## 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.

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.

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 well-accepted 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 control-amenable 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.

## 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 monopole-like moving pistons. It will be assumed that *K* ≥ 1 actuators are available for control purposes. To facilitate the following observer and control design and analysis, a block diagram is also provided in
**Figure 3**
.

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), *x* ∈
*x _{ak}
* ∈

*k*= 1 , … ,

*K*, and the heat source is located at

*x*),

_{f}*t*∈

_{≥0}denotes nondimensional time,

*p*(

*x*,

*t*) ∈

*u*(

*x*,

*t*) ∈

*M*∈

*c*∈

*L*

_{0}∈

*γ*∈

By using the nondimensionalized variables defined in Eqs. (1) and (2), the thermoacoustic system with *K* actuators can be expressed as

In the expressions in Eqs. (3) and (4), *ζ* ∈
*ζp* physically expresses losses resulting from the effects of friction and thermo-viscous damping, and *α _{ak}
* ∈

*α*=

_{ak}*S*/

_{ak}*S*for

*k*= 1 , … ,

*K*. The nondimensional heat release rate

where

In Eq. (6), *d _{w}
*,

*L*, and

_{w}*T*∈

_{w}*ρ*∈

*T*∈

*λ*is thermal conductivity;

*c*∈

_{v}*τ*∈

The acoustic pressure *p* and velocity *u* inside the duct can be expressed as a superposition of the duct natural modes as

where *N* ∈ *ℕ* denotes the number of modes considered in the numerical discretization.

The actuation signal *v _{ak}
* ∈

*k*monopole-like source (e.g., a loudspeaker) [28] can now be expressed as

^{th}where *
* and

In Eq. (10), *ζ* represents the overall damping in the system [29, 30]. For the model under consideration here, the terms *p* and

## 3. Control-oriented model derivation

To facilitate the presentation of the main ideas, we consider a thermoacoustic system with two modes (i.e., *N* = 2). However, the theoretical development presented here can be directly extended to address *N* modes.

In the following discussion, the vector of modes (i.e., the state vector) will be annotated as *η*(*t*) ≜ [*η*
_{1}(*t*), *η*
_{2}(*t*)]*
^{T}
*, and define

Assuming that ∣*u _{f}
* (

*t*−

*τ*)∣ < 1/3, the heat release rate can be approximated as

where **Ψ**
*
_{f}
* ≜

**Ψ**(

*x*).

_{f}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), *η*(*t*) = [*η*
_{1}(*t*), *η*
_{2}(*t*)]*
^{T}
* ∈

^{2}is a vector containing the natural modes,

*τ*(

_{d}*t*) ∈

^{2}is a general unknown bounded disturbance. To facilitate the control development in the following analysis, the dynamic equation in Eq. (14) is rewritten in the control-oriented form:

where the uncertain constant terms **C**
_{
d
} and **K** ∈
^{2 × 2} are defined as

Also, in Eq. (16), the uncertain constant control input gain matrix

where **v**(*t*) ∈
^{2} is a subsequently defined auxiliary control signal.

**Remark 1** *Note that* Eq. (19) *highlights one of the primary challenges in the control design presented in this chapter. Specifically, the input-multiplicative parametric uncertainty in*
*presents a nontrivial control design challenge, which will be mitigated in the proposed control method through the use of a novel Lyapunov-based adaptive law.*

**Assumption 1** The unknown nonlinear disturbance *τ _{d}
*(

*t*) satisfies

where *ζ* ∈

## 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, norm-bounded 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. Open-loop 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 *N* = 2 mode scenario, which will be shown to regulate the modes *η*
_{1}(*t*) and *η*
_{2}(*t*) to zero in the sense that

To mathematically describe the regulation control objective, an auxiliary regulation error signal **r**(*t*) ∈
^{2} is defined as

where *α* ∈
**M**, and using Eq. (16), the regulation error dynamics can be expressed as

To address the case where the constant matrices **D**
_{
d
} = **C**
_{
d
} + **M**
*α*, **K**, and

In Eqs. (24) and (25),
**Y**
_{2}(**v**) ∈
^{2 × p
2
} are measurable regression matrices, and *θ*
_{1} ∈
^{
p
1
} and *θ*
_{2} ∈
^{
p
2
} are vectors containing the uncertain constant parameters in **D**
_{
d
}, **K**, and
*p*
_{1} and *p*
_{2} ∈ *ℕ* denote the number of uncertain parameters in the vectors *θ*
_{1} and *θ*
_{2}, respectively.

To facilitate the subsequent Lyapunov-based adaptive control law development to compensate for the input-multiplicative uncertain matrix
*θ*
_{2} is defined via the linear parameterization:

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.

**Assumption 2** The unknown nonlinear term

where *ρ*(·) ∈
**z**(*t*) ∈
^{4} is defined as

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. Closed-loop error system

Based on the open-loop error system in Eq. (27), the control input **v**(*t*) is designed as

where *k _{s}
* ∈

*β*∈

^{2 × 2}is a positive-definite, diagonal control gain matrix. In Eq. (31), sgn(·) denotes a vector form of the standard signum function.

After substituting the control input expression in Eq. (34) into the open-loop dynamics in Eq. (27), the closed-loop error system is obtained as

where

Based on Eq. (32) and the subsequent stability analysis, the parameter estimates

where Γ_{1} ∈
^{
p
1 ×}
^{ p
1
} and Γ_{2} ∈
^{
p
2 ×}
^{ p
2
} are positive-definite adaptation gains.

**Remark 2** *The function* proj(·) *in* Eq. (34) *denotes a normal projection algorithm, which ensures that the following inequalities are satisfied:*

*where*
*represent known, constant lower and upper bounds of the elements of*
*and*
*respectively. In the current result, the use of the* proj(·) *function is primarily motivated by the desire to avoid singularities in the matrix estimate and facilitate the matrix inverse calculation in* Eq. (31).

To facilitate the following stability analysis, the control gain matrix *β* will be selected to satisfy the sufficient condition:

where *ζ* is introduced in Eq. (20) and *λ*
_{min}{·} denotes the minimum eigenvalue of the argument.

## 5. Stability analysis

**Theorem 1** *The control law in* Eq. (31) *with adaptive laws defined as in* Eq. (34) *ensures asymptotic regulation of the thermoacoustic modes η*
_{1}(*t*) *and η*
_{2}(*t*) *in the sense that*

*provided that k _{s} is selected as sufficiently large (see the subsequent stability proof) and β is selected to satisfy inequality* (36).

*Proof.* Let

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 **r**
*
^{T}
* sgn(

**r**) = |

**r**| was utilized. After using inequality (36), the upper bound in Eq. (42) can be expressed as

where *λ*
_{0} ≜ min {*α*, 1} and the triangle inequality (i.e., |**r**| ≥ ‖**r**‖ ∀**r** ∈ *
*) was utilized. Based on Eq. (43),

*c*, inside the set

The expressions in Eqs. (38) and (43) can be used to prove that *η*(*t*), **r**(*t*),
*η*(*t*) and **r**(*t*) ∈
_{∞} in
**v**(*t*) ∈
_{∞} in
*η*(*t*),
**v**(*t*) ∈
_{∞} in
**Y**
_{2}(**v**) ∈
_{∞} in
*η*(*t*), **r**(*t*),
**Y**
_{2}(**v**) ∈
_{∞} in
*η*(*t*) and **r**(*t*) are uniformly continuous in
**z**(*t*) is uniformly continuous in
*η*(*t*), **r**(*t*),
*V*(*t*) ∈
_{∞} in
**z**(*t*) ∈
_{∞} ∩
_{2} in
**z**(*t*)‖→0 as *t*→∞. Hence, ‖*η*(*t*)‖→0 as *t*→∞ in
*k _{s}
*—a semiglobal result.

## 6. Sliding-mode 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
*x _{s}
*. Then, the system can be rewritten as

where

where

and the output matrix **C** is determined by the sensor choice and its location. The output equation for the velocity sensor case is given by

and thus

For the pressure sensor case, we have

and hence

It is assumed that the sensor location *x _{s}
* is chosen such that the systems (45) and (46) subject to

*τ*= 0 are observable, i.e.,

where **E** = **A** + **B**.

It can be shown that there exists a coordinate transformation of the forms **z** = [*z*
_{1}, **z**
_{2}]*
^{T}
* =

**Tx**,

*z*

_{1}∈

**z**

_{2}∈

^{3}, such that the systems (45) and (46) in the new variables take the following form:

where

The estimate
**z** will now be generated via the observer equation

Then, the error dynamics

Partition the system above as

The following result can now be stated:

**Theorem 2** *Let*
*and*
*, and assume that the pair*
*is observable. Then, the observer gain L*
_{1} *can be chosen such that the system* (57) *is asymptotically stable for all τ* ∈ [0, *τ _{max}
*]

*with τ*

_{max}defined by:where
**Q** is any symmetric, positive-definite matrix such that **P** is a symmetric, positive-definite matrix **P** solution of the Lyapunov matrix equation:

and **Q**
_{1} is the square root of the matrix **Q**, i.e.,

*Proof.* From Eq. (58), sliding mode exists in an area:

Condition (63) guarantees sliding in Eq. (58) along the manifold

Substitution of Eq. (64) into Eq. (59) yields

Using the fact that the pair
**L**
_{2} can be chosen such that the eigenvalues of the matrices
*τ* ≤ *τ*
_{max}. This implies that

## 7. Simulation results

A numerical simulation was created for two modes and two actuators (i.e., *N* = *K* = 2) to demonstrate the performance of the control law described by Eqs. (26), (31), and (34). The simulation utilizes the dynamics described in Eq. (16) for a system with two modes. The physical parameters used in the simulation are given in
**Table 1**
.

The initial conditions for the modes were selected as *η*
_{1}(0) = 0.07 and *η*
_{2}(0) = 0. The initial conditions of the velocity and pressure parameters at the location of the heat source are *u*(*x _{f}
*, 0) = − 0.1646 and

*p*(

*x*, 0) = 0 for

_{f}*x*= 0.7.

_{f}**Figure 4**shows the open-loop velocity and pressure perturbations at the heat source in the absence of control actuation.

In closed-loop operation, the adaptation gain matrices used in the simulation were selected as **Γ**
_{1} = 0.1**I**
_{8} and **Γ**
_{2} = 0.001**I**
_{4}. The value for the gain *α* was selected as 1, and the control gain matrices *β* and **k**
*
_{s}
* were selected as

*β*= diag {2.43, 2.1} ,

**k**

*= diag {4.5, 2.4}. The results of 20 Monte Carlo-type simulations for the closed-loop operation are shown in*

_{s}**Figures 5**–

**10**.

**Figure 5**shows the time evolution of the velocity and pressure values at the heat source location during closed-loop controller operation.

**Figure 6**shows the time history of the modes

*η*

_{1}(

*t*) and

*η*

_{2}(

*t*) during closed-loop operation. The results clearly show the capability of the proposed robust and adaptive control law to drive the states to zero. The commanded control signals are shown in

**Figure 7**. The control actuation remains within the reasonable limits throughout closed-loop operation.

**Figures 8**–

**10**show the time responses of the elements of the parameter estimate vectors

## 8. Conclusion

A robust and adaptive nonlinear control method is presented, which asymptotically regulates thermoacoustic oscillations in a Rijke-type system in the presence of dynamic model uncertainty and unknown disturbances. To demonstrate the methodology, a well-accepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopole-like 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 input-multiplicative 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 Lyapunov-based 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 sliding-mode observer design is developed, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunov-based stability analysis is provided to prove that the proposed closed-loop active thermoacoustic control system achieves asymptotic (zero steady-state error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop control system under 20 different sets of operating conditions.