmodes to exponentially decay. However, for realistic thermoacoustic systems where the eigenmodes are nonorthogonal, controlling only the dominant eigenmode can result in the excitation of other modes as a result of the coupling between the acoustic modes. To address this challenge, a transient growth controller is presented in [22–24], which achieves strict dissipativity. Experimental or numerical empirical methods for thermoacoustic oscillation control have been widely considered, but more systematic approaches such as robust and adaptive control have gained popularity in more recent research. Active linear control methods have been widely investigated for applications considering simplified thermoacoustic dynamic models [1, 25]. However, by leveraging the tools of nonlinear control, effective suppression control of thermoacoustic oscillations can be achieved

regulate the potentially catastrophic effects of thermoacoustic oscillations.

level of control actuation in response to sensor stimuli.

oscillations have been presented in recent research literature.

over a wide range of operating conditions and dynamic model uncertainty.

Figure 1. A functional implementation diagram of a thermoacoustic oscillation control system, including a microphone for sensing and a loudspeaker for actuation.

Physically speaking, thermoacoustic oscillation suppression can be achieved by disrupting the inherent dynamic coupling between the unsteady heat release and the acoustic waves. By designing active control systems to alter the interaction between the acoustic waves and the unsteady heat release, the amplitude of the thermoacoustic oscillations can be forced to decrease, instead of increase. Additional challenges in designing control systems for thermoacoustic oscillations can be incurred as a result of parametric model uncertainty and unmodeled operating conditions. The recent result in [26] presents a nonlinear active control method, which is proven to asymptotically regulate thermoacoustic oscillations in Rijke-type systems that do not include parametric model uncertainty and unmodeled nonlinearities. The design of active closed-loop control systems for thermoacoustic oscillation suppression that achieve reliable performance over a wide range of operating conditions and model uncertainty remains very much an open problem.

In this chapter, an observer-based nonlinear active closed-loop control method is presented, which achieves asymptotic suppression of self-excited thermoacoustic oscillations in a Rijke-type system, where the system dynamic model includes unmodeled nonlinearities and parametric uncertainty in the system dynamics and actuator dynamics. To achieve the result, a wellaccepted thermoacoustic model is utilized, which employs arrays of sensors and monopole-like actuators. To facilitate the control design, the original dynamic equations are recast in a controlamenable form, which explicitly includes the effects of unmodeled, nonvanishing external disturbances and linear time delay. A sliding-mode observer-based nonlinear control law is then derived to regulate oscillations in the thermoacoustic system. A primary challenge in the control design is the presence of input-multiplicative parametric uncertainty in the control-oriented model. This challenge is handled through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunov-based adaptive law. A rigorous Lyapunov-based stability analysis is used to prove that the closed-loop system achieves asymptotic regulation of a thermoacoustic system consisting of multiple modes. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop active thermoacoustic oscillation suppression system.
