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 η1(t) and η2(t) to zero in the sense that

$$\|\|\eta(t)\|\|\,\|\dot{\eta}(t)\|\|\to 0.\tag{21}$$

To mathematically describe the regulation control objective, an auxiliary regulation error signal r(t) ∈ R<sup>2</sup> is defined as

$$\mathbf{r} = \dot{\eta} + \alpha \eta\_{\prime} \tag{22}$$

where α∈ R denotes a positive, constant control gain. After taking the time derivative of Eq. (22), multiplying the result by M, and using Eq. (16), the regulation error dynamics can be expressed as

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems http://dx.doi.org/10.5772/intechopen.70683 169

$$\mathbf{M}\dot{\mathbf{r}} = -\mathbf{D}\_d \dot{\eta} - \mathbf{K}\eta + \mathbf{h}(\eta, \dot{\eta}) + \mathcal{B}\mathbf{v} + \tau\_d. \tag{23}$$

To address the case where the constant matrices Dd = Cd + Mα, K, and B are uncertain, the dynamics can be linearly parameterized as

<sup>B</sup><sup>v</sup> ¼ �2<sup>γ</sup> <sup>X</sup>

168 Adaptive Robust Control Systems

novel Lyapunov-based adaptive law.

4. Control development

4.1. Open-loop error system

signal r(t) ∈ R<sup>2</sup> is defined as

expressed as

K

<sup>α</sup>akRkΦð Þ xak <sup>Ψ</sup><sup>T</sup>ð Þ xak " #

where v(t) ∈ R<sup>2</sup> is a subsequently defined auxiliary control signal.

Assumption 1 The unknown nonlinear disturbance τd(t) satisfies

where ζ∈ R denotes a positive bounding parameter.

<sup>η</sup> <sup>þ</sup> <sup>2</sup><sup>γ</sup> <sup>X</sup> K

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

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).

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

To mathematically describe the regulation control objective, an auxiliary regulation error

where α∈ R denotes a positive, constant control gain. After taking the time derivative of Eq. (22), multiplying the result by M, and using Eq. (16), the regulation error dynamics can be

will be shown to regulate the modes η1(t) and η2(t) to zero in the sense that

k¼1

<sup>α</sup>akSkΦð Þ xak <sup>Φ</sup><sup>T</sup>ð Þ xak <sup>M</sup> " #

k k τdð Þt ≤ ζ, ∀t ≥ 0, (20)

k k ηð Þt , k k η\_ð Þt ! 0: (21)

r ¼ η\_ þ αη, (22)

η\_ (19)

k¼1

$$\mathbf{Y}\_1 \theta\_1 = -\mathbf{D}\_\mathbf{d} \dot{\eta} - \mathbf{K} \eta\_\prime \tag{24}$$

$$
\mathbf{Y}\_2 \theta\_2 = \mathcal{B} \mathbf{v}.\tag{25}
$$

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 θ2, respectively.

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

$$
\mathbf{Y}\_2 \dot{\theta}\_2 = \dot{\mathcal{B}} \mathbf{v}.\tag{26}
$$

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 openloop error dynamics can be expressed as

$$\mathbf{M}\dot{\mathbf{r}} = \mathbf{Y}\_1\theta\_1 + \mathbf{h}(\eta, \dot{\eta}) + \mathbf{Y}\_2\hat{\theta}\_2 + \hat{\mathcal{B}}\mathbf{v} + \tau\_{d\nu} \tag{27}$$

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

$$
\hat{\theta}\_2 \triangleq \theta\_2 - \hat{\theta}\_2. \tag{28}
$$

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 hð Þ η; η\_ can be upper bounded as

$$\|\|\mathbf{h}(\eta, \dot{\eta})\|\| \le \rho(\|\|\mathbf{z}\|) \|\mathbf{z}\|\,\tag{29}$$

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 as

$$\mathbf{z}(t) = \begin{bmatrix} \eta^T(t) \ \mathbf{r}^T(t) \end{bmatrix}^T. \tag{30}$$

In Eq. (29), k�k 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 hð Þ η; η\_ .

#### 4.2. Closed-loop error system

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

$$\mathbf{v} = \widehat{\mathcal{B}}^{-1} \left( -\mathbf{Y}\_1 \widehat{\theta}\_1 - (k\_s + 1)\mathbf{r} - \beta \text{sgn}\left(\mathbf{r}\right) - \eta \right), \tag{31}$$

where ks∈ R denotes a positive, constant control gain and β ∈ R<sup>2</sup> � <sup>2</sup> 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

$$\mathbf{M}\dot{\mathbf{r}} = \mathbf{Y}\_1 \ddot{\theta}\_1 + \mathbf{Y}\_2 \ddot{\theta}\_2 + \mathbf{h}(\eta, \dot{\eta}) - \eta - (k\_\circ + 1)\mathbf{r} - \beta \text{sgn}\left(\mathbf{r}\right) + \tau\_d \tag{32}$$

where <sup>θ</sup>~1ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>p</sup><sup>1</sup> is the parameter estimate mismatch defined as

$$
\bar{\theta}\_1 \triangleq \theta\_1 - \dot{\theta}\_1. \tag{33}
$$

5. Stability analysis

satisfy inequality (36).

be expressed as

where the fact that r

Eq. (43), <sup>V</sup>\_ <sup>≤</sup> � <sup>c</sup>k k<sup>z</sup> <sup>2</sup>

Eq. (42) can be expressed as

Proof. Let V η;r; θb1; θb2; t

Theorem 1 The control law in Eq. (31) with adaptive laws defined as in Eq. (34) ensures asymptotic

provided that ks is selected as sufficiently large (see the subsequent stability proof) and β is selected to

� � <sup>∈</sup> <sup>R</sup> be defined as the nonnegative function:

After taking the time derivative of Eq. (38) and using Eq. (32), V t \_ ð Þ can be expressed as

<sup>T</sup> <sup>Y</sup>1θ~<sup>1</sup> <sup>þ</sup> <sup>Y</sup>2θ~<sup>2</sup> � � � <sup>θ</sup>~<sup>T</sup>

where Eq. (22) was utilized. After substituting the adaptive laws in Eq. (34) and canceling

By using inequalities of Eqs. (20) and (29), the expression in Eq. (40) can be upper bounded as

� � � k k<sup>r</sup> <sup>2</sup> � <sup>β</sup><sup>r</sup>

After completing the squares for the parenthetic terms in Eq. (41), the upper bound on V t \_ ð Þ can

<sup>2</sup> � <sup>β</sup>j jþ<sup>r</sup> <sup>ζ</sup>k k� <sup>r</sup> ks k k� <sup>r</sup> <sup>ρ</sup>ð Þ k k<sup>z</sup>

V t \_ ð Þ <sup>≤</sup> � <sup>λ</sup><sup>0</sup> � <sup>ρ</sup><sup>2</sup>ð Þ k k<sup>z</sup>

where <sup>λ</sup>0<sup>≜</sup> min {α, 1} and the triangle inequality (i.e., |r<sup>|</sup> <sup>≥</sup> <sup>k</sup>r<sup>k</sup> <sup>∀</sup>r<sup>∈</sup> <sup>R</sup><sup>n</sup>

4ks � �

, for some positive constant c, inside the set R, where R is defined as

þr

k k ηð Þt ! 0 as t ! ∞ (37)

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

<sup>T</sup> <sup>h</sup>ð Þ� <sup>η</sup>; <sup>η</sup>\_ <sup>η</sup> � ð Þ ks <sup>þ</sup> <sup>1</sup> <sup>r</sup> � <sup>β</sup>sgn ð Þþ <sup>r</sup> <sup>τ</sup><sup>d</sup> � �

<sup>T</sup> <sup>h</sup>ð Þ� <sup>η</sup>; <sup>η</sup>\_ ð Þ ks <sup>þ</sup> <sup>1</sup> <sup>r</sup> � <sup>β</sup>sgn ð Þþ <sup>r</sup> <sup>τ</sup><sup>d</sup>

2ks

k k<sup>z</sup> <sup>2</sup>

<sup>T</sup>sgn(r)=|r| was utilized. After using inequality (36), the upper bound in

� �<sup>2</sup>

k kz

<sup>1</sup> Γ�<sup>1</sup> 1 \_ <sup>θ</sup>b<sup>1</sup> � <sup>θ</sup>~<sup>T</sup>

� �: (40)

<sup>2</sup> <sup>θ</sup>~2: (38)

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

171

<sup>T</sup> sgn ð Þþ <sup>r</sup> <sup>ζ</sup>k k<sup>r</sup> : (41)

k k<sup>z</sup> <sup>2</sup>

) was utilized. Based on

, (42)

<sup>þ</sup> <sup>ρ</sup><sup>2</sup>ð Þ k k<sup>z</sup> 4ks

, (43)

, (39)

<sup>2</sup> Γ�<sup>1</sup> 2 \_ θb2

regulation of the thermoacoustic modes η1(t) and η2(t) in the sense that

V tð Þ<sup>≜</sup> <sup>1</sup> 2 <sup>η</sup><sup>T</sup><sup>η</sup> <sup>þ</sup> 1 2 r <sup>T</sup>Mr <sup>þ</sup> 1 2 θ~T <sup>1</sup> Γ�<sup>1</sup> <sup>1</sup> <sup>θ</sup>~<sup>1</sup> <sup>þ</sup> 1 2 θ~T <sup>2</sup> Γ�<sup>1</sup>

V t \_ ðÞ¼ <sup>η</sup><sup>T</sup>ð Þþ <sup>r</sup> � αη <sup>r</sup>

V t \_ ðÞ¼�αη<sup>T</sup><sup>η</sup> <sup>þ</sup> <sup>r</sup>

V t \_ ð Þ <sup>≤</sup> � α ηk k<sup>2</sup> � ksk k<sup>r</sup> <sup>2</sup> � <sup>ρ</sup>ð Þ k k<sup>z</sup> k k<sup>z</sup> k k<sup>r</sup>

common terms, V t \_ ð Þ can be expressed as

V t \_ ð Þ <sup>≤</sup> � α ηk k<sup>2</sup> � k k<sup>r</sup>

Based on Eq. (32) and the subsequent stability analysis, the parameter estimates θb1ð Þt and θb2ð Þt are generated online according to the adaptive laws:

$$
\dot{\hat{\boldsymbol{\theta}}}\_1 = \text{proj}(\Gamma\_1 \mathbf{Y}^T \mathbf{r}), \qquad \dot{\hat{\boldsymbol{\theta}}}\_2 = \text{proj}(\Gamma\_2 \mathbf{Y}\_2^T \mathbf{r}), \tag{34}
$$

where Γ1∈ Rp<sup>1</sup> � <sup>p</sup><sup>1</sup> and Γ2∈ Rp<sup>2</sup> � <sup>p</sup><sup>2</sup> 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:

$$
\underline{\theta\_1} \le \widehat{\theta}\_1 \le \overline{\theta}\_{1\prime} \qquad \underline{\theta\_2} \le \widehat{\theta}\_2 \le \overline{\theta}\_{2\prime} \tag{35}
$$

where θ1, θ1, θ<sup>2</sup> and θ<sup>2</sup> ∈ R represent known, constant lower and upper bounds of the elements of θb1ð Þt and θb2ð Þt , 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:

$$
\lambda\_{\min} \{ \beta \} > \zeta \tag{36}
$$

where ζ is introduced in Eq. (20) and λmin{�} denotes the minimum eigenvalue of the argument.
