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) ≜[η1(t), η2(t)]<sup>T</sup> , and define

$$\boldsymbol{\Psi}(\mathbf{x}) = \begin{bmatrix} \cos \left( \pi \mathbf{x} \right), \cos \left( 2 \pi \mathbf{x} \right) \end{bmatrix}^T,\tag{11}$$

$$\boldsymbol{\Phi}(\mathbf{x}) = \begin{bmatrix} \sin \left( \pi \mathbf{x} \right), \sin \left( 2 \pi \mathbf{x} \right) \end{bmatrix}^{\mathrm{T}}.\tag{12}$$

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

$$
\dot{Q}\_s(\mathbf{x}\_f, t - \tau) \approx \frac{\sqrt{3}\mathcal{K}}{2} \Psi\_f^T \eta(t - \tau),
\tag{13}
$$

where Ψf≜ Ψ(xf).

By following a derivation procedure similar to that presented in [24], the dynamics of the duct natural modes can be expressed as

$$\mathbf{M}\ddot{\boldsymbol{\eta}} + \left(\mathbf{D} - (\boldsymbol{\gamma} - 1)\boldsymbol{\%}\boldsymbol{\%}\boldsymbol{\sigma}\mathbf{O}\_{\boldsymbol{\prime}}\boldsymbol{\Psi}\_{\boldsymbol{\prime}}^{T}\right)\dot{\boldsymbol{\eta}} + \left(\mathbf{M}^{-1} + (\boldsymbol{\gamma} - 1)\boldsymbol{\%}\boldsymbol{\%}\boldsymbol{\Theta}\boldsymbol{\Theta}\_{\boldsymbol{\prime}}\mathbf{W}\_{\boldsymbol{\prime}}^{T}\right)\boldsymbol{\eta} - \mathbf{h}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}}) - \boldsymbol{\tau}\_{d} = \boldsymbol{\mathcal{B}}\mathbf{v}, \tag{14}$$

where

Q\_ <sup>s</sup> ¼ K

assumed to be inviscid, perfect, and nonconductive.

the duct natural modes as

The actuation signal vak∈ R of the k

η\_j

consideration here, the terms p and <sup>∂</sup><sup>u</sup>

<sup>j</sup><sup>π</sup> ¼ �2ð Þ <sup>γ</sup> � <sup>1</sup> <sup>Q</sup>\_

be expressed as

η€j jπ

þ jπη<sup>j</sup> þ ζ<sup>j</sup>

the duct walls [29–31].

where

166 Adaptive Robust Control Systems

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

" # r

r

<sup>3</sup> <sup>þ</sup> ufð Þ <sup>t</sup> � <sup>τ</sup> <sup>∣</sup>

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πλcvρ<sup>0</sup>

ffiffiffi 1 3

dw 2

, (5)

: (6)

ð Þt , (7)

ð Þt , (8)

αakva xak ð Þ ; t sin ð Þ jπxak : (10)

th monopole-like source (e.g., a loudspeaker) [28] can now

vak ¼ Rku xð Þþ ak Skp xð Þ ak , (9)

X K

k¼1

<sup>∂</sup><sup>x</sup> are taken to be zero at the ends of the duct. Moreover, it

∣ 1

r

<sup>K</sup> <sup>¼</sup> <sup>2</sup>Lw Tw � <sup>T</sup><sup>0</sup>

p xð Þ¼� ; <sup>t</sup> <sup>X</sup>

u xð Þ¼ ; <sup>t</sup> <sup>X</sup>

expressions in Eqs. (4)–(8), the discretized governing equations are obtained as

� � ffiffiffiffiffiffiffi 3u<sup>0</sup> <sup>p</sup> <sup>S</sup>γp<sup>0</sup>

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

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

sin ð Þ jπx <sup>j</sup><sup>π</sup> <sup>η</sup><sup>j</sup>

cos ð Þ jπx η<sup>j</sup>

N

j¼1

N

j¼1

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

where R<sup>k</sup> and S<sup>k</sup> ∈ R are dimensionless control parameters of the actuators. After using the

In Eq. (10), ζ represents the overall damping in the system [29, 30]. For the model under

is assumed that no acoustic energy is dissipated in the thermal and viscous boundary layers at

� � � <sup>2</sup><sup>γ</sup>

<sup>s</sup> xf ; <sup>t</sup> � <sup>τ</sup> � � sin <sup>j</sup>πxf

$$\mathbf{M} = \text{diag}\left\{\frac{1}{\pi}, \frac{1}{2\pi}\right\}, \quad \mathbf{D} = \text{diag}\left\{\frac{\zeta\_1}{\pi}, \frac{\zeta\_2}{2\pi}\right\}. \tag{15}$$

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 rewritten in the control-oriented form:

$$\mathbf{M}\ddot{\boldsymbol{\eta}} + \mathbf{C}\_{\mathbf{d}}\dot{\boldsymbol{\eta}} + \mathbf{K}\boldsymbol{\eta} - \mathbf{h}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}}) - \boldsymbol{\tau}\_d = \mathcal{B}\mathbf{v},\tag{16}$$

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

$$\mathbf{C\_d} = \mathbf{D} - (\gamma - 1)\mathbb{K}\sqrt{3}\tau \mathbf{Op}\_f \mathbf{W}\_{f, \prime}^T \tag{17}$$

$$\mathbf{K} = \mathbf{M}^{-1} + (\boldsymbol{\gamma} - 1)\boldsymbol{\mathcal{K}}\sqrt{3}\boldsymbol{\mathfrak{op}}\_f \boldsymbol{\Psi}\_f^T. \tag{18}$$

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

$$\mathcal{B}\mathbf{v} = -2\gamma \left[ \sum\_{k=1}^{K} a\_{ik} \mathcal{R}\_k \mathbf{O}(\mathbf{x}\_{ik}) \mathbf{W}^T(\mathbf{x}\_{ik}) \right] \eta + 2\gamma \left[ \sum\_{k=1}^{K} a\_{ik} \mathcal{S}\_k \mathbf{O}(\mathbf{x}\_{ik}) \mathbf{O}^T(\mathbf{x}\_{ik}) \mathbf{M} \right] \dot{\eta} \tag{19}$$

where v(t) ∈ R<sup>2</sup> 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 B 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 τd(t) satisfies

$$\|\|\pi\_d(t)\|\| \le \zeta, \qquad \forall t \ge 0,\tag{20}$$

Mr\_ ¼ �Ddη\_ � Kη þ hð Þþ η; η\_ Bv þ τd: (23)

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

Y1θ<sup>1</sup> ¼ �Ddη\_ � Kη, (24)

Y2θ<sup>2</sup> ¼ Bv: (25)

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

169

Y2θb<sup>2</sup> ¼ Bbv: (26)

<sup>θ</sup>~<sup>2</sup> <sup>≜</sup> <sup>θ</sup><sup>2</sup> � <sup>θ</sup>b2: (28)

k k hð Þ η; η\_ ≤ ρð Þ k kz k kz , (29)

: (30)

Mr\_ <sup>¼</sup> <sup>Y</sup>1θ<sup>1</sup> <sup>þ</sup> <sup>h</sup>ð Þþ <sup>η</sup>; <sup>η</sup>\_ <sup>Y</sup>2θ~<sup>2</sup> <sup>þ</sup> <sup>B</sup>b<sup>v</sup> <sup>þ</sup> <sup>τ</sup>d, (27)

To address the case where the constant matrices Dd = Cd + Mα, K, and B are uncertain, the

In Eqs. (24) and (25), <sup>Y</sup>1ð Þ <sup>η</sup>; <sup>η</sup>\_ <sup>∈</sup> <sup>R</sup><sup>2</sup>�p<sup>1</sup> and <sup>Y</sup>2(v)<sup>∈</sup> <sup>R</sup><sup>2</sup> � <sup>p</sup><sup>2</sup> are measurable regression matrices, and θ1∈ Rp<sup>1</sup> and θ2∈ Rp<sup>2</sup> are vectors containing the uncertain constant parameters in Dd, K, and B. The constants p<sup>1</sup> and p2∈ ℕ denote the number of uncertain parameters in the vectors θ<sup>1</sup> and

To facilitate the subsequent Lyapunov-based adaptive control law development to compensate for the input-multiplicative uncertain matrix <sup>B</sup>, an estimate <sup>θ</sup>b2ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>p</sup><sup>2</sup> of the uncertain vector

In Eq. (26), <sup>B</sup><sup>b</sup> ð Þ<sup>t</sup> <sup>∈</sup> <sup>ℝ</sup><sup>2</sup>�<sup>2</sup> denotes a time-varying estimate of the uncertain constant matrix <sup>B</sup>. By adding and subtracting the term Bb ð Þt vð Þt in Eq. (23) and using Eqs. (24) and (26), the open-

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 non-

where <sup>ρ</sup>(�) <sup>∈</sup> <sup>R</sup> is a positive, globally invertible nondecreasing function and <sup>z</sup>(t) <sup>∈</sup> <sup>R</sup><sup>4</sup> is defined

<sup>T</sup>ð Þ<sup>t</sup> � �<sup>T</sup>

<sup>z</sup>ðÞ¼ <sup>t</sup> <sup>η</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>r</sup>

where <sup>θ</sup>~2ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>p</sup><sup>2</sup> denotes the parameter estimate mismatch, which is defined as

Assumption 2 The unknown nonlinear term hð Þ η; η\_ can be upper bounded as

In Eq. (29), k�k denotes the standard Euclidean norm of the vector argument.

dynamics can be linearly parameterized as

θ<sup>2</sup> is defined via the linear parameterization:

loop error dynamics can be expressed as

linearities present in the system dynamics.

θ2, respectively.

as

where ζ∈ R denotes a positive bounding parameter.
