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 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

$$
\mu = \frac{\tilde{u}}{\mu\_0}, \quad p = \frac{\tilde{p}}{\gamma M\_0 p\_0}, \quad \dot{Q}\_s = \frac{\dot{\tilde{Q}}\_s}{\gamma p\_0 \mu\_0}, \tag{1}
$$

∂u ∂t þ ∂p

> <sup>s</sup>δ x � xf � � <sup>þ</sup> <sup>γ</sup>

In the expressions in Eqs. (3) and (4), ζ∈ R denotes a damping coefficient; thus, the term ζp physically expresses losses resulting from the effects of friction and thermo-viscous damping, and αak∈ R represents a dimensionless area ratio that can be explicitly defined as αak = Sak/S for

Figure 3. A block diagram illustrating the main components of the proposed robust and adaptive thermoacoustic

Figure 2. A control-oriented schematic of a combustion system with actuators modeled as monopole-like moving pistons.

<sup>s</sup> ∈ R is explicitly defined as [19].

k =1, … ,K. The nondimensional heat release rate Q\_

oscillation control system.

X K

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

k¼1

∂p ∂t

þ ζp þ

∂u

<sup>∂</sup><sup>x</sup> <sup>¼</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>Q</sup>\_

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup>, (3)

αakvakδð Þ x � xak , (4)

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165

$$\mathbf{x} = \frac{\tilde{\mathbf{x}}}{L\_0}, \quad t = \frac{\tilde{t}c\_0}{L\_0}, \quad \frac{\delta \{\mathbf{x} - \mathbf{x}\_f\}}{L\_0} = \tilde{\delta} \left(\tilde{\mathbf{x}} - \tilde{\mathbf{x}}\_f\right), \tag{2}$$

where the above tilde notation denotes the dimensional quantities and the subscript 0 denotes the mean values. In Eqs. (1) and (2), x ∈ R denotes the location along the duct (the actuators are located at xak∈ R, for k =1, … , K, and the heat source is located at xf), t∈ R≥<sup>0</sup> denotes nondimensional time, p(x, t)∈ R is the acoustic pressure, u(x, t) ∈ R denotes the velocity, M ∈ R is the Mach number, c∈ R is the speed of sound, L0∈ R is the length of the duct, and γ∈ R is the ratio of specific heats.

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

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems http://dx.doi.org/10.5772/intechopen.70683 165

$$
\frac{\partial \mu}{\partial t} + \frac{\partial p}{\partial x} = 0,
\tag{3}
$$

$$\frac{\partial p}{\partial t} + \zeta p + \frac{\partial u}{\partial \mathbf{x}} = (\gamma - 1)\dot{Q}\_s \delta(\mathbf{x} - \mathbf{x}\_f) + \gamma \sum\_{k=1}^{K} \alpha\_{ak} v\_{ak} \delta(\mathbf{x} - \mathbf{x}\_{ak}),\tag{4}$$

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

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

Consider the system shown in Figure 2, where the actuators are modeled as multiple monopolelike 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

To facilitate the subsequent model development, nondimensional system variables are defined as

, Q\_

<sup>s</sup> <sup>¼</sup> ~\_ Qs γp0u<sup>0</sup>

, (1)

<sup>¼</sup> <sup>~</sup><sup>δ</sup> <sup>~</sup><sup>x</sup> � <sup>x</sup>~fÞ, (2)

γM0p<sup>0</sup>

, <sup>δ</sup> <sup>x</sup> � xf L0

where the above tilde notation denotes the dimensional quantities and the subscript 0 denotes the mean values. In Eqs. (1) and (2), x ∈ R denotes the location along the duct (the actuators are located at xak∈ R, for k =1, … , K, and the heat source is located at xf), t∈ R≥<sup>0</sup> denotes nondimensional time, p(x, t)∈ R is the acoustic pressure, u(x, t) ∈ R denotes the velocity, M ∈ R is the Mach number, c∈ R is the speed of sound, L0∈ R is the length of the duct, and γ∈ R is

By using the nondimensionalized variables defined in Eqs. (1) and (2), the thermoacoustic

, p <sup>¼</sup> <sup>~</sup><sup>p</sup>

, t <sup>¼</sup> <sup>~</sup>tc<sup>0</sup> L0

<sup>u</sup> <sup>¼</sup> <sup>u</sup><sup>~</sup> u0

<sup>x</sup> <sup>¼</sup> <sup>~</sup><sup>x</sup> L0

thermoacoustic oscillation suppression system.

2. Thermoacoustic system model

in [22–24, 27].

provided in Figure 3.

164 Adaptive Robust Control Systems

the ratio of specific heats.

system with K actuators can be expressed as

Figure 2. A control-oriented schematic of a combustion system with actuators modeled as monopole-like moving pistons.

Figure 3. A block diagram illustrating the main components of the proposed robust and adaptive thermoacoustic oscillation control system.

In the expressions in Eqs. (3) and (4), ζ∈ R denotes a damping coefficient; thus, the term ζp physically expresses losses resulting from the effects of friction and thermo-viscous damping, and αak∈ R represents a dimensionless area ratio that can be explicitly defined as αak = Sak/S for k =1, … ,K. The nondimensional heat release rate Q\_ <sup>s</sup> ∈ R is explicitly defined as [19].

$$\dot{Q}\_s = \mathcal{K} \left[ \sqrt{|\frac{1}{3} + u\_f(t-\tau)|} - \sqrt{\frac{1}{3}} \right],\tag{5}$$

3. Control-oriented model derivation

, and define

extended to address N modes.

η(t) ≜[η1(t), η2(t)]<sup>T</sup>

where Ψf≜ Ψ(xf).

where

relationship:

natural modes can be expressed as

rewritten in the control-oriented form:

3 <sup>p</sup> <sup>τ</sup>Φ<sup>f</sup> <sup>Ψ</sup><sup>T</sup> f � �η\_ <sup>þ</sup> <sup>M</sup>�<sup>1</sup> <sup>þ</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>K</sup> ffiffiffi

<sup>M</sup>η€ <sup>þ</sup> <sup>D</sup> � ð Þ <sup>γ</sup> � <sup>1</sup> <sup>K</sup> ffiffiffi

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

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

In the following discussion, the vector of modes (i.e., the state vector) will be annotated as

Ψð Þ¼ x ½ � cos ð Þ πx ; cos 2ð Þ πx

Φð Þ¼ x ½ � sin ð Þ πx ; sin 2ð Þ πx

ffiffiffi 3 <sup>p</sup> <sup>K</sup> <sup>2</sup> <sup>Ψ</sup><sup>T</sup>

By following a derivation procedure similar to that presented in [24], the dynamics of the duct

� �, <sup>D</sup> <sup>¼</sup> diag <sup>ζ</sup><sup>1</sup>

In Eq. (14), <sup>η</sup>(t)=[η1(t), <sup>η</sup>2(t)]T<sup>∈</sup> <sup>R</sup><sup>2</sup> is a vector containing the natural modes, <sup>h</sup>ð Þ <sup>η</sup>; <sup>η</sup>\_ <sup>∈</sup> <sup>R</sup><sup>2</sup> is an unknown nonlinear function, and τd(t) ∈ R<sup>2</sup> is a general unknown bounded disturbance. To facilitate the control development in the following analysis, the dynamic equation in Eq. (14) is

Cd <sup>¼</sup> <sup>D</sup> � ð Þ <sup>γ</sup> � <sup>1</sup> <sup>K</sup> ffiffiffi

<sup>K</sup> <sup>¼</sup> <sup>M</sup>�<sup>1</sup> <sup>þ</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>K</sup> ffiffiffi

Also, in Eq. (16), the uncertain constant control input gain matrix B ∈R<sup>2</sup>�<sup>2</sup> is defined via the

3 <sup>p</sup> <sup>Φ</sup><sup>f</sup> <sup>Ψ</sup><sup>T</sup> f � �<sup>η</sup> � <sup>h</sup>ð Þ� <sup>η</sup>; <sup>η</sup>\_ <sup>τ</sup><sup>d</sup> <sup>¼</sup> <sup>B</sup>v, (14)

> π ; ζ2 2π

Mη€ þ Cdη\_ þ Kη � hð Þ� η; η\_ τ<sup>d</sup> ¼ Bv, (16)

3 <sup>p</sup> <sup>τ</sup>Φ<sup>f</sup> <sup>Ψ</sup><sup>T</sup>

> 3 <sup>p</sup> <sup>Φ</sup><sup>f</sup> <sup>Ψ</sup><sup>T</sup>

Assuming that ∣uf(t � τ)∣ < 1/3, the heat release rate can be approximated as

<sup>s</sup> xf ; <sup>t</sup> � <sup>τ</sup> � � <sup>≈</sup>

Q\_

<sup>M</sup> <sup>¼</sup> diag <sup>1</sup>

where the uncertain constant terms Cd and K ∈ R<sup>2</sup> � <sup>2</sup> are defined as

π ; 1 2π T, (11)

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167

<sup>T</sup>: (12)

<sup>f</sup> ηð Þ t � τ , (13)

� �: (15)

<sup>f</sup> , (17)

<sup>f</sup> : (18)

where

$$\mathcal{K} = \frac{2L\_w \left(T\_w - \overline{T}\_0\right)}{\sqrt{3\mu\_0} S \gamma p\_0} \sqrt{\pi \lambda c\_v \rho\_0} \frac{d\_w}{2}. \tag{6}$$

In Eq. (6), dw, Lw, and Tw ∈ R denote the diameter, length, and temperature of the heated wire, respectively; ρ∈ R denotes air density; T∈ R is temperature; λ is thermal conductivity; cv∈ R denotes the specific heat capacity at constant volume; and τ∈ R represents the time delay between the initial velocity field perturbation (i.e., the actuation) and the resulting effect on the heat release. Readers are referred to [22] for details on the numerical values of the physical parameters used for the thermoacoustic model being considered in this chapter. The gas is assumed to be inviscid, perfect, and nonconductive.

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

$$p(\mathbf{x}, t) = -\sum\_{j=1}^{N} \frac{\sin\left(j\pi\mathbf{x}\right)}{j\pi} \eta\_j(t),\tag{7}$$

$$\mu(\mathbf{x}, t) = \sum\_{j=1}^{N} \cos \left( j \pi \mathbf{x} \right) \eta\_j(t), \tag{8}$$

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

The actuation signal vak∈ R of the k th monopole-like source (e.g., a loudspeaker) [28] can now be expressed as

$$
\sigma\_{ak} = \mathcal{R}\_k \mu(\mathbf{x}\_{ak}) + \mathcal{S}\_k p(\mathbf{x}\_{ak}),
\tag{9}
$$

where R<sup>k</sup> and S<sup>k</sup> ∈ R are dimensionless control parameters of the actuators. After using the expressions in Eqs. (4)–(8), the discretized governing equations are obtained as

$$\frac{\ddot{\eta}\_{j}}{\dot{\eta}\pi} + j\pi\eta\_{j} + \zeta\_{j}\frac{\dot{\eta}\_{j}}{\dot{\eta}\pi} = -2(\gamma - 1)\dot{Q}\_{s}(\mathbf{x}\_{f}, t - \tau)\sin\left(j\pi\mathbf{x}\_{f}\right) - 2\gamma\sum\_{k=1}^{K} a\_{ik}\boldsymbol{\upsilon}\_{a}(\mathbf{x}\_{ak}, t)\sin\left(j\pi\mathbf{x}\_{ak}\right). \tag{10}$$

In Eq. (10), ζ represents the overall damping in the system [29, 30]. For the model under consideration here, the terms p and <sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>x</sup> are taken to be zero at the ends of the duct. Moreover, it is assumed that no acoustic energy is dissipated in the thermal and viscous boundary layers at the duct walls [29–31].
