Physical parameters.
Abstract
Adaptive nonlinear control of selfexcited oscillations in Rijketype thermoacoustic systems is considered. To demonstrate the methodology, a wellaccepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopolelike actuators. To facilitate the derivation of the adaptive control law, the dynamic model is recast as a set of nonlinear ordinary differential equations, which are amenable to control design. The controloriented nonlinear model includes unknown, unmeasurable, nonvanishing disturbances in addition to parametric uncertainty in both the thermoacoustic dynamic model and the actuator dynamic model. To compensate for the unmodeled disturbances in the dynamic model, a robust nonlinear feedback term is included in the control law. One of the primary challenges in the control design is the presence of inputmultiplicative parametric uncertainty in the dynamic model for the control actuator. This challenge is mitigated through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunovbased adaptive control law. To address practical implementation considerations, where sensor measurements of the complete state are not available for feedback, a detailed analysis is provided to demonstrate that system observability can be ensured through judicious placement of pressure (and/or velocity) sensors. Based on this observability condition, a slidingmode observer design is presented, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunovbased stability analysis is provided to prove that the proposed closedloop active thermoacoustic control system achieves asymptotic (zero steadystate error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlotype simulation results are also provided, which demonstrate the performance of the proposed closedloop control system under various sets of operating conditions.
Keywords
 thermoacoustics
 robust
 adaptive
 nonlinear control
1. Introduction
Rijketype 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 Rijketype 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 landbased 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 Rijketype 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 closedloop 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 closedloop thermoacoustic oscillation suppression system. The two primary strategies for achieving closedloop active control of thermoacoustic oscillations include (1) using a monopolelike 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.
Standard linear control systems for thermoacoustic oscillation suppression are based on stabilizing the closedloop 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.
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 Rijketype systems that do not include parametric model uncertainty and unmodeled nonlinearities. The design of active closedloop 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 observerbased nonlinear active closedloop control method is presented, which achieves asymptotic suppression of selfexcited thermoacoustic oscillations in a Rijketype 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 monopolelike 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 slidingmode observerbased nonlinear control law is then derived to regulate oscillations in the thermoacoustic system. A primary challenge in the control design is the presence of inputmultiplicative parametric uncertainty in the controloriented model. This challenge is handled through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunovbased adaptive law. A rigorous Lyapunovbased stability analysis is used to prove that the closedloop system achieves asymptotic regulation of a thermoacoustic system consisting of multiple modes. Numerical Monte Carlotype simulation results are also provided, which demonstrate the performance of the proposed closedloop active thermoacoustic oscillation suppression system.
2. Thermoacoustic system model
The thermoacoustic system model that will be utilized in this chapter consists of a horizontal Rijke tube with multiple actuators. The model is identical to that studied in our previous work in [22–24, 27].
Consider the system shown in
Figure 2
, where the actuators are modeled as multiple monopolelike moving pistons. It will be assumed that
To facilitate the subsequent model development, nondimensional system variables are defined as
where the above tilde notation denotes the dimensional quantities and the subscript 0 denotes the mean values. In Eqs. (1) and (2),
By using the nondimensionalized variables defined in Eqs. (1) and (2), the thermoacoustic system with
In the expressions in Eqs. (3) and (4),
where
In Eq. (6),
The acoustic pressure
where
The actuation signal
where
In Eq. (10),
3. Controloriented model derivation
To facilitate the presentation of the main ideas, we consider a thermoacoustic system with two modes (i.e.,
In the following discussion, the vector of modes (i.e., the state vector) will be annotated as
Assuming that ∣
where
By following a derivation procedure similar to that presented in [24], the dynamics of the duct natural modes can be expressed as
where
In Eq. (14),
where the uncertain constant terms
Also, in Eq. (16), the uncertain constant control input gain matrix
where
where
4. Control development
In this section, a rigorous regulation error system development will be utilized to develop a nonlinear control system, which will be proven to effectively compensate for the inherent parametric uncertainty in the dynamic model of the thermoacoustic system in addition to the uncertain actuator model. Moreover, the proposed controller compensates for unmodeled, normbounded disturbances present in the dynamic model (e.g., the disturbances could represent unmodeled nonlinearities resulting from time delays due to the finite heat release rate).
4.1. Openloop error system
The robust and adaptive nonlinear control design presented here is motivated by the desire to eliminate the transient growth of acoustical energy in a thermoacoustic dynamic system. To present the control design methodology, we consider a simplified
To mathematically describe the regulation control objective, an auxiliary regulation error signal
where
To address the case where the constant matrices
In Eqs. (24) and (25),
To facilitate the subsequent Lyapunovbased adaptive control law development to compensate for the inputmultiplicative uncertain matrix
In Eq. (26),
where
The error dynamics in Eq. (27) are now in a form amenable for the design of a robust and adaptive control law, which compensates for the parametric uncertainty and unmodeled nonlinearities present in the system dynamics.
where
In Eq. (29), ‖·‖ denotes the standard Euclidean norm of the vector argument.
Assumption 2 is mild in the sense that inequality (29) is satisfied for a wide range of nonlinear function
4.2. Closedloop error system
Based on the openloop error system in Eq. (27), the control input
where
After substituting the control input expression in Eq. (34) into the openloop dynamics in Eq. (27), the closedloop error system is obtained as
where
Based on Eq. (32) and the subsequent stability analysis, the parameter estimates
where Γ_{1} ∈
To facilitate the following stability analysis, the control gain matrix
where
5. Stability analysis
After taking the time derivative of Eq. (38) and using Eq. (32),
where Eq. (22) was utilized. After substituting the adaptive laws in Eq. (34) and canceling common terms,
By using inequalities of Eqs. (20) and (29), the expression in Eq. (40) can be upper bounded as
After completing the squares for the parenthetic terms in Eq. (41), the upper bound on
where the fact that
where
The expressions in Eqs. (38) and (43) can be used to prove that
6. Slidingmode observer design
In practical thermoacoustic systems, the full state of the dynamic system is not directly measurable, and so it must be estimated through direct sensor measurements of velocity and pressure. This section presents an observer design, which is utilized to estimate the complete state of the system. The necessary observability condition can easily be satisfied through judicious sensor placement.
Let
where
where
and the output matrix
and thus
For the pressure sensor case, we have
and hence
It is assumed that the sensor location
where
It can be shown that there exists a coordinate transformation of the forms
where
The estimate
Then, the error dynamics
Partition the system above as
The following result can now be stated:
where
and
Condition (63) guarantees sliding in Eq. (58) along the manifold
Substitution of Eq. (64) into Eq. (59) yields
Using the fact that the pair
7. Simulation results
A numerical simulation was created for two modes and two actuators (i.e.,

1.025 kg/m^{3} 

0.0328 W/m K 

719 J/kg K 

1.4 

1 m 

2.5 m 

344 m/s 

0.3 m/s 

295 K 

1680 K 

0.5 × 10^{−3} m 

1.56 × 10^{−3} m 

8.69 × 10^{4} Pa 

0.01 

0.0440 

0.1657 
The initial conditions for the modes were selected as
In closedloop operation, the adaptation gain matrices used in the simulation were selected as
8. Conclusion
A robust and adaptive nonlinear control method is presented, which asymptotically regulates thermoacoustic oscillations in a Rijketype system in the presence of dynamic model uncertainty and unknown disturbances. To demonstrate the methodology, a wellaccepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopolelike actuators. To facilitate the derivation of the adaptive control law, the dynamic model is recast as a set of nonlinear ordinary differential equations, which are amenable to control design. To compensate for the unmodeled disturbances in the dynamic model, a robust nonlinear feedback term is included in the control law. One of the primary challenges in the control design is the presence of inputmultiplicative parametric uncertainty in the dynamic model for the control actuator. This challenge is mitigated through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunovbased adaptive control law. To address practical implementation considerations, where sensor measurements of the complete state are not available for feedback, a detailed analysis is provided to demonstrate that system observability can be ensured through judicious placement of pressure (and/or velocity) sensors. A slidingmode observer design is developed, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunovbased stability analysis is provided to prove that the proposed closedloop active thermoacoustic control system achieves asymptotic (zero steadystate error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlotype simulation results are also provided, which demonstrate the performance of the proposed closedloop control system under 20 different sets of operating conditions.
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