5. Stability analysis

Assumption 2 is mild in the sense that inequality (29) is satisfied for a wide range of nonlinear

�Y1θb<sup>1</sup> � ð Þ ks þ 1 r � βsgn ð Þ� r η � �

Mr\_ <sup>¼</sup> <sup>Y</sup>1θ~<sup>1</sup> <sup>þ</sup> <sup>Y</sup>2θ~<sup>2</sup> <sup>þ</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>d, (32)

<sup>θ</sup>b<sup>2</sup> <sup>¼</sup> proj <sup>Γ</sup>2Y<sup>T</sup>

θ<sup>1</sup> ≤ θb<sup>1</sup> ≤ θ1, θ<sup>2</sup> ≤ θb<sup>2</sup> ≤ θ2, (35)

� � > ζ, (36)

<sup>θ</sup>~<sup>1</sup> <sup>≜</sup> <sup>θ</sup><sup>1</sup> � <sup>θ</sup>b1: (33)

<sup>2</sup> <sup>r</sup> � �, (34)

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

After substituting the control input expression in Eq. (34) into the open-loop dynamics in

Based on Eq. (32) and the subsequent stability analysis, the parameter estimates θb1ð Þt and θb2ð Þt

Remark 2 The function proj(�) in Eq. (34) denotes a normal projection algorithm, which ensures that

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

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

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

λmin β

<sup>θ</sup>b<sup>1</sup> <sup>¼</sup> proj <sup>Γ</sup>1Y<sup>T</sup><sup>r</sup> � �, \_

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.

, (31)

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

<sup>v</sup> <sup>¼</sup> <sup>B</sup><sup>b</sup> �<sup>1</sup>

Eq. (27), the closed-loop error system is obtained as

are generated online according to the adaptive laws:

the following inequalities are satisfied:

\_

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

function hð Þ η; η\_ .

170 Adaptive Robust Control Systems

function.

Eq. (31).

the sufficient condition:

4.2. Closed-loop error system

Theorem 1 The control law in Eq. (31) with adaptive laws defined as in Eq. (34) ensures asymptotic regulation of the thermoacoustic modes η1(t) and η2(t) in the sense that

$$\|\eta(t)\| \to 0 \qquad\text{as}\qquad t \to \infty \tag{37}$$

provided that ks is selected as sufficiently large (see the subsequent stability proof) and β is selected to satisfy inequality (36).

Proof. Let V η;r; θb1; θb2; t � � <sup>∈</sup> <sup>R</sup> be defined as the nonnegative function:

$$V(t) \triangleq \frac{1}{2}\boldsymbol{\eta}^{T}\boldsymbol{\eta} + \frac{1}{2}\mathbf{r}^{T}\mathbf{M}\mathbf{r} + \frac{1}{2}\tilde{\theta}\_{1}^{T}\Gamma\_{1}^{-1}\tilde{\theta}\_{1} + \frac{1}{2}\tilde{\theta}\_{2}^{T}\Gamma\_{2}^{-1}\tilde{\theta}\_{2}.\tag{38}$$

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

$$\begin{split} \dot{V}(t) = \boldsymbol{\eta}^{T}(\mathbf{r} - \alpha \boldsymbol{\eta}) + \mathbf{r}^{T} \{ \mathbf{h}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}}) - \boldsymbol{\eta} - (k\_{\mathrm{s}} + 1)\mathbf{r} - \beta \text{sgn} \,(\mathbf{r}) + \tau\_{d} \} \\ + \mathbf{r}^{T} \{ \mathbf{Y}\_{1} \tilde{\boldsymbol{\theta}}\_{1} + \mathbf{Y}\_{2} \tilde{\boldsymbol{\theta}}\_{2} \} - \tilde{\boldsymbol{\theta}}\_{1}^{T} \Gamma\_{1}^{-1} \dot{\tilde{\boldsymbol{\theta}}}\_{1} - \tilde{\boldsymbol{\theta}}\_{2}^{T} \Gamma\_{2}^{-1} \dot{\tilde{\boldsymbol{\theta}}}\_{2} \end{split} \tag{39}$$

where Eq. (22) was utilized. After substituting the adaptive laws in Eq. (34) and canceling common terms, V t \_ ð Þ can be expressed as

$$\dot{V}(t) = -a\eta^T \eta + \mathbf{r}^T \{\mathbf{h}(\eta, \dot{\eta}) - (k\_s + 1)\mathbf{r} - \beta \text{sgn}\left(\mathbf{r}\right) + \tau\_d\}.\tag{40}$$

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

$$\dot{V}(t) \le -\alpha \|\eta\|^2 - \left(k\_s \|\mathbf{r}\|^2 - \rho(\|\mathbf{z}\|) \|\mathbf{z}\| \|\mathbf{r}\|\right) - \|\mathbf{r}\|^2 - \beta \mathbf{r}^T \text{sgn}(\mathbf{r}) + \zeta \|\mathbf{r}\|.\tag{41}$$

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

$$\dot{V}(\mathbf{t}) \le -\alpha \|\eta\|^2 - \|\mathbf{r}\|^2 - \beta |\mathbf{r}| + \zeta \|\mathbf{r}\| - k\_s \left( \|\mathbf{r}\| - \frac{\rho(\|\mathbf{z}\|)}{2k\_s} \|\mathbf{z}\| \right)^2 + \frac{\rho^2(\|\mathbf{z}\|)}{4k\_s} \|\mathbf{z}\|^2,\tag{42}$$

where the fact that r <sup>T</sup>sgn(r)=|r| was utilized. After using inequality (36), the upper bound in Eq. (42) can be expressed as

$$\dot{V}(t) \le -\left(\lambda\_0 - \frac{\rho^2(\|\mathbf{z}\|)}{4k\_s}\right) \|\mathbf{z}\|^2,\tag{43}$$

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> ) was utilized. Based on Eq. (43), <sup>V</sup>\_ <sup>≤</sup> � <sup>c</sup>k k<sup>z</sup> <sup>2</sup> , for some positive constant c, inside the set R, where R is defined as

$$\mathcal{R} \triangleq \left\{ \mathbf{z} \middle| \mathbf{z} < \rho^{-1} \left( 2\sqrt{\lambda\_0 k\_s} \right) \right\}. \tag{44}$$

<sup>y</sup> <sup>¼</sup> u xs ð Þ¼ ; <sup>t</sup> <sup>Ψ</sup><sup>T</sup>

<sup>y</sup> <sup>¼</sup> p xs ð Þ¼� ; <sup>t</sup> <sup>Φ</sup><sup>T</sup>

<sup>C</sup> <sup>¼</sup> <sup>0</sup><sup>1</sup>�<sup>2</sup> �Φ<sup>T</sup>

rank <sup>C</sup><sup>T</sup> <sup>E</sup><sup>T</sup>C<sup>T</sup> <sup>E</sup><sup>T</sup> � �<sup>2</sup>

It can be shown that there exists a coordinate transformation of the forms z = [z1, z2]

, and <sup>G</sup><sup>~</sup> <sup>¼</sup> TG .

The estimate <sup>b</sup><sup>z</sup> of the the state <sup>z</sup> will now be generated via the observer equation

It is assumed that the sensor location xs is chosen such that the systems (45) and (46) subject to

C<sup>T</sup> E<sup>T</sup> � �<sup>3</sup>

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

<sup>s</sup> 0<sup>1</sup>�<sup>2</sup>

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

<sup>C</sup> <sup>¼</sup> <sup>Ψ</sup><sup>T</sup>

and thus

and hence

τ = 0 are observable, i.e.,

where E = A + B.

where <sup>A</sup><sup>~</sup> <sup>¼</sup> TAT�<sup>1</sup>

, <sup>B</sup><sup>~</sup> <sup>¼</sup> TBT�<sup>1</sup>

Then, the error dynamics <sup>z</sup> <sup>¼</sup> <sup>z</sup> � <sup>b</sup><sup>z</sup> can be written as

\_

The following result can now be stated:

Partition the system above as

\_

z\_

and z2∈ R<sup>3</sup>

For the pressure sensor case, we have

<sup>s</sup> η, (49)

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173

<sup>s</sup> η\_, (51)

<sup>T</sup> =Tx, z1∈ R,

� �: (50)

<sup>s</sup> <sup>M</sup>: � �: (52)

<sup>C</sup><sup>T</sup> h i <sup>¼</sup> <sup>4</sup>, (53)

<sup>z</sup>\_ðÞ¼ <sup>t</sup> Az <sup>~</sup> ð Þþ <sup>t</sup> Bz <sup>~</sup> ð Þþ <sup>t</sup> � <sup>τ</sup> Gu<sup>~</sup> ð Þ<sup>t</sup> , (54)

<sup>b</sup>zðÞ¼ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>b</sup>zð Þþ <sup>t</sup> <sup>B</sup>~bzð Þþ <sup>t</sup> � <sup>τ</sup> Gu<sup>~</sup> ð Þþ <sup>t</sup> <sup>L</sup>sgn y tð Þ� <sup>b</sup>z1ð Þ<sup>t</sup> � �: (56)

<sup>z</sup>1ðÞ¼ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>11</sup>z1ðÞþ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>12</sup>z2ð Þþ <sup>t</sup> <sup>B</sup>~11z1ð Þþ <sup>t</sup> � <sup>τ</sup> <sup>B</sup>~12z2ð Þ� <sup>t</sup> � <sup>τ</sup> <sup>L</sup><sup>1</sup> sgn ð Þ <sup>z</sup>1ð Þ<sup>t</sup> , (58)

<sup>2</sup>ðÞ¼ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>21</sup>z1ð Þþ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>22</sup>z2ð Þþ <sup>t</sup> <sup>B</sup>~21z1ð Þþ <sup>t</sup> � <sup>τ</sup> <sup>B</sup>~22z2ð Þ� <sup>t</sup> � <sup>τ</sup> <sup>L</sup><sup>2</sup> sgn ð Þ <sup>z</sup>1ð Þ<sup>t</sup> : (59)

<sup>z</sup>\_ðÞ¼ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>z</sup>ð Þþ <sup>t</sup> <sup>B</sup><sup>~</sup> <sup>z</sup>ð Þ� <sup>t</sup> � <sup>τ</sup> <sup>L</sup>sgn ð Þ <sup>z</sup>1ð Þ<sup>t</sup> : (57)

y tðÞ¼ z1ð Þt , (55)

The expressions in Eqs. (38) and (43) can be used to prove that <sup>η</sup>(t), <sup>r</sup>(t), <sup>θ</sup>~1ð Þ<sup>t</sup> , and <sup>θ</sup>~2ð Þ<sup>t</sup> <sup>∈</sup>L<sup>∞</sup> in R and Eq. (22) can then be used to prove that η\_ð Þt ∈L<sup>∞</sup> in R. Given that η(t) and r(t)∈L<sup>∞</sup> in R, Eq. (31) can be used along with Eq. (35) to prove that v(t)∈L<sup>∞</sup> in R. Since η(t), η\_ð Þt , and v(t) ∈L<sup>∞</sup> in R, Y1ð Þ η; η\_ and Y2(v)∈L<sup>∞</sup> in R. Given that η(t), r(t), η\_ð Þt , Y1ð Þ η; η\_ , and Y2(v) ∈L<sup>∞</sup> in R, Eq. (32) can be used along with Eq. (35) to show that r\_ð Þt ∈L<sup>∞</sup> in R. Since η\_ð Þt and r\_ð Þt ∈L<sup>∞</sup> in R, η(t) and r(t) are uniformly continuous in R. It then follows from Eq. (30) that <sup>z</sup>(t) is uniformly continuous in <sup>R</sup>. Given that <sup>η</sup>(t), <sup>r</sup>(t), <sup>θ</sup>~1ð Þ<sup>t</sup> , and <sup>θ</sup>~2ð Þ<sup>t</sup> <sup>∈</sup>L<sup>∞</sup> in <sup>R</sup>, <sup>V</sup>(t)∈L<sup>∞</sup> in <sup>R</sup>, and Eq. (43) can be integrated to prove that Ð <sup>∞</sup> <sup>0</sup> k k <sup>z</sup>ð Þ<sup>t</sup> <sup>2</sup> dt∈L<sup>∞</sup> in R. Thus, z(t) ∈L<sup>∞</sup> ∩L<sup>2</sup> in R. Barbalat's lemma can now be invoked to prove that kz(t)k!0 as t!∞. Hence, kη(t)k!0 as t!∞ in R, where the set R can be made arbitrarily large by increasing the control gain ks—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 <sup>x</sup> <sup>¼</sup> <sup>η</sup><sup>T</sup>; <sup>η</sup>\_ <sup>T</sup><sup>M</sup> � �<sup>T</sup> denote the state. For simplicity in the subsequent development, only one sensor is assumed to be available at location xs. Then, the system can be rewritten as

$$\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{x}(t-\tau) + \mathbf{G}\mathbf{u}(t),\tag{45}$$

$$y = \mathbb{C}\mathbf{x},\tag{46}$$

where

$$\mathbf{A} = \begin{bmatrix} \mathbf{0} & \mathbf{M}^{-1} \\ -\mathbf{M}^{-1} & -\mathbf{D} \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ -\mathbf{W} & \mathbf{0} \end{bmatrix}, \quad \mathbf{G} = \begin{bmatrix} \mathbf{0} \\ \mathbf{I} \end{bmatrix}, \tag{47}$$

where

$$\mathbf{W} = \sqrt{3}(\gamma - 1)\mathcal{K}\mathbf{O}\_f \Psi\_{f'}^T \tag{48}$$

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

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$$y = \mu(\mathbf{x}\_s, t) = \Psi\_s^T \eta\_\prime \tag{49}$$

and thus

<sup>R</sup> <sup>≜</sup> <sup>z</sup>j<sup>z</sup> <sup>&</sup>lt; <sup>ρ</sup>�<sup>1</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffi

and Eq. (43) can be integrated to prove that Ð <sup>∞</sup>

6. Sliding-mode observer design

judicious sensor placement.

where

where

semiglobal result.

172 Adaptive Robust Control Systems

The expressions in Eqs. (38) and (43) can be used to prove that <sup>η</sup>(t), <sup>r</sup>(t), <sup>θ</sup>~1ð Þ<sup>t</sup> , and <sup>θ</sup>~2ð Þ<sup>t</sup> <sup>∈</sup>L<sup>∞</sup> in R and Eq. (22) can then be used to prove that η\_ð Þt ∈L<sup>∞</sup> in R. Given that η(t) and r(t)∈L<sup>∞</sup> in R, Eq. (31) can be used along with Eq. (35) to prove that v(t)∈L<sup>∞</sup> in R. Since η(t), η\_ð Þt , and v(t) ∈L<sup>∞</sup> in R, Y1ð Þ η; η\_ and Y2(v)∈L<sup>∞</sup> in R. Given that η(t), r(t), η\_ð Þt , Y1ð Þ η; η\_ , and Y2(v) ∈L<sup>∞</sup> in R, Eq. (32) can be used along with Eq. (35) to show that r\_ð Þt ∈L<sup>∞</sup> in R. Since η\_ð Þt and r\_ð Þt ∈L<sup>∞</sup> in R, η(t) and r(t) are uniformly continuous in R. It then follows from Eq. (30) that <sup>z</sup>(t) is uniformly continuous in <sup>R</sup>. Given that <sup>η</sup>(t), <sup>r</sup>(t), <sup>θ</sup>~1ð Þ<sup>t</sup> , and <sup>θ</sup>~2ð Þ<sup>t</sup> <sup>∈</sup>L<sup>∞</sup> in <sup>R</sup>, <sup>V</sup>(t)∈L<sup>∞</sup> in <sup>R</sup>,

<sup>0</sup> k k <sup>z</sup>ð Þ<sup>t</sup> <sup>2</sup>

Barbalat's lemma can now be invoked to prove that kz(t)k!0 as t!∞. Hence, kη(t)k!0 as t!∞ in R, where the set R can be made arbitrarily large by increasing the control gain ks—a

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

Let <sup>x</sup> <sup>¼</sup> <sup>η</sup><sup>T</sup>; <sup>η</sup>\_ <sup>T</sup><sup>M</sup> � �<sup>T</sup> denote the state. For simplicity in the subsequent development, only one

, <sup>B</sup> <sup>¼</sup> 0 0

<sup>p</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>K</sup>Φ<sup>f</sup> <sup>Ψ</sup><sup>T</sup>

and the output matrix C is determined by the sensor choice and its location. The output

�W 0 � �

x\_ðÞ¼ t AxðÞþ t Bxð Þþ t � τ Guð Þt , (45)

y ¼ Cx, (46)

, (47)

<sup>f</sup> , (48)

, <sup>G</sup> <sup>¼</sup> <sup>0</sup> I � �

sensor is assumed to be available at location xs. Then, the system can be rewritten as

<sup>A</sup> <sup>¼</sup> 0 M�<sup>1</sup> �M�<sup>1</sup> �<sup>D</sup>

equation for the velocity sensor case is given by

" #

<sup>W</sup> <sup>¼</sup> ffiffiffi 3 λ0ks

n o � � <sup>p</sup> : (44)

dt∈L<sup>∞</sup> in R. Thus, z(t) ∈L<sup>∞</sup> ∩L<sup>2</sup> in R.

$$\mathbf{C} = \begin{bmatrix} \Psi\_s^T & \mathbf{0}\_{1\times 2} \end{bmatrix}. \tag{50}$$

For the pressure sensor case, we have

$$y = p(\mathbf{x}\_s, t) = -\boldsymbol{\Phi}\_s^I \,\dot{\eta}\_{\prime} \tag{51}$$

and hence

$$\mathbf{C} = \begin{bmatrix} \mathbf{0}\_{1 \times 2} & -\mathbf{0}\_{s}^{T}\mathbf{M}. \end{bmatrix}. \tag{52}$$

It is assumed that the sensor location xs is chosen such that the systems (45) and (46) subject to τ = 0 are observable, i.e.,

$$\text{rank}\left[\mathbf{C}^{T}\;\mathbf{E}^{T}\mathbf{C}^{T}\;\left(\mathbf{E}^{T}\right)^{2}\mathbf{C}^{T}\;\left(\mathbf{E}^{T}\right)^{3}\mathbf{C}^{T}\right] = \mathbf{4},\tag{53}$$

where E = A + B.

It can be shown that there exists a coordinate transformation of the forms z = [z1, z2] <sup>T</sup> =Tx, z1∈ R, and z2∈ R<sup>3</sup> , such that the systems (45) and (46) in the new variables take the following form:

$$\dot{\mathbf{z}}(t) = \tilde{\mathbf{A}}\mathbf{z}(t) + \tilde{\mathbf{B}}\mathbf{z}(t-\tau) + \tilde{\mathbf{G}}\mathbf{u}(t),\tag{54}$$

$$y(t) = z\_1(t),\tag{55}$$

where <sup>A</sup><sup>~</sup> <sup>¼</sup> TAT�<sup>1</sup> , <sup>B</sup><sup>~</sup> <sup>¼</sup> TBT�<sup>1</sup> , and <sup>G</sup><sup>~</sup> <sup>¼</sup> TG .

The estimate <sup>b</sup><sup>z</sup> of the the state <sup>z</sup> will now be generated via the observer equation

$$
\dot{\hat{\mathbf{z}}}(t) = \mathbf{\tilde{A}}\hat{\mathbf{z}}(t) + \mathbf{\tilde{B}}\hat{\mathbf{z}}(t-\tau) + \mathbf{\tilde{G}u}(t) + \mathbf{L}\text{sgn}\left(y(t) - \hat{z}\_1(t)\right). \tag{56}
$$

Then, the error dynamics <sup>z</sup> <sup>¼</sup> <sup>z</sup> � <sup>b</sup><sup>z</sup> can be written as

$$
\dot{\overline{\mathbf{z}}}(t) = \overset{\circ}{\mathbf{A}} \overline{\mathbf{z}}(t) + \overset{\circ}{\mathbf{B}} \overline{\mathbf{z}}(t - \tau) - \mathbf{L} \text{sgn} \left( \overline{z}\_1(t) \right). \tag{57}
$$

Partition the system above as

$$
\dot{\tilde{z}}\_1(t) = \tilde{A}\_{11}\overline{z}\_1(t) + \tilde{\mathbf{A}}\_{12}\overline{z}\_2(t) + \tilde{B}\_{11}\overline{z}\_1(t-\tau) + \tilde{\mathbf{B}}\_{12}\overline{z}\_2(t-\tau) - L\_1 \operatorname{sgn}\left(\overline{z}\_1(t)\right), \tag{58}
$$

$$
\dot{\overline{\mathbf{z}}}\_{2}(t) = \tilde{\mathbf{A}}\_{21}\overline{z}\_{1}(t) + \tilde{\mathbf{A}}\_{22}\overline{\mathbf{z}}\_{2}(t) + \tilde{\mathbf{B}}\_{21}\overline{z}\_{1}(t-\tau) + \tilde{\mathbf{B}}\_{22}\overline{\mathbf{z}}\_{2}(t-\tau) - \mathbf{L}\_{2}\text{sgn}\left(\overline{z}\_{1}(t)\right). \tag{59}
$$

The following result can now be stated:

Theorem 2 Let <sup>E</sup>~<sup>12</sup> <sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>12</sup> <sup>þ</sup> <sup>B</sup>~<sup>12</sup> and <sup>E</sup>~<sup>22</sup> <sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>22</sup> <sup>þ</sup> <sup>B</sup>~22, and assume that the pair <sup>E</sup>~22; <sup>E</sup>~<sup>12</sup> h i is observable. Then, the observer gain L<sup>1</sup> can be chosen such that the system (57) is asymptotically stable for all τ∈ [0, τmax] with τmax defined by:

$$\tau\_{\text{max}} = \frac{1}{2\sqrt{\lambda\_{\text{max}} \left( \mathbf{Q}\_1^{-T} \mathbf{F}^T \mathbf{P} \mathbf{H} \mathbf{Q}^{-1} \mathbf{H}^T \mathbf{P} \mathbf{F} \mathbf{Q}\_1^{-1} \right)}},\tag{60}$$

where <sup>F</sup> <sup>¼</sup> <sup>E</sup>~<sup>22</sup> � <sup>L</sup>2E~12=L1, <sup>H</sup> <sup>¼</sup> <sup>B</sup>~<sup>22</sup> � <sup>L</sup>2B~12=L1, and <sup>Q</sup> is any symmetric, positive-definite matrix such that P is a symmetric, positive-definite matrix P solution of the Lyapunov matrix equation:

$$\mathbf{F}^T \mathbf{P} + \mathbf{P} \mathbf{F} = -\mathbf{Q} \tag{61}$$

The initial conditions for the modes were selected as η1(0) = 0.07 and η2(0) = 0. The initial conditions of the velocity and pressure parameters at the location of the heat source are u(xf, 0) = 0.1646 and p(xf, 0) = 0 for xf = 0.7. Figure 4 shows the open-loop velocity and

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175

In closed-loop operation, the adaptation gain matrices used in the simulation were selected as Γ<sup>1</sup> = 0.1I<sup>8</sup> and Γ<sup>2</sup> = 0.001I4. The value for the gain α was selected as 1, and the control gain matrices β and k<sup>s</sup> were selected as β = diag {2.43, 2.1} , k<sup>s</sup> = diag {4.5, 2.4}. The results of 20 Monte Carlo-type simulations for the closed-loop operation are shown in Figures 5–10. Figure 5 shows the time evolution of the velocity and pressure values at the heat source location during

ρ 1.025 kg/m<sup>3</sup> λ 0.0328 W/m K

cv 719 J/kg K γ 1.4 L<sup>0</sup> 1 m Lw 2.5 m c 344 m/s u<sup>0</sup> 0.3 m/s T<sup>0</sup> 295 K Tw 1680 K dw 0.5 <sup>10</sup><sup>3</sup> <sup>m</sup> <sup>S</sup> 1.56 <sup>10</sup><sup>3</sup> <sup>m</sup>

<sup>P</sup><sup>0</sup> 8.69 104 Pa <sup>α</sup>a<sup>1</sup> <sup>=</sup> <sup>α</sup>a<sup>2</sup> 0.01 ζ<sup>1</sup> 0.0440 ζ<sup>2</sup> 0.1657

Figure 4. Time response of the velocity u(t) and pressure p(t) at the heat source location during open-loop (uncontrolled)

Table 1. Physical parameters.

operation.

pressure perturbations at the heat source in the absence of control actuation.

and Q<sup>1</sup> is the square root of the matrix Q, i.e.,

$$\mathbf{Q}\_1^I \mathbf{Q}\_1 = \mathbf{Q}.\tag{62}$$

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

$$L\_1 > |\tilde{A}\_{11}\overline{z}\_1(t) + \check{\mathbf{A}}\_{12}\overline{\mathbf{z}}\_2(t) + \check{B}\_{11}\overline{z}\_1(t-\tau) + \check{\mathbf{B}}\_{12}\overline{z}\_2(t-\tau)|.\tag{63}$$

Condition (63) guarantees sliding in Eq. (58) along the manifold <sup>z</sup><sup>1</sup> <sup>¼</sup> 0; thus <sup>b</sup><sup>η</sup> ! <sup>η</sup>. According to the equivalent control method [32], the system in sliding mode behaves as if L<sup>1</sup> sgn ð Þ z<sup>1</sup> is replaced by its equivalent value ð Þ L<sup>1</sup> sgn ð Þ z<sup>1</sup> eq which can be calculated from subsystem (58) assuming <sup>z</sup><sup>1</sup> <sup>¼</sup> 0 and \_ z<sup>1</sup> ¼ 0. Hence,

$$(L\_1 \operatorname{sgn}(\overline{z}\_1))\_{\text{eq}} = \tilde{\mathbf{A}}\_{12} \overline{\mathbf{z}}\_2(t) + \tilde{\mathbf{B}}\_{12} \overline{\mathbf{z}}\_2(t-\tau). \tag{64}$$

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

$$
\dot{\mathbf{\bar{z}}}\_2 = \left(\tilde{\mathbf{A}}\_{22} - \mathbf{L}\_2 \tilde{\mathbf{A}}\_{12}/\mathbf{L}\_1\right) \mathbf{\bar{z}}\_2(t) + \left(\tilde{\mathbf{B}}\_{22} - \mathbf{L}\_2 \tilde{\mathbf{B}}\_{12}/\mathbf{L}\_1\right) \mathbf{\bar{z}}\_2(t-\tau). \tag{65}
$$

Using the fact that the pair E~22; E~12=L<sup>1</sup> h i is observable, the observer gain <sup>L</sup><sup>2</sup> can be chosen such that the eigenvalues of the matrices <sup>E</sup>~<sup>22</sup> � <sup>L</sup>2E~12=L<sup>1</sup> have negative real parts. Thus, the subsystem (65) is asymptotically stable for <sup>τ</sup> <sup>≤</sup> <sup>τ</sup>max. This implies that <sup>z</sup> ! 0 and <sup>b</sup>zðÞ¼ <sup>t</sup> <sup>z</sup>ð Þ<sup>t</sup> , and hence, <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>x</sup>.

#### 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 η1(0) = 0.07 and η2(0) = 0. The initial conditions of the velocity and pressure parameters at the location of the heat source are u(xf, 0) = 0.1646 and p(xf, 0) = 0 for xf = 0.7. 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<sup>1</sup> = 0.1I<sup>8</sup> and Γ<sup>2</sup> = 0.001I4. The value for the gain α was selected as 1, and the control gain matrices β and k<sup>s</sup> were selected as β = diag {2.43, 2.1} , k<sup>s</sup> = diag {4.5, 2.4}. The results of 20 Monte Carlo-type simulations for the closed-loop operation are shown in Figures 5–10. Figure 5 shows the time evolution of the velocity and pressure values at the heat source location during


Table 1. Physical parameters.

Theorem 2 Let <sup>E</sup>~<sup>12</sup> <sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>12</sup> <sup>þ</sup> <sup>B</sup>~<sup>12</sup> and <sup>E</sup>~<sup>22</sup> <sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>22</sup> <sup>þ</sup> <sup>B</sup>~22, and assume that the pair <sup>E</sup>~22; <sup>E</sup>~<sup>12</sup>

<sup>τ</sup>max <sup>¼</sup> <sup>1</sup>

λmax Q�<sup>T</sup>

Q<sup>T</sup>

Condition (63) guarantees sliding in Eq. (58) along the manifold <sup>z</sup><sup>1</sup> <sup>¼</sup> 0; thus <sup>b</sup><sup>η</sup> ! <sup>η</sup>. According to the equivalent control method [32], the system in sliding mode behaves as if L<sup>1</sup> sgn ð Þ z<sup>1</sup> is replaced by its equivalent value ð Þ L<sup>1</sup> sgn ð Þ z<sup>1</sup> eq which can be calculated from subsystem (58)

that the eigenvalues of the matrices <sup>E</sup>~<sup>22</sup> � <sup>L</sup>2E~12=L<sup>1</sup> have negative real parts. Thus, the subsystem (65) is asymptotically stable for <sup>τ</sup> <sup>≤</sup> <sup>τ</sup>max. This implies that <sup>z</sup> ! 0 and <sup>b</sup>zðÞ¼ <sup>t</sup> <sup>z</sup>ð Þ<sup>t</sup> , and hence, <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>x</sup>.

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

2

for all τ∈ [0, τmax] with τmax defined by:

174 Adaptive Robust Control Systems

and Q<sup>1</sup> is the square root of the matrix Q, i.e.,

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

z<sup>1</sup> ¼ 0. Hence,

<sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>22</sup> � <sup>L</sup>2A<sup>~</sup> <sup>12</sup>=L<sup>1</sup> � �

h i

physical parameters used in the simulation are given in Table 1.

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

z\_

Using the fact that the pair E~22; E~12=L<sup>1</sup>

7. Simulation results

equation:

assuming <sup>z</sup><sup>1</sup> <sup>¼</sup> 0 and \_

observable. Then, the observer gain L<sup>1</sup> can be chosen such that the system (57) is asymptotically stable

where <sup>F</sup> <sup>¼</sup> <sup>E</sup>~<sup>22</sup> � <sup>L</sup>2E~12=L1, <sup>H</sup> <sup>¼</sup> <sup>B</sup>~<sup>22</sup> � <sup>L</sup>2B~12=L1, and <sup>Q</sup> is any symmetric, positive-definite matrix such that P is a symmetric, positive-definite matrix P solution of the Lyapunov matrix

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H<sup>T</sup>PFQ�<sup>1</sup> 1 <sup>q</sup> � � , (60)

<sup>L</sup><sup>1</sup> <sup>&</sup>gt; <sup>∣</sup>A<sup>~</sup> <sup>11</sup>z1ð Þþ <sup>t</sup> <sup>A</sup><sup>~</sup> <sup>12</sup>z2ð Þþ <sup>t</sup> <sup>B</sup>~11z1ð Þþ <sup>t</sup> � <sup>τ</sup> <sup>B</sup>~12z2ð Þ <sup>t</sup> � <sup>τ</sup> <sup>∣</sup>: (63)

ð Þ <sup>L</sup><sup>1</sup> sgn ð Þ <sup>z</sup><sup>1</sup> eq <sup>¼</sup> <sup>A</sup><sup>~</sup> <sup>12</sup>z2ðÞþ <sup>t</sup> <sup>B</sup>~12z2ð Þ <sup>t</sup> � <sup>τ</sup> : (64)

<sup>z</sup>2ð Þþ <sup>t</sup> <sup>B</sup>~<sup>22</sup> � <sup>L</sup>2B~12=L<sup>1</sup>

<sup>F</sup><sup>T</sup><sup>P</sup> <sup>þ</sup> PF ¼ �Q, (61)

<sup>1</sup> Q<sup>1</sup> ¼ Q: (62)

� �z2ð Þ <sup>t</sup> � <sup>τ</sup> : (65)

is observable, the observer gain L<sup>2</sup> can be chosen such

<sup>1</sup> F<sup>T</sup>PHQ�<sup>1</sup>

h i

is

Figure 4. Time response of the velocity u(t) and pressure p(t) at the heat source location during open-loop (uncontrolled) operation.

closed-loop controller operation. Figure 6 shows the time history of the modes η1(t) and η2(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 θb1ð Þt and θb2ð Þt during closed-loop controller operation.

Figure 7. Commanded control inputs ν1(t) and ν2(t) during closed-loop controller operation.

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

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

177

Figure 8. Time response of the adaptive parameter estimates θb11ð Þt , θb12ð Þt , θb13ð Þt , and θb14ð Þt during closed-loop controller

operation.

Figure 5. Time response of the velocity u(t) and pressure p(t) during closed-loop controller operation at the heat source location.

Figure 6. Time response of the oscillation modes η1(t) and η2(t) during closed-loop controller operation.

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems http://dx.doi.org/10.5772/intechopen.70683 177

Figure 7. Commanded control inputs ν1(t) and ν2(t) during closed-loop controller operation.

closed-loop controller operation. Figure 6 shows the time history of the modes η1(t) and η2(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

Figure 5. Time response of the velocity u(t) and pressure p(t) during closed-loop controller operation at the heat source

Figure 6. Time response of the oscillation modes η1(t) and η2(t) during closed-loop controller operation.

vectors θb1ð Þt and θb2ð Þt during closed-loop controller operation.

location.

176 Adaptive Robust Control Systems

Figure 8. Time response of the adaptive parameter estimates θb11ð Þt , θb12ð Þt , θb13ð Þt , and θb14ð Þt during closed-loop controller operation.

8. Conclusion

Author details

References

1999;394:51-72

of Fluid Mechanics. 2005;37:151-182

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

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

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

179

closed-loop control system under 20 different sets of operating conditions.

William MacKunis\*, Mahmut Reyhanoglu and Krishna Bhavithavya Kidambi

identification, and modeling. IEEE Control Systems. 2015;35(2):57-77

ities. Progress in Energy and Combustion Science. 1993;19(1):1-29

Physical Sciences Department, Embry-Riddle Aeronautical University, Daytona Beach, FL, USA

[1] Epperlein J, Bamieh B, Astrom K. Thermoacoustics and the Rijke tube: Experiments,

[2] McManus KR, Poinsot T, Candel SM. A review of active control of combustion instabil-

[3] Dowling AP. A kinematic model of a ducted flame. Journal of Fluid Mechanics.

[4] Dowling AP, Morgans AS. Feedback control of combustion oscillations. Annual Review

\*Address all correspondence to: mackuniw@erau.edu

Figure 9. Time response of the adaptive parameter estimates θb15ð Þt , θb16ð Þt , θb17ð Þt , and θb18ð Þt during closed-loop controller operation.

Figure 10. Time response of the adaptive parameter estimates θb21ð Þt , θb22ð Þt , θb23ð Þt , and θb24ð Þt during closed-loop controller operation.
